CBSE Class 10 Maths HOTs Polynomials Set 06

Say True (T) or False (F).

Question. Each polynomial has at least one zeroes.
Answer: F

Question. Zeroes of polynomial \( x^2 - 4 \) are equal in magnitude but opposite in sign.
Answer: T

Question. A cubic polynomial has 6 zeroes.
Answer: F

Question. Degree of zero polynomial is zero.
Answer: F

Question. Number zero itself is known as zero polynomial.
Answer: T

Question. Sum of zeroes in polynomial \( x^2 - ax + a \) is same as the product of its zeroes.
Answer: T

Question. Polynomial \( (x^4 - 1) \) has only two real zeroes.
Answer: T

Question. We can not find real zeroes of polynomial \( x^4 + 16 \).
Answer: T

Question. Polynomials \( (x^3 + 1) \) and \( (x^3 - 1) \) each has only one real zeroes.
Answer: T

Question. Graph of polynomial \( x^2 + 4x + 4 \) meets x-axis at two points.
Answer: F

Fill in the Blank.

Question. A monomial has ________ term/terms.
Answer: only one

Question. Least number of real zeroes of a cubic polynomial can be have is ________.
Answer: one

Question. Maximum number of distinct real zeroes of a polynomial is equal to ________ of the polynomial.
Answer: degree

Question. Polynomial \( ax^2 + c \) has two zeroes which are ________ but opposite in sign.
Answer: equal in magnitude

Question. If all zeroes of a polynomial are zero then its degree is ________.
Answer: not defined

Question. Degree of remainder in the division of polynomials is always ________ than the degree of divisor.
Answer: less

Question. *Graph of constant polynomial never meet ________ and parallel to ________.
Answer: x-axis, x-axis

Question. Graph of polynomial \( (x - 1)(x - 4) \) will intersect x axis exactly ________ points.
Answer: at two distinct

Question. Product of zeroes of any polynomial of the form \( ax^2 + bx + a \) is always ________.
Answer: one

Question. Sum of zeroes of any polynomial of the form \( ax^2 + c \) is always ________.
Answer: zero

Question. Graph of polynomial of type \( ax^2 + ax + a \) meets x axis at ________ points.
Answer: none of the

Question. Degree of ________ polynomial is not defined.
Answer: zero polynomial

* In a quadratic polynomial if \( a = c \) then zeroes are reciprocal of each other so their product is one.

MCQs with more than one correct options.

Question. Graph of a quadratic polynomial can meet the x-axis at :
(a) 4 points
(b) 3 points
(c) 2 points
(d) one point
Answer: (c) 2 points, (d) one point

Question. Value of p for which polynomial \( 3x^2 - px + 3 \) has equal zeroes is
(a) 6
(b) 12
(c) -6
(d) -12
Answer: (a) 6, (c) -6

Question. Sum of two zeroes of a polynomial \( x^3 + 6x^2 + cx + d \) is 2, then value of third zeroes can not be
(a) - 8
(b) - 4
(c) - 2
(d) - 1
Answer: (b) - 4, (c) - 2, (d) - 1

Question. If 2 is a zeroes of polynomial \( x^2 - k^2 \) then value of \( k \) is :
(a) 2
(b) 1
(c) -1
(d) -2
Answer: (a) 2, (d) -2

Question. The value 'p' for which polynomial \( x^2 - (p^2 - 9)x + 8 \) has zeroes equal in magnitude but opposite in sign is :
(a) 9
(b) -9
(c) 3
(d) -3
Answer: (c) 3, (d) -3

Question. If \( \alpha \) and \( \beta \) are zeroes of polynomial \( x^2 - 2x + 1 \), then product of zeroes of a polynomial having zeroes \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is
(a) \( \alpha \beta \)
(b) \( \frac{1}{\alpha \beta} \)
(c) 0
(d) 1
Answer: (a) \( \alpha \beta \), (b) \( \frac{1}{\alpha \beta} \), (d) 1

Question. If one of the zeroes of polynomial \( a^2x^2 + x + b^2 \) is -1, then 'a' and 'b' are related with relation is :
(a) \( a^2 + b^2 = 0 \)
(b) \( a^2 + b^2 - 1 = 0 \)
(c) \( a^2 + b^2 + 1 = 0 \)
(d) \( a^2 + b^2 = -1 \)
Answer: (c) \( a^2 + b^2 + 1 = 0 \), (d) \( a^2 + b^2 = -1 \)

Question. Zeroes of polynomial \( abx^2 + (b^2 - ac)x - bc \) are
(a) \( -\frac{b}{a} \)
(b) \( \frac{c}{a} \)
(c) \( \frac{c}{b} \)
(d) \( -\frac{a}{b} \)
Answer: (a) \( -\frac{b}{a} \), (c) \( \frac{c}{b} \)

Question. The value of 'k' for which the zeroes of polynomial \( kx^2 + 4x + 4 \) are \( \alpha \) and \( \beta \) related to \( \alpha^2 + \beta^2 = 24 \) is :
(a) 1
(b) -1
(c) \( \frac{3}{2} \)
(d) \( \frac{2}{3} \)
Answer: (b) -1, (d) \( \frac{2}{3} \)

Question. If \( \alpha, \beta \) and \( \gamma \) are the zeroes of a polynomial \( x^3 - 5x^2 - 2x + 24 \), such that \( \alpha \beta = 12 \) then difference of first two zeroes can be:
(a) 1
(b) -1
(c) 0
(d) None of the options
Answer: (a) 1, (b) -1