CBSE Class 10 Mathematics Polynomials MCQs Set J

Question. If the zeroes of the quadratic polynomial \(x^2 + (a+3)x + b\) are 3 and - 4, then
(a) \(a = 2, b = 6\)
(b) \(a = -2, b = -12\)
(c) \(a = 3, b = 4\)
(d) \(a = 4, b = -3\)
Answer: B

Question. Which one among the following statements is incorrect?
(a) Graph of a linear polynomial is a straight line whereas the graph of a quadratic polynomial has one of the two shapes of parabola either open upwards \(\cup\) or open downwards \(\cap\).
(b) The shape of the parabola depends on the value of 'a' of the quadratic polynomial \(ax^2 + bx + c\).
(c) The zeroes of a quadratic polynomial \(ax^2 + bx + c, a \neq 0\) are y coordinates of the points where the parabola \(y = ax^2 + bx + c\) intersects the y-axis.
(d) A real number m is a zero of the polynomial p(x) if p(m) = 0
Answer: C

Question. A polynomial of degree n has ________
(a) two zeroes
(b) n zeroes
(c) atleast n zeroes
(d) atmost n zeroes
Answer: D

Question. If one zero of the quadratic polynomial \(x^2 + 5x + k\) is 3 then second zero of this polynomial is ______
(a) 5
(b) -3
(c) -5
(d) -8
Answer: D

Question. If the zeroes of a quadratic polynomial \(ax^2 + bx + c\) are both negative, then
(a) a is positive., b and c are negative
(b) a is negative/ b and c are positive
(c) a and c are negative, b is positive
(d) a, b and c all have the same sign
Answer: D

Question. If \((3 + \sqrt{3})\) is one of the zeroes of the quadratic polynomial \(x^2 + mx + 6\) then find the second zero.
(a) \(-\sqrt{3}\)
(b) \(3 - \sqrt{3}\)
(c) \(3 + \sqrt{3}\)
(d) \(\sqrt{3}\)
Answer: B

Question. For a quadratic polynomial \(2x^2 - 8x + b\), sum of its roots is 4 and one of the roots is \(\frac{4+\sqrt{2}}{2}\), then the value of b is______
(a) 3
(b) 6
(c) 7
(d) 8
Answer: C

Question. If the zeroes of the quadratic polynomial \(p(x) = abx^2 - (b^2 - ac)x - bc\) are \(\alpha\) & \(\beta\), then
(a) \(\alpha = \frac{-b^2}{a}\) and \(\beta = \frac{-c^2}{b}\)
(b) \(\alpha = \frac{a}{b}\) and \(\beta = \frac{b}{c}\)
(c) \(\alpha = \frac{b}{a}\) and \(\beta = \frac{-c}{b}\)
(d) \(\alpha = \frac{-a}{b}\) and \(\beta = \frac{c}{b}\)
Answer: C

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

Question. If the zeroes of the quadratic polynomial \(ax^2 - x - b\) are \(\frac{-3}{2}\) and \(\frac{5}{3}\), then
(a) a = 15, b = 6
(b) a = 6, b = 15
(c) a = 12, b = 4
(d) a = 4, b = 12
Answer: B

Question. If \(p(x) = 25x^2 - 15x - a\) where \(\alpha\) and \(\beta\) are the zeroes of the polynomial, also if it is given that \(\alpha^3 + \beta^3 = \frac{63}{125}\), then
(a) a = 5
(b) roots are \(\frac{-1}{5}\) and \(\frac{4}{5}\)
(c) a = 3
(d) roots are \(\frac{1}{5}\) and \(\frac{-2}{5}\)
Answer: B

Question. If two zeroes of the cubic polynomial \(px^3 + qx^2 + rx + s\) are 0, then the third zero is _________
(a) \(\frac{p}{q}\)
(b) \(-\frac{p}{q}\)
(c) \(-\frac{q}{p}\)
(d) 0
Answer: C

Question. If one of the zeroes of a cubic polynomial of the form \(x^3 + ax^2 + bx + c\) is the negative of the other, then
(a) a is of negative sign and b and c are of positive sign
(b) b is of negative sign and a and c are of positive sign
(c) a and c are of opposite signs and b is of negative sign
(d) a and b are of opposite signs and c is of positive sign
Answer: C

Question. If all the zeroes of the cubic polynomial \(x^3 + cx^2 + dx + b\) are equal, then
(a) cd = 9b
(b) bd = 8b
(c) cd = 6b
(d) bd = 8b
Answer: A

Question. If p and q are the zeroes of the polynomial \(bx^2 + cx + a\), value of \(\frac{1}{p^3} + \frac{1}{q^3}\)
(a) \(\frac{3abc - c^3}{ab^2}\)
(b) \(\frac{3abc + c^3}{ab^2}\)
(c) \(\frac{3abc - c^3}{a^2b}\)
(d) None of these
Answer: D

Question. If the zeroes of the polynomial \(6x^2 + 7\sqrt{3}x - 15 = 0\) are \(\alpha\) & \(\beta\), then
(a) \(\alpha = \frac{-\sqrt{3}}{2}\) & \(\beta = \frac{5\sqrt{3}}{3}\)
(b) \(\alpha = -\sqrt{3}\) & \(\beta = 5\sqrt{3}\)
(c) \(\alpha = \frac{\sqrt{3}}{2}\) & \(\beta = \frac{-5\sqrt{3}}{3}\)
(d) \(\alpha = 5\sqrt{3}\) & \(\beta = -\sqrt{3}\)
Answer: C

Question. If \(\alpha\) & \(\beta\) are the zeroes of the quadratic polynomial \(3x^2 - 11x + 6\), then find the polynomial whose zeroes are \((2\alpha + \beta)\) and \((\alpha + 2\beta)\)
(a) \(k\left(x^2 - 5x + \frac{270}{9}\right)\), k is any non-zero real number
(b) \(k\left(x^2 - 11x + \frac{260}{9}\right)\), k is any non-zero real number
(c) \(k(3x^2 - 3x + 26)\), k is any non-zero real number
(d) \(k(2x^2 - 5x + 27)\), k is any non-zero real number
Answer: B

Question. If \(\alpha, \beta\) & \(\gamma\) are the roots of the equation \(x^3 - 4x^2 - 53x + 168\) then the relation between their roots is _______
(a) \(3\alpha + \beta = 2\gamma\)
(b) \(3\alpha + 4\beta = 4\gamma\)
(c) \(3\alpha + \beta = 4\gamma\)
(d) \(\alpha + 2\beta = \gamma\)
Answer: C

Question. What must be subtracted from \(6x^4 + 16x^3 + 15x^2 - 8x + 9\), so that it is exactly divisible by \(3x^2 + 5x - 2\)?
(a) \(-19x + 15\)
(b) \(19x + 16\)
(c) \(13x + 19\)
(d) \(19x - 15\)
Answer: A

Question. If \(p(x) = x^3 - 10x^2 + 31x - 30\) and \(q(x) = x^3 - 12x^2 + 41x - 42\), then find the LCM of the polynomials p(x) and q(x).
(a) \(x^4 - 17x^3 + 101x^2 - 247x + 210\)
(b) \(x^3 - 36x^2 + 90x + 105\)
(c) \(x^4 + 18x^3 - 95x^2 + 234x - 119\)
(d) \(x^3 - 18x^2 + 108x + 114\)
Answer: A

Question. What should be added to \(\frac{1}{x^2 - 12x + 32}\) to get \(\frac{1}{x^2 - 11x + 30}\)
(a) \(\frac{2x^2 - 25x + 96}{(x - 6)(x - 5)(x - 4)(x - 8)}\)
(b) \(\frac{2x^2 - 25x - 66}{(x - 6)(x - 5)(x - 4)(x - 8)}\)
(c) \(\frac{2x^2 - 25x + 66}{(x - 6)(x - 5)(x - 4)(x - 8)}\)
(d) \(\frac{2}{(x - 6)(x - 5)(x - 4)(x - 8)}\)
Answer: C

Question. If \(p(x) = x^2 + x + 1\) and \(q(x) = x^3 - x + 1\), then the HCF of \(p(a) - p(b)\) and \(q(a) - q(b)\) is
(a) \(a + b + 1\)
(b) \(a - b + 1\)
(c) \(a - b\)
(d) \(a + b\)
Answer: C

Question. If \((x^2 + x - 1)\) is a factor of \(x^4 + 9x^3 + qx^2 - 8x + 5\) then find the values of p and q.
(a) \(p = -3, q = 4\)
(b) \(p = 4, q = -3\)
(c) \(p = 2, q = -4\)
(d) \(p = -4, q = 2\)
Answer: B

Question. If the zeroes of the algebraic expression \(3ax^2 + x(3b + 5a) + 5b\) are \(-\frac{3}{7}\) and \(-\frac{5}{3}\), then find the value of \(\frac{a}{b}\).
(a) \(\frac{1}{3}\)
(b) \(\frac{4}{5}\)
(c) \(\frac{7}{3}\)
(d) 3
Answer: B

Question. If degree of both p(x) and [p(x) + q(x)] is 15 then degree of q(x) can be
(a) 12
(b) 10
(c) 15
(d) any one of the options
Answer: D

Question. If the LCM of p(x) and q(x) is \(a^9 - b^9\) then their HCF can be
(a) \((a - b)\)
(b) \((a^2 + b^2 + ab)\)
(c) \(a^6 + b^6 + a^3b^3\)
(d) All the options
Answer: D

Question. If \(m = \frac{a+1}{a-1}\) and \(n = \frac{a-1}{a+1}\), then \(m^2 + n^2 - 3mn\) is equal to
(a) \(\frac{-a^4 + 18a^2 - 1}{a^4 - 2a^2 + 1}\)
(b) \(\frac{a^4 - 9a^2 + 3}{a^4 + 2a^2 + 1}\)
(c) \(\frac{a^4 + 9a^2 - 3}{a^4 - 2a^2 + 1}\)
(d) \(\frac{-a^4 + 16a^2 + 1}{a^4 - 2a^2 + 1}\)
Answer: A

Question. Solve \(\frac{p^2(q - r)^2}{(p+r)^2 - q^2} + \frac{q^2 -(p - r)^2}{(p + q)^2 - r^2} + \frac{r^2 - (p - q)^2}{(q+r)^2 - p^2}\)
(a) \(\frac{1}{p + q + r}\)
(b) \(p + q + r\)
(c) 0
(d) 1
Answer: D

Question. Find the value of a - b so that \(8x^4 + 14x^3 - ax^2 + bx + 2\) is exactly divisible by \(4x^2 + 3x - 2\).
(a) 4
(b) 6
(c) 9
(d) -3
Answer: C

Question. If two zeroes of the polynomial \(f(x) = x^4 - 2x^3 - 18x^2 - 6x + 45\) are \(-\sqrt{3}\) and \(\sqrt{3}\), then find the sum of other two zeroes.
(a) 0
(b) -1
(c) -2
(d) 1
Answer: C

Question. If the zeroes of the polynomial \(x^3 - 15x^2 + 66x - 80\) are \(\alpha, \beta\) & \(\gamma\) and it is also given that \(2\beta = \alpha + \gamma\) then
(a) \(\alpha = 4\)
(b) \(\gamma = 3\)
(c) \(\gamma = 7\)
(d) \(\alpha = 2\)
Answer: D

Question. If \(\alpha\) & \(\beta\) are the zeroes of the polynomial \(x^2 + 6x - k\) such that \(2\beta + \alpha = 11\) then k is equal to
(a) 18
(b) -23
(c) 391
(d) -391
Answer: C

Question. If p and q are the zeroes of the quadratic polynomial \(f(x) = cx^2 + ax + b\) then the value of \(p^4 + q^4\) is ______
(a) \(\frac{(a^2 - 2bc)^2 - b^2c^2}{c^4}\)
(b) \(\frac{(a^2 - 2bc)^2 - 2b^2c^2}{c^4}\)
(c) \(\frac{(b^2 - 2ac)^2 - a^2c^2}{c^4}\)
(d) \(\frac{(b^2 - 2ac)^2 - 2a^2c^2}{c^4}\)
Answer: B

Question. If on dividing the polynomial \(f(x) = x^3 - 4x^2 + 7x - 9\) by a polynomial g(x), the quotient q(x) and the remainder r(x) are \((x - 3)\) and \((2x - 3)\) respectively, the polynomial g(x) is ____
(a) \(x^2 + x + 1\)
(b) \(x^2 - x + 2\)
(c) \(2x^2 + x + 1\)
(d) \(2x^2 - x + 2\)
Answer: B

Question. If the zeroes of the polynomial \(f(x) = ax^3 + 3bx^2 + 3cx + d\) are in A.P. then \(2b^3 + a^2d\) is equal to _______
(a) \(a^2bc\)
(b) \(3abc\)
(c) \(2b^2ac\)
(d) \(abc\)
Answer: B

Question. If \(f(x) = 3x^4 + 6x^3 - 2x^2 - 10x - 5\) and two of its zeroes are - 1, - 1, then the other two zeroes are _______
(a) \(\sqrt{\frac{3}{5}}, -\sqrt{\frac{3}{5}}\)
(b) \(\sqrt{\frac{2}{5}}, -\sqrt{\frac{2}{5}}\)
(c) \(\sqrt{\frac{5}{3}}, -\sqrt{\frac{5}{3}}\)
(d) \(\sqrt{\frac{5}{4}}, -\sqrt{\frac{5}{4}}\)
Answer: C