CBSE Class 10 Polynomials Sure Shot Questions Set 08

Very Short Answer Type Questions

Question. Factorize each of the following expression : \( x^2 - x - 42 \)
Answer: \( (x + 6) (x - 7) \)

Question. Factorize each of the following expression : \( 6 - 5y - y^2 \)
Answer: \( (6 + y) (1 - y) \)

Question. Factorize each of the following expression : \( a^2 + 46a + 205 \)
Answer: \( (a + 41) (a + 5) \)

Question. Factorize each of the following expression : \( ab + ac - b^2 - bc \)
Answer: \( (a - b) (b + c) \)

Question. Factorize each of the following expression : \( p^4 - 81q^4 \)
Answer: \( (p + 3q) (p - 3q) (p^2 + 9q^2) \)

Question. Use remainder theorem to find remainder, when \( p(x) \) is divided by \( q(x) \) in following questions : \( p(x) = 2x^2 - 5x + 7, q(x) = x - 1 \)
Answer: 4

Question. Use remainder theorem to find remainder, when \( p(x) \) is divided by \( q(x) \) in following questions : \( p(x) = x^9 - 5x^4 + 1, q(x) = x + 1 \)
Answer: -5

Question. Use remainder theorem to find remainder, when \( p(x) \) is divided by \( q(x) \) in following questions : \( p(x) = 2x^3 - 3x^2 + 4x - 1, q(x) = x + 2 \)
Answer: -37

Short Answer Type Questions

Question. Find positive square root of \( 36x^2 + 60x + 25 \)
Answer: \( 6x + 5 \)

Question. Simplify : \( \sqrt{2a^2 + 2\sqrt{6}ab + 3b^2} \)
Answer: \( (\sqrt{2}a + \sqrt{3}b) \)

Question. \( (x^2 + 4y)^2 + 21(x^2 + 4y) + 98 \)
Answer: \( (x^2 + 4y + 7) (x^2 + 4y + 14) \)

Question. Find the value of \( k \) if \( (x - 2) \) is a factor of \( 2x^3 - 6x^2 + 5x + k \).
Answer: -2

Question. Find the value of \( k \) if \( (x + 3) \) is a factor of \( 3x^2 + kx + 6 \).
Answer: 11

Question. \( p(x) = 3x^6 - 7x^5 + 7x^4 - 3x^3 + 2x^2 - 2, q(x) = x - 1 \)
Answer: [Answer not provided in key]

Question. For what value of \( k \) is \( y^3 + ky + 2k - 2 \) exactly divisible by \( (y + 1) \) ?
Answer: 3

Long Answer Type Questions

Question. If \( x + 1 \) and \( x - 1 \) are factors of \( mx^3 + x^2 - 2x + n \), find the value of \( m \) and \( n \).
Answer: \( m = 2, n = -1 \)

Question. Find the zeros of the polynomial \( f(x) = 2x^2 + 5x - 12 \) and verify the relation between its zeroes and coefficients.
Answer: \( -4, \frac{3}{2} \)

Question. Find the zeroes of the polynomial \( f(x) = x^2 - 2 \) and verify the relation between its zeroes and coefficients.
Answer: \( -\sqrt{2}, \sqrt{2} \)

Question. Obtain the zeroes of the quadratic polynomial \( \sqrt{3}x^2 - 8x + 4\sqrt{3} \) and verify the relation between its zeroes and coefficients.
Answer: \( 2\sqrt{3}, \frac{2}{\sqrt{3}} \)

Question. Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes as 2, –7 and –14 respectively.
Answer: \( x^3 - 2x^2 - 7x + 14 \)

Question. Find a cubic polynomial whose zeroes are 3, 5 and – 2.
Answer: \( x^3 - 6x^2 - x + 30 \)

Question. Divide \( 5x^3 - 13x^2 + 21x - 14 \) by \( (3 - 2x + x^2) \) and verify the division algorithm.
Answer: quotient \( = 5x - 3 \), Remainder \( = -5 \)

Question. What real number should be subtracted from the polynomial \( (3x^3 + 10x^2 - 14x + 9) \) so that \( (3x - 2) \) divides it exactly?
Answer: 5

Question. Find all the zeroes of \( (2x^4 - 3x^3 - 5x^2 + 9x - 3) \), it being given that two of its zeroes are \( \sqrt{3} \) and \( -\sqrt{3} \).
Answer: \( \sqrt{3}, -\sqrt{3}, 1, \frac{1}{2} \)

Question. If \( \left( x + \frac{1}{x} \right) = 3 \), then find value of \( \left( x^2 + \frac{1}{x^2} \right) \).
Answer: 7

Question. If \( \left( x - \frac{1}{x} \right) = \frac{1}{2} \), then find \( \left( 4x^2 + \frac{4}{x^2} \right) \).
Answer: 9

Question. If \( \left( x + \frac{1}{x} \right) = 4 \), then find \( \left( x^4 + \frac{1}{x^4} \right) \).
Answer: [Answer not provided in key]

Question. If \( (x - 2) \) is a factor of \( (x^2 + 3qx - 2q) \), then find the value of \( q \).
Answer: -1

Question. If \( x^3 + 6x^2 + 4x + k \) is exactly divisible by \( (x + 2) \), then find the value of \( k \).
Answer: -8

Question. Let \( f(x) = x^3 - 6x^2 + 11x - 6 \). Then, which one of the following is not factor of \( f(x) \) ?
(a) \( x - 1 \)
(b) \( x - 2 \)
(c) \( x + 3 \)
(d) \( x - 3 \)
Answer: (c) \( x + 3 \)

Question. If \( x^{100} + 2x^{99} + k \) is divisible by \( (x + 1) \), then find the value of \( k \).
Answer: 1

Question. On dividing \( (x^3 - 6x + 7) \) by \( (x + 1) \), find the remainder.
Answer: 12

Question. Find the value of expression \( (16x^2 + 24x + 9) \) for \( x = -\frac{3}{4} \).
Answer: 0

Question. If \( 2x^3 + 5x^2 - 4x - 6 \) is divided by \( 2x + 1 \), then find remainder.
Answer: -3

Question. If \( p(x) = x^2 - 2x - 3 \), then find (i) \( p(3) \); (ii) \( p(-1) \)
Answer: (i) 0, (ii) 0

Question. Find the zeros of the quadratic polynomial \( (6x^2 - 7x - 3) \) and verify the relation between its zeros and coefficients.
Answer: \( \frac{3}{2}, -\frac{1}{3} \)

Question. Find the zeros of the quadratic polynomial \( (5u^2 + 10u) \) and verify the relation between the zeros and the coefficients.
Answer: -2, 0

Question. Find the quadratic polynomial whose zeros are \( \frac{2}{3} \) and \( \frac{-1}{4} \). Verify the relation between the coefficients and the zeros of the polynomial.
Answer: \( 12x^2 - 5x - 2 \)

Question. Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.
Answer: \( (x^2 - 8x + 12) \), {6, 2}

Question. Find the quadratic polynomial, the sum of whose zeros is -5 and their product is 6. Hence, find the zeros of the polynomial.
Answer: \( (x^2 + 5x + 6) \), {-3, -2}

Question. Find the quadratic polynomial, the sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.
Answer: \( (x^2 - 1) \), {1, -1}

Question. Find a quadratic polynomial whose one zero is \( 5 + \sqrt{7} \).
Answer: \( x^2 - 10x + 18 \)

Question. On dividing \( (x^3 - 3x^2 + x + 2) \) by a polynomial \( g(x) \), the quotient and remainder are \( (x - 2) \) and \( (-2x + 4) \) respectively. Find \( g(x) \).
Answer: \( x^2 - x + 1 \)

Question. If the polynomial \( (x^4 + 2x^3 + 8x^2 + 12x + 18) \) is divided by another polynomial \( (x^2 + 5) \), the remainder comes out to be \( (px + q) \). Find the value of \( p \) and \( q \).
Answer: \( p = 2, q = 3 \)

Question. Obtain all zeros of the polynomial \( (2x^3 - 4x - x^2 + 2) \), if two of its zeros are \( \sqrt{2} \) and \( -\sqrt{2} \).
Answer: \( \sqrt{2}, -\sqrt{2}, \frac{1}{2} \)

Question. If 1 and –2 are two zeros of the polynomial \( (x^3 - 4x^2 - 7x + 10) \), find its third zero.
Answer: 5

Question. Find all the zeros of the polynomial \( (2x^4 - 11x^3 + 7x^2 + 13x - 7) \), it being given that two if its zeros are \( (3 + \sqrt{2}) \) and \( (3 - \sqrt{2}) \).
Answer: \( (3 + \sqrt{2}), (3 - \sqrt{2}), \frac{1}{2}, -1 \)

Question. If \( \alpha, \beta \) are the zeros of the polynomial \( f(x) = x^2 - 5x + k \) such that \( \alpha - \beta = 1 \), find the value of \( k \).
Answer: 6

Question. Show that the polynomial \( f(x) = x^4 + 4x^2 + 6 \) has no zero.
Answer: [Verification provided in textbook]

Question. Use remainder theorem to find the value of \( k \), it being given that when \( x^3 + 2x^2 + kx + 3 \) is divided by \( (x - 3) \), then the remainder is 21.
Answer: -9