CBSE Class 9 Maths Polynomials MCQs Set 03

Question: The polynomial p(x) = x³ – 5x²– x + 5 is such that p(1) = 0 and p(– 1) = 0. Which of these is equivalent to p(x) ?

a) (x – 1)(x + 5)
b) (x – 1)(x + 1)(x + 5)
c) (x – 1)(x + 1)(x – 5)
d) (x + 1)(x – 5) 
Answer: c

Question: Which of the following is a quadratic polynomial ?
a) x + 2
b) x² + 2
c) x³ + 2
d) 2x + 2
Answer: b

Question: The coefficient of ‘x’ in the expansion of(x+3)³ is
a) 1
b) 27
c) 9
d) 18
Answer: b

Question: Which of the following is not a polynomial?
a) 3
b) 3x² + √5x
c) x² +1/x²  – 4
d) πt⁷ – 3t² + 4
Answer: c

Question: Degree of zero polynomial is : 
a) 0
b) 1
c) any natural number
d) not defined
Answer: d

Question: The polynomial (4x – 3)is a factor of the polynomial q(x) = 4x³ + x² – 11x + 2r. What is the value of r ? 
a) 2
b) 3
c) 4
d) 11 
Answer: b

Question: If p(x) = x + 3, then p(x) + p(-x) is equal to
a) 2x
b) 3
c) 0
d) 6
Answer: d

Question: The degree of constant function is
a) 0
b) 3
c) 1
d) 2
Answer: a

Question: Which of the following is a cubic polynomial?
a) x³ + 3x² – 4x + 3
b) x²+ 4x – 7
c) 3x² + 4
d) 3(x²  + x + 1)
Answer: a

Question: If x+y+z = 0, then x³+y³+z³ is
a) 3xyz
b) xyz
c) 2xyz
d) 0 
Answer: a

Question: The maximum number of terms in a polynomial of degree 10 is :
a) 9
b) 10
c) 11
d) 1
Answer: c

Question: The coefficient of x in the expansion (x + 3)³ is :
a) 1
b) 9
c) 18
d) 27
Answer: d

Question: Which of the following is a polynomial in x ?
a) x + 1/x
b) x² + √x
c) x +√2x² + 1 
d) √3x+ 1
Answer: c

Question: (x + 1) is a factor of the polynomial
a) x³+x²-x+1
b) x²+x³+x+1
c) x⁴+3x³+3x²+x+1
d) x⁴+x³+x²+x+1
Answer: b 

Question: The coefficient of x² in (3x + x³) (x + 1/x) is :
a) 3
b) 1
c) 4
d) 2
Answer: c

Question: Consider the expression x^(m2-1) + 3x^m/2 where m is a constant. For what value of m, will the expression be a cubic polynomial?
a) – 2
b) – 1
c) 1
d) 2
Answer: d

Question: Which of these is a zero of the polynomial p(y) = 3y³ – 16y – 8 ?
a) – 8
b) – 2
c) 0
d) 2
Answer: b

Question: Degree of which of the following polynomials is zero :
a) x
b) 15
c) y
d) x + x²
Answer: b

Question: A cubic polynomial is a polynomial with degree :
a) 1
b) 3
c) 0
d) 2
Answer: b

Question: The value of p for which x + p is a factor of x²+ px + 3 – p is :
a) 1
b) – 1
c) 3
d) – 3 
Answer: c

Question: Fill in the blanks: The highest power of the variable in a polynomial is called the ________ of the polynomial.
Answer: degree

Question: Fill in the blanks: A polynomial containing one non-zero term is called a ________.
Answer: monomial

Question: Evaluate the following by using identities: 0.54x0.54 - 0.46x0.46
Answer: We have,
0.54 x 0.54 - 0.46 x 0.46
= (0.54)² - (0.46)² = (0.54 + 0.46)(0.54 - 0.46) = 1 x 0.08 = 0.08

Question: Write (-3x + y + z)2 in the expanded form:
Answer: We have,
(-3x + y + z)²
Using identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
= (-3x)² + y² + z² + 2 (-3x) y + 2 y z + x 2 x (-3x)
= 9x² + y² + z² - 6xy + 2yz - 6zx

Question: Evaluate: 185x185 - 15x15
Answer: 185 x 185 - 15 x 15
 (185)² - (15)² [Using a² - b² -(a - b)(a + b) ]
(185 + 15)(185 - 15)
200 x 170 = 34000

Question: If the polynomials az³ + 4z² + 3z - 4 and z³ - 4z + a leave the same remainder when divided by z - 3, find the value of a.
Answer: .Let p(z) = az³ + 4z² + 3z - 4
And q(z) = z³ - 4z + a
As these two polynomials leave the same remainder, when divided by z - 3, then p(3) =
q(3).
p(3) = a(3)³ + 4(3)² + 3(3) - 4
= 27a + 36 + 9 - 4
Or p(3) = 27a + 41
And q(3) = (3)³ - 4(3) + a
= 27 - 12 + a = 15 + a
Now, p(3) = q(3)
⇒27a + 41 = 15 + a
⇒26a = -26; a = -1
Hence, the required value of a = -1.

Question: Evaluate 105x108 without multiplying directly.
Answer: 11. 105x108=(100+5)(100+8)
Using identity (x+a)(x+b)=x2+(a+b)x+ab
We get, 105x108 = 100² + (5 + 8)100 + 5x8
=10000+1300+40=11340

Question: Factorize : (a² - b²)³ + (b² - c²)³ + (c² - a²)³
Answer: ^{Let x = a2 - b2, y = b2 - c2 and z = c2 - a2. Then,
x + y + z = a2 - b2 + b2 - c2 + c2 - a2 = 0
x3 + y3 + z3 = 3xyz
⇒(a2 - b2)3 + (b2 - c2)3 + (c2 - a2)3 = 3(a2 - b2)(b2 - c2)(c2 - a2)
⇒(a2 - b2)3 + (b2 - c2)3 + (c2 - a2)3 = 3(a + b)(a - b)(b + c)(b - c)(c + a)(c - a)
⇒(a2 - b2)3 + (b2 - c2)3 + (c2 - a2)3 = 3(a + b)(b + c)(c + a)(a - b)(b - c)(c - a)}

Question: If both x - 2 and x-1/2 are factors of px2 + 5x + r, show that p = r.
Answer: Let f(x) = px2 + 5x + r be the given polynomial. Since x - 2 and x-1/2 are factors of f(x).
f(2) = 0 and f(1/2) = 0
⇒pX2² + 5X2 + r = 0 and p(1/2)² + 5X1/2 + r = 0
⇒4p + 10 + r = 0 and P/4+5/2 + r = 0
⇒4p + r = -10 and p+4r+10/4
⇒4p + r = -10 and p + 4r + 10 = 0
⇒4p + r = -10 and p + 4r = -10
⇒4p + r = p + 4r [RHS of the two equations are equal]
⇒3p = 3r⇒p = r

Question: Simplify (x + y)³ - (x - y)³ - 6y(x + y) (x - y).
Answer: (x + y)³ - (x - y)³ - 6y(x + y)(x - y)
= (x + y)³ - (x - y)³ - 3.2y(x + y)(x - y)
= (x + y)³ - (x - y)³ - 3 (x + y - x + y)(x + y)(x - y)
= [x + y - x + y)]³ [∴a³ - b³ - 3ab(a - b) = (a - b)³]
=(2y)³ = 8y³

Case based MCQs

Read the following passage and answer the questions.
Teacher wrote of polynomial x² + kx + 8 on the blackboard and said that value of this polynomial at x = –3 is –1. 

Question: Value of the polynomial at x = –5 is

a) –3
b) 3
c) –7
d) 7 
Answer: b

Question: Value of the polynomial at x = 3 is
a) 1
b) 18
c) 25
d) 35
Answer: d

Question: What type of polynomial is it?
Answer: Quadratic 

Question: Value of k is
a) 6
b) –6
c) 9
d) –9 
Answer: a

Question: Zeroes of this polynomial are
a) 4 and 2
b) 4 and –2
c) –4 and 2
d) –4 and –2 
Answer: d

Read the following passage and answer the questions.
Teacher asked the students to state the value of ‘a’, if on dividing the polynomial y³ – 4y + a by (y – 3), remainder is 31. 

Question: What will be the remainder, if the polynomial is divided by (x + 2)?
a) 32
b) 16
c) 12
d) 8
Answer: b

Question: Which of the following responses of the students is correct?
a) 16
b) 13
c) –13
d) –16
Answer: a

Question: What will be remainder if the polynomial is divided by (x – 5)?
a) 115
b) 117
c) 119
d) 121 
Answer: d

Question: What type of a polynomial is it?
Answer: Cubic

Question: What will be the remainder, if the polynomial is divided by (x + 5)?
a) – 87
b) – 89
c) – 91
d) – 93 
Answer: b