CBSE Class 10 Mathematics Polynomials Worksheet Set 10

VERY SHORT ANSWER TYPE QUESTIONS

Question. If the zeroes of the quadratic polynomial x² + (a + 1)x + b are 2 and – 3, then
(a) a = – 7, b = – 1
(b) a = 5, b = – 1
(c) a = 2, b = – 6
(d) a = 0, b = – 6
Answer. (d) a = 0, b = –6

Question. If one zero of the polynomial P(x) = 5x² + 13x + K is reciprocal of the other, then value of k is
(a) 0
(b) 5
(c) 1/6
(d) 6
Answer. (b) 5

Question. What should be added to the polynomial x² – 5x + 4, so that 3 is the zero of the resulting polynomial:
(a) 1
(b) 2
(c) 4
(d) 5
Answer. (b) 2

Question. If α and 1/α are zeros of 4x² – 17x + k – 4, find the value of k.
Answer. k = 8

Question. What will be the number of real zeros of the polynomial x² + 1?
Answer. 0

Question. If α and β are the zeros of the polynomial
ƒ(x) = x² + x + 1, then 1/α + 1/β =
Answer. – 1

Question. What will be the number of zeros of a linear polynomial p(x) if its graph (i) passes through the origin. (ii) doesn’t intersect or touch x-axis at any point?
Answer. (i) 1 (ii) 0

Question. What will be the number of zeros of the polynomials whose graphs are parallel to (i) y-axis (ii) x-axis?
Answer. (i) 1 (ii) 0

Question. If a quadratic polynomial ƒ(x) is not factorizable into linear factors, then it has no real zero. (True/False)
Answer. True

Question. If a quadratic polynomial ƒ(x) is a square of a linear polynomial, then its two zeros are coincident. (True/False).
Answer. True

SHORT ANSWER TYPE (I) QUESTIONS

Question. If zeros of x² – kx + 6 are in the ratio 3 : 2, find k.
Answer. – 5, 5

Question. Find a quadratic polynomial whose zeros are (3+√5)/5 and (3-√5)/5.
Answer. α + β = 6/5, αβ = 4/25, 25x² - 30x + 4

Question. For what value of k, x² – 4x + k touches x-axis.
Answer. 4

Question. If α and β are zeros of the polynomial t² – t – 4, form a quadratic polynomial whose zeros are 1/α and 1/β .
Answer. 4t² + t – 1

Question. If one zero of the quadratic polynomial (k² + k)x² + 68x + 6k is reciprocal of the other, find k.
Answer. 5

SHORT ANSWER TYPE (II) QUESTIONS

Question. If zeros of the polynomial ax² + bx – c, a ≠ 0 are additive inverse of each other then what is the value of b?
Answer. b = 0

Question. Obtain zeros of 4√3x² + 5x – 2√3 and verify relation between its zeroes and coefficients.
Answer. -2/√3, √3/4

Question. Find the value of k such that 3x² + 2kx + x – k – 5 has the sum of zeros as half of their product.
Answer. 1

Question. If x⁴ + 2x³ + 8x² + 12x + 18 is divided by (x² + 5) , remainder comes out to be (px + q) , find values of p and q.
Answer. p = 2, q = 3

Question. –5 is one of the zeros of 2x² + px – 15, zeroes of p(x² + x) + k are equal to each other. Find the value of k.
Answer. 7/4

LONG ANSWER TYPE QUESTIONS

Question. Find all zeros of the polynomial 2x³ + x² – 6x – 3 if two of its zeroes are √3 and – √3.
Answer. √3, – √3, -(1/2)

Question. If sum of the zeros of 5x² + (p + q + r)x + pqr is zero, then find p³ + q³ + r³.
Answer. Product of the zeros = 3 pqr

Question. If two zeros of x⁴ – 6x³ – 26x² + 138 x – 35 are (2±√3), find other zeroes.
Answer. – 5, 7

Question. If α and β are zeros of the polynomial x² + 4x + 3, find the polynomial whose zeros are 1 + β/α and 1 + α/β
Answer. x² - (16/3)x + 16/3 or 1/3(3x² - 16x + 16)

Question. Form a quadratic polynomial one of whose zero is 2 + √5 and sum of the zeros is 4.
Answer. 2-√5

Question. Obtain all zeros of the polynomial 2x⁴ – 2x³ – 7x² + 3x + 6 if two factors of this polynomial are (x±√3/2).
Answer. 2, -1, ±√3/2

Question. If (x + 2) is a factor of x² + px + 2q and p + q = 4 then what are the values of p and q?
Answer. p = 4, q = 0

Question. If the polynomial x⁴ – 3x³ – 6x² + kx – 16 is exactly divisible by x² – 3x + 2, then find the value of k.
Answer. x² – 3x + 2 = (x – 2) (x – 1)
P(1) = 0, K = 24.

Question. What should be subtracted from x³ – 3x² + 6x – 15, so that it is completely divisible by (x – 3)?
Answer. 3