X Mathematics Practice Paper

Transcription

1 Brilliant Public School, Sitamarhi X Mathematics Practice Paper Session : Rajopatti,Dumra Road,Sitamarhi(Bihar),Pin Ph ,Mobile:

2 Mathematics for Class 10 Q 1 Why is 7x11x13+7 a composite integer? 1. Real Numbers Q 2 Without actual division, state whether is a terminating or a non terminating rational number. Q 3 Identify whether 16 is a rational number. Q 4 what is the cojugate number of 2+ 5? Q 5 Express 140 as the product of its prime factors. Q 6 Determine whether 875/10 3 is a terminating or a non-terminating decimal. Q 7 HCF of two integers 26, 91 is 13, what will be its LCM? Q 8 If f(x) is divisble by q(x), what will be the value of r(x), where f(x) = g(x)q(x) + r(x)? Q 9 Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. Q 10 Use the division algorithm to find the quotient q(y) and remainder r(y), when f(y) = 8y 4 12y 3 2y y 4 is divided by g(y) = 2y 2 3y + 1. Q 11 State the following: (i) Euclid s Division Lemma with boundary conditions. (ii) Fundamental Theorem of Arithmetic. Q 12 Find the least number that is divisible by all numbers between 1 and 10 (both inclusive). X Mathematics C.B.S.E. Practice Papers Page 1

3 Q 13 Prove that if a and b are odd positive integers then a 2 +b 2 is even integer. Q 14 Express the positive integers 72 and 90 as the product of its prime factors. Q 15 Find the HCF of 144 and 60 by prime factorisation method. Q 16 Find the LCM of 60 and 144 by prime factorisation method. Q 17 There is a circular path around a sports field. Ankit takes 18 minutes to drive one round of the field, while Ankita takes 12 minutes for the same. Suppose they both start at the same point, at the same time and go in the same direction. After how many minutes will they meet again at the starting point? Q 18 A bakery seller has 210 vanilla pastries and 390 chocolate pastries. She wants to stack them in such a way that each stack has the same number, and they can take up the least area of the tray. What is the number of pastries that can be placed in each stack for this purpose? Q 19 Write the condition for terminating of a rational number. And hence, find whether the rational number (13/3125) has a terminating decimal or non-terminating repeating decimal. Q 20 For any positive real number, prove that there exists an irrational number y such that 0<y<x. Q 21 Prove that there is no natural number for which 4 n ends with the digit zero. Q 22 Find the HCF of 480 and 404 by prime factorisation method. Hence, find their LCM. Q 23 Express the positive integers 180 and 360 as the product of its prime factors. Q 24 Prove that 5+ 3 is an irrational number. Q 25 Prove that 3+ 2 is an irrational number. Q 26 State Euclid s division algorithm. X Mathematics C.B.S.E. Practice Papers Page 2

4 Q 27 Prove that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q+1, where q is some integer. Q 28 Two tankers contain 2340 litres and 3825 litres of petrol respectively. Find the maximum capacity of the container that can measure the petrol of either tanker in exact number of times. Q 29 Show that 3 2 is irrational. Q 30 Show that 5 3 is irrational. Q 31 Prove that 2 is irrational number? Q 32 Express as the product of its prime factors. Q 33 Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method. Q 34 Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM. Q 35 Find the LCM and HCF of 6 and 20 by the prime factorisation method. Q 36 A sweet seller has 420 Kaju burfis and 130 Badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of burfis that can be placed in each stack for this purpose? Q 37 Use Euclid s division algorithm to find the HCF of 4052 and Q 38 Find the largest number that will divide 2053 and 967 and leaves a remainder of 5 and 7 respectively. Q 39 Express the positive integers 144 & 180 as the product of its prime factors. X Mathematics C.B.S.E. Practice Papers Page 3

5 Q 40 In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required if in each room same numbers of participants are to be seated and all of them being in the same subject. Q 41 Prove that 3 is irrational number. Q 42 Prove that 5 is irrational number. Q 43 Theorem: If a and b are positive integers such that a = bq + r, then every common divisor of a and b is a common divisor of b and r. Q 44 Prove that one of every three consecutive positive integers is divisible by 3. Q 45 Prove that if x and y are odd positive integers then x 2 +y 2 is even but not divisible by 4. Q 46 Use Euclid s division algorithm to find the HCF of 210 and 55. Q 47 Show that any positive integer is of the form 3q or 3q+1 or 3q+2 for some integer q. Q 48 Prove that n 2 -n is divisible by 2 for every positive integer n. Q 49 Show that the square of any positive integer is of the form 3m or 3m+1 for some integer m. Most Important Questions Q 1 Find the HCF of 17 and 6. Q 2 Find the prime factors of Q 3 State the fundamental theorem of Arithmetic. Q 4 What does Euclid s division lemma state? Q 5 Check whether 6 n will end with the digit 0 for any natural number n. Q 6 Find the HCF and LCM of 26 and 91 using fundamental theorem of arithmetic. X Mathematics C.B.S.E. Practice Papers Page 4

6 Q 7 There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same time and the same point, and go in the same direction. After how many minutes will they meet again at the starting point? Q 8 Using Euclid s division lemma find the HCF of 420 and 130. Q 9 Show that any positive odd integer is of the form 6q+ 1, or 6q+ 3 or 6q+5 where q is some integer. Q 10 Find the HCF and LCM of 72, 120, 360 using prime factorization method. Q 11 Given the HCF (306, 657)=9, find LCM (306, 657). Q 12 Using Euclid s lemma find the length of the longest tape needed to measure a room of length, breadth, height of 520 cm, 480 cm, 750cm respectively. Q 13 In a school parade, 616 students are to march behind a band of 32 senior students, both these group of students have to march in the same number of columns. what is the maximum number of columns in which they can march? X Mathematics C.B.S.E. Practice Papers Page 5

7 2. Polynomials Q 1 If the sum of the squares of the roots of the equation x 2 + 2x p = 0 is 8, find the value of p. Q 2 If one root of the equation 6x x+ m = 0 is reciprocal of the other, find the value of m. Q 3 If f(x) = ax 3 + bx 2 + cx + d, a 0, then what will be the sum of zeroes? Q 4 If the sum of the zeroes of the polynomial f(x) = 2x 3-3kx 2 + 4x - 5 is 6, then find the value of k. Q 5 Find a quadratic polynomial, if the sum and the product of the zeroes are 4 and 4 respectively. Q 6 The product of two zeroes of the polynomial f(x) = 2x 3 + 6x 2-4x + 9 is 3, then find its third zero. Q 7 If (x + 2)(2x - 1)(3x - 2) = 0, find the zeroes of the polynomial. Q 8 Find a quadratic polynomial, the sum and product of its zeroes are 1 and 6 respectively. Q 9 Find a quadratic equation, whose roots are(1+ 5) and (1-5). Q 10 Find a quadratic polynomial, the sum and product of its zeroes are 8 and 15 respectively. Q 11 Find a quadratic polynomial, the sum and product of whose zeroes are -7 and 7 respectively. Q 12 Find a quadratic polynomial, the sum and product of whose zeroes are -5 and 4 respectively. Q 13 Find a quadratic polynomial, the sum and product of whose zeroes are 3 and 2 respectively. Q 14 Find the quadratic polynomials, the sum and product of whose zeroes are 4 and 1 respectively. X Mathematics C.B.S.E. Practice Papers Page 6

8 Q 15 Q 16 If, are the roots of the equation 25x 2-10x + 1= 0, find the value of Q 17 If the zeroes of the polynomial f(x) = x 3-3x + x + 1 are a b, a and a + b find a, b. Q 18 If and are the zeroes of the polynomials f(x) = x 2 - px + q, find the value of Q 19 If and are the zeroes of a quadratic polynomial such that + = 24 and - = 8, then find the quadratic polynomial. Q 20 Q 21 Q 22 Q 23 If and are the zeroes of polynomial 9x 2-3x - 2, evaluate Q 24 A quadratic polynomial 2x 2 - mx + n has and as its two zeroes. Evaluate 2 + ( )2. X Mathematics C.B.S.E. Practice Papers Page 7

9 Q 25 Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time, and the product of its zeroes as 2, - 7 and 14 respectively. Q 26 If, are the roots of the equation 3x 2-4x + 1 = 0, find the value of Q 27 If the zeroes of the polynomial f(x) = x 3-3x 2 + x + 1 are a b, a and a + b, find the values of a and b. Q 28 If and are the zeroes of the quadratic polynomial f(x) = ax 2 + bx + c, then evaluate Q 29 If and are the zeroes of the polynomial p(x) = x 2-16x + 63, then find the value of Q 30 If the sum of the squares of the zeroes of a quadratic polynomial x 2 18x + p is 180, find the value of p. Q 31 Q 32 Q 33 If one root of the quadratic equation 2x 2 + px + 4 = 0 is 2, then find the other root and also find the value of p. Q 34 Q 35 Find a quadratic polynomial, whose zeroes are (-3+ 3) and (-3-3). X Mathematics C.B.S.E. Practice Papers Page 8

10 Q 36 If and are the roots of the equation ax 2 + bx + c = 0, find the value of ( ). Q 37 Q 38 Q 39 Form a quadratic polynomial whose one zero is 4+ 7 and the sum of zeroes is 8. Q 40 Q 41 If p and q are such that the quadratic polynomial px 2-5x + q = 0 has 10 as sum of the roots and also the product of the roots, find the values of p and q. Q 42 X Mathematics C.B.S.E. Practice Papers Page 9

11 Q 43 Q 44 Q 45 If sum of the squares of zeroes of the polynomial f(t) = t 2 8t + p is 40, find the value of p. Q 46 If and are the zeroes of the quadratic polynomial g(x) = x 2 - (a + 12)x + 3(3a + 4), such that ( + ) = ( ), then find the value of a. Q 47 Find a cubic polynomial whose zeroes are 3, 8 and 1. Q 48 Find a cubic polynomial whose zeroes are m, n and r such that (m + n + r) = - 9, (mn + nr + rm) = 6 and mnr = 56. Q 49 Find a cubic polynomial with the sum, sum of the product of it zeroes taken two at a time, and product of its zeroes as 2, -41 and 42 respectively. Q 50 If and are the zeroes of the quadratic polynomial f(x) = 4x 2 + 4x + 4, such that = 2u, find the value of u. Q 51 Divide -x 3 + 3x 2-3x + 5 by -x 2 + x - 1 and verify the division algorithm. Q 52 Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2 respectively. Q 53 Find the values of a and b so that x 4 + x 3 + 8x 2 + ax + b is divisible by x Q 54 Find the condition that the zeroes of the polynomial f(x) = x 3 - px 2 + qx - r may be a - d, a and a + d. X Mathematics C.B.S.E. Practice Papers Page 10

12 Q 55 Find the zeroes of the polynomial f(x) = x 3-5x 2 16x + 80, if its two zeroes are equal in magnitude but opposite in sign. Q 56 Q 57 Find the zeroes of the quadratic polynomial 8x x and verify the relationship between the zeroes and the coefficients of the polynomial. Q 58 Q 59 Q 60 If the square of the difference of the zeroes of the polynomial f(x) = x 2 + kx + 85 is equal to 144, evaluate the value of k. Q 61 Evaluate the values of b if m and n are the zeroes of the polynomial q(y) = by 2 35y + 12 and m 2 + n 2 = 1. Q 62 Check whether the polynomial l(x) = x 2 5x + 1 is a factor of the polynomial m(x) = 4x 4 13x 3 31x x 10 by dividing m(x) by l(x). Q 63 The zeroes of a quadratic polynomial p(x) = 2x 2 + x + n are and. Find the value of n if X Mathematics C.B.S.E. Practice Papers Page 11

13 Q 64 If and are the zeroes of the polynomial f(x) = 6x 2 + x 12, then find a quadratic polynomial whose zeroes are + 2 and + 2. Q 65 Find a polynomial whose zeroes are the reciprocals of the zeroes of the polynomial f(y) = 9y 2 18y + 8. Q 66 If the zeroes of the polynomial x 3-3x 2 + x + 1 are (a b), a and (a + b), find a and b. Q 67 Find the zeroes of the quadratic polynomial f(x) = x 2 + 5x + 6 and verify the relationship between the zeroes and the coefficients of the polynomial. Q 68 Find the zeroes of the quadratic polynomial f(x) = abx 2 + (b 2 ac)x - bc and verify the relationship between the zeroes and the coefficients of the polynomial. Q 69 Q 70 What are the quotient and the remainder when 3x 4 + 5x 3 7x 2 + 2x + 2 is divided by x 2 + 3x + 1? Q 71 If and are the zeroes of the polynomial g(y) = y 2 - y - 6, then find a quadratic polynomial whose zeroes are and Q 72 Q 73 Use the division algorithm to find the quotient q(y) and the remainder r(y) when f(y) = 12y y 2 20y 10 is divided by g(y) = 3y 2 + 2y - 5. Q 74 If (Z 3) is a factor of Z 3 + az 2 + bz + 18 and a + b = -7, find a and b. X Mathematics C.B.S.E. Practice Papers Page 12

14 Q 75 Find a quadratic polynomial whose zeroes are (2 +1) and (2 + 1), if and are the zeroes of the polynomial P(x) = 2x 2 7x + 6. Q 76 Q 77 Find the zeroes of quadratic polynomial f(x) = x 2 3x 28 and verify the relationship between the zeroes and the coefficients of the polynomial. Q 78 Find the zeroes of the quadratic polynomial f(x) = x 2 +7x+12 and verify the relationship between the zeroes and the coefficients of the polynomial. Q 79 Find the zeroes of the quadratic polynomial f(t)= t 2 15 and verify the relationship between the zeroes and the coefficient of the polynomial. Q 80 If and are the zeroes of the polynomial f(x) = x 2-5x + k such that - = 1, find the value of k. Q 81 Q 82 X Mathematics C.B.S.E. Practice Papers Page 13

15 Q 83 If and are the zeroes of the quadratic polynomial f(x) = 9x 2-9x + 2, then find the quadratic polynomial whose zeroes are ( + ) 2 and ( - ) 2. Q 84 Find the zeroes of the polynomial f(t) = t 3 3t 2-25t + 75, if its two zeroes are equal in magnitude but opposite in sign. Q 85 The product of two of the zeroes of the polynomial g(t) = t 3 + 3t 2 10t 24 is -6. Find the zeroes of g(t) Q 86 The zeroes of the cubic polynomial f(t) = t 3 6t 2 13t + 42 are (a-d),a and (a+d). Find its zeroes. Q 87 If the zeroes of the cubic polynomial g(x) = x 3 + 3x 2 13x 15 are (m n), m and (m + n), then find the values of m and n. Q 88 Divide the polynomial P(t) = 6t t 2 13t + 1 by the polynomial g(t) = 3t 1. Find the quotient and the remainder. Q 89 Divide the polynomial P(t) = 2t 3 11t t 4 by the polynomial g(t) = t 2 2t +1 and verify the division algorithm. Q 90 eroes of the quadratic olynomial p(x) = 2x 2 3x 5, then find the Polynomials whose zeroes are (2 / ) and (2 / ). Q find the values of k. Q 92 Divide 3x 2 x 3 3x + 5 by x 1 x 2, and verify the division algorithm. Q 93 Find the zeroes of the quadratic polynomial x 2-2x-8 and verify the relationship between the zeroes and the coefficients. Q 94 Divide 3x 3 + x 2 + 2x + 5 by 1+2x+ x 2, and verify the division algorithm. Q 95 Find all the zeroes of 2x 4 3x 3 3x 2 +6x 2, if two of its zeroes are 2 and 2. Q 96 Divide x 4 -x 3 +3x 2 +3 by x 2 -x+1 and verify the division algorithm. X Mathematics C.B.S.E. Practice Papers Page 14

16 Q 97 If and are the zeroes of the polynomial f(x)=x 2 +x-2, find the value of (1/ )-(1/ 2 + 2). Q 98 Find the zeroes of the quadratic polynomial x 2 +7x+12 and find the value of (1/ )-(1/ ) and 2 2. Q 99 Find the zeroes of the polynomial f(x) = x 3-5x 2-2x + 24, given that the product of its 2 zeroes is 12. Marks (5) Q 100 Find the zeroes of the polynomial f(u) = 4u 2 + 8u and verify the relationship between the zeroes and its coefficients. Marks (5) Q 101 If and are the zeroes of the quadratic polynomial 2x 2-5x +7, find a polynomial whose zeroes are and Marks (6) Q 102 If the polynomial f(x) = x 4-6x x 2-25x + 10 is divided by another polynomial x 2-2x + k, the remainder comes out to be x + a, find k and a. Marks (6) Q 103 If two zeroes of the polynomial f(x) = x 4-6x 3-26x x - 35 are (2 + 3) and (2-3), find other zeroes. Marks (6) Q 104 Find the zeroes of the polynomial f(x) = x 3-5x 2-2x + 24, given that the product of its two zeroes is 12. Marks (6) Q 105 Verify that 3, -1, - are the zeroes of the cubic polynomial p(x) = 3x 3-5x 2-11x - 3 and then verify the relationship between the zeroes and the coefficients. Marks (6) Q 106 Use the division algorithm to find the quotient q(t) and the remainder r(t) when f(t) = 8t 3 38t t +5 is divided by g(t) = 4t 3. Marks (6) Q 107 Apply the division algorithm to find the quotient and remainder on dividing f(y) by g(y) as given below: f(y) = y 4 3y 2 +4y + 5, g(y) = y y Marks (6) Q 108 Find the zeroes of the cubic polynomial m(x) = x 3 3x 2 13x + 15, it being given that 1 is one of the zeroes of m(x). X Mathematics C.B.S.E. Practice Papers Page 15

17 Marks (6) Q 109 By applying division algorithm prove that the polynomial g(x) = x 2 + 3x + 1 is a factor of the polynomial f(x) = 3x 4 + 5x 3-7 x 2 + 2x + 2. Marks (6) Q 110 It is given that 2 and - 2 are two zeroes of the polynomial f(y) = 2y 4-3y 3-3y 2 + 6y - 2, find all the zeroes of f(y). Marks (6) Q 111 Q 112 Marks (6) Marks (6) Q 113 Find the polynomial g(x), if q(x) = x 2 is the quotient and r(x) = -2x + 4 is the remainder when f(x) = x 3 3x 2 + x + 2 is divided by g(x). Marks (6) Q 114 What must be subtracted from the polynomial f(y) = 8y y 3 2y 2 + 7y 8 so as to make it exactly divisible by g(y) = 4y 2 + 3y 2? Marks (6) Q 115 A remainder r(x) = (x + a) is obtained when the polynomial f(x) = x 4 6x x 2 25x + 10 is divided by the polynomial g(x) = x 2 2x + k. Find the values of k and a. Marks (6) Most Important Questions Q 1 If sum of roots of a equation is 1 and their product is 6. Write the equation. Q 2 Form the equation whose roots are 6 and -1. Q 3 Find the quadratic equation whose roots are X Mathematics C.B.S.E. Practice Papers Page 16

18 Q 4 Q 5 Q 6 Q 7 Q 8 Q 9 Find all the Zeroes of 2x 4-3x 3-3x 2 + 6x - 2 if two of its zeroes are 2 and - 2. Q 10 Divide 3x 2 - x 3-3x + 5 by x x 2 and verify the division algorithm. Dividend = - x 3 + 3x 2-3x + 5 Divisor = - x 2 + x - 1, on dividing, we have, quotient = x- 2, remainder = 3, According to the question, we have, Q 11 Verify that 1,-1,-2 are the zeroes of the cubic polynomial p(x)= x 3 + 2x 2 - x - 2, and verify the relationship between the zeroes and the coefficients. Q 12 Find the zeroes of the polynomial x 3 5x 2 2x + 24 if give that the product of its two zeroes is 12. X Mathematics C.B.S.E. Practice Papers Page 17

19 Q 13 Find a quadratic polynomial, the sum and product of Zeroes are -3 and 2 respectively. Q 14 What will be the remainder of? Q 15 The G.C.D. of two polynomials (x 2 + ax - 28) (x + 5) and (x 2 + 8x + b)(x - 4) is (x - 4)(x + 5). Find the value of a and b. Q 16 Find the G.C.D. of the polynomials (2x 2-2x- 4) and 4(x 3-8). Q 17 Find the G.C.D. and L.C.M. of the following polynomials p(x) = 6(x - 2)(x 2 + x - 6) and, q(x) = 3(x 2 + 4x - 12). Q 18 Find the G.C.D. of the polynomials (x 2-1) and (x 2-2x + 1). Q 19 Find the G.C.D. of the polynomials (x 2-4) and (x 2)(x + 1). Q 20 Find the L.C.M. of the given polynomials 8(x 3 x 2 + x) and 28(x 3 + 1). Q 21 Write the discriminate of the quadratic equation 4x 2 ax + 2 = 0. Q 22 Polynomials of degree n having numbers of Zeros. Q 23 f(x) = 3x 2 + 2x + 5 is a polynomials of variable and of degree. Q 24 What will be the coefficient of x 3 in 9x 3-5x+20. Q 25 Factorise: Q 26 Factorise the polynomial x 2 + 2x - 6 into two linear factor. Q 27 Factorise the polynomial x 2 + 6x Q 28 Find the value of the quadratic equation 2x 2-3x - 2 at x = 1 and x = -2. Q 29 Find the value of the cubic polynomial equation x 3-6x x - 6 at x = 1, 2 and 3. Q 30 Show that x = 1 is a root of the polynomial 3x 3 4x 2 + 8x 7. Q 31 Find zero of the polynomial 2x 2 8. Q 32 The zeroes of the quadratic polynomial x 2 + 7x + 10 are (a) 2, 5 (b) 2, 5 (c) 2, 5 (d) 2, 5 Q 33 The zeroes of the polynomial x 2 3 are (a) - 3, - 3 (b) 3, - 3 (c) 3, 3 (d) 3, -3 Q 34 Write the degrees of each of the following polynomials. (i) 7x 3 + 4x 2-3x + 12, (ii) 12 - x + 2x 3, (iii) 5x - 2, (iv) 7 X Mathematics C.B.S.E. Practice Papers Page 18

20 3. Pair of Linear Equations in Two Variables Q 1 If 5x + 7y = 3 and 15x + 21y = k represent coincident lines, then find the value of k. Q 2 Solve the following pair of linear equations. 4/x + 5y = 7 3/x + 5y = 5. Q 3 Sum of two numbers in 48 and their difference is 20. Find the numbers. Q 4 If the difference of two numbers is 26 and one number is three times the other, find the numbers. Q 5 Use elimination method to find all possible solutions of the following pair of linear equations: 2x + 3y = 8 4x + 6y = 7. Q 6 Solve the systems of linear equations. 2x + y -11 =0 x - y 1=0 Q 7 Determine the value of u so that the following pair of linear equations have no solution. (3u + 1)x + 3y - 2 = 0 (u 2 + 1)x + (u - 2)y - 5 = 0. Q 8 Solve the following pair of linear equations. 2x + y = 3 2x 3y = 7 Q 9 If one number is thrice the other and their sum is 60, then find the numbers. Q 10 The denominator of a fraction is 7 more than the numerator. If 5 is added to each, the value of the resulting fraction is 1/2. Find the original fraction. Q 11 Solve the systems of linear equations x -2y = 5 2x +3y = 10 X Mathematics C.B.S.E. Practice Papers Page 19

21 Q 12 Solve the systems of linear equations. x + y = 6 x - y = 2 Q 13 Solve the systems of linear equations. x + y = 3 2x +5y = 12 Q 14 Solve the systems of linear equations. 3x + y 5 =0 2x - y 5=0 Q 15 Q 16 Solve the systems of linear equations. 2x + y = 6 2x - y +2 = 0 Q 17 The sum of two numbers is 36 and their difference is 14. Find the numbers. Q 18 Solve the pair of linear equations by substitution method. x + y = 14, x y = 4. Q 19 Solve the systems of linear equations. 3x + 2y = 7 2x + 3y = 5 Q 20 Solve the systems of linear equations. 0.4x +0.3 y = x - 0.2y = 0.8 Q 21 Find the values of k for which the system of linear equations has unique solution. 2x + 3y = 7, kx + 9y = 28. Q 22 Find the values of k for which the system of linear equations has no solution. 2x + 3y = 7, kx + 9y = 28. X Mathematics C.B.S.E. Practice Papers Page 20

22 Q 23 In triangle ABC, C = 3 B = 2( A + B). Find the three angles. Q 24 Solve the following pair of linear equations using cross multiplication method. ax + by = a - b bx - ay = a + b Q 25 The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 75 and for a journey of 15 km, the charge paid is Rs 110. How much does a person have to pay for travelling a distance of 25 km? Q 26 (a) Solve: 217x + 131y = (i) 131x + 217y = (ii) (b) For what value of u the system of equations 3x + 5y = 0 ux + 10 y = 0 has unique solution. Q 27 Solve by cross multiplication, the following pair of linear equations: x +y = 7 5x + 12y = 7 Q 28 Q 29 Solve the pair of linear equations 3x + 5 y 13 =0 2x - 5 y 7=0 and hence find the value of m for which y = mx 3. X Mathematics C.B.S.E. Practice Papers Page 21

23 Q 30 Q 31 Q 32 Q 33 The two forces are acting on a body such that their maximum and minimum value of resultant are 17 N and 13 N respectively. Find the values of forces. Q 34 Find the values of m and n for which the system of linear equations has infinite number of solutions. 2x + 3y = 7, 2mx + (m+n)y = 28 Q 35 Q 36 4 chairs and 3 tables cost Rs and 5 chairs and 2 tables cost Rs Find the cost of a chair and a table separately. X Mathematics C.B.S.E. Practice Papers Page 22

24 Q spy pens and 53 spy pencils together cost Rs 3200, while 53 spy pens and 37 spy pencils together cost Rs Find the cost of a pen and that of a pencil. Q 38 In a two digit number, the unit s digit is twice the ten s digit. If 27 is added to the number, the digits interchange their places. Find the number. Q 39 The sum of the digits of a two digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number. Q 40 I am three times as old as my son. Five years later, I shall be two and a half times as old as my son. How old am I and how old is my son? Q 41 Taxi charges in a city comprise of a fixed charge together with the charge for the distance covered. For a journey of 10 km, the charge paid is Rs 75 and for a journey of 15 km, the charge paid is Rs 110. What will a person have to pay for travelling a distance of 25 km? Q 42 A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was Rs after 4 year of service and Rs after 10 years of service, what was his starting salary and what is the annual increment? Q 43 Five years hence, father s age will be three times the age of his son. Five years ago, father was seven times as old as his son. Find their present ages. Q 44 A man sold a chair and a table together for Rs There is a profit of 25% on the chair and 10% on the table. By selling them together for Rs 1535, he could have made a profit of 10% on the chair and 25% on the table. Find the cost price of each. Marks (6) Q 45 On selling a tea set at 5% loss and a lemon set at 15% gain, a crockery seller gained Rs 7. If he sells the tea set at 5% gain and the lemon set at 10% gain, the gain is Rs 13. Find the actual price of the tea set and the lemon set. Marks (6) Q 46 A boat goes 30 km upstream and 44 km downstream is 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of boat in still water. Marks (6) Q 47 The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there? Marks (6) Q 48 (a) For what value of will the following pair of equations have infinitely many solutions? ux + 3y - (u - 3) = 0 12x + uy - u = 0 (b) For what value of p does the pair of equations has unique solution? 4x + py + 8 = 0 2x +2y + 2 = 0 Marks (6) X Mathematics C.B.S.E. Practice Papers Page 23

25 Q 49 Solve: Marks (6) Q 50 Solve the following pair of linear equations: Marks (6) Q 51 Solve the given pair of linear equations by using the method of elimination. Marks (6) Q 52 Solve the given pair of equations by using the method of substitution. 2x + 3y = 9 3x + 4y = 5. Marks (6) Most Important Questions Q 1 Solve the following system of equation by substitution method x- 2y = 5 2x + 3y =10 Q 2 Solve the following system equation x+ y =6 and x y =2. Q 3 Solve the following system of equations: x 2y =8 5x 10y =10 Q 4 5 pens and 6 pencils cost Rs.9 and 3 pens and 2 pencils cost Rs. 5. Find the cost of 1 pen and 1 pencil. Q 5 The coach of a cricket team buys 7 bats and 6 balls for Rs Later he buys 3 bats and 5 balls for Rs Find the cost of 1 bat and 2 balls. Q 6 The sum of two numbers is 8.If their sum is four times their difference, find the numbers. X Mathematics C.B.S.E. Practice Papers Page 24

26 Q 7 The sum of digits of a two-digit number is 15. The number obtained by reversing the order of the digits exceeds the given number by 9. Find the number. Q 8 Seven times a two-digit number is equal to four times the number obtained by reversing the digits.if the difference between the digits is 3. Find the number. Q 9 The sum of the numerator and the denominator of a fraction is 12. If the denominator is increased by 3 the fraction becomes 1/2. Find the fraction. Q 10 A father is 3 times as old as his son. After 12 years, his age will be twice as that of his son then. Find their present ages. Q 11 Solve the following the system of equations 8x + 5y = 9 3x + 2y = 4 by equating the coefficients Q 12 Solve the following system of equation 7(y +3) 2 (x+2) =14 4(y 2) +3(x 3) =2 Q 13 Solve the following system of equations Q 14 In the following system of equations, verify whether there exist unique solution, no solution or infinitely many solution. x 2y = 8 5x 10y =10 Q 15 Check whether the following system of equation has unique solution or not 3x 6y =8 2x + 12y = 10 Q 16 Find the value of k for which the following system of equation will have unique solution x+ 2y = 3 5x + ky +7 =0 Q 17 For what value of K will the following system of equation have infinitely many solution? Kx 2y + 6 =0 4x 3y + 9 =0 Q 18 For what values of k will the following system of equations be inconsistent 4x + 6y =11 2x + ky = 7 Q 19 For what values of a, the system of equations will have no solution? ax + 3y = a x + ay = a Q 20 Determine the value of k so that the following system of equations have no solution: (3k + 1)x + 3y 2 = 0 (k 2 + 1)x + (k 2 )y 5 = 0 Q 21 Solve: a(x + y ) + b(x - y) = a 2 - ab + b 2 a(x + y) - b(x - y) = a 2 + ab + b 2 Q 22 Solve the following system of equations ax by = a 2 + b 2 x + y = a - b X Mathematics C.B.S.E. Practice Papers Page 25

27 Q 23 Solve the following system of equations: Q 24 Solve the following system of equation Q 25 The taxi charges in a city comprises of a fixed charge together with the charge for the distance covered. For a journey of 10 km the charge paid is Rs 75 and for a journey of 15km the charge paid is Rs.110. What will a person have to pay for traveling a distance of 25 km. Q 26 A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream. Q 27 Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars? Q 28 Solve the system of equations x+ y =10 x y=4 graphically. Q 29 Solve the system of equation graphically: 3x + 6y = 3900 x + 3y = 1300 Q 30 Radha went to a shop and purchased 2 pencils and 3 erasers for Rs. 9. Her friend Kavita saw the new variety of pens and pencils and purchased 4 pencils and 6 erasers for Rs. 18. Represent the situation graphically. Q 31 The path of a bus A is given by the equation x + 2y - 4 = 0 and the path B of another bus B is given by the equation 2x + 4y 12 = 0. Represent the situation graphically. Q 32 Solve graphically the system of equations x + y = 3 3x 2y = 4 Q 33 Show the system of equations graphically 2x + 4y = 10 3x + 6y = 12 Q 34 Represent the situation graphically: 3x y = 2 9x 3y = 6 Q 35 Solve the following system of linear equations graphically: 2x y 4 =0 x + y + 1 =0 Find the points where the line meets the y axis. Q 36 Solve the following system of equations graphically: x + 3y = 6 2x 3y = 12 X Mathematics C.B.S.E. Practice Papers Page 26

28 Q 1 Solve the equation 3x 2-2x - 1 = Quadratic equation Q 2 Product of two consecutive positive integers is 240. Find the integers. Q 3 Solve the equation x 2-9 = 0. Q 4 Find the roots of the equation x 2-3x - 18 = 0. Q 5 Find the roots of the quadratic equation (x + 6) 2 = 64. Q 6 Find the values of k for which the given equation has equal roots. x 2 + k(4x + k -1) + 2 = 0 Q 7 Using quadratic formula solve the equation for x. abx 2 + (b 2 - ac)x - bc = 0 Q 8 Solve by factorisation method. Q 9 Solve the quadratic equation by factorisation method. Q 10 If one root of the quadratic equation 2x 2 + kx - 6 = 0 is 2, find the value of k. Also find the other root. Q 11 The product of two successive multiples of 5 is 300. Determine the multiples. X Mathematics C.B.S.E. Practice Papers Page 27

29 Q 12 Q 13 Solve the quadratic equation: x 2-8=0 by factorisation method. Q 14 Solve the quadratic equation: x 2 =3x by factorisation method. Q 15 Solve the quadratic equation: x 2 +6x+5=0 by factorisation method. Q 16 For what value of k, the quadratic equation 2x 2 -kx+3=0 have 3 as one of the roots? Q 17 Formulate a quadratic equation whose roots are two consecutive positive integers, such that their product is 72. And hence, find its roots. Q 18 Determine the nature of the roots of the quadratic equation: x 2 +x+1=0. Q 19 Using quadratic formula, solve the equation 6x 2 +x-2=0. Q 20 If one root of the quadratic equation 2x 2 +kx -6 =0 is 2, find the value of k. Q 21 If one of the roots of the quadratic equation x 2 + mx + 24 = 0 is 1.5, then find the value of m. Q 22 Divide 16 into two positive numbers such that twice the square of the larger part exceeds the square of the smaller part by 164. Q 23 If 4 is a root of the quadratic equation x 2 + px - 4 = 0 and the quadratic equation x 2 + px + k = 0 has equal roots, find the value of k. Q 24 If the equation 4x 2 + x(p+1) + 1 = 0 has exactly two equal roots, then find the values of p. X Mathematics C.B.S.E. Practice Papers Page 28

30 Q 25 A two digit number is such that the product of its digit is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number. Q 26 Q 27 Solve the quadratic equation: x x-6=0 by factorisation method. Q 28 Find the roots of the equation 5x 2-6x-2=0 by the method of completing the square. Q 29 If x=2 and x=3 are roots of the equation 3x 2-2kx+2m=0, find the value of k and m. Q 30 Find the values of k for the quadratic equation Kx(x-2) + 6 = 0 having two equal roots. Q 31 Find the root of the equation: Q 32 The area of a right angled triangle is 600 cm 2. If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle. Q 33 Two pipes running together can fill a cistern in (40/13) minutes. If one pipe takes 3 minutes more than the other to fill it, find the time taken by each pipe to fill the cistern. Q 34 Seven years ago Varun s age was five times the square of Swati s age. Three years hence Swati's age will be two fifth of Varun s age. Find their present ages. Q 35 A plane left 30 minutes later than the schedule time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed. X Mathematics C.B.S.E. Practice Papers Page 29

31 Q 36 Some students arranged a picnic.the budget for food was Rs.240.Because four students of the group failed to go, the cost of food to each student got increased by Rs 5.How many students went for the picnic? Q 37 A sheet of aluminium costs Rs 2oo. If the sheet was 5m longer and each metre of the sheet costs Rs 2 less, the cost of the sheet would have remained unchanged. How long is the sheet of aluminium? Q 38 If one root of the quadratic equation x 2 +kx -6 =0 is 1, find the value of k. Also find the other root. Q 39 If the roots of the equation (b-c) x 2 +(c-a) x+ (a-b)=0 are equal, then prove that 2b= a+c. Q 40 Find the two consecutive natural numbers whose sum of square is 313. Q 41 One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son s age. Find their present ages. Q 42 Q 43 A piece of cloth costs Rs 2oo. If the piece was 5m longer and each metre of cloth costs Rs 2 less, the cost of the piece would have remained unchanged. Calculate (i) How long is the piece of the cloth? (ii) What is original rate per metre? Most Important Questions Q 1 Check whether the following are quadratic equations: X Mathematics C.B.S.E. Practice Papers Page 30

32 Q 2 Solve (x + 7)(x 3) = 0 Q 3 Solve x 2-3x - 18 = 0. Q 4 Solve 3x 2 + 4x - 7 = 0. Q 5 Solve 5x 2-17x = -6. Q 6 Solve x 2-9 = 8x. Q 7 Solve. Q 8 Solve 5x 2 + x - 4 = 0 by the method of completing the square. Q 9 Solve 2x 2-3x - 7 by the method of completing the square. Q 10 Find the roots of 3x 2-7x + 4 = 0 by quadratic formula. Q 11 Find the nature of the roots of quadratic equation 3x 2 + 2x + 7 = 0. Q 12 Find the values of k for which the quadratic equation 5x 2-7x + k = 0 has real and distinct roots. Q 13 Find the values of k for which the quadratic equation 2x 2 - (k + 2)x + k = 0 has real and equal roots. Q 14 Find two successive even natural numbers whose squares have the sum 452. Q 15 The sum of two numbers is 12. If the sum of their reciprocal is 12/35, Find the two numbers. Q 16 A two-digit number is such that the product of the digits is 18. When 27 is reduced from this number the digits interchange their places. Determine the number. Q 17 If the length of a rectangle is 17 cm more than the breadth and its area is 168 sq.cm, find its dimensions. Q 18 A local train travels a distance of 450km at uniform speed. Due to some problem, speed had been 15 km/hr less. Train took 1 hour 30 minutes more to cover the same distance. Find the speed of the local train. Q 19 The difference of the squares of two numbers is 45. The Square of the smaller number is 4 times the larger number. Determine the numbers. Q 20 The sum of the first even natural numbers is 420, Find the value of n. X Mathematics C.B.S.E. Practice Papers Page 31

33 5. Arithmetic Progression Q 1 The sum of three numbers of an AP is 15. Find its first term. Q 2 Find the sum of first 20 natural numbers. Q 3 The first term and the common difference of an AP are 4 and 3 respectively. Find first four terms of the AP. Q 4 Find the arithmetic mean of 8 and 18. Q 5 The nth term of an AP is 7 4n. Find its common difference. Q 6 If the first term of an AP is a and the common difference is d, what will be the 4th term of the series? Q 7 The first and last terms of an AP are 5 and 45 respectively and the sum is 400. Find the number of terms. Q 8 The 17 th term of an AP exceeds its 10 th term by 7. Find the common difference. Q 9 Find the sum of the first 40 positive integers divisible by 6. Q 10 Find the number of terms in an AP in which the first term is 5, common difference is 3 and the last term is 83. Q 11 Find the sum of first 40 positive integers divisible by 3. Q 12 Check whether 301 is a term of the list of numbers 5,11,17,23, Q 13 Find the 20 th term from the last term of AP:3,8,13,,253. Q 14 Find the sum of odd numbers between 0 and 50. Q 15 Find the sum of first 150 positive integers. X Mathematics C.B.S.E. Practice Papers Page 32

34 Q 16 For what value of n, are the n th terms of two APs: 63,65,67, and 3,10,17, equal? Q 17 How many two-digit numbers are divisible by 4. Q 18 Find the 11 th term from the last of the AP: 10,7,4, -62. Q 19 Which term of the sequence 4,9,14,19,..is 124. Q 20 Determine 10 th term from end of the sequence 4,9,14,...,254. Q 21 How many terms are there in the sequence 3,6,9,12,,111. Q 22 If 2x, (x+10), (3x+2) are in A.P., Find the value of x. Q 23 If t n = 2n + 1, find the sum of first n terms of the AP. Q 24 Find the sum of the first n natural numbers. Q 25 The 3rd term of AP is 40 and 13th term is 0. Find the common difference. Q 26 If the sum of first n terms of an AP is S n =4n-n 2, Find AP and n th term. Q 27 The 2nd term of an AP is nine times the 5th term and the sum of the first eight terms is 56. Find the first term and the common difference. Q 28 The sum of the first 30 terms of an AP is If its last term is 98, find the first term and the common difference of the given AP. Q 29 The 3rd and the 9th terms of an AP are 4 and -8 respectively. Which term of the AP will be 0? Q 30 Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73. X Mathematics C.B.S.E. Practice Papers Page 33

35 Q 31 How many terms of the AP 3, 5, 7, 9, must be added to get the sum 120? Q 32 Find the ratio of the sum of the first 24 and 36 terms of the AP: 5, 8, 11, 14, Q 33 How many terms are there in the AP: 6, 10, 14, 18,, 174? Q 34 Find the sum of 7 th and 10th term of the series: 63, 58, 53, 48, Q 35 In the sum of n successive odd natural numbers starting from 3 is 48, find the value of n. Q 36 How many terms of AP : 24,21,18,...; must be taken so that their sum is 78? Q 37 Shobha dutta started working in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In how many years did her income reach Rs 7000? Find her total income during this time. Q 38 If m times the mth term of an A.P. is equal to n times its nth term, find its (m + n)th term. Q 39 The sum of the first 30 terms of an A.P. is 1635.if its last term is 98,find the sum of first 20 terms. Q 40 Which term of the A.P. 5,15,25, is 130 more than its 31 st term? Q 41 If the m th term of an A.P. be 1/n and n th term is 1/m, then find (mn) th term. Q 42 How many numbers of two digits are divisible by 7? Q 43 The sum of three numbers in A.P. is -3, and their product is 8. Find the numbers. Q 44 The sum of Rs.280 is to be used to award four prizes. If each prize after the first is Rs.20 less than its preceding prize, find the value of each of the prizes. Q 45 Mr. X started work in 2011 at an annual salary of 4,00,000 in a reputed company and he received a year. In which year his annual salary will be 16,00,000? 50,000 increament each X Mathematics C.B.S.E. Practice Papers Page 34

36 Most Important Questions Q 1 Which of the following progressions are APs? If they form AP, find the common difference d and write three more terms. (a) 7, 13, 19, 25,... (b) 15, 11, 7, 3,... (c) 0.3, 0.33, 0.333, ,... (d) Q 2 Write first three terms of the AP, when the first term a and the common difference d are given as follows: (a) a = -7, d = 3/2 (b) a = 2.25, d = (c) a = 2 + 2, d = 3 2 Q 3 Identify the first term and the common difference of each of the following A.P s and write next three terms: (a) 12, 9, 6,... (b) Q 4 Write the expressions for k th terms of the following A.P's and find the 100 th term of each of the following : (a) 5, 8, 11, 14,... (b) 4, 0, 4, 8.. Q 5 If the n th term of a progression is given by a n = 7 2n, show that it is an A.P. Q 6 Find the 35 th term of an A.P whose first term is 76 and common difference is 6. Q 7 The 10 th term of an A.P is 15 and the 6 th term is 1. Find the AP. Q 8 Which term of the A.P? Q 9 Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12. Q 10 Find the sum of the first 10 terms of the following A.P: 193, 189, 185,... Q 11 Find the sum of the following A.P. s: X Mathematics C.B.S.E. Practice Papers Page 35

37 Q 12 Find the sum of the following A.P. s:. Q 13 If the sum of the first 14 terms of an AP is 175 and its first term is 7, find the 21 st term. Q 14 The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. Q 15 How many terms of the AP: 33, 27, 21, must be taken so that their sum is - 96? Q 16 The sum of the series of the terms in AP is 128. If the first term is 2 and the last term is 14, Find the common difference. Q 17 Find the sum of all two digit multiples of 3. Q 18 Find the sum of all numbers between 100 and 200 which are divisible by 7. Q 19 The sum of 30 terms of a series in AP whose last term is 98, is 1635, find the first term and the common difference. Q 20 The third term of an AP is 7, and the seventh term is 2 more than 3 times the third term. Find the first term, the common difference and the sum of first 20 terms. X Mathematics C.B.S.E. Practice Papers Page 36

38 Q 1 In figure if AD BC prove that AB 2 + CD 2 = BD 2 + AC Triangles Q 2 ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove that AE 2 + CD 2 = AC 2 + DE 2. Q 3 Any point X inside the DEF is joined to its vertices. From a point P in DX, PQ is drawn parallel to DE meeting XE at Q and QR is drawn parallel to EF meeting XF in R. Prove that PR DF. Q 4 The perimeters of two similar triangles are 36 cm and 48 cm respectively. If one side of the first triangles is 9 cm, what is the corresponding side of the other triangle? Q 5 In the figure given below, DE BC. If AD=x cm, DB=x-2 cm, AE=x-1 cm, then find the value of x. X Mathematics C.B.S.E. Practice Papers Page 37

39 Q 6 In the figure given below, if AB QR, then find the length of PB. Q 7 and CD=5cm, then find the value of BC. X Mathematics C.B.S.E. Practice Papers Page 38

40 Q 9 In this figure, DE BC. If AD=x, DB=x-5, AE=x+5 and EC=x-2, then find the value of x. Q 10 In the given Q 11 X Mathematics C.B.S.E. Practice Papers Page 39

41 Q 12 In the given figure, DE BC such that AE=(1/4)AC. If AB= 6 cm, then find the value of AD. Q 14 The areas of two similar triangles are 169 cm 2 and 121 cm 2 respectively. If the longest side of the larger triangle is 26 cm, then find the longest side of the smaller triangle. Q 15 In triangle ABC, D and E are mid-points of AB and AC respectively. Find the ratio of the area of triangle ADE and area of triangle ABC. Q 16 In two similar triangles ABC and PQR, if their corresponding altitudes AD and PS are in the ratio 4 : 9, find the ratio of the areas of triangles ABC and PQR. 2 X Mathematics C.B.S.E. Practice Papers Page 40

42 Q 18 If triangle ABC is similar to triangle DEF such that BC = 3 cm, EF = 2 cm and the area of triangle ABC is equal to 81 cm 2, find the area of triangle DEF (in cm 2 ). Q 19 In figure ABC is a right triangle, right angled at B, medians AD and CE are of respective lengths 5 cm and 2 length of AC. 5 cm. Find the Q 20 ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let P be the length of perpendicular form C on AB prove that (i) cp = ab (ii) Q 21 In the figure given below, if AB QR, then find the length of PR. X Mathematics C.B.S.E. Practice Papers Page 41

43 Q 22 Q 23 In equilateral triangle ABC, if AD BC, then prove that 3AB 2 = 4AD 2 -point of CA. Prove that BD/CD=BF/CE. Q 25 The hypotenuse of a right triangle is 6 m more than the twice of the shortest side. If the third side is 2 m less than the hypotenuse, then find the sides of the triangle. Q 26 From the given figure, express x in terms of a, b, c. X Mathematics C.B.S.E. Practice Papers Page 42

44 Q 27 In the given figure, if XY AX/AB. AC and XY divides the triangular region ABC into two parts equal in area, then find the value of Q 28 In the given fig. AB MN, if PA = x 2, PM = x; PB = x 1 and PN = x + 2, find the value of x. Q 29 The perimeters of two similar triangles are 36 cm and 48 cm respectively. If one side of the first triangle is 9 cm, what is the corresponding side of the other triangle? Q 30 ABC is an isosceles triangle is which AB=AC=10cm and BC=12. PQRS is a rectangle inside the isosceles triangle. Given PQ=SR=y cm, PS=QR=2x. Prove that x = 6 -(3/4)y. X Mathematics C.B.S.E. Practice Papers Page 43

45 Q 31 In trapezium ABCD, AB DC and DC = 2AB; FE drawn parallel to AB cuts AD in F and BC in E, such that BE/EC = 3/4. Diagonal DB intersects FE at G. Prove that 7FE = 10AB. Q 32 A Point O in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D prove that OB 2 + OD 2 = OC 2 + OA 2. Q 33 ABC is a triangle in which AB =AC and D is any point in BC. Prove that AB 2 - AD 2 = BD.CD. Q 34 D, E and F are respectively mid-points of the sides of BC, CA and AB of ABC. Find the ratio of the areas of DEF and ABC. Q 35 Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side, together with twice the square of the median which bisects the third side. Q 36 If ABC is an obtuse angled triangle, obtuse angled at B and if AD AC 2 =AB 2 + BC 2 +2BCxBD. CB then Prove that Q 37 In a right triangle ABC, right angled at C, P and Q are points of the sides CA and CB respectively that divide these sides in the ratio 2: 1. Prove that (i) 9AQ 2 = 9AC 2 +4BC 2 (ii) 9BP 2 = 9BC 2 + 4AC 2 (iii) 9 (AQ 2 +BP 2 ) = 13AB 2 Q 38 ABC is a right-angled isosceles triangle, right-angled at B. AP, the bisector of BAC, intersects BC at P. Prove that AC 2 = AP 2 + 2(1+ 2) BP 2. Q 39 ABC is a right triangle, right-angled at B. Let D and E be any points on AB and BC respectively. Prove that AE 2 +CD 2 =AC 2 +DE 2. X Mathematics C.B.S.E. Practice Papers Page 44

46 Q 40 Prove that in a right triangle, square of the hypotenuse is equal to the sum of the squares of the other two sides. Q 41 In ABC, B=90. Let D and E be any points on AB and BC respectively. Then prove that AE 2 +CD 2 =AC 2 +DE 2. Q 42 In the figure, AB, EF and CD are each perpendicular to BD.Prove that (1/x)+(1/y)=(1/z). Q 43 Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle. Q 44 A girl of height 120 cm is walking from the base of a lamp-post at a speed to 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. Q 45 In PQR, PM QR, then prove that PR 2 =PQ 2 +QR 2-2QM.QR. X Mathematics C.B.S.E. Practice Papers Page 45

47 Q 46 A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. Marks (5) Q 47 Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. Marks (5) Q 48 Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Use the above theorem to prove that PR 2 = PQ 2 + QR 2-2QM.QR Marks (5) Most Important Questions Q 1 ABCD is a trapezium in which AB DC and its diagonal intersect each other at O. Show that Q 2 In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB=AC. If AD BC and EF AC, prove that ABD ECF X Mathematics C.B.S.E. Practice Papers Page 46

48 Q 3 In the given figure Q 4 In the figure ABC and DBC are two triangles on the same base BC. Prove that Q 5 In the given figure ABE ACD, prove that ADE ABC. X Mathematics C.B.S.E. Practice Papers Page 47

49 Q 6 In the given figure DE is parallel BC and AD = 4x - 3, AE = 8x - 7, BD = 3x - 1 and CE = 5x - 3, find the value of x. Q 7 ABC is an equilateral triangle of side 2a. Find each of its altitudes. Q 8 Give two different examples of pair of (i) Similar Figures (ii) Non-similar figures Q 9 In the given figure BD AC and CE AB, prove that Q 10 Any point X inside DEF is joined to its vertices. From a point P in DX, PQ is drawn parallel to DE meeting XE at Q and QR is drawn parallel to EF meeting XF in R Prove that PR DF. Q 11 Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of the corresponding medians. Q 12 ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Find the ratio of the area of triangles ABC and BDE. Q 13 In a trapezium ABCD, AB DC and DC = 2AB; FE drawn parallel to AB cuts AD in F and BC in E, such that. Diagonal DB intersects FE at G. Prove that 7FE = 10 AB. X Mathematics C.B.S.E. Practice Papers Page 48

50 Q 14 In the given figure if XY AC and XY divides the triangular region ABC into two parts equal in area. Determine. Q 15 In the given figure,m and N are points on sides AB and AC of triangle ABC such that AM = 4 cm, MB = 8cm, AN = 6 cm and NC = 12 cm. Prove that BC = 3MN. Q 16 Prove that the equilateral triangles described on the two sides of a right-angled triangle on the hypotenuse in terms of their areas. Q 17 In figure ABC is a right triangle, right angled at B. Medians AD and CE are of respective lengths 5 cm and length of AC. cm. Find the Q 18 Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. Q 19 A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. Q 20 ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove that X Mathematics C.B.S.E. Practice Papers Page 49

51 Q 21 ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular form C on AB prove that Q 22 In Figure if AD BC, prove that AB 2 + CD 2 = BD 2 + AC 2. Q 23 ABC is a triangle in which AB = AC and D is any point in BC. Prove that Q 24 In the given figure a point O is in the interior of a rectangle ABCD is joined with each of the vertices A,B, C and D prove that Q 25 Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Use the above theorem in the figure to prove that PR 2 = PQ 2 + QR 2-2QM.QR Q 26 Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12m, find the distance between their tops. Q 27 In the given figure O is a point in the interior of the triangle ABC, OE AC, OF AB, OD BC Show that OA 2 + OB 2 + OC 2 - OD 2 - OE 2 - OF 2 = AF 2 + BD 2 + CE 2 X Mathematics C.B.S.E. Practice Papers Page 50

52 7. Coordinate Geometry Q 1 Find the coordinates of the centroid of ABC with vertices A(0, 6), B(8, 12) and C(8, 0). Q 2 Find the coordinates of centre of circle if the coordinates of end points of its diameter are P(x 1, y 1 ) and Q(x 2, y 2 ). Q 3 Find the distance of a point P(x, y) from origin. Q 4 Find the relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear. Q 5 Is (0, 0) the mid-point of (0, 1) and (0, -1)? Q 6 Find the distance between the points A(4, 5) and B(-3, 2). Q 7 Find the coordinates of point P on y-axis which is equidistant from A(-5, -2) and B(3, 2). Q 8 Find the coordinates of the point of intersection of two distinct lines if both of them pass through the origin. Q 9 What will be the area of a triangle whose vertices are P(x 1, y 1 ), Q(x 2, y 2 ) and R(x 3, y 3 )? Q 10 If A(1, 4), B(3, 0) and C(2, 1) are the vertices of a triangle, then find the length of the median through C. Q 11 Find the value of k if the points A(2, 3), B(4, k) and C(6, 3) are collinear. Q 12 In what ratio does the point ( 4, 6) divide the line segment joining the points A( 6, 10) and B(3, 8)? Q 13 Find the coordinates of the point which divides the line segment joining the points (4, 3) and (8, 5) in the ratio 3 : 1 internally. Q 14 Find a point on the y-axis which is equidistant from the points A(6, 5) and B( 4, 3). Q 15 Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5). X Mathematics C.B.S.E. Practice Papers Page 51

53 Q 16 Let the points P and Q lies on x-axis and y-axis, respectively. If the abscissa of point P is -12 and the ordinate of point Q is -16, then find the length of the line segment PQ. Q 17 Find the coordinates of the point which divides the line segment joining the points (4, 3) and (8, 5) in the ratio 3 : 1 internally. Q 18 Find the area of a triangle formed by the points A(5, 2), B(4, 7) and C (7, 4). Q 19 Find the distance between the point (3,4) and the origin. Q 20 Find the distance between the point P(a+b,a-b) and Q(a-b,-a-b). Q 21 Find the distance between the points (acosa+bsina,0) and Q(0,asinA-BcosA). Q 22 Show that the points (1,-1),(5,2) and (9,5) are collinear. Q 23 Find the coordinates of the centroid of a triangle whose vertices are (1,6) (-1,2) and (3,1). Q 24 Find the area of the triangle whose vertices are A(3,2),B(11,8) and (8,12). Q 25 Q 26 Find a relation between x and y such that the point (x, y) is equidistant from the points (5, 1) and (1, 5). Q 27 Q 28 Find the ratio in which the line 3x + y 9 =0 divides the line segment joining the points A(1, 3) and B(2, 7). X Mathematics C.B.S.E. Practice Papers Page 52

54 Q 29 Find the area of the triangle formed by joining the mid points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of the area of the triangle formed to the area of the given triangle. Q 30 If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p. Q 31 The vertices of a quadrilateral PQRS are P(1, 1), Q(7, -3), R(12, 2) and S(7, 21), find its area. Q 32 Find the area of a triangle formed by joining the mid point of the sides of triangle formed by the points A(5, 2), B(4, 7) and C (7, 4).Find the ratio of there areas. Q 33 Show that the points (1, 7), (4, 2), ( 1, 1) and ( 4, 4) are the vertices of a square. Q 34 Do the points P(3, 2), Q( 2, 3) and R(2, 3) form a triangle? If so, name the type of triangle formed. Q 35 If A(5, -1), B(-3, -2) and C(-1, 8) are the vertices of triangle ABC, find the length of median through A and the coordinates of the centroid. Q 36 If the points A(6, 1), B(8, 2), C(7, 4) and D(2p, 3) are the vertices of a parallelogram, taken in order, find the value of p. Q 37 If A( 5, 7), B( 4, 5), C( 1, 6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD. Q 38 Prove that points (a,b+c), (b,c+a) and (c,a+b) are collinear. Q 39 Find the ratio in which the line 3x+y-9=0 divides the segment joining the points (1,3) and (2,7). Q 40 In what ratio does the line x+y=0 divide the line segment joining the points (2,-3) and (5,6)? X Mathematics C.B.S.E. Practice Papers Page 53

55 Q 41 If the points A(2,1) and B(1,-2) are equidistant from the point P(x,y). Prove that x+3y=0. Q 42 Find the value of x if the distance between the points (x,-1) and (3,2) is 5. Q 43 The coordinates of one end point of the diameter of a circle are (4,-1) and coordinates of the centre of circle are (1,-3). Find the radius and coordinates of the other end of the diameter. Q 44 Q 45 The three vertices of a parallelogram taken in order are (-1,0),(3,1) and (2,2) respectively. Find the coordinates of the forth vertex and hence find its area. Q 46 Show that the points (2, 8), (5, 3), (0, 0) and ( 3, 5) are the vertices of a square. Q 47 Find the coordinates of the points of trisection of the line segment joining the points A(2, 2) and B( 7, 4). Marks (5) Q 48 Prove that the points (-3, 0), (1, -3) and (4, 1) are the vertices of an isosceles right angled triangle. Find the area of this triangle. Marks (6) Most Important Questions Q 1 Find the distance between the following pair of points. (i) (-5, 3), (3, 1) (ii) (4, 5), (-3, 2) Q 2 The distance between A(1, 3) and B(x, 7) is 5. Calculate the possible values of x. Q 3 What points on the x-axis are at a distance of 5 units from the point (5, -4)? Q 4 Find the perimeter of the triangle formed by the points (5, 0), (4, -2) and (2, -1). Q 5 A point P lies on x-axis and another point Q lies on y-axis. (i) Write the ordinate of point P. (ii) Write the abscissa of point Q. (iii) If the abscissa of point P is -12 and the ordinate of point Q is -16, calculate the length of the line segment PQ. Q 6 A point is equidistant from A (-6, 4) and B (2, -8). Find its co-ordinates, if its abscissa and ordinate are equal. X Mathematics C.B.S.E. Practice Papers Page 54

56 Q 7 What point on y-axis is equidistant from the points (7, 6) and (-3, 4)? Q 8 A square has two opposite vertices at (2, 3) and (4, 1). Find the length of the side of the square. Q 9 Calculate the co-ordinates of the point P which divides the line joining A(-3, 3) and B(2, -7) internally in the ratio 2 : 3. Q 10 Find the co-ordinates of the points of trisection of the line segment joining the points (3, -3) and (6, 9). Q 11 In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7, 6)? Q 12 In what ratio the line joining the points (4, 2) and (3, -5) is divided by the x-axis? Also, find the co-ordinates of the point of division. Q 13 The line joining the points A(-3, -10) and B(-2, 6) is divided by the point P such that PB : AB =1 : 5. Find the co-ordinates of P. Q 14 Points A, B, C and D divides the line segment joining the points (5, -10) and the origin in five equal parts. Find the co-ordinates of A, B, C and D. Q 15 Show that the line segment joining the points (-5, 8) and (10, -4) is trisected by co- ordinate axes. Q 16 In what ratio does the point (1/2, -3/2) divide the line segment joining the points (3, -5) and (-7, 9)? Q 17 Show by section formula, that the points (3, -2), (5, 2) and (8, 8) are collinear. X Mathematics C.B.S.E. Practice Papers Page 55

57 8. Introduction of Trigonometry Q 1 In ABC, Find the value of sina = Q 2 In ABC, secant of A =.. Q 3 Fill in the blank: 1 + tan 2 A = Q 4 Express sina in terms of cota. Q 5 Fill in the blank: cos 2 A + = 1 Q 6 Fill in the blank: Cot(90 - A) = Q 7 X Mathematics C.B.S.E. Practice Papers Page 56

58 Q 8 In ABC, right angled at B, if AB = 4 and BC = 3, find the values of sina and tana. Q 9 Q 10 If coseca = 2, find the value of tana. Q 11 ABC is right angled at B and A = C. Is cosa = cosc? Q 12 If xcosa ysina = a, xsina + ycosa = b, prove that x 2 +y 2 =a 2 +b 2. Q 13 If sina =1/2, show that 3cosA-4cos 3 A = 0. Q 14 If 7sin 2 A+3cos 2 Q 15 If A, B are acute angles and sina= cosb, then find the value of A+B. Q 16 If tana =5/6 & A +B =90, what is the value of cotb. Q 17 If 2x=secA and 2/y=tanA, then find the value of 2(x 2-1/y 2 ). Q 18 Simplify: sinxcos 2 x-sinx. Q 19 Simplity: sina+cotacosa. Q 20 If secx= 2, find other trigonometric ratios. Q 21 If sinx=-1, find other trigonometric ratios. X Mathematics C.B.S.E. Practice Papers Page 57

59 Q 22 Simplify: (3-3sinx) (3+3sinx). Q 23 Simplify: tan 2 x-tan 2 xsin 2 x. Q 24 Simplify: sec 2 x(1-sin 2 x)+secb(sinb/tanb) Q 25 In ABC, right angled at B, BC = 3 and AC = 6. Determine BCA and BAC. Q 26 (SinA+CosecA) 2 +(CosA+SecA) 2 =7+tan 2 A+cot 2 A. Q 27 Q 28 If tana + sina = m and tana - sina = n, prove that (m 2 - n 2 ) 2 = 16mn. Q 29 Q 30 Q 31 sin 1/2 xcosx-sin 5/2 xcosx=cos 3 xsin 1/2 x Q 32 Prove that: cot 2 A/(1+cosecA)=(1-sinA)/sinA Q 33 Prove that: cot 2 A/(1+cosecA)=(1-sinA)/sinA Q X Mathematics C.B.S.E. Practice Papers Page 58

60 Q 35 Prove that: secx+tanx=cosx/(1-sinx) Q 36 sec 4 A-tan 4 A=1+2tan 2 A. Q 37 sec 6 x(secxtanx)-sec 4 x(secxtanx) =sec 5 xtan 3 x Q 38 Q 39 If SecA+tanA=p, then prove that sina=(p 2-1)/(p 2 +1) Q 40 Prove that: sina/(seca+tana-1)+cosa/(coseca+cota-1) = 1 Q 41 Q 42 Q 43 X Mathematics C.B.S.E. Practice Papers Page 59

61 Q 44 Prove that: - Q 45 Q 46 Evaluate: (sec29 /cosec61 )+2cot8 cot17 cot45 cot73 cot82 -(sin sin 2 52 ) Q 47 Prove that sinx/(cotx+cosecx)=2+sinx/(cotx-cosecx). Q 48 (sinx+1-cosx)/(cosx-1+sinx)=(1+sinx)/cosx Q 49 Marks (6) Most Important Questions Q 1 In the given figure of triangle ABC, find the value of sin of A. Q 2 Express sin A in terms of cot A. Q 3 In a ABC, right angled at B, if AB = 4 cm and BC = 3 cm, find the value of sin Aand tan A. X Mathematics C.B.S.E. Practice Papers Page 60

62 Q 4 If cosec A = 2, then find the value of tan A. Q 5 In a ABC, right angled at Band A = C. Is cos A = cos B? Q 6 Q 7 Q 8 In a ABC, right angled at B, BC = 3 cm and AC = 6 cm. Determine BCA and BAC. Q 9 If in a triangle ABC, AB = 3 cm, BC = 4 cm and ABC = 90, then find the values of sin C, cos C and tan C. Q 10 Q 11 Q 12 Q 13 In ABC right-angled at B, BC = 3, AC = 6 determine BCA and BAC. Q 14 In the fgiven ABC, find the secant of A. X Mathematics C.B.S.E. Practice Papers Page 61

63 Q 15 Fill in the blank: cos 2 A+...= 1 Q 16 Fill in the blank: 1+tan 2 A=... Q 17 Q 18 Q 19 Q 20 Prove that. Q 21 Prove that. Q 22 Q 23 Q 24 Q 25 Q 26 Q 27 If sin 4 + sin 2 = 1, then prove that tan 4 - tan 2 = 1. X Mathematics C.B.S.E. Practice Papers Page 62

64 Q 28 Q 29 Fill in the blank cos (90 -A)=... Q 30 Fill in the blank: cot(90 -A)=... Q 31 In a ABC, right angled at Band A = C. Is cos A = cos B? Q 32 Q 33 Q 34 Q 35 An equilateral triangle is inscribed in a circle of radius 6 cm, find its side. Q 36 Q 37 Q 38 Q 39 Q 40 Q 41 X Mathematics C.B.S.E. Practice Papers Page 63

65 9. Some Applications of Trigonometry Q 1 The angle of elevation of the top of a tower is 30 observation.. Find the relation between height h and distance x of the tower from point of Q 2 A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 30. Q 3 The height of a tower is 12 m. What is the length of its shadow when sun s altitude is 45? Q 4 The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30. Find the height of the tower. Q 5 If the height of a boy is equal to his shadow at any time of the day. Find the angle of elevation of sun from the head of the shadow. Q 6 A log reaches a point on a wall which is 20 m above the ground and its foot is 20 by the log with the wall. 3 m away from the wall. Find the angle made Q 7 Find the length of the ladder making an angle of 45 with a wall and whose foot is 7 m away from the wall. Q 8 If the altitude of sun is at 30, then find the height of the vertical tower that will cast a shadow of length 20 m. Q 9 The ratio of the length of a rod and its shadow is 1:1. What is the angle of elevation of sun? Q 10 A lamp post 5 3 m high casts a shadow 5 m long on the ground. Find the sun s elevation at this point. Q 11 If the ratio of the height of a girl and the length of her shadow is 3:1, what is the angle of the elevation of sun? Q 12 From a point on the ground, 20 m away from the foot of a building, the angle of elevation of the top of building is 60. What is the height of the building? X Mathematics C.B.S.E. Practice Papers Page 64

66 Q 13 A stick makes an angle of 30 with the ground. If the foot of the ladder is 5 m away from the wall, find the length of the ladder. Q 14 The height of a tower is 31 m. Calculate the length of its shadow when sun s altitude is 45. Q 15 Find the angle of elevation of sun if at any time the height of a tree is 3 times the length of its shadow. Q 16 The length of a string between a kite and a point on the ground level is 90 metres. If the string makes an angle level such that sin =8/15, how high is the kite? Assume that there is no slack in the string. with the ground Q 17 A kite is flying at a height of 75 metres from the ground level, attached to a string inclined at 60 to the horizontal. Find the height of the tower. Q 18 A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60. Find the length of the string, assuming that there is no slack in the string. Q 19 A tower stands vertically above from the ground. From a point on the ground which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60 o. Find the height of the tower. Q 20 An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45 o. What is the height of the chimney? Q 21 A bridge across a river makes an angle of 45 o with the river bank. If the length of the bridge across the river is 50 m, what is the width of the river? Q 22 A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 30. Q 23 From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45 and 60 respectively. Find the height of the tower. Q 24 A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60 and from the same point the angle of elevation of the top of the pedestal is 45. Find the height of the pedestal. X Mathematics C.B.S.E. Practice Papers Page 65

67 Q 25 The angles of elevation of the top of a tower from two points at a distance of 16 m and 8 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 12. Q 26 The pilot of a helicopter flying at an altitude of 1200 metres sees two ships sailing towards him in the same direction. The angles of depression of the ships as observed by him are 60 and 30. Find the distance between the two ships. Q 27 The angles of depression of a point on the ground as viewed from a window of a building and from the top of the building are 30 and 45 respectively. Calculate the height of the building if the height of the window is 20 metres. Q 28 The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays meet the ground at an angle of 60. Find the angle between the sun rays and the ground at the time of longer shadow. Q 29 Two men are on opposite sides of a tower. The angles of elevation of the tower as seen by them are 30 and 60 respectively. If the first man is 100 m away from the tower, how far is the other man from the tower? Q 30 The length of the string between a kite and a point on the ground is 120 m. If the string makes an angle such that tan =3/4, how high is the kite? with the level ground Q 31 Prove that if the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains unchanged. Q 32 Two ships are sailing in the sea towards a lighthouse. The angles of depression of the two ships are observed as 60 and 45 respectively. If the distance between the two ships is 100 m, find the height of the lighthouse. Q 33 The shadow of a tower standing on a level ground is found to be 40 m longer when the sun altitude is 30 o than when it is 60 o. Find the height of the tower. Q 34 From a point on the ground 40 m away from the foot of a tower, the angle of elevation of the top of the tower is 30. The angle of elevation of the top of a water tank (on the top of the tower) is 45. Find the height of the tower and depth of the tank. X Mathematics C.B.S.E. Practice Papers Page 66

68 Q 35 Determine the height of a mountain if the elevation of its top at an unknown distance from the base is 30 and at a distance 10 km further off the mountain, along the same line, the angle of elevation is 15. [Use tan15 =0.27] Q 36 At a point, the angle of elevation of a tower is such that its tangent is 5/12. On walking 240 m nearer to the tower, the tangent of the angle of elevation becomes ¾. Find the height of the tower. Q 37 The angle of elevation of a jet plane from a point A on the ground is 60. After a flight of 15 seconds, the angle of elevation changes to 30. If the jet plane is flying at a constant height of m, find the speed of the jet plane. Q 38 If the angle of elevation of a cloud from a point h metres above a lake is and the angle of depression of its reflection in the lake is, prove that the height of the cloud is h(tan +tan )/(tan -tan ). Q 39 A person standing on the bank of a river observes that the angle of elevation of the top of the tower standing on the opposite bank is When he moves 40 m away from the bank, he finds the angle of elevation to be 30. Find the height of the tree and the width of the river. (use 3= 1.732) Q 40 As observed from the top of a lighthouse, 100 m high above sea level, the angle of depression of a ship sailing directly towards it changes from 30 0 to Determine the distance travelled by the ship sailing directly towards it when the angle changes from 30 0 to Determine the distance travelled by the ship during the period of observation. (use 3= 1.732) Q 41 The angles of elevation of the top of a tower from two points at distances a and b metres from the base and in the same straight line with it are complementary, prove that the height of the tower is (ab) metres. Marks (5) Q 42 From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be 30 and45. Find the height of the hill. Marks (5) Q 43 From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30 o and 45 o, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. Marks (6) X Mathematics C.B.S.E. Practice Papers Page 67

69 Q 44 Two pillars of equal height are on either side of a road, which is 100 m wide. The angles of elevation of the top of the pillars are 60 and 30 at a point on the road between the pillars. Find the position of the point between the pillars and the height of each pillar. Marks (6) Q 45 A round balloon of radius r subtends an angle at the eye of the observer while the angle of elevation of its centre is. Prove that the height of the centre of the balloon is r sin cosec ( /2). Marks (6) Most Important Questions Q 1 A tower is 10 3 m high. Find the angle of elevation of its top from a point 10m away from its foot. Q 2 A tower stands vertically from the ground. From a point on the ground which is 15 m away from the foot of the tower the angle of elevation of the top of the tower is found to be 60 o. Find the height of the tower. Q 3 An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45 o. What is the height of the chimney? Q 4 A bridge across a river makes an angle of 45 o with the river bank. If the length of the bridge across the river is 50m, what is the width of the river? Q 5 5 A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 30. Q 6 A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60. Find the height of the tower. Q 7 The shadow of a tower standing on a level ground is found to be 40 m longer when the sun altitude is 30 o than when it is 60 o. Find the height of the tower. Q 8 From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30 o and 45 o, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. X Mathematics C.B.S.E. Practice Papers Page 68

70 Q 9 Two ships are sailing in the sea on either side of a lighthouse. The angles of depression of the two ships are observed as 60 0 and 45 0 respectively. If the distance between the two ships is 100 m find the height of the lighthouse. Q 10 From a point on the ground 40m away from the foot of a tower, the angle of elevation of the top of the tower is The angle of elevation of the top of a water tank (on the top of the tower) is 45 O. Find the height of the tower and depth of the tank. Q 11 The angles of elevation of the top of a tower from two points at distances a and b meters from the base and in the same straight line with it are complementary, prove that the height of the tower is ab meters. Q 12 From the top of the hill, the angles of depression of two consecutive kilometer stones due east are found to be 30 and45. Find the height of the hill. Q 13 Two pillars of equal height and on either side of a road, which is 100 m wide. The angles of elevation of the top of the pillars are 60 and 30 at a point on the road between the pillars. Find the position of the point between the pillars and the height of each pillar. Q 14 At what angle is the height of an object and the length of the shadow equal? Q 15 A round balloon of radius r subtends an angle at the eye of the observer while the angle of elevation of its centre is. Prove that the height of the centre of the balloon is r sin cosec( /2). X Mathematics C.B.S.E. Practice Papers Page 69

71 10. Circles Q 1 In figure, AQ and AR are tangents from A to the circle with centre O. P is a point on the circle. Prove that AB + BP = AC + CP. Q 2 Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. Q 3 In figure, ABC is circumscribing circle. Find the length of BC. Q 4 In a circle a point P is 13 cm away from centre and length of tangent drawn from P to circle is 12 cm. Find the diameter of circle. Q 5 If PA and PB are two tangents from external point P to a circle with centre O and APB = 35, find the angle OAB. Q 6 From a point A the length of the tangent to a circle is 8 cm and distance of A from the center is 10 cm. What is the diameter of circle? Q 7 From a point P a tangent is drawn to circle of diameter 48 cm. The point P is situated at a distance of 25 cm from center O of the circle then find the length of tangent. X Mathematics C.B.S.E. Practice Papers Page 70

72 Q 8 From a point Q the length of tangent to circle is 24 cm and distance Q from the center is 25 cm then find the area of circle. Q 9 What is the length of a chord of a circle of radius 17 cm which is at a distance of 8 cm from the centre? Q 10 In the given figure, if PQR=67 and SPR=70 and RP is the diameter of the circle, then find the value of QRS. Q 11 In this figure, triangle ABC is circumscribing a circle. Find the length of BC. X Mathematics C.B.S.E. Practice Papers Page 71

73 Q 12 From the figure given below, find the the perimeter of triangle PDC. Q 13 In figure, O is the centre of the Circle.AP and AQ two tangents drawn to the circle. B is a point on the tangent QA and PAB = 125, Find POQ. Q 14 Two tangents PA and PB are drawn to the circle with center O, such that APB=120 o. Prove that OP=2AP. Q 15 In the given fig OPQR is a rhombus, three of its vertices lie on a circle with centre O If the area of the rhombus is 32 3 cm 2. Find the radius of the circle. X Mathematics C.B.S.E. Practice Papers Page 72

74 Q 16 In figure, O is the centre of a circle. The area of sector OAPB is 5/18 of the area of the circle. Find the value of x. Q 17 In figure, a circle touches all the four sides of a quadrilateral ABCD with AB=6cm, BC=7cm and CD=4 cm. Find AD. Q 18 PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the distance of chord PQ from centre. Q 19 A circle is touching the side BC of ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = (Perimeter of ABC). Q 20 Prove that the parallelogram circumscribing a circle is a rhombus. X Mathematics C.B.S.E. Practice Papers Page 73

75 Q 21 A point P is 13 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 12 cm. Find the radius of the circle. Q 22 In figure, if AB = AC, prove that BE = EC. Q 23 ABC is right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle. Q 24 Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ = 2 OPQ. Q 25 Prove that the tangents drawn at the ends of a diameter of a circle are parallel. X Mathematics C.B.S.E. Practice Papers Page 74

76 Q 26 Prove that the parallelogram circumscribing a circle is rhombus. Q 27 Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre. Q 28 :- A circle is inscribed in a triangle ABC having sides 8cm, 10cm and 12cm as shown in the figure. Find AD, BE and CF. Q 29 PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length of TP. Q 30 Let PQ is the tangent at a point R on the circle with centre O. If TRQ=30, then find the value of PRS. X Mathematics C.B.S.E. Practice Papers Page 75

77 Q 31 A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm an 6 cm respectively. Find the sides AB and AC. Q 32 Prove that the lengths of the tangents drawn from an external point to a circle are equal.using the above theorem, prove that: If quadrilateral ABCD is circumscribing a circle, then AB+CD=AD+BC. Q 33 PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. Q 34 Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ=2 OPQ. Q 35 Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above,do the following: In figure,o is the centre of the two concentric circles. AB is a chord of the larger circle touchin g the smaller circle at C.Prove that AC=BC. X Mathematics C.B.S.E. Practice Papers Page 76

78 Q 36 The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle, BD is tangent to the smaller circle touching it at D. Find the length of AD. Q 37 From a point P, two tangents PA and PB are drawn to a circle with center O. If OP is equal to the diameter of the circle, prove that triangle PAB is equilateral triangle. Q 38 if a,b,c are the sides of a right triangle where c is the hypotenuse prove that the radius of the circle which touches the sides of sides of the triangle are given by r= a+b-c/2. Q 39 If three circles of radius "a" each, are drawn such that each touches the other two. Prove that the area included between them is equal to 4/25 a 2 (use = 3.14 and 3 = 1.73) Most Important Questions Q 1 Find the length of tangent drawn to a circle with radius 7 cm from a point 25 cm away from the centre of the circle. Q 2 A point P is 5 cm from the centre of a circle, the radius of the circle is 3cm. find the length of the tangent drawn to the circle from the point P. Q 3 O is the centre of the circle and PT is the tangent drawn from the point P to the circle. Secant PAB passes through the centre O of the circle. If PT = 15cm and PA = 9 cm, then find the radius of the circle. Q 4 Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. Q 5 O is the centre of the circle of radius 6 cm and PT is the tangent drawn from the point P to the circle. Secant PAB passes through the centre O of the circle. If PT = 8cm, then find the OP. Q 6 In the given figure, O is the centre of two concentric circles with radii of bigger and smaller circle are 5 cm and 3 cm respectively. AB is the chord of the bigger circle such that it touches the smaller circle at point P. Find the length of the chord.< Q 7 O is the centre of the circle and PT is the tangent drawn from the point P to the circle. Secant PAB passes through the centre O of the circle. If PT = 16 cm, radius of circle is 12 cm, then find PA. Q 8 In the given figure, O is the centre of two concentric circles with radii of bigger and smaller circle are 10 cm and r cm respectively. AB is the chord of the bigger circle such that it touches the smaller circle at point P. Length of the chord is 16 cm. Find the r. X Mathematics C.B.S.E. Practice Papers Page 77

79 Q 9 O is the centre of the circle and PT is the tangent drawn from the point P to the circle. Secant PQB passes through the centre O of the circle such that PQ = 2 cm and radius of the circle is 3 cm. Find the length of the tangent PT. Q 10 In the given figure O is the centre of the circle. PQ and PR are tangents drawn to the circle. If QPR= 70 then find QOR. Q 11 In the given figure O is the centre of the circle. PQ and PR are tangents drawn to the circle. If QPR. QOR = 160 o, then find Q 12 In the given figure O is the centre of the circle. PQ and PR are tangents drawn to the circle. Show that PQRS is a cyclic quadrilateral. Q 13 Two tangent segments BC and BD are drawn to a circle with centre O such that ÐCBD = 120 o. Prove that OB = 2BC. Q 14 A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC Q 15 The incircle of a ABC touches the sides BC CA and AB at D, E and F respectively. Prove that AF + BD + CE = AE + CD + BF =( 1/2)(Permeter of ABC) Q 16 In the given figure, ABO = 30. Find AOB and OBT. Q 17 A circle is inscribed in ABC having sides AB = 15 cm, BC = 9 cm, AC = 12 cm. find AD, BE and CF. Q 18 A circle is inscribed in ABC having sides AB = 15 cm, BC = 9 cm, AC = 12 cm. find AD, BE and CF. Q 19 A ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 5 cm and 4 cm respectively. Find the sides AB and AC. X Mathematics C.B.S.E. Practice Papers Page 78

80 11. Constructions Q 1 Construct an isosceles triangle whose base is 8 cm and altitude 4 cm. Then construct another triangle whose sides are 1 times the corresponding sides of the isosceles triangle. Q 2 Draw a triangle ABC with sides BC = 7 cm, B = 45, A = 105. Then construct a triangle whose sides are 4/3 times the corresponding sides of ABC. Q 3 Draw a right triangle in which the sides (other than hypotenuse) are of length 4 cm and 3 cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle. Q 4 Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. Q 5 Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle. Q 6 Draw a right triangle in which the sides other than hypotenuse are of length 4cm and 3cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle. Q 7 Draw a line segment of length 8.1 cm and divide it in the ratio 4:5. Measure the two parts. Marks (5) Q 8 Draw a circle of radius 8 cm. From a point 12 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Marks (5) Q 9 Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60. Marks (5) Most Important Questions Q 1 Draw a pair of tangents from a point 7 cm away from the centre of the circle whose radius is 3 cm. Q 2 Draw a pair of tangents from a point 6 cm away from the centre of the circle whose radius is 4 cm. Q 3 Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 120. Q 4 Draw a pair of tangents to a circle of radius 7 cm which are inclined to each other at an angle of 60. Justify the construction. X Mathematics C.B.S.E. Practice Papers Page 79

81 Q 5 Construct a right triangle ABC right angled at B with sides 6 cm and 8 cm. BD is perpendicular from B on AC. Construct a circle through B, C and D and tangents from A to this circle. Q 6 Construct a tangent to a circle of radius 3 cm from a point on the concentric circle of radius 5 cm. Q 7 Draw a line segment 7 cm. Taking P as centre, draw a circle of radius 4 cm. and taking Q as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. Q 8 Draw a line segment 8 cm. Taking P as centre, draw a circle of radius 4 cm. and taking Q as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. Q 9 Draw a line segment AB of length 8.8cm and divide into two segments AP and PB such that point P divides the line segment AB in the ratio of AP : PB = 5 : 4. Q 10 Draw a line segment AB of length 7.3 cm and divide into two segments AP and PB such that point P divides the line segment AB in the ratio of AP : PB = 2 : 5. Q 11 Draw a line segment of length 6.2 cm and divide it in the ratio 5 : 3. Justify the construction. Q 12 Construct a similar triangle whose sides are 2/3 to an isosceles triangle of base 5 cm and altitude 3 cm. Q 13 Construct a triangle ABC, with AB = 3 cm and B = 45 and construct a triangle whose sides are similar to of the corresponding sides of the triangle ABC. Q 14 Construct a triangle ABC with BC = 3 cm, C = 45 and B = 60 and construct a triangle whose sides are similar to of the corresponding sides of the triangle ABC. Q 15 Construct a right angled triangle with sides 3 cm and 4 cm. construct another triangle similar to it whose sides are 3/2 times the corresponding sides of first triangle. Q 16 Construct a triangle ABC with BC= 5 cm B= 60 and A= 90, then construct a triangle whose sides are 4/7 of the corresponding sides of triangle ABC. X Mathematics C.B.S.E. Practice Papers Page 80

82 12. Areas Related to Circles Q 1 The circumference of a circle exceeds its diameter by 16.8 cm. Find the radius of circle. Q 2 The length of minute hand of a clock is 21 cm. Find the angle swept by the clock in 2 minutes. Q 3 A paper is in the form of a rectangle ABCD in which AB = 40 cm and BC = 28 cm. A semi-circular portion with BC as diameter is cut off. Find the area of a remaining part. Q 4 A play ground has the shape of a rectangle, with two semi-circles on its smaller sides as diameter, added to its outside. If the sides of the rectangle are 34 m and 24 m, find the area of any semi-circular playground. Q 5 A sector is cut from a circle of radius 28 cm. The angle of the sector is 120. Find the length of its arc and area. Q 6 A chord of a circle of radius 14 cm subtends a right angle at the centre. What is the area of the minor sector? Q 7 The length of minute hand of a clock is 21 cm. find the area swept by the clock in 2 minutes. Q 8 If the radius of circle is 25 cm then find the length of arc which subtends an angle of 115 at centre. Q 9 The area enclosed between the concentric circle is 770 sqcm. If the radius of the outer circle is 21 cm, find the radius of the inner circle. Q 10 If the radius of sector is 14 cm, find the perimeter of this sector if centre angle is 45. Q 11 Find the area of a quadrant of a circle whose circumference is 22 cm. X Mathematics C.B.S.E. Practice Papers Page 81

83 Q 12 Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. Q 13 The radii of two circles are 19 m and 9m respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles. Q 14 If the perimeter of a semi-circular protractor is 66 cm, find the diameter of the protractor. Q 15 A copper wire, when bent in the form of a square, when encloses an area of 484 cm 2. If the same wire is bent in the form of a circle, find the radius of circle. Q 16 A wire is looped in the form of a circle of radius 28 cm. It is re-bent into a square form. Determine the length of the side of square. Q 17 A bicycle wheel makes 5000 revolutions in moving 11 km. Find the diameter of the wheel. Q 18 A wheel had diameter 84 cm. Find how many complete revolutions must it take to cover 792 metres. Q 19 The circumference of two circles are in the ratio 2:3. Find the ratio of their areas. Q 20 The side of a square is 10 cm. Find the area of circumscribed and inscribed circles. Q 21 The perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm. Find the area of the sector. Q 22 A copper wire, when bent in the form of a square, encloses an area of 484 sq cm. If the same wire is bent in the form of a circle, find the area enclosed by it. Q 23 The area of a circle inscribe in a equilateral triangle is 154 sq cm. Find the perimeter of the triangle. Q 24 A rectangular park is 120 m by 100 m. At the centre of the park there is a circular lawn. The area of park excluding the lawn is 8700 square metres. Find the radius of the lawn. X Mathematics C.B.S.E. Practice Papers Page 82

84 Q 25 The area enclosed between the concentric circles is 770 sq cm. If the radius of the outer circle is 21 cm, find the radius of the inner circle. Q 26 A chord AB of a circle, of radius 14 cm makes an angle of 60 at the centre of the circle. Find the area of the minor segment of the circle. Q 27 A chord of a circle subtends an angle of at the centre of the circle. The area of the minor segment cut off by the chord is oneeighth of the area of the circle. Prove that Q 28 Four equal circles, each of radius 7 cm touch each other as shown in the figure. Find the area included between them. Q 29 A road which is 7 m wide surrounded a circular park whose circumference is 352 m. Find the area of the road. Q 30 Two circles touch internally. The sum of the sum of their areas is 116 cm 2 and distance between their centres is 6 cm. Find the sum of radii of the circles. Q 31 Find the area of the sector of a circle with radius 4 cm and of angle 30. Also, find the area of the corresponding major sector. Q 32 The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find the area of the sector. Q 33 In a circle of radius 35 cm, an arc subtends an angle of 72 at the centre. Find the length of the arc and area of the sector. X Mathematics C.B.S.E. Practice Papers Page 83

85 Q 34 A chord AB of a circle of radius 15 cm makes an angle of 60 at the centre of the circle. Find the area of the major and minor segment. Q 35 Q 36 The square ABCD is divided into five equal parts, all having same area. The central part is circular and lines AE, GC, BF and HD lie along the diagonal AC and BD of the square. If AB = 22 cm, find the circumference of the circle. Q 37 In figure,two circular flower beds have been shown on two sides of a square lawn ABCD of side 56 cm. If the centre of each circular flower bed is the point of intersection O of the diagonal of the square lawn, find the sum of the areas of the lawn and the flower beds. X Mathematics C.B.S.E. Practice Papers Page 84

86 Q 38 Find the area of the shaded region in the figure,where ABCD is a square of side 14 cm. Q 39 A gulab jamun, when ready for eating, contains sugar syrup of about 30% of its volume. Find approximately how much syrup would be found in 45 such gulab jamuns, each shaped like a cylinder with two hemispherical ends, if the complete length of each of them is 5 cm and its diameter is 2.8cm. Q 40 In figure,pq=24 cm, PR=7 cm and O is the centre of the circle. Find the area of the shaded region. Q 41 In figure, ABC is a right-angled triangle, B=90,AB=28 cm and BC=21 cm. With AC as diameter a semicircle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded region. X Mathematics C.B.S.E. Practice Papers Page 85

87 Q 42 In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in figure. Find the area of the design (shaded region). Q 43 ABCD is a field in the shape of a trapezium. AB DC and ABC=90, DAB=60. Four sectors are formed with centres A,B,C and D. The radius of each sector is 17.5 m. Find the (i) total area of the four sectors. (ii)area of remaining portion given that AB=75 m and CD=50 m. Q 44 Marks (5) X Mathematics C.B.S.E. Practice Papers Page 86

88 Most Important Questions Q 1 Q 2 Q 3 Find the area of a circle whose circumference is 22 cm. Q 4 The cost of fencing a circular field at the rate of Rs 24 per meter is Rs The field is to be plouged at the rate of Rs per m 2. Find the cost of ploughing the field. Q 5 A race track is of the form of a ring whose inner circumference is 352 m and the outer circumference is 396 cm. Find the width of the track. Q 6 A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheels is 60 cm, calculate the speed per hour with which boy is cycling. Q 7 A sector is cut from a radius 21 cm. The angle of the sector is 150. Find the length of the arc and area of the sector. Q 8 The length of a minute hand of a clock is 14 cm. Find the area swept by the minute hand in one minute. Q 9 In a circle with centre O and radius 5 cm. AB is a chord of length 5 3 cm. Find the area of the sector AOB. Q 10 A sector is cut off from a circle of radius 28 cm. The angle of the sector is 120. Find the length of arc and area of sector. Q 11 The area of a circle inscibed in an equilateral triangle is 154 sq.cm. Find the perimeter of the triangle. Q 12 A road which is 7 cm wide surrounded by a circular park whose circumference is 352 m. Find the area of the road. Q 13 The area enclosed between the concentric circles is 770 sq cm. If the radius of the outer circle is 21 cm. Find the radius of the inner circle. Q 14 Q 15 X Mathematics C.B.S.E. Practice Papers Page 87

89 Q 16 The minute hand of a clock is 10 cm long. Find the area of the face of the clock described by the minute hand between 9 A.M. and 9: 35 A.M. Q 17 Find the area of the sector of the circle with radius 4 cm and of angle 30. Also find the area of the corresponding major sector ( = 3.14) Q 18 Q 19 The inner and outer diameters of ring I of a dartboard are 32 cm and 34 cm respectively and those of rings II are 19 cm and 21 cm respectively. What is the total area of these two rings. Q 20 A paper is of the form of the rectangle ABCD in which AB = 20 cm and BD= 14 cm. A semi-circular portion with BC as diameter is cut off. Find the area of the remaining part. Q 21 A circular grassy plot of land, 42 m in diameter, has a path 3.5 m wide running round it on the outside. Find the cost of gravelling the path at Rs 4 per square metre. Q 22 Q 23 X Mathematics C.B.S.E. Practice Papers Page 88

90 Q 24 The diagram shows a sector of a circle of radius r cm containing an angle of the sector is 50 cm. Prove that = (360/ )(25/r -1) degree the area of sector is A sq cm and perimetre Q 25 A chord AB of a circle of radius 14 cm makes an angle of 60 at the centre of the circle. Find the area of the minor segment of the circle Q 26 Four equal circles each of radius 7 cm touch each as shown in figure. Find the area included between them. X Mathematics C.B.S.E. Practice Papers Page 89

91 13. Surface Areas and Volumes Q 1 Two cubes each of 12 cm edge are joined end to end. Find the surface area of the resulting cuboid. Q 2 Three cubes of edge 6 cm each are joined end to end. Find the surface area of the resulting cuboid. Q 3 A solid sphere of radius 6 cm is melted and then cast into small spherical balls each of diameter 0.6 cm. Find the number of balls thus obtained. Q 4 How many spherical bullets can be made out of a solid cube of lead whose edge measures 55 cm, each bullet being 10 cm in diameter? Q 5 Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube. Q 6 The radius of the base of cone is 8 cm and height of cone is 36 cm. Find the volume of cone. Q 7 Find the curved surface area of a cone whose radius is 6 cm and height is 8 cm. Q 8 Radius of hemisphere is 2.1 cm, find the total surface area of hemisphere. Q 9 The diameters of lower and upper ends of a frustum are 16 cm and 40 cm respectively. The height is 16 cm. Find its volume. Q 10 The slant height of a frustum is 10 m and diameters of lower and upper circular ends are 14 m and 26m respectively. Find curved surface area of frustum. Q 11 The dimensions of a metallic cuboid are: 100 cm, 80 cm, 64 cm. It is melted and recast into a cube. Find the surface area of the cube. Q 12 Determine the ratio of the volume to that of a sphere which will exactly fit inside the cube. Q 13 Radius and height of a cone are 2.1 cm and 7 cm respectively. Find its volume. Q 14 Radius and height of a cylinder are 2.1 cm and 7 cm respectively. Find its volume. X Mathematics C.B.S.E. Practice Papers Page 90

92 Q 15 A solid iron pole having cyliderical portion 110 cm high and base diameter 12 cm. Find the mass of the pole, givent that the mass of 1 cm 3 of iron is 8 gm. Q 16 The difference between outside and inside surface area of cylindrical metallic pipe is 44m 2.If the pipe is made of 99 cm 3 of metal, find the sum of outer and inner radii of the pipe. Q 17 A solid toy of 8 cm 3 is dipped in a cylindrical vessel of radius 2 cm and height 4 cm which is full of water, find the volume of remaining volume of water in vessel. Q 18 A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8 cm edge is immersed in it completely. Calculate the height by which water will rise in the cylinder. Q 19 A sphere of diameter 7 cm is dropped in a right circular cylinder vessel partly filled with water. The diameter of the cylindrical vessel is 14 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel? Q 20 A hemispherical bowl of internal diameter 40 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 4 cm and height 8 cm. How many bottles are required to empty the bowl? Q 21 A conical vessel whose internal radius is 6 cm and height is 25 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height to which the water rises. Q 22 The radii of the circular ends of a conical bucket which is 49 cm high, are 35 cm and 14 cm. Find the capacity of the bucket. Q 23 Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 10 cm. Q 24 Three metal cubes whose edges measure 3 cm, 4 cm and 5 cm respectively are melted to form a single cube. Find the edge of the new cube and also find its surface area. Q 25 Find the volume of the largest right circular cone that can be fitted in a cube whose edge is 14 cm. Q 26 A solid sphere of radius 6 cm is melted and recast into small spherical balls each of diameter 0.6 cm. Find the number of balls thus obtained. X Mathematics C.B.S.E. Practice Papers Page 91

93 Q 27 How many spherical bullets can be made out of a solid cube of lead whose edges measure 55cm, each bullet being 10 cm in diameter? Q 28 Determine the ratio of the volume of cube to that of a sphere which will exactly fit inside the cube. Q cubic dm of aluminium is to be drawn into a cylindrical wire 0.50 cm in diameter. Find the length of the wire. Q 30 Water in a canal, 30 dm wide and 12 dm deep is flowing with velocity of 10 km/hr.how much area will it irrigate in 30 minutes, if 8 cm of standing water is required for irrigation. Q 31 An iron pillar has lower part in the form of a right circular cylinder and the upper part in the form of a right circular cone. The radius of the base of each of the cone and a cylinder is 8 cm. The cylindrical part is 240 cm high and conical part is 36 cm high. Find the weight of the pillar, if 1 cm 3 of iron weighs 8 grams. Q 32 The interior of a building is in the form of a right circular cylinder of diameter 4.2 m and height 4 m surmounted by a cone. The vertical height of cone is 2.1 m. Find the outer surface and volume of the building. Q 33 A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8/9 of the curved surface of the whole cone, find the ratio of the line segments into which the altitude of the cone is divided by the plane. Q 34 A toy is in the form of a cone surmounted on a hemisphere. The diameter of the base of the cone is 6 cm and its height is 4 cm. Find the cost of painting the toy at the rate of Rs. 5 per 1000 sq cm. Q 35 From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base Radius 6 cm, is hollowed out. Find the volume of the remaining solid correct to two places of decimals.also, find the total surface area of the remaining solid. Q 36 The area of an equilateral triangle is 49 3 cm 2.Taking each angular point as centre,circles are drawn with radius equal to half the length of the side of the triangle.find the area of triangle not included in the circles.[take 3=1.73] Q 37 A well of diameter 3m and 14m deep is dug. The earth taken out of it, has been evenly spread all around it in the shape of a circular ring of width 4 m to form an embankment.find the height of the embankment. X Mathematics C.B.S.E. Practice Papers Page 92

94 Q glass spheres each of radius 2 cm are packed in a cuboidal box of internal dimensions 16 cm x 8 cm x 8 cm and then the box is filled with water. Find the volume of water filled in the box. Q 39 The slant height of the frustum of a cone is 4 cm and the circumferences of its circular ends are 18 cm and 6cm.Find the curved surface area of the frustum. Q 40 A toy is in the form of a cone mounted on a hemisphere of diameter 7 cm. The total height of the toy is 14.5 cm. Find the volume and total surface area of the toy. Q 41 The diameter of bottom of a frustum of right circular cone is 10cm, and that of the top is 6 cm and height is 5 cm. Find out the area of total surface of the frustum. Q 42 A cylindrical bucket,32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. Q 43 A circus tent is cylindrical upto a height of 3 m and conical above it. If the diameter of the base is 105 m and the vertical height of the conical part is 7.26 m, find the total canvas used in making the tent. Marks (5) Q 44 A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are 5 cm and 13 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Find the surface area of the toy if the total height of the cone is 30 cm. Marks (5) Q 45 A right circular cylinder just enclosed a sphere of radius r. Find the surface area of the sphere and also curved surface area of the cylinder. Also, find their ratio. Marks (6) Most Important Questions Q 1 A solid is composed of a cylinder with hemi-spherical ends. If the length of the solid is 108 cm and the diameter of the hemispherical ends is 36 cm, find the cost of polishing the surface of the solid at the rate of 10 paise per square cm. Q 2 A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of the cylinder is 24 m. The height of the cylindrical portion is 11m while the vertex of the cone is 16 m above the ground. Find the canvas required for the tent. Q 3 A vessel is in the form of an inverted cone of height 8 cm and radius of the top, which is open, is 5 cm, It is filled with water upto the brim. When lead shots, spherical in shape with radius 0.5 cm are dropped in the water, one fourth of the water flows out. Find the number of lead shots dropped in the water. Q 4 The sweet Chamcham which is ready for eating, contains sugar syrup upto 30% of its volume. Find approximately how much sugar syrup would be found in 45 Chamchams shaped like cylinder with hemi-spherical ends if the complete length of each chamcham is 5 cm and its diameter is 2.8 cm. X Mathematics C.B.S.E. Practice Papers Page 93

95 Q 5 A hemispherical depression is cut out from one face of the cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. Q 6 A medicine capsule is in the shape of a cylinder with two hemispherical ends. The length of the entire capsule is 14mm and the diameter of the capsule is 5 mm. Find the surface area of the capsule. Q 7 A tent is in the shape of a cylinder surmounted by a conical top. If the height and the diameter of the cylindrical part are 2.1 m and 4 m, the slant height of the top is 2.5 m, find the area of the canvas used for making the tent. Q 8 An orange juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion, which reduced the capacity of glass. If the height of the glass was 10 cm, find out the apparent capacity of the glass and what was the actual capacity of glass? Q 9 A solid toy is in the form of a right circular cylinder with a hemispherical shape at one end and a cone at the other end. Their common diameter is 4.2 cm and the height of the cylindrical and the conical portion are 12 cm and 7 cm respectively. Find the volume of the solid toy. Q 10 Manav made a bird bath for his garden in the shape of a cylinder with a hemispherical depression at one end. The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total surface area of the bird bath if it is placed on the ground. Q 11 An iron pillar has lower part in the form of a right circular cylinder and the upper part in the form of a right circular cone. The radius of the base of each of the cone and a cylinder is 8 cm. The cylindrical part is 240 cm high and conical part is 36 cm high. Find the weight of the pillar if 1 cm 3 of iron weighs 8 grams. Q 12 The radii of the circular ends of a frustum of a cone of height 6 cm are 14 cm and 6 cm respectively. Find the lateral surface area and the total surface area of the frustum of the cone. Q 13 The perimeters of the ends of the frustum are 48 cm and 36 cm. If the height of the frustum be 11 cm, find the volume. Q 14 A right triangle whose sides are 3 cm and 4 cm, is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. Q 15 A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many such bottles are necessary to empty the bowl? Q 16 A tent consists of a frustum of a cone capped by a cone. If the radii of the ends of the frustum be 16 m and 10 m, the height of the frustum is 8 meters and the slant height of the conical cap is 12 meters, find the number of square meters of canvas required for the tent. ( = 22/7) Q 17 The dimensions of a metallic cuboid are 100 cm x 80 cm x 64 cm. It is melted and recast into a cube. Find the surface area of the cube. Q 18 A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of the first two balls are 1.5 cm and 2 cm respectively. Determine the diameter of the third ball. Q 19 A bucket is in the form of a frustum of a cone which holds litres of water. The radii of the top and bottom are 28 cm and 21 cm respectively. Find the height of the bucket. Q 20 The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27 of the volume of the given cone, at what height above the base is the section made? Q 21 A solid toy in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 6 cm. Determine: (i) the volume of the toy. (ii) surface area of the toy. (iii) the difference of the volumes of the cylinder and the toy, when a right circular cylinder circumscribes the toy. X Mathematics C.B.S.E. Practice Papers Page 94

96 Q 22 A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream. Q 23 Mohan and his wife Geeta are making jaggery out of sugarcane juice. They have processed the sugarcane juice to make the molasses, which is poured into moulds in the shape of a frustum of a cone having the diameters of its two circular faces as 30 cm and 35 cm and the vertical height of the mould is 14 cm. If each cm 3 of molasses has mass about 2 gram, find the mass of molasses that can be poured into each mould. X Mathematics C.B.S.E. Practice Papers Page 95

97 14. Statistics Q 1 Find the value of x, if the mode of the following data is 25: 15, 20, 25, 18, 14, 15, 25, 15, 18, 16, 20, 25, 20, x, 18 Q 2 The total number of marks scored by class in test is given below. Find the mean. Below 20 4 Below Below Below Below Q 3 Find the median class of the following data: Marks Obtained Frequency Q 4 Find the mean by direct methods for the following data: Classes Frequency Q 5 Find the median for the following data: Classes Frequency Q 6 Find the mode for the following data: Classes Frequency Q 7 X Mathematics C.B.S.E. Practice Papers Page 96

98 Q 8 Find the mean of all factors of 24. Q 9 The mean, of x 5y, x 3y, x y, x + y, x + 3y & x + 5y is 12. Find the value of x. Q 10 The mean of 10 observations is 25. If one observation, namely 25, is deleted, find the new mean. Q 11 The average marks scored by girls is 68 and that of the boys is 62. The average marks of the whole class is 64. Find the ratio of the girls & boys in the class. Q 12 The mean of 9 observations is 36. If the mean of the first 5 observations is 32 & that of the last 5 observations is 39 then find the fifth observation. Q 13 The mean of the values of 1,2,3---n is (n+1)/2, find the mean of x,2x,3x, nx. Q 14 Consider the data given below: Q 15 The ogive given below shows the marks out of 50 obtained by a group of students in an examination. Find the median mark from ogive. X Mathematics C.B.S.E. Practice Papers Page 97

99 Q 16 The ogive given below shows the marks out of 50 obtained by a group of students in an examination. Find the number of students who got more than 80% marks? Q 17 Q 18 Find the mean of a discrete frequency distribution x i /f i ; i = 1, 2, 3,..,n. Q 19 X Mathematics C.B.S.E. Practice Papers Page 98