# CLASS - X Mathematics (Real Number)

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1 CLASS - X Mathematics (Real Number) is a (a) Composite number (c) Prime number (b) Whole number (d) None of these. For what least value of n a natural number, ( 4) n is divisible by 8? (a) 0 (b)-1 (c) 1 (d) No value of n is possible 3. The sum of a rational and an irrational is (a) Rational (b) Irrational (c) Both (a)&(c) (d) Either (a) or (b) 4. HCF of two numbers is 113, their LCM is It one number is 904. The other number is: (a) 7719 (b) 7119 (c) 7791 (d) Show that every positive even integer is of the from q and that every positive odd integer is the four q+1 for some integer q. 6. Show that any number of the form 4 n, nen can never end with the digit Use Euclid s division algorithm to find the HCF of 405 and Given that HCF of two numbers is 3 and their LCM is If one of the numbers is 161, find the other. 9. Find the greatest of 6 digits exactly divisible by 4, 15 and Prove that the square of any positive integer is of the form 4q or 4q+1 for some integer cartoons of coke can and 90 cartoons if Pepsi can are to be stacked in a canteen It each stack is of the same height and is to contain cartoons of the same Drink. What would be the greaten number of cartoons each stack would have 1. Prove that Product of three consecutive positive integers is divisible by 6.

2 Class X- Mathematics (Real Number) 1. A lemma is an axiom used for proving (a) other statement (b) no statement (c) Contradictory statement (d) none of these. If HCF of two numbers is 1, the Two number are called relatively or (a) Prime, co-prime (b) Composite, prime (c) Both (a) and (b) (d) None of these is (a) a terminating decimal number (c) an irrational number is (a) a rational number (c) an irrational number (b) a rational number (d) Both (a) and (b) (b) a non terminating decimal number (d) both (a)&(c) 5. Show that every positive odd integer is of the form (4q+1) or (4q+3) for same inter q. 6. Show that any number of the form 6 x, x N can never end with the digit 0 7. Find HCF and LCM of 18 and 4 by the prime factorization method. 8. The HCF of two numbers is 3 and their LCM is If one of the number is 161, find the other 9. Prove that (3-5) is irrational. 10. Prove that if x and y are odd positive integers then x +y is even but not divisible by Show that one and only one out of n, (n+) or (n+4) is divisible by 3, where n N 1. Use Euclid s division lemma to show that the square of any positive integer of the from 3m or (3m+1) for some integer q

3 Class X - Mathematics (Real Numbers) 1. The smallest composite number is: is (a) 1 (b) (c) 3 (d) 4 (a) an integer (c) a rational number (b) an irrational number (d) None of there, 3. π is (a) rational (b) irrational (c) both (a)&(b) (d) neither rational nor irrational 4. (+ 5) is (a) rational (b) irrational (c) An integer (d) Not real 5. Prove that the square of any positive integer of the form 5g+1 is of the same form 6. Use Euclid s division algorithm to find the HCF of 405 and Find the largest number which divides 45 and 109 leaving remainder 5 in each case 8. A shop keeper has 10 litres of petrol, 180 litres of diesel and 40 litres of kerosene. He wants to sell oil by filling the three kinds of oils in tins of equal capacity. What should be the greatest capacity of such a tin 9. Prove that in n is not a rational number, if n is not perfect square 10. Prove that the difference and quotient of ( 3+ 3) and ( 3 3) are irrational 11. Show that (n -1) is divisible by 8, if n is an odd positive integer 1. Use Euclid division lemma to show that cube of any positive integer is either of the form 9m. (9m+1) or 9m+8

4 Class X - Mathematics (Polynomials) 1. Which of the following is polynomial? 1 (a) x 6 x + (b) x + x (c) 5 x 3x + 1 (d) none of these 4 3. Polynomial x + 3x 5x 5x + 9x + 1is a (a) Linear polynomial (b) quadratic polynomial (c) cubic polynomial (d) Biquadratic polynomial 3. If αand βare zero s of x + 5x + 8then the value of ( α + β ) is (a) 5 (b) -5 (c) 8 (d) The sum and product of the zeros of a quadratic polynomial are and -15 respectively. The quadratic polynomial is 5. (a) x x + 15 (b) x x 15 (c) x + x 15 (d) x + x + 15 Find the quadratic polynomial where sum and product of the zeros one a and 1 a. 6. If αand βare the zeroes of the quadratic polynomial ( ) value of 1 α + 1 αβ β f x = x x 4, find the 7. If the square of the difference of the zeroes of the quadratic polynomial ( ) f x = x + px + 45is equal to 144, find the value of p. 8. Divide ( 6x 3 6x 1 x ) + by ( 7 3x) + 9. Apply division algorithms to find the quotient q(x) and remainder r(x) an dividing 3 f x x x x g x x x = , = f(x) by g(x) where ( ) ( ) 10. If two zeroes of the polynomial zeroes. 4 3 x x x x are ± 3, find the other 11. What must be subtracted from the polynomial ( ) 4 3 that the resulting polynomial is exactly divisible by ( ) f x = x + x 13x 1x + 1so g x = x 4x What must be added to divisible by 3x x x 5x 11x 3x x so that it may be exactly

5 Class X - Mathematics (polynomials) 1. If P(x)= x -3x+5,3x+5,then P(-1) is equal to (a) 7 (b) 8 (c) 9 (d) 10. Zeroes of P(x) = x -x-3 are (a) 3 and 1 (b) 3 and -1 (c)-3 and -1 (d) 1 and If αand βare the zeros of x +5x-10, them the value of αβis (a) 5 (b) 5 (c)-5 (d) 5 4. A quadratic polynomial, the sum and product of whore zeros are 0 and 5respectively is (a) x + 5 (b) x - 5 (c) x -5 (d) None of these 5. Find the value of k such that the quadratic polynomial x -(k+6) x+(k+1) has sum of the zeros is half of their product 6. If αand βare the zeroes of the quadratic polynomial f(x) = x -p(x+1)-c, show that (α+1) ( β+1)=1-c 7. If the sum of the zeroes of the quadratic polynomial f(t) = kt +t+3k is equal to their product, find the value if k 8. Divide (x 4-5x+6) by (-x) 9. Find all the zeroes of the polynomial f(x) = x 4-3x 3-3x+6x-, if being given that two of its zeroes are and On dividing x 3-3x +x+ by a polynomial g(x) the quotient and the remainder were (x-) and -x+4 respectively find g(x) 11. Find all zeroes of f(x) = x 3-7x +3x+6 if its two zero one - 3 and 3 1. Obtain all zeroes of the polynomial f(x)= x 4 +x 3-14x -19x-6, if two of its zeros are - and -1

6 Class X - Mathematics (Polynomials) 1. Degree of polynomial y 3 -y - 1 3y + is (a) 1 (b) (c) 3 (d) 3. Zeroes of P(x) = x +9x-35 are (a) 7 and 5 (b) -7 and 5 (c) 7 and 5 (d) 7 and 3. The quadratic polynomial whore zeros are 3 and -5 is (a) x +x-15 (b) x +3x-8 (c) x -5x-15 (d) None of these 4. If αand βare the zeros of the quadratic polynomial P(x) = x -px+q, then the value of α + β is equal to (a) p -q (b) p q (c) q -p (d) none of these 5. Find the zeros of the polynomial p(x) = 4 3x +5x - 3 and verify the relationship b/w the zeros and its coefficients 6. Find the value of k so that the zeroes of the quadratic polynomial 3x -kx+14 are in the ratio 7:6 7. If one zero of the quadratic polynomial f(x) = 4x -8kx-9 is negative of the other, find the value of k. 8. Cheek whether the polynomial (t -3) is a factor of the polynomial t 4 +3t 3 -t -9t-1 by Division method 9. Obtain all other zeroes of 3x 4 +6x 3 -x -10x-5. If two of its zeroes are 5 3 and If the polynomial x 4-6x 3 +16x -5x+10 id divided by another polynomial x -x+k, the remainder comes out to be (x+a), find k and a 11. Find the value of k for which the polynomial x 4 +10x 3 +5x +15x+k is exactly divisible by (x+7) 1. If α, and β are the zeros of the polynomial f(x) = x +px+q form polynomial whore zeros are (α+ β) and (α- β)

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