CBSE Class 10 Mathematics Polynomials MCQs Set 08

Question. The value of \(x\), for which the polynomials \(x^2 - 1\) and \(x^2 - 2x + 1\) vanish simultaneously, is
(a) 2
(b) \(-2\)
(c) \(-1\)
(d) 1
Answer: (d) 1
The expressions \((x - 1)(x + 1)\) and \((x - 1)(x - 1)\) which vanish if \(x = 1\)

Question. If \(\alpha\) and \(\beta\) are zeroes and the quadratic polynomial \(f(x) = x^2 - x - 4\), then the value of \(\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta\) is
(a) \(\frac{15}{4}\)
(b) \(\frac{-15}{4}\)
(c) 4
(d) 15
Answer: (a) \(\frac{15}{4}\)
Given that, \(f(x) = x^2 - x - 4\)
\(\alpha + \beta = 1\) and \(\alpha\beta = -4\)
We have, \(\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta = \frac{\alpha + \beta}{\alpha\beta} - \alpha\beta = -\frac{1}{4} + 4 = \frac{15}{4}\)

Question. The value of the polynomial \(x^8 - x^5 + x^2 - x + 1\) is
(a) positive for all the real numbers
(b) negative for all the real numbers
(c) 0
(d) depends on value of \(x\)
Answer: (a) positive for all the real numbers
Let \(f(x) = x^8 - x^5 + x^2 - x + 1\)
For \(x = 1\) or 0, \(f(x) \geq 1 > 0\)
For \(x < 0\), each term of \(f(x)\) is Positive and so first \(f(x) > 0\).
Hence, \(f(x)\) is Positive for all real \(x\).

Question. On dividing \(x^3 - 3x^2 + x + 2\) by a polynomial \(g(x)\), the quotient and remainder were \(x - 2\) and \(-2x + 4\) respectively, then \(g(x)\) is equal to
(a) \(x^2 + x + 1\)
(b) \(x^2 + 1\)
(c) \(x^2 - x + 1\)
(d) \(x^2 - 1\)
Answer: (c) \(x^2 - x + 1\)
Here, Dividend = \(x^3 - 3x^2 + x + 2\)
Quotient = \(x - 2\)
Remainder = \(-2x + 4\) and
Divisor = \(g(x)\)
Since, dividend = Divisor \(\times\) Quotient + Remainder
So, \(x^3 - 3x^2 + x + 2 = g(x) \times (x - 2) + (-2x + 4)\)
\(g(x) \times (x - 2) = x^3 - 3x^2 + x + 2 + 2x - 4\)
\(g(x) = \frac{x^3 - 3x^2 + 3x - 2}{x - 2} = \frac{(x - 2)(x^2 - x + 1)}{(x - 2)} = x^2 - x + 1\)

Question. If \(x = 0.\bar{7}\), then \(2x\) is
(a) 1.4
(b) 1.5
(c) \(1.5\bar{4}\)
(d) 1.45
Answer: (b) 1.5
\(10x = 7.\bar{7}\)
or \(x = 0.\bar{7}\)
Subtracting, \(9x = 7\)
\(x = \frac{7}{9}\)
\(2x = \frac{14}{9} = 1.555.......... = 1.\bar{5}\)

Question. The difference between two numbers is 642. When the greater is divided by the smaller, the quotient is 8 and the remainder is 19, then find the sum of cube of numbers.
(a) 391322860
(b) 319322860
(c) 319322680
(d) 391223860
Answer: (a) 391322860
Let one number be \(x\). Then, another number be \(642 + x\).
Difference between two numbers = 642
By division algorithm, Dividend = Divisor \(\times\) Quotient + Remainder
Here, Dividend = \(642 + x\), divisor = \(x\), quotient = 8 and remainder = 19
\(642 + x = 8x + 19\)
\(x - 8x = 19 - 642\)
\(-7x = -623\)
\(x = \frac{-623}{-7} = 89\)
Then, other number = \(642 + x = 642 + 89 = 731\)
Hence, the required numbers are 89 and 731.
\(89^3 + (731)^3 = 704969 + 390617891 = 391322860\)

Question. Lowest value of \(x^2 + 4x + 2\) is
(a) 0
(b) 2
(c) \(-2\)
(d) 4
Answer: (c) \(-2\)
\(x^2 + 4x + 2 = (x^2 + 4x + 4) - 2 = (x + 2)^2 - 2\)
Lowest value = \(-2\)
When, \(x + 2 = 0\)

Question. If \(a^3 - 3a^2b + 3ab^2 - b^3\) is divided by \((a - b)\), then the remainder is
(a) \(a^2 - ab + b^2\)
(b) \(a^2 + ab + b^2\)
(c) 1
(d) 0
Answer: (d) 0

Question. A quadratic polynomial when divided by \(x+2\) leaves a remainder of 1 and when divided by \(x-1\), leaves a remainder of 4. What will be the remainder if it is divided by \((x+2)(x-1)\)?
(a) 1
(b) 4
(c) \(x + 3\)
(d) \(x - 3\)
Answer: (c) \(x + 3\)

Question. If the sum of the zeroes of the polynomial \(f(x) = 2x^3 - 3kx^2 + 4x - 5\) is 6, then the value of k is
(a) 2
(b) \(-2\)
(c) 4
(d) \(-4\)
Answer: (c) 4
Sum of the zeroes = \(\frac{3k}{2}\)
\(6 = \frac{3k}{2}\)
\(k = \frac{12}{3} = 4\)

FILL IN THE BLANK

Question. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most .......... zeroes.
Answer: 3

Question. A .......... is a polynomial of degree 0.
Answer: Constant

Question. If \( \alpha \) and \( \beta \) are the zeroes of the quadratic polynomial \( ax^2 + bx + c \), then \( \alpha + \beta = \frac{-b}{..........} \) and \( \alpha\beta = \frac{c}{..........} \)
Answer: a, a

Question. If \( \alpha, \beta, \gamma \) are the zeroes of the cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), then \( \alpha + \beta + \gamma = \frac{-b}{..........} \)
Answer: a

Question. The highest power of a variable in a polynomial is called its ..........
Answer: Degree

Question. A liner polynomial is represented by a ..........
Answer: Straight line

Question. Zero of a polynomial is always ..........
Answer: zero

Question. A polynomial of degree n has at the most .......... zeroes.
Answer: n

TRUE/FALSE

Question. A polynomial of degree n has exactly n zeros.
Answer: True

Question. 3, -1, 1/3 are the zeroes of the cubic polynomial \( p(x) = 3x^3 - 5x^2 - 11x - 3 \)
Answer: True

Question. Number of zeros that polynomial \( f(x) = (x - 2)^2 + 4 \) can have is three.
Answer: False

Question. A cubic polynomial has atleast one zero.
Answer: False

Question. \( \frac{1}{\sqrt{5}}x^2 + 1 \) is a polynomial
Answer: False, because the exponent of the variable is not a whole number.

Question. \( (z - 1) \) is a factor of \( g(z) = 2z^3 - 2 \).
Answer: True

Question. Degree of a zero polynomial is not defined.
Answer: True

Question. For polynomials \( p(x) \) and any non-zero polynomial \( g(x) \), there are polynomials \( q(x) \) and and \( r(x) \) such that \( p(x) = g(x)q(x) + r(x) \), where \( r(x) = 0 \) or \( \text{degree } r(x) < \text{degree } g(x) \).
Answer: True

Question. A polynomial having two variables is called a quadratic polynomial.
Answer: False

Question. Sum of zeroes of quadratic polynomial \( = -\frac{(\text{coefficient of } x)}{(\text{coefficient of } x^2)} \)
Answer: True

MATCHING QUESTIONS

Question. Match the Zeroes in Column-I with the Quadratic polynomial in Column-II.
Column-I (Zeroes): (A) 3 and -5, (B) \( 5 + \sqrt{2} \) and \( 5 - \sqrt{2} \), (C) -9 and 1/9, (D) 5 and -5
Column-II (Quadratic polynomial): (p) \( x^2 - 25 \), (q) \( x^2 + 2x - 15 \), (r) \( x^2 + (80/9)x - 1 \), (s) \( x^2 - 10x + 21 \)
Answer: (A) - q, (B) - s, (C) - r, (D) - p.

Question. Match the Polynomial in Column-I with the Remainder in Column-II.
Column-I (Polynomial): (A) \( \frac{x^3 - 3x^2 + x + 2}{x^2 - x + 1} \), (B) \( \frac{x^3 - 3x^2 + 5x - 3}{x + 2} \), (C) \( \frac{x^4 - 6x^3 + 16x^2 - 25x + 10}{x^2 - 2x + 5} \), (D) \( \frac{x^4 - 3x^2 + 4x + 5}{x^2 - x + 1} \)
Column-II (Remainder): (p) 8, (q) \( x - 5 \), (r) -33, (s) \( -2x + 4 \)
Answer: (A) - s, (B) - r, (C) - q, (D) - p.

Question. Match the Polynomials in Column-I with their Zeroes in Column-II.
Column-I (Polynomials): (A) \( 4 - x^2 \), (B) \( x^3 - 2x^2 \), (C) \( 6x^2 - 3 - 7x \), (D) \( -x + 7 \)
Column-II (Zeroes): (p) 7, (q) -2, (r) 2, (s) 3/2, (t) 0, (u) -1/3
Answer: (A) - (r, q), (B) - (r, t), (C)- (s, u), (D)- p.

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : \( x^3 + x \) has only one real zero.
Reason : A polynomial of nth degree must have n real zeroes.
(a) A
(b) B
(c) C
(d) D
Answer: (c)
Reason is false [a polynomial of nth degree has at most n zeroes]. \( x^3 + x = x(x^2 + 1) \) has only one real zero \( x = 0 \).

Question. Assertion : The sum and product of the zeros of a quadratic polynomial are \( -\frac{1}{4} \) and \( \frac{1}{4} \) respectively. Then the quadratic polynomial is \( 4x^2 + x + 1 \).
Reason : The quadratic polynomial whose sum and product of zeros are given is \( x^2 - (\text{sum of zeros})x + \text{product of zeros} \).
(a) A
(b) B
(c) C
(d) D
Answer: (a)

Question. Assertion : If both zeros of the quadratic polynomial \( x^2 - 2kx + 2 \) are equal in magnitude but opposite in sign then value of k is \( \frac{1}{2} \).
Reason : Sum of zeros of a quadratic polynomial \( ax^2 + bx + c \) is \( -\frac{b}{a} \).
(a) A
(b) B
(c) C
(d) D
Answer: (d)
Sum of zeros \( = 0 \implies -\frac{(-2k)}{1} = 0 \implies 2k = 0 \implies k = 0 \). So, A is incorrect but R is correct.

Question. Assertion : Degree of a zero polynomial is not defined.
Reason : Degree of a non-zero constant polynomial is '0'.
(a) A
(b) B
(c) C
(d) D
Answer: (b)

Question. Assertion : The graph \( y = f(x) \) is shown in figure, for the polynomial \( f(x) \). The number of zeros of \( f(x) \) is 4.
Reason : The number of zero of the polynomial \( f(x) \) is the number of point of which \( f(x) \) cuts or touches the axes.
(a) A
(b) B
(c) C
(d) D
Answer: (c)
As the number zero of polynomial \( f(x) \) is the number of points at which \( f(x) \) cuts (intersects) the x-axis. In the given figure it is 4. So A is correct but R is incorrect as it mentions 'axes' plural (should be x-axis).