Real Numbers Key Points

Transcription

1 . Euclid's division lemma :- Real Numbers Key Points For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b.. Euclid s division algorithms : HCF of any two positive integers a and b. With a>b is obtained as follows : Step : pply Euclid s division lemma to a and b to find q and r such that a = bq + r. 0 r > b Step : If r = 0, HCF (a, b), = b if r 0, apply Euclid s lemma to b & r 3. The Fundamental Theorem of rithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorizationi is unique, apart from the order in which the prime factors occur. p 4. Let x = q, q 0 to be a rational number, such that the prime factorization of q is of the form m 5 n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating. p 5. Let x = q, q 0 be a rational number, such that the prime factorization of q is not of the form m 5 n, where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 6. p is irrational, which p is a prime. number is called irrational if it cannot be written in the form p q where p and q are integers and q 0 Mark questions : Real Numbers Questions. number, which we can not write in the form of p q where p and q are integers and q 0 then write, what we call this number.. How many prime numbers are there between and Write whether rational number 9 has a terminating decimal expansion or nonterminating 343 repeating decimal. 4. Write the L.C.M. of the numbers 60 and 7. (300) (Maths Xth class)

2 5. Find HCF LCM for the numbers 00 and Which of the following rational numbers have terminating decimal expansion 37, 43, Express in the form of p q to is lowest term. 8. Write the H.C.F. of the numbers 3. 5 and Express 0.3 in the form of 0. Find the L.C.M. of the numbers 3 3 and 3 3. p q. Fill in teh blank of the following 945 = Write decimal representation of the rational number 3. If x = and y = 5 3. Write whether sum of two irrational number x and y is a rational or irrational number. 4. Whether the product of two irrational numbers (+ 5) and ( 5) is rational or irrational number. 5. The L.C.M. and H.C.F. of two numbers are 80 and 6 respectively. If one of the number is 30. Write the other number Marks questions 6. State fundamental theorem of the rithmetic. 7. Write two irrational numbers between and. 8. Write two rational numbers between and Decimal expansion of two real numbers is given as (i) (ii) State whether they are rational or irrational numbers. 0. Using Euclid s division alogrithm. Find HCF of 35 and 5.. Find H.C.F. and L.C.M. of numbers 40, 70 and 90.. State Euclid s division Lemma. 3. Find x and y in the following diagram. X Y 3 5 (30) (Maths Xth class)

3 4. Explain why is a composite. 5. Find the largest number whether divides 45 and 09 leaving remainder 5 in each case. 6. n army group of 308 members is to march behind an army band of 4 numbers in a parade. The two groups are to march in the same number of columns. What is the maximum number of column is which they can march. 3 Marks Questions 7. Prove that 5 is an irrational number. 8. Prove that 5 3 is an irrational number. 9. Find the L.C.M. and H.C.F. of the numbers 306 and 657 and veryfy that L.C.M. H.C.F. = Pruduct of two numbers. 30. Find the HCF of 867 and 55; by using Euclid s division alogrithm. 3. Show that 8 n cannot end with the digit O for any natural number n. 3. Divide x 4 3x + 4x + 5 by x x + and verify the division alogrithm. 33. The length, breath and height of a room are 8m 5 cm, 6m 75 and 4 m 50 cm respectively. Determine the longest rod wich can measure the three dimensions of the room exactly. 34. Find the largest number that will divide 398, 436 and 540 leaving remainder 7, and 3 respectively. 35. Find two rational and two irrational numbers between and 3.. Irrational number Non-terminating Rational nswers (30) (Maths Xth class)

4 4. Rational Non terminating recurring decimal 8. (> and <), p q, 3 ; but q 0 9. (a) Irrational (b) Rational x = 30; y = 5 4. Hint : 3 (7 + ) is a factor except , Q = x + x Remainder = Hint = and 3 =.73 We can take two rational number between and 3 e.g..5 = 3,.6 = 8 5 For irrational number <.<.<3 so <.< 3 Polynomials (303) (Maths Xth class)

5 Key Poins. Polynomials of degrees, and 3 are called linear, quadratic and cubic polynomials respectively.. quadratic polynomial in x with real coefficient is of the form ax + bx + c, where a, b, c are real number with a o. 3. The zeroes of a polynomial p(x) are precisely the x - coordinates of the points where the graph of y = p(x) intersectes of the x-axis i.e. x = a is a zero of polynomial p(x) if p (a) = polynomial can have at most the same number zeros as the degree of polynomial. 5. For quadriatic polynomial ax + bx + c (a 0) Sum of roots = b a Product of roots = c a 6. The division alogrithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that :- p(x) = g(x).q (x) + r(x), g(x) 0 wether r(x) = 0 or degree of r(x) < degree of g(x) Polynomials Mark Questions - (Q. No. 0 under HOTS). The graph of y = p(x) is shown in figure write the number of zeroes of p(x) y x x y. If x = is zero of polynomial, x 3x + k write the value of k. 3. If α, β are zeroes of quadratic polynomial x + 5x + 0. Write the value of α + β. 4. Write the degree of polynomial x 4 x 3 + 5x Write the product of zeroes of the quadratic polynomial x x + (304) (Maths Xth class)

6 6. Write the polynomial p(x) whose zeroes are and. 7. Write the quadratic polynomial the product and sum of zeroes are 3 and How many maximum zeroes can be polynomial of degree three have. 9. For what value of k, ( 4) is zero of polynomial x x (k+) 0*. Write the zeroes of the polynomial 5x x 6. Marks question (Q. No. 8 under HOTS). Find the zeroes of following quadratic polynomials and verify the relation between the zeroes and the coefficient of the polynomials (a) p(x) = x +7x + 0 (b) (c) q(x) = x +5x+3 p(x) = 6x 3 7x (d) q(s) = s 3. Find the quadratic polynomial whose zeroes are 3+ and 3 3. Find the zeroes of quadric polynomial x +4 x+6 4. Find the quadratic polynomial whose zeroes are and Find the zeroes of polynomeal p(x) = x. 6. Find the quadratic polynomial whose zeroes are 3 and Find the quadratic polynomial whose product and sum of zeroes are 7, and 3. 8*. If α, β are the zeroes of the polynomial p(x) = x 7x +3. Find the value of α + β. 3 marks questions (Q. No. 9, 3 and 6 under HOTS) 9*. Find all the zeroes of x 4 3x 3 3x. If it is given that two of its zeroes are and. 0. Divide 4x 4 + x 3 x 3x + 4 by x + 5x 3 and veryfy the division algorithm.. Find the value of p for which the polynomial x 3 + 4x px + 8 is exactly divisible by x.. If x+a is a factor of x + ax 0 find the value of a. 3*. Find all zeroes of the polynomial x 4 + x 3 7x 5x + 0. If its two zeroes are 5 and Find the quadratic polynomial sum of whose zeroes is 8 and product is. Hence find the zeroes of the polynomial. 5. Using division algorithm find quotient and remainder on dividing p(x) by g(x) if (a) p(x) = (x) 3 + 3x 5x+6, g(x) = x 3 (b) p(x) = x 3 + 4, g (x) = x+ 6*. If two zeroes of the polynomial x 4 6x 3 6x + 38 x 35 are ± 3. Find other zeroes. (305) (Maths Xth class)

7 nswers. 3. k = x x 7. x + 3x Three 9. k = 9 0. /3, 3/5. (a) (, 5) (b) (, 3/) (e) ( 3, 3 ) (d) ± 3. x 6x , 3 5. ± 6. x 5x 7. x + 3x /4 9.,.,, 0. (x + 5x 3) (x + x) + 4. p = 6. a = 3.., 5, 5 4. x 8x +, zeroes 6, 5. (a) x + 3x +, , 7 (b) x x +, 3 (306) (Maths Xth class)

8 Pair of Linear Equation in two variable Key points. The most general form of a pair of linear equations is : a x + b y + c = 0 a x ± b y + c = 0 Where a, a, b, b, c, c are real numbers and a + b 0, a +b 0. The graph of a pair of linear equations in two variables is represented by two lines ; (i) If the lines intersect at a point, the pair of equations is consistent. The point of intersection gives the unique solution of the equation. (ii) If the lines coincide, then there are infinitely many solutions. The pair of equations is consistent. Each point on the line will be a solution. (iii) If the lines are parallel, the pair of the linear equations has no solution. The pair of linear equations is inconsistent. 3. If a pair of linear equations is given by a x + b y + c = 0 and a x + b y + c = 0 (i) a a (ii) a a b b the pair of linear equations is consistent. (Unique solution) b = c b c (iii) a b = = c a b c many solutions). the pair of linear equations is inconsistent (No solution) the pair of linear equations is dependent & consistent (infinitely Pair of linear equation in two variables Mark question. Express y is terms of x in the equation. 3x y = 5. For what value of k the pair of linear equation? kx y = 3 3x + y = 5 has unique solutions 3. For what value of m the pair of linear equation and represent parallel lines? 3x + my 8 = 0 3x 5y + 7 = 0 4. For what value of k the following pair of linear equation x + 3y = 7 4x + ky = 4 has infinite many solutions. (307) (Maths Xth class)

9 5. Write the condition for which pair of linear equations. a x + b y + c = 0 a x + b y + c = 0 has no solution 6. The difference between two number is 36. One number is four times of other. Form pair of linear equation of this word problem. 7. How many solution of the equation 5x 4y + 6 = 0 are possible. 8. Write value of x if : x + y = 5 x y = 3 Marks Questions (Q. No. 7, 8 and 9 under HOTS) Solve for x and y (qn. 9-5) 9. x 4y = x + y + 3 = 0 x + + y + = 0. 4x 3y 8 = 0 6x y 9 3 = 0. x + 3 y = 3 5 x 4 y = 3. 3x + 43y = 7 43x + 3y = x + 4 y = 4 x + y = 5. x + 3y = 3x + y = For what value of p will be the following pair of linear equations have unique solutions. 3x = 4 y y = 3 px (308) (Maths Xth class)

10 7*. Solve for x and y, by cross multiplication method ax + by + a = 0 bx + ay + b = 0 Solve for x and y 8*. 3(x+y) = 7xy 9*. 0. 3(x+3y) = xy x a + y b = ax by = a b 44 x + y + 30 x y = 0 55 x + y + 40 x y = 3 3 marks questions (Q. No. 8, 3, 3, 33 and 35 under HOTS). Gaphically show that the system of linear equation 4x + 6y - 0 = 0 x + 3y + 3 = 0 has no solution. Determine graphically whether the system of linear equation 3x + y = 5 3x y = has unique solution. 3. Show graphically that the following linear equations have infinite solution y = 4x 6 x = y Solve graphically for x and y, x y = 4, x + y + = 0 Find the points of x-axis where the lines intersect. 5. number consists of two digits whose sum is 9. If 7 is added to the number the digit are reversed. Find the number. 6. The ratio of income of and is 9:7 and the ratiio of their expenditure is 4:3 if each of them saves Rs. 000 yearly. Find their annual income. 7. fraction become when is substracted from numerator and is added in denominator. It becomes when 7 is substracted from numerator and is substracted from denominator. 3 Find the fraction. 8*. person travels 600 km partly by train and partly by car. He takes 8 hours, if he travels (309) (Maths Xth class)

11 0 km. by train and rest by car. He takes 0 minutes longer if he travels 00 km by train and the rest by the car. Find the speed of the train and the car separately. 9. The taxi charges in a city comporised of a fixed charge for st km. together with the charge for distance covered. For a journey of 5 km the charge paid is Rs. 5 and for a journey of 7 km the charged for paid is Rs. 99. What a person has to pay for a distance of 50 km. 30. Place and are 80 km a part from each other on a high way. car starts from and other from at the same time. If they move in same direction they meat in 8 hours. If they move in opposite direction they meet in hour 0 minutes. Find the speed of the cars. 3*. Solve the following pair of linear equations. px + qx = p q qx py = p + q 3*. The students of a class are made to stand in rows. If 4 students are extra in a row, there would be rows less. If 4 students are less in a row there would be 4 more rows. Find the number of students be in the class. 33*. Solve for x and y ax b by a = a + b ax by = ab 34. father s age is thrice the sum of ages of two children. fter five year his age wil be twice the sum of children s ages. How old is father at present? 35*. Sum of two numbers is 6 and the sum of their reciprocals is. Find these numbers boat goes 6 km. upstream and 4 km. down stream in 6 hours. lso it covers km upstream and 36 km down stream in the same time. Find the speed of boat in still water and that of the stream men and boys can finish a piece of work in 5 days, while 6 men and 8 boys can finish it in 7 days. Find the time taken by man alone and that by boy alone to finish the same work.. y = 3x 5. k 6 3. m = 5 4. k = 6 5. a a = b b 6. x y = 36 x 4y = 0 + c c 7. Infinite solution nswers (30) (Maths Xth class)

12 8. x = 4 y = 9. x = y = 3 0. x = 7, y = 3. x = 3/ y = /3. x = y = 3 3. x =, y = 4. x = y = /5 5. x = y 3 6. p 3/ 7. x = y = 0 8. x = y = 3/ 9. x = a y = b 0. x = 8 y = 3 5. Number = (a) Rs (b) Rs Speed of train 60 km/h 30. Speed of car = 80 k/b 3. x = y = 3. Number of student 96 Here let no. of rows = y No. of students = x Total students = xy (x )(x+4) = xy (x+4)(x 4) = xy 33. x = b y a years 35. and Speed of boat = 8 km/b Speed of stream = 4km/b 37. Man - 70 days, oys - 40 days (3) (Maths Xth class)

13 Quadratic Equation Key Points. The equation ax + bx + c = 0, a 0 is the standard form of a quadratic equation, where a, b, c are real numbers.. real number α is said to be a root of the quadratic equation ax + bx + c = 0. If 9x + bx + c = 0, the zeroes of the quadratic polynomial ax + bx + c and the roots of the quadratic equation ax + bx + c = 0 are the same. 3. If we can factorize ax + bx + c = 0, a 0 into a product of two linear factors, then the roots of the quadratic equation ax + bx + c = 0 can be found by equating each factors to zero. 4. quadratic equation can also be solved by the method of completing the square. 5. quadratic formula : the roots of a quadratic equation ax + bx + c = 0 are given by b ± b 4ac provided that b a 4ac 0 6. quadratic equation ax + bx + c = 0 has :- (i) Two distinct and real roots if b 4ac > 0 (ii) Two equal and real roots, if b 4ac = 0 (iii) Two roots are not real, if b 4ac < 0 Quadratic Equations Questions Mark questions (Q. No. 9 and 0 under HOTS). The product of two consecutive odd integers is 63. Represent this in form of mathematical equation.. Write the discriminant of the quadratic equation 3x 5x =0 3. If x = is a root of the equation 3x 5x + k = 0. Write the value of k. 4. For what value of k quadratic equation x kx + 4 = 0 has equal roots. 5. Write the nature of the roots of equation 4x x + 8 = 0 6. Show that x = 3 is the solution of the quadratic equation x + 6x + 9 = 0 7. Write the value of x in equation x 4 = 0 8. Form a quadratic equation whose roots are 3 and 4. 9*. For what value of m for which x = /3 is a solution of mx x = 0 0*. For what value of p the quadratic equation x 6x + p = 0 has real roots. (3) (Maths Xth class)

14 Marks questions (Q, No. 9 and 0 under HOTS). Solve the following quadratic equations 3y + (6 + 4a) y + 8a = 0. Find the value of a and b such that x =, x = are the solution of the quadratic equation. x + ax + b = 0 3. Find the roots of equation, x + x = 3 4. Find the value of p if equation x + px + 3 = 0 has two equal roots. 5. Find the value of k for which equation 5kx + 8x + = 0 has two equal roots. 6. Find the roots of equation a b x + (b a ) x = 0 7. Find the roots of equation x +7x + 5 = 0 8. Divide 5 in to two parts sucvh athat their product is *. If the roots of the equation (b c)x + (c a)x + (a b) = 0 are equal then prove that b = a+c 0*. Find k so that equation [k+4]x + (k+)x + = 0 has equal roots. 3 marks question (Q. No. 8, 9 and 30 under HOTS). Solve the following quadratic equation by the method of completing square. (a) x 5x + 3 = 0 (b) ax + bx + c = 0. Solve the following quadratic equation by using quadratic formula. abx + (b ac) x bc = 0 3. Find the discriminant of the equation 3x x + 3 roots, find the roots if they are real. Solve the following equations (4 30) 4. p x + (p q )x q = x x + x 3 x 4 = 3 3 x + x + x x + = 5 6, x x+ + x+ = 4 x+4 6. x + 5 3x + 6 = 0 7. x + x = 6 x x, 4 x 0,. 8*. 3a x + 8abx + 4b = 0 a 0 = 0 and hence find the nature of (33) (Maths Xth class)

15 9*. x x + + x + x 30*. = a + b + x = a + b + x 6. Marks questions (Q. No. 3, 34, 35 adn 39 under HOTS) 3*. two digit number is such that the product of digit is 35, when 8 is added to the number the digits interchange their places. Find the number. 3. train travels 360 km at uniform speed. If the speed had been 5 km/h more it would have taken hour less for the same journey. Find the speed of the train. 33. Find two numbers whose sum is 7 and product is 8. 34*. motorboat whose speed is 9 km/h is still water goes km. down stream and comes back in a total time 3 hours. Find the speed of the stream. 35*. The hypotenuse of right angled triangle is 6cm more than twice the shortest side. If the third side is cm less than the hypotenuse find the sides of the triangle. 36. Sum of two number is 5, if sum of their recipocal is 3. Find the numbers Rs were divided equally among a certain number of students. Had there been 0 more students, each would have got Rs. 60 less. Find the original number of students. 38. In a class test sum of Kamal s marks in Mathematics and English is 40. Had he got 3 marks more in mathematics and 4 marks less in English, the product of his marks would have been 360. Find his marks in two subject separately. 39*. Solve for x 9x 9 (a+b) x+ (a + 5ab + b ) = y a reduction of Rs. per kg in the price of sugar. Ram Lal can kg sugar more for Rs. 44. Find the original price of sugar per kg. 4. n aeroplane takes an hour less for a journey of 00 km. if the speed is increased by 00 km/h from its usual speed. Find the usual speed.. (x )(x+) = 63. D = k = 4. K = = 4 5. D 0 real number 6. 3 is solution 7. x = ± nswers (34) (Maths Xth class)

16 8. x + x = 0 9. m = 6 0. p < 9., 4a 3. a =, b = , 4. p = ± 6 5. k = 8/ a, b 8. 9, 4 5, 9. proof 0. (a) 3. (b) x = b + b 4ac a x =. c/b, b/a b b 4ac a , 3 4. q, p 5. +, 6. 3, 3/ 7. 3, b a, b 3a 9. 4, x = a, b (35) (Maths Xth class)

17 km/h 33. 3, k/h cm, 4 cm, 0 cm 36. 5, students 38. Maths - Eng Maths - Eng. = 8 a+b 3, a+b Rs km/h (36) (Maths Xth class)

18 rithmetic Progression Key Points. Sequence : set of numbers arranged in some definite order and formed according to some rules is called a sequence.. Progression : The sequence that follows a certain pattern is called progression. 3. rithmetic progression : sequence in which the difference obtained by substracting from any term its preceeding term is consistent throughout, is called on arithmetic sequence or arithmetic progression (.P.) The general form of an.p. is a, a+d, a+d,... (a : first term, d : common difference) 4. General Term : if a is the first term and d is common difference in an.p., then n th term (general term) is given by a n = a + (n )d 5. SUM OF n TERMS OF N.P. : If a is the first term and d is the common difference of an.p., then sum of first n terms is given by S n = n { a+(n )d} If l is the last term of a finite.p., then the sum is given by s n = n {a +l} 6. (i) If a n is given, then common difference d = a n a n (ii) If s n is given, then n th term is given by a n = s n s n iii) if a, b, c are in.p., then b = a + c (iv) If a sequence has n terms, its r th term from the end = (n r + ) th term from the beginning. Mark Questions. If n th term of an.p. is 5 3n, write the common difference of this.p.. Which term of the.p. 7, 3,, 5,... is 73? 3. If 5, k 3, 9 are in.p., then write the value of k 4. Is 0, a term of the.p. 5,, 9,...? 5. Write 3th term of the.p. 3, 8, Write n th term of the.p. 5,,, Is 7 7, 7 7, 7 3,....P.? If yes, write the common difference The first term of an.p. is 3 and sixth term is 3. Write common difference o the.p. 9. Write the first term and common difference of the.p. 7.3, 6.9, Write first three terms of an.p., whose second term is 4 and common difference is. (37) (Maths Xth class)

19 . Write the sum of first 0 natural numbers.. Is, 8, 8, 3... an.p.? If yes, then write next two terms. 3. Write the missing terms of the.p. 3,,, 3, 4. Write 9th term from the end of the.p. 7,, 5,..., If the sum of n terms of an.p. is n, write its n th term. 6. For what value of m the numbers m, m, m+ are in.p.? (m 0) m 7. The sum of 6 th and 7 th terms of an.p. is 39 and common difference is 3. Write its first term. 8. The sum of 3 numbers is.p. is 30. If the greatest number is 3, write its common difference. 9. Write an.p. whose third term is 6 and the difference of the 9 th term from th term is. 0. Write the sum of first n even natural number. Marks Questions (Q. No. 36 to 40 under HOTS). If 9, 4, 9,... is an.p., then find a 30 a 0.. Find the.p. whose second term is 0 and the sixth term exceeds the fourth term by. 3. The sum of 3 rd and 7 th terms of an.p. is 4 and the sum of 5 th and 9 th terms is 34. Find the first term and common difference of the.p. 4. Which term of the.p. 4, 38, 35,... is the first negative term? 5. If the sum of first n terms of an.p. is n + n, then find n th term and common difference of the.p. 6. How many terms of.p., 0, 8... should be taken so that their sum is zero? 7. Find the sum of odd positive integers less than If 9 times of 9 th term is equal to 8 times the 8th term of an.p. Find its 7th term. 9. Which term of.p. 5, 3,, 9... will be 48 less than its 9 th term. 30. How many two digits numbers between 4 and 0 are divisible by 6? 3. Find an.p. whose 3rd term is 3 and 6 th term is. 3. The angles of triangle are in.p. If the smallest angle is one fifth the sum of other two angles. Find the angles. 33. Nidhi, starts a game and scores 00 points in the first attempt and she increases the points by 40 in each attempt. How many points will she score in teh 30th attempt? 34. Find k, if the given value of x is the k th term of the.p. 3, 7,,..., x = nurag saves Re. on day, Rs. on day, Rs. 3, on day 3 and so on. How much money will he save in the month of feb. 00. (38) (Maths Xth class)

20 36*. Find an.p. of 8 terms, whose first term is 37*. For an.p. a, a, a 3,... if a b 4 7 = 3, then find a a and last term is *. The fourth term of an.p. is equal to 3 times the first term and the seventh term exceeds twice the third term by. Find the first term and the common difference of the.p. 39*. If nd, 3 st and last term of an.p. are 3 4, and 3 terms in the.p respectively. Find the number of 40*. For what value of n, are the n th terms of two.p.s, 0, 8,... and 68, 70, 7... equal? lso find the term. 3 Marks Questions (Q. No. 56 to 60 under HOTS) 4. Find the sum of.p If p th and q th term of an.p. are q and p respectively, then find the sum of pq terms. 43. Find the sum of the first 40 terms of an.p., whose n th term is 3 n. 44. If n th term of an.p. is 4, common difference is and sum of n terms is 4, then find first term and number of terms. 45. Find the sum of all the three digits numbers each of which leaves the remainder 3 when divided by The sum of first six terms of an.p. is 4. The ratio of the 0th term to the 30th term is :3. Find first term and th term of the.p. 47. The sum of three numbers in.p. is 4 and their product is 440. Find the numbers. 48. The sum of n terms of two.p. s are in the ratio 3n + 8 : 7n + 5. Find the ratio of their th terms. 49. The sum of first 6 terms of an.p. is 58 and sum of next 6 terms is 55. Find the first term and common difference of the.p. 50. The sum of first 8 terms of an.p. is 40 and sum of first 4 terms is 996. Find the.p. 5. If p th, q th and r th terms of an.p. are l,m and n respectively. then proe that p(m n) + q (n l) + r(l m) = 0 5 Find the number of terms of the.p. 57, 54, 5... so that their sum is 570. Expain the double answer. 53. If the sum of first 0 terms of an.p. is one third of the sum of next 0 terms. If first term is, then find the sum of first 30 terms. 54. picnic group for Manali consists of students whose ages are in.p., the common diference being 3 months. If the youngest students Rohit is just years old and the sum of ages of all the students is 375 years. Find the number of students in the group. 55. The digits of a three digits positive number are in.p. and the sum of digits is 5. On (39) (Maths Xth class)

21 subtracting 594 from the number the digits are inerchanged, find the number. 56*. If the roots of the equation a(b c)x + b(c a)x + c (a b) = 0 are equal, then show that a, b, c are in.p. 57*. If the sum of m terms of an.p. is n and the sum of n terms is m, then show that sum of (m + n) terms is (m +n). 58*. The sum of 5 th and 9 th terms of an.p. is 8 and their product is 5. Find the sum of first 8 terms of the.p. 59*. If m th and n th terms of an.p. are a & b respectively, then show that the sum of its (m+n) terms is m + n { a + b + a b m n } 60*. nita arranged balls in rows to form an equilateral triangle. The first row consists of one ball, the second of two balls, and so on. If 669 more balls are added, then all the balls can be arranged in the shape of a square and each of its sides then contains 8 ball less than each side of the triangle. Determine the initial number of balls, nita has k = 5 4. NO n = 8 nswers 7. Yes, common difference = , common difference = , 4, Yes, 5, 6 3. and n 6. m = , 0 and 3 (30) (Maths Xth class)

22 9. 4, 0, 6, n + n , 0, 6, First term = 3, d = th term 5. n th term = 4n, common difference = terms Zero 9. 3 th term , 8, 3, , 60 0 and k = 35. Rs , 5 6, 7 6, (Hint. a + 3d a + 6d = 3 ) 38. First term = Common difference = 40. n =, term = (pq + ) 44. First term = 8 total terms = First term = th term = 47. 5, 8, 48. 7:6 (3) (Maths Xth class)

23 49. First term = 3 common difference = , 0, 3, (hint : a n = a + (n ) d) 5. 9 or 0 (0th term is zero) studetns (Hint. : in quadratic equation, D = 0) (for equal roots) 57. Hint : s n = n {a + (n )d} 58. 5, 45, {d = ± } 59. Hint : s n = n {a +(n )d} balls. Trignometry (3) (Maths Xth class)

24 Key Points. Trignometrical Ratios :- In C, = 90 0 for angle C sin = perpendicular Hypotenuse cos = ase Hypotenuse tan = Perpendicular ase cot = ase Perpendicular Hypotenuse ase Perpendicular sec. = Hypotenuse ase. Reciprocal Relations : sinθ = cosθ = tanθ = cosec = cosec θ, cosecθ = sin θ' sec θ, sec θ = cos θ' cot θ, cotθ = tan θ' Hypotenuse Perpendicular 3. Quotient Relations : tanθ = 4. Identities : Sin θ, cos θ' cos θ cot θ = sin θ' sin θ + cos θ = sin θ = cos θ and cos θ = sin θ + tan θ = sec θ tan θ = sec θ and sec θ tan θ = +cot θ = cosec θ cot θ = cosec θ and cosec θ cot θ = 5. TRIGNOMETRIC RTIOS OF SOME SPECIFIC NGLES : sin 0 3 cos 3 0 (33) (Maths Xth class)

25 tan 0 3 cosec Not defined sec 3 cot Not defined 3 3 Not defined 3 Not defined Trignometric ratios of complementary angles : sin (90 0 θ) = cosθ cos(90 0 θ) = sin θ tan (90 0 θ) = cot θ cot (90 0 θ) = tan θ sec 90 0 θ) = cosec θ cosec (90 0 θ) = sec θ 7. Line of sight :- The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. 8. ngle of elevation : The angle of elevation is the angle formed by the line of sight with the horizontal when it is above the horizontal level i.e. the case when we raise our head to look at the object. 9. ngle of depression : The angle of depression is the angle formed by the line of sight with the horizontal when it is below the horizontal i.e. case when we lower our head to look at the object. mark questions. Write tan in terms of sin. In PQR, Q = 90 0 and sin R = 3 5, what is the value of cos P? 3. If and are acute angles and sin = Cos, then write the value of Write the value of 7sec cot If sinθ =, write the value of sinθ + cosecθ. 6. Write the value of sin6 0 sin8 0 cos6 0 cos8 0 (34) (Maths Xth class)

26 7. Express cosec tan 6 0 in terms of trignometrical ratios of angles between 0 0 and If 4 cot θ = 3, then write the value of tanθ + cot θ 9. If sin θ cos θ = 0, o 0 < θ< 90 0, then write the value of θ 0. What is the value of sin 4 0 cos 49 0?. Write the value of cot sec Write the value of sin sin If θ = 30 0, then write the value of sinθ + cos θ 4 Write the value of sin(90 0 θ) cosθ + cos(90 0 θ) sinθ. 5. If tan (3x 5 0 ) =, then write the value of x. 6. In C, write sin + in terms of angle C. 7. Write the value of tan (55 0 θ) cot ( θ) 8. If tan θ + cot θ = 5, then what is the value of tan θ + cot θ? 9. If θ = 30 0, then write the value of tan θ 0. If θ = 45 0, then what is the value of cosec θ + 3sec θ? Marks questions (Question No. 36 to 40 under HOTS). If θ = 30 0, find the falue of tan θ + tan θ. If sin (+) = and cos ( ) = 3, 00 (+) 90 0,, then find the values of and. 3. If sinθ = cos(θ 36 0 ), θ and θ 36 0 are acute angles. Find the value of θ 4. If θ = 30 0, then verify; sin3θ = 3sin θ 4 sin 3 θ 5. If tan (3 0 + θ) = cotθ, θ and (3 0 + θ) are acute angles, find the values of θ 6. Simplify : tan cos sec cos If tan θ =, then find the value of tan θ + tan θ 8. If 4 cot θ = 3, find the value of 3 cos θ + 4sin θ 5 cos θ 3 sinθ 9. Prove that, sec 4 θ sec θ = tan θ + tan 4 θ 30. If sin θ + sin θ =, then find the value of cos θ + cos 4 θ 3. Find the value of (35) (Maths Xth class)

27 0 sin cos 8 3. Find the value of tan 73 cot sin 8. sec 6 7sec 3 7 cot sin cos cos 53. cosec sec 3 7 cot Find the value of sin60 0 geometrically 34. Find the value of 0 cosec ( 90 θ) tanθ tan 30. sec 5, sin (cos 40 + cos 50 ) 3(cosec 70 tan 0 ) If tan (+) = 3 and tan( ) = 3, 00 (+) 90 0, >, then find the value of cos ( 3) 36*. Find the value of sin sin sin sin sin *. In MNR, N = 90 0, MN = 8cm, RN MN = 7 cm. Find the value of sinr, tanr and secm. 38*. If sin(+) = sin cos + cos sin, then find the values of sin75 0 and cos *. If sin (3x 5 0 ) = 3, then find the value of sin (x +0 0 ) + tan (x+5 0 ). 40*. If x = m sinα. cosβ, y = m sinα sinβ and z = m cos α,then prove that x +y +z = m 3 Marks questions (question No. 56 to 60 under HOTS) 4. Prove that 4. Find the value of cos tan + cos = sin cos cot tan (90 0 θ) cot θ sec (90 0 θ) cosec θ (cot 7 sec 63 ) cot 6 cot4 cot45 cot49 cot sec 4 sin tan 30 0 sin 6 + sin Prove that cosec cot sin = sin cosec cot 44. If sec θ + tan θ = 4, then prove that cos θ = 8 7 (36) (Maths Xth class)

28 45. Prove that sec θ + sec θ + sec θ + = cosec θ sec θ 46. Prove that (sinθ + cosecθ) + (cosθ + secθ) = tan θ + cot θ + 7 sec θ cos θ+ 47. Prove that = + sin θ sec θ cos θ cos θ 48. Pove that ( + cot cosec ) ( + tan + Sec ) = 49. Prove that ( + tan θ ) ( + cot θ ) = sin θ sin Prove that cos 8 θ sin 8 θ = (cos θ sin θ) ( sin θ. cos θ) 5. If a sin = b cos and a sin 3 + b cos 3 = sincos, then prove that a +b = 5. Prove that (sin 6 + cos 6 ) 3 (sin 4 + cos 4 ) + = If cosθ sinθ = sinθ, then prove that : cosθ + sinθ = cosθ 54. If secθ = x Prove that cosec cosec + 4x, then prove that secθ + tanθ = x or x = cot cos cot + cos 56*. If cos α sin α = tan β, then prove that cos β = cosα 57*. If cosecθ - sinθ = m 3 and secθ cosθ = n 3, then prove that m 4 n + m n 4 = 58*. If x = tan + sin and y = tan sin, then prove that x y = 4 xy 59*. If sinα = α sinβ and tan α = b tan β, then prove that cos α = a b 60*. If sinθ + sin θ =, then prove that cos θ + 3cos 0 θ + 3 cos 8 θ + cos 6 θ + cos 4 θ + cos θ = + sin θ θ 6 MRKS QUESTIONS (Question No. 76 to 80 under HOTS) 6. The shadow of a tower standing on a level ground is found to be 60 m shorter when the Sun s altitude is 60 0 then when it is 30 0, find the height of the tower. 6. The angles of elevation of a bird from a point on the ground is 60 0, after 50 seconds flight the elevation changes to If the bird flying at the height 500 3m. Find the speed of the bird. 63. From a point on the ground the angles of elevation of the bottom and the top of a water tank kept at the top of 30 m. high building are 45 0 and 60 0 respectively. Find the height of (37) (Maths Xth class)

29 the water tank. 64. tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 60 0 with the ground. The distance from the foot of the tree to the point where the top touches the ground is 5 m. Find total height of the tree. 65. From a window (0 m high above the ground) o a house in a street. The angles of elevation and depression of the top and the foot of an other house opposite side of street are 60 0 and 45 0 respectively. Find the height of the opposite house. 66. light pole 4 m high is fixed on the top of a building, the angle of elevation of the top of the pole observed from a point p on the ground is 60 0 from the top of the building is Find the height of the building. 67. n aeroplane at an altitude of 00 meters observes the angles of depression of opposite points on the two banks of a river to be 60 0 and 45 0, find the width of the river. 68. The angles of elevation of the top of a pole from two points P and Q at distances of x and y respectively from the base and in the same straight line with it, are complementary. Prove that the height of the pole is xy. 69. The angle of elevation of a bird from a point metres above a lake is 30 0 and the angle of depression of its reflection in the lake is Find the distance of the bird from the point of observation. 70. The angle of elevation of the top of a 0 metres tall building from a point P on the ground is flag is hoisted at the top of the building and the angle of elevation of the flag staff from P is Find the length of flag staff and the distance of the building from P. 7. Nikita standing on a bank of a river observes that the angle subtended by a tree on the opposite bank is 60 0, when she retires 30 metres from the bank, she finds the angle to be 30 0, find the breadth of the river and height of the tree. 7. man, on a cliff, observes a boat at an angle of depression of 30 0, which is approaching the shore to the point on the immediately beneath the observer with a uniform speed, minutes later, the angle of depression of the boat is found to be Find the time taken by the boat to reach the shore. 73. man on the deck of a ship, 8 metres above water level, observes that the angle of elevation and depression respectively of the top and bottom of a cliff are 60 0 and Find the distance of the cliff from the ship and height of the cliff. 74. The angle of depression of the top and bottom of a 0 metres tall building from the top of a tower are 30 0 and 45 0 respectively. Find the height of the tower and distance between building and tower. 75. n aeroplane when 3000 metres high, passes vertically above another aeroplane at an instant when the angle of elevation of two aeroplanes from the same point on the ground are 60 0 and 45 0 respectively. Find the vertical distance between the two planes. 76*. t the foot of the mountain the elevation of its summit is fter ascending 000 metres towards the mountain at an inclination of 30 0, the elevation is Calculate the height of the mountain. ( 3 =.73) 77*. From an aeroplane vertically above a straight horizontal plane, the angle of depression of two consecutive kilometre stones on the opposite sides of the aeroplane are found to be θ (38) (Maths Xth class)

30 and α. Show that the height of the aeroplane is tan θ. tanα tan θ + tanα 78*. t a point P on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 3 5. On walking 9 metres away from P the tangent of the angle is 4. Find the height of the tower. 79*. round balloon of radius r subtends on an angle θ at the eye of the observer while the angle of elevation of its centre is α. prove that the height of the centre of the balloon is r sinα cosec θ 80*. boy standing on a horizontal plane, finds a bird flying at a distance of 00 metres from him at an elevation of girl, standing on the roof of 0 metres high building, finds the angle of elevation of the same bird to be oth the boy and girl are on opposite sides of the bird. Find the distance of the bird from the girl.. tan = sin sin nswers. cos p = = Zero 7. sec3 0 + cot θ = Zero x = 0 6. cos c (39) (Maths Xth class)

31 7. Zero = 60 0, = θ = (Hint. : 3θ = 90 0 ) 5. θ = Hint : sec θ = tan θ sin R = 8 7, tan R = 8 5 sec M = , 3+ (hint : = 450, = 30 0 or = 30 0, = 45 0 ) (330) (Maths Xth class)

32 (Hint : m = cos θ sinθ 3 n = sin θ cosθ (Hint : m + n = tanθ) m n = sinθ) 60. (Hint : sinθ = cos θ (a+b) 3 = a 3 + 3a b + 3ab + b 3 ) metres 6. 0 metres/sec ( 3 ) metres 64. 5(+ 3) metres 65. 0( 3+) metres 66. ( 3+) metres (3+ 3) metres 68. (Hint : complementary angles) metres 70. Length of flag staff = 0 ( ) metres Distance of the building = 0 3 metres 7. 5 metres, 5 3 metres 7. 8 minutes metres, 7 metres 74. 5(3+ 3) metres, 5(3+ 3) metres (3 3) metres metres metres metres (33) (Maths Xth class)

33 Co-ordinate Geometry Key Points. The length of a line segment joining & is the distance between two points (x, y ) and (x, y ) is { (x x ) + (y y) }. The distance of a point (x, y) from the origin is ( x +y ). The distance of P from x-axis is y units and from y-axis is x-units. 3. The co-ordinates of the points p(x, y) which divides the line segment joining the points (x, y ) and (x,y ) in the ratio m : m are ( m x +m x m +m, m y +m y m +m ) we can take ratio as k:, k = m m 4. The mid-points of the line segment joining the points P(x, y ) and Q(x, y ) is ( x +x y +y ) 5. The area of the triangle formed by the points (x, y ), (x, y ) and (x 3, y 3 ) is the numeric value of the expressions [x (y y 3 ) + x (y 3 y ) + x 3 (y y )] 6. If three points are collinear then we can not draw a triangle, so the area will be zero i.e. x (y y 3 ) + x (y 3 y ) + x 3 (y y ) = 0, Mark questions (Question 8- are under HOTS). What is the distance between (a o) and (o b).. What is the midpoint of the line segment joining the points (3, 4) and (, 6)? 3. What is the value of a and b if (, 3) is the mid point of the line segment joining (, a) and (b, )? 4. What is the area of the triangle joining the points (, 4), (, 0) and (, )? 5. is the diameter of a circle with centre at origin. What are the co-ordinates of if co-ordinates of point are (3, 4)? 6. What is the length of the side of the rhombus (, ), (, 5), C( 5, 4) and D( 6, 3)? 7. In the adjoining figure what is the length of? ( 3, 3) (, 0) (33) (Maths Xth class)

34 8. What is the value of x if (3, 5) and (7, ) are equidistant from T(x, o)? 9. What is the value of y if ar ( C) = 0 and co-ordinates of vertices are (, ), (y, 6), C(, 3)? 0. Given a circle with centre at origin and radius 5 units. State where the point (5, 7) lies?. line is drawn through p(4, 6) parallel to x-axis what is the distance of the line from x- axis? marks questions - (Question 6-8 are under HOTS). Find x if the distance between the points (x, ) and (3, 4) be 8 units. 3. Find the point on y-axis which is equidistant from the points (, 5) and (, 3). 4. Find the co-ordinates of the point which divides the line segment joining the points (, 3) and (, 7) in the ratio 3:4. 5. and are the points (, ) and (, 3). Find the co-ordinates of a point G on the line-segment such that G G = The mid point of the line segment joining the points (5, 7) and (3, 9) is also the mid point of the line segment joining the points (8, 6) and (a, b). Find a, b. 7. Find the distance between the points (a, b) and (b, a), if a b = 4 8. Find the ratio in which the point (, 5) divides the line segment joining the points (5, 5) and (9, 0). 9. Find the point of trisection of the line segment joining the points ( 3, 4) and (, ). 0. NICE is a parallelogram whose three vertices taken in order are ( 3, ), (, ) and (3, 3). Find the co-ordinate of the fourth vertex.. Find the point which is 3 4 of the way from (3, ) to (, 5).. Prove that we can draw the line passing through the points (0, ), (3, 5) and (6, 9). (Show that points are collinear). 3. Find the area of the triangle whose vertices are (, ), ( 3, 5) and (, 7). 4. Find the value of k if (k, ), (5, 5) and (0, 7) are Collinear. 5. The vertex of the triangle C are (, 3), (, ) and C(5, ). Find the length of the median drawn from the vertex. 6. C is an isosceles triangle with = C and vertex is on y-axis. If the co-ordinates of vertex and C are ( 5, ) and (3, ) respectively then find the co-ordinates of vertex. 7. The point K(, ) lies on the perpendicular bisector of line segment joining the points E(6, 8) and F(, 4). What is the distance of the point K from the line segment EF. 8. Point P(k, 3) is the mid point of. If the distance = 5 units and co-ordinates of are ( 3, 5) then find the value of k. 3 marks questions - (Questions 40-4 are under HOTS) (333) (Maths Xth class)

35 9. Find the abscissa of a point whose ordinate is 4 and which is at a distance of 5 units from (5, 0). (5, 3) 30. In figure, CD is a median from the vertex C on the side of C, P is the point on CD such that DP = unit. Find DP PC. D P 3. If ( 3, ), (x, y) and C(, 4) are the vertices of an isosceles triangle with = C. Find the value of (x+y). C (3, ) (7, 3) 3. Find the ratio in which the line 3x + y = divides the line segment joining the points (, 3) and (, 7). 33. Prove that the figure obtained on joining the mid points of parallelogram PQRS is a square where P(, 0). Q(5, 3), R(, 7) and S(, 4). lso find the sum of the diagonals. 34. point P on the x-axis divides the line segment joining the points (4, 5) and (, 3) in certain ratio. Find the co-ordinates of point P. 35. In right angled triangle C, = 90 0 and = 34 unit. The co-ordinates of points and C are (4, ) and (, y) respectively. If the ar ( C) = 7 unit then find the value of y. 36. If the point (6, 4) divides the line segment joining L (a,b) and M (8,5) in the ratio :5 then find the value of a and b. lso find the co-ordinates of the mid point of ML. 37. The vertices of quadrilateral CD are ( 5, 7), ( 4, 5), C(, 6) and D(4, 5). Find the area of the quadrilateral CD. 38. In figure D and E are the mid-points of the side C and respectively. Find the length of DE. E (, ) 39. Find the value of y such that ar ( C) = 4 units and co-ordinates of the vertices are (, ), (3, y) and C(5, ) 40. If the point P(3, 4) is equidistant from the points (a+b, b a) and (a b, a+b) then prove that 3b 4a = 0 4. If the area of the quadrilateral PQRS is zero where P(, ), Q( 5, 6) R(7, 4) and 5(h, ) are the vertices then find the value of h. State are the points really making quadrilateral. 4. In figure find the radius of the circle. (334) (Maths Xth class) ( 6, ) D C (4, )

36 y x 3 x y NSWERS. ( a + b ). (7, 5) 3. a = 5 b = 4. Zero 5. ( 3, 4) unit 7. 5 unit 8. x = 9. y = 0. Outside. 6 unit. x =, 5 3. (0, ) 4. ( 0 7, 33 7 ) 5. ( 7, 8 7 ) 6. a = 0, b = unit (335) (Maths Xth class)

37 8. : 9. ( 5 3, ), ( 3, 0) 0. (, 3). ( 3 4, 4) 3. 5 Sq. units 4. k = unit 6. (0, ) 7. 5 unit 8. k = 0, 6 9., : : ( 7 8, 0) a = 5 5 b = 8 5 ( 66 0, 43 0 ) square unit y unit 4. h = 3, NO units Triangles (336) (Maths Xth class)

38 . Similar triangles : Key Points Two triangles are said to be similar in their corresponding angles are equal and their corresponding sides are proportional.. Criteria for Similarity :- in C and DEF (i) similarity C ~ DEF when = D, = E and C = F (ii) SS similarity : C ~ DEF when DE = C EF = C ND = D DF (iii) SS similarity : C ~ DEF DE = C DF = C EF 3. The proofs of the following theorems can be asked in the examination :- (i) (ii) asic proportionality Theorems : If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio. The ratio of the area of two similar triangles is equal to the square of the ratio of their corresponding sides. (iii) Pythagoras theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. (iv) Converse of Pythagoras Theorem : In a triangle, if the square of ne side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is a right angle. Mark Questions Similar Triangles P. In fig() what PS SQ if ST QR, PT = 8 cm and PR = 0 cm. S T. In fig () is C ~ QP Q (Fig. ) R 3. In fig () if QC = 0 cm is it possible that PQ C? 8cm P 0 cm 5cm cm Q (Fig. ) (337) (Maths Xth class) C

39 4. In XYZ, P&Q are points on XY and XZ respectively XP XY =, XQ = 6 cm, QZ = 3 cm. 3 What type of linesegments PQ & YZ are. 5. In PQR, S&T are points on PQ and PR such that ST QR, PS = cm, SQ = 3 cm, PR = 5 cm. What is the length of TR. 6. n isoscels triangle C is similar to PQR. C = = 4 cm, PQ = 0 cm and C = 6 cm. What is the length of PR? 7. In fig (3), l m. what is the measurement of x? x y (Fig. 3) l m 8. In fig (4) C ~ PQR. What is the value of x? 4 5 R C P x Q (Fig. 4) 9. In fig (5), if DE C, what is the values of x? x 8 x D E (Fig. 5) C 0. If the ratio of the corresponding sides of two similar triangles in 4:5. What is the ratio of their areas? P. In fig (6) DE QR and DE = QR. How many times is PQ of 4 PD. (This can be asked as write PQ : QD) D E (338) (Maths Xth class) Q (Fig. 6) R

40 . The length of median of an equilateral riangle is 3 cm. What is the length of its sides? 3. In two triangles C and PQR if = Q and PQ = PR, what is the value of QR? 4. Measurement of three sides of a triangle are a, 0 a, 3a. What is the measurement of the angle opposite to the longest side? 0 cm 5. If fig (7), DE C what is the value of DE. P D x 3 cm (Fig. 7) E cm C Marks Questions 6 cm 6. In fig (8) find SR. Q S 9 cm (Fig. 8) R 7. In PQR, RS PQ, QRS = P, PS = 5 cm, SR = 8cm. Find PQ 8. Two similar triangles C and PC are made on opposite sides of the same base C. Prove that = P 9. In fig (9) CD is a rectangle. DE and F are two triangles. Such that E = F. Prove that D E = F D F C (Fig. 9) E 0. In fig. (0) DE C. If D = 3 8 ar ( DE). Find ar ( C) D E. In figure (0) DE C, DE = 3 cm, C = 90 cm and ar ( DE) = 30 cm. Find ar (trap CED) (Fig. 0) C (339) (Maths Xth class)

41 . mit is standing at a point on the ground 8 m away from a house. mobile network tower is fixed on the root of the house. Finds that the top and bottom of the tower are 7 m and 0 m away from the point. Find the heights of the tower and house. 3. In a right angled triangle right angle at, C = 3. Find C. 4. In a right angled triangle PRO, PR is the hypotanous and the other two sides are of length 6 cm and 8 cm. Q is a point outside the triangle such that PQ = 4 cm, PQ = 6 cm. What is the measure of RPQ? How many such triangles PQR are possible? 5. and CD of a quadrilateral CD and right angles. Prove that D = + D + CD. 6. In figure () C is isosceles with = C. Prove taht M CN = MP NP M N P (Fig. ) C E D 7. Find the length of the diagonal of rectangle CDE (fig (i)) if C = DCF. C 5 cm 0 cm (Fig. (i)) F 8. In fig. ( (ii) EF is a rectangle. C is the mid point of D. If = 6 cm, De = 9 cm, D = 4 cm. E = 5 cm. Prove that CE = In fig. (3) Find the value of x if PQ C (Fig. 3) C 30. PQRS is a trapezium. SQ is a diagonal. E & F are two points on PQ and RS respectively interesting SQ at G. Prove that SG QE = QG SF. 3x P x+ x+ Q 4x+ F (Fig. (ii)) C E D (340) (Maths Xth class)

42 3. In figure (4) prove that DE C. lso find the ratio of ar ( DE) or (trap CED). 4 6 Where D is the mid point of C. D x (Fig. 4) E.5 C 3. In C, EF C, such that EF passes through the controid G. Find, where D is the mid point of C. C 33. In fig. (5) D E and D DC = CE E. Prove that PDCE is a parallelogram. D P (Fig. 5) E 34. In a quadrilateral CD, + D = Prove that C + C = D + C. 35. In fig. 5(i) PQR and S are points on the sides of quadrilateral CD such that these points divides the sides, C, CD and D in the ratio :. Prove that PQRS is a parallelogram. 36. Equiangular triangles are drawn on sides of right angled triangle in which perpendicular is double of the base. Show that the area of the triangle on the hypotenuse is the sum of the areas of the other two triangles. 37. In a rhombus prove that four times the square of any sides is equal to sum of squares of its diagonals. 38. CD is a rectangle in which length is double of its breadth two equilateral triangles are drawn one each on length and breadth. Find the ratio of their areas. R Q S (Fig. 5 (i)) D R C 39. In fig. (6) EF = FE E is the mid point of C. Prove that D CE = F CD. E (Fig. 6) 40. Prove that if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Rider s based on above theorem C D (34) (Maths Xth class)

43 C (i) in the adj. fig. (8) DE, D EF. Find CD. D F E (ii) CD is a parallelogram (see fig.9. ) Prove that DP PQ = DC Q (iii) Find x, if DE C (fig. 0) D (Fig. 9) D 3x 9 x 4 E 8 4 P C Q (Fig. 8) (Fig. 0) C (iv) CD is a trapezium. Find value of x (fig. ) D C 3 x 3 3x 9 x 5 (Fig. ) 4. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Riders : Use above result to prove the following ar C ar (trap CED) = 6 49, C DE What is the length of the altitudes to the bigger triangles if length of altitude to smaller triangle is 8 cm (fig. ) D C (Fig. ) E (34) (Maths Xth class)

44 4. In a triangle, if the square of one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle. Use above result to prove the following Rider;s (i) (ii) Dinesh, Naresh and shu are standing in such a way that distance between Dinesh and Naresh is p meter. Naresh and shu are at a distance ( q ) m from ech other. Dinesh and shu are ( p + q ) m apart. What type of triangle they are forming. Three sticks of length (a ) cm, a cm and (a+) cm are joined with their end pts. to from a triangle. Do they form a right triangle. Show. 43. In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Rider based on aboe theorm. mar and shok are two friends standing at a corner of a rectangular garden. They wanted to drink wter. mar goes due north at a speed of 50 m/min. and shok due west at a speed of 60 m/min. They travel for 5 minutes. mar reaches the tap and drink water. ut somebody told shok to go towards point C. s there was to other tap. Find the min. distance shok has to travel to reach point C. (fig. 3) C (Fig. 3) nswers. 4. NO 3. No 4. Parrallel 5. 9 cm 6. 0 cm cm 9. 4 cm 0. 6: (343) (Maths Xth class)

45 4. Right angle cm cm. 9 m, 6 m Right angle, Two : (i) CF C, (ii) ---, (iii), (iv) 8 or cm 4. (i) right angled triangle (iv) yes m (344) (Maths Xth class)

46 Statistics Key Points. The mean for grouped data can be found by : (i) The direct method = X = fixi fi (ii) The assumed mean method = X (iii) The step deviation method = X fidi = a +, where d fi i = x i a fiui = a + h, where u fi i = x a i h. The mode for the grouped data can be found byusing the formula :- mode = l + f f0 f f f 0 h l = lower limit of the model class. f = frequency of the model class f 0 = frequency of the proceeding class of the model class. f = frequency of the succeeding class of the model class h = size of the class interval. Model class - class interval with highest frequency. 3. The median for the grouped data can be found by using the formula :- median = l + n Cf f h l = lower limit of the median class. n = number of observations Cf = cumulative frequency of class interval preceeding the median class. f = frequency of median class. h = class size. 4. Imperical Formula :- Mode = 3 median - mean 5. Cumulative frequency curve or an Ogive :- (i) Ogive is the graphical representation of the cumulative frequency distribution. (345) (Maths Xth class)

47 (ii) Less than type Ogive :- * Construct a cumulative frequency table * Mark the upper class limit on the x = axis. (iii) More than type Ogive :- * Construct a frequency table * Mark the lower class limit on the x-axis. (iv) To obtain the median of frequency distribution from the graph :- * Locate point of intersection of less than type Ogive and more than type Ogive :- Draw a perpendicular from this point to x-axis. * The point at which it cuts the x-axis gives us the median. Statistics I mark Questions (Question-5 are under HOTS) What is the median of the following distribution,3,6,0,,4,8,,5 What is the Mean if Median= 4 and Mode= 3 What is the mean of x, x+, x+, x+3, x+4 4 The following table shows the frequenay distribution of the marks of 50 students What is the value of y. class interval frequency 8 3 y 0 5 Write the class mark of the class interval teacher ask the student to find the average marks obtauned by most of the Students. What the student will find: Mean, Mode or Median. 7 What is the mode of the following data., 0,,, 3,, 4, 5,, 0, 8 In the following distribution, Write the modal class Class interval Frequency For the frequency distribution fi=40 and fixi=440 What is the mean of the distribution. 0 teacher ask the student to find the average marks obtained by the class student in Mathematics What the student will find: Mean Mode or Median. (346) (Maths Xth class)

48 The following data is arranged in the ascending order,,,3,7x,7x+,6,6,8,0 if the median of data is 4.5,what is the value of x What is the value of the median of the data using the following qraph of less then ogive and More than ogive 3 The following More than ogive Shows the weight of 40 Student of a class. What is the lower limit of the Median class. (347) (Maths Xth class)

49 4. From the cumulative frequency table Write the frequency of the class 0-30 Marks Number of student less than 0 less than 0 4 less than less than less than Following is a commulative frequency curve for the marks obtained by 0 Student as shown in find the Median marks obtained by the student. 6 Marks Questions (Question No.4, 5, 6, 7 are under HOTS) 6. Find the mean of the following frequency distribution: Class interval Frequency Find the value of p, if the mean of the following distribution is 0 x p 3 f 3 4 5p 6 8. The mean of the following frequency distribution is 6.8 and the sum of all the frequencies is 50. Find the values of p and q. Class-Interval Frequency 5 p 0 q 7 8 (348) (Maths Xth class)