CBSE Class 9 Mathematics Polynomials VBQs

Question. Given that x = 2 is a solution of x³ – 7x + 6 = 0. The other solutions are ______.
(A) –1, 3
(B) 1, – 3
(C) 1, –2
(D) –1, –2
Answer : B

Question. Find the remainder when the expression 3x³ + 8x² – 6x + 1 is divided by x + 3.
(A) 1
(B) 10
(C) 6
(D) 0
Answer : B

Question. When p(x) = x³ + ax² + 2x + a is divided by (x + a), the remainder is _____.
(A) 0
(B) a
(C) –a
(D) 2a
Answer : C

Question. The product (a + b) (a – b) (a² – ab + b²) (a² + ab + b²) is equal to _____.
(A) a⁶ + b⁶
(B) a⁶ – b⁶
(C) a³ – b³
(D) a³ + b³ 
Answer : B

Question. The value of (x – a)³ + (x – b)³ + (x – c)³ –3(x – a)(x – b)(x – c), when a + b + c = 3x is ______.
(A) 3
(B) 2
(C) 1
(D) 0
Answer : D

Question. Value of R, if a²−19a−25/a−7 = a − 12 + R/a−7 is ____.
(A) –109
(B) –88
(C) –84
(D) –64
Answer : A

Question. If (x + 2) and (x – 1) are factors of (x³ + 10x² + mx + n), then the value of m, n respectively are _______.
(A) –5, 5
(B) 7, 18
(C) 7, –18
(D) –5, –18
Answer : C

Question. Length, breadth and height of a cuboidal tank are (x – 3y)m, (x + 3y)m and (x² + 9y²)m respectively. Find the volume of the tank.
(A) (x³ + 3xy + 27y3)m³
(B) (x⁴ + 2x²y² + 81y4)m³
(C) (x⁴ – 81y4)m³
(D) (x⁴ + 81y4)m³
Answer : C

Question. A rectangular field has an area (35x² + 13x – 12)m². What could be the possible expression for length and breadth of the field?
(A) (5x + 4)m and (7x – 3)m
(B) (3x + 9)m and (7x – 12)m
(C) Both (A) and (B)
(D) None of these
Answer : A

Question. Vikas has ₹(x³ + 2ax + b), with this money he can buy exactly (x – 1) jeans or (x + 1) shirts with no money left. How much money Vikas has, if x = 4?
(A) ₹ 80
(B) ₹ 120
(C) ₹ 30
(D) ₹ 60
Answer : D

Question. x¹² – y¹² =
(A) (x – y)(x² + xy + y²)(x + y)(x² – xy + y²) (x² + y²)(x⁴ – x²y² + y⁴)
(B) (x + y)(x² – xy + y²)(x + y)(x² – xy + y²) (x² + y²)(x⁴ – x²y² + y⁴)
(C) (x + y)(x² + xy – y²)(x + y)(x² – xy + y²) (x² + y²)(x⁴ – x²y² + y⁴)
(D) (x – y)(x² – xy + y²)(x + y)(x² – xy + y²) (x² + y²)(x⁴ – x²y² + y⁴)
Answer : A

Question. If x a-b/a+b, y = b-c/b+c, z = c-a/c+a, then the value of (1+x)(1+y)(1+z)/(1-x)(1-y)(1-z) is ______.
(A) abc
(B) a²b²c²
(C) 1
(D) –1
Answer : C

Question. If x² – 1 is a factor of ax⁴ + bx³ + cx² + dx + e, then
(A) a + b + e = c + d
(B) a + b + c = d + e
(C) b + c + d = a + e
(D) None of these
Answer : D

Question. If a, b, c are all non-zeroes and a + b + c = 0, then a²/bc + b²/ca + c²/ab = ______.
(A) 0
(B) 1
(C) 2
(D) 3
Answer : D

Question. If (x + k) is a common factor of f(x) = (x² + px + q) and g(x) = (x² + lx + m), then the value of k is ______.
(A) l + p
(B) m – q
(C) l−p/m−q
(D) m−q/l−p
Answer : D

Question. Which of the following statements is INCORRECT?
(A) Every non-zero constant polynomial has zero roots.
(B) Zero polynomial has zero root.
(C) Every linear polynomial has exactly one root.
(D) If x – a is the root of p(x) = 0, then p(a) = 0.
Answer : B

Question. Santosh has ₹ (x³ – 3x² + 4x + 50). He want to buy chocolates each of cost ₹ (x – 3). After buying maximum number of chocolates with his money, how much money is left with him?
(A) ₹ 50
(B) ₹ 40
(C) ₹ 62
(D) ₹ 20
Answer : C

Question. When (x³ – 2x² + px – q) is divided by (x² – 2x – 3), the remainder is (x – 6). The values of p and q respectively are ______.
(A) –2, – 6
(B) 2, – 6
(C) –2, 6
(D) 2, 6
Answer : C

Question. Match the following.

(A) (P) → (iii); (Q) → (iv); (R) → (i); (S) → (ii)
(B) (P) → (ii); (Q) → (iv); (R) → (i); (S) → (iii)
(C) (P) → (iii); (Q) → (i); (R) → (iv); (S) → (ii)
(D) (P) → (iii); (Q) → (ii); (R) → (i); (S) → (iv)
Answer : C

Question. Study the given statements.
Statement-I : (a²−b²)³ + (b²−c²)³ + (c²−a²)³ / (a+b) + (b+c) + (c+a) = (a + b) (b + c) (c + a)
Statement-II : a² + b² + c² – ab – bc – ca = 1/2[(a-b)² + (b-c)² + (c-a)²]
Which of the following options holds?
(A) Both Statement-I and Statement-II are true.
(B) Statement-I is true but Statement-II is false.
(C) Statement-I is false but Statement-II is true.
(D) Both Statement-I and Statement-II are false.
Answer : C

Question. The value of k for which (x + 2) is a factor of (x + 1)⁷ + (3x + k)³ is ______.
(A) –7
(B) 7
(C) –1
(D) – 6 – 3^(7/3)
Answer : B

Question. The remainder when x⁴ – y⁴ is divided by x – y is ______.
(A) 0
(B) x + y
(C) x² – y²
(D) 2y⁴
Answer : A

Question. Area of a rectangular field is (2x³ – 11x² – 4x + 5) sq. units and side of a square field is (2x² + 4) units.
Find the difference between their areas (in sq. units).
(A) 4x⁴ – 2x³ – 27x² – 4x + 11
(B) 4x⁴ – 2x³ + 27x² + 4x + 11
(C) 4x⁴ + 27x² + 4x – 11
(D) 4x⁴ + 2x³ + 27x² + 4x + 11
Answer : B

Question. If (5x² + 14x + 2)² – (4x² – 5x + 7)² is divided by (x² + x + 1), then quotient ‘q’ and remainder ‘r ’ respectively, are ______.
(A) (x² + 19x – 5), 0
(B) 9(x² + 19x – 5), 0
(C) (x² + 19x – 5), 1
(D) 9(x² + 19x – 5), 1
Answer : B

Question. Select the CORRECT statement.
(A) If x = √3+1/√3−1 + √3−1/√3+1 + √3−2/√3+2, then the value of x² +(39/x)² is 110.
(B) Every integer is a whole number.
(C) Between two rational numbers, there exist infinite number of integers.
(D) None of these
Answer : D