CBSE Class 10 Mathematics Quadratic Equations MCQs Set 09

Question. The quadratic equation \(ax^2 - 4ax + 2a + 1 = 0\) has repeated roots, if \(a =\)
(a) 0
(b) 1/2
(c) 2
(d) 4
Answer: B

Question. The roots of the equation \(2x - \frac{3}{x} = 1\) are
(a) \(\frac{1}{2}, -1\)
(b) \(\frac{3}{2}, 1\)
(c) \(\frac{3}{2}, -1\)
(d) none of these
Answer: C

Question. If roots of the quadratic equation \(3ax^2 + 2bx + c = 0\) are in the ratio \(2 : 3\), then which of the following statements is true?
(a) \(8ac = 25b\)
(b) \(8ac = 9b^2\)
(c) \(8b^2 = 9ac\)
(d) \(8b^2 = 25ac\)
Answer: D

Question. A rope of 16 m is divided into two parts such that twice the square of the greater part exceeds the square of the smaller part by 164. Then greater and smaller parts are respectively
(a) 11 m, 5 m
(b) 9 m, 7 m
(c) 12 m, 4 m
(d) 10 m, 6 m
Answer: D

Question. The two roots of a quadratic equation are 2 and – 1. The equation is
(a) \(x^2 + 2x - 2 = 0\)
(b) \(x^2 + x + 2 = 0\)
(c) \(x^2 - 2x + 2 = 0\)
(d) \(x^2 - x - 2 = 0\)
Answer: D

Question. The roots of the equation \(x^2 + 5x + 5 = 0\) are
(a) \(\frac{-5 \pm \sqrt{5}}{2}\)
(b) \(\frac{5 \pm \sqrt{5}}{2}\)
(c) \(\frac{-3 \pm \sqrt{5}}{2}\)
(d) \(\frac{3 \pm \sqrt{5}}{2}\)
Answer: A

Question. \(ax^2 + bx + c = 0, a > 0, b = 0, c > 0\) has
(a) two equal roots
(b) one real roots
(c) two distinct real roots
(d) no real roots
Answer: D

Question. If the equation \(ax^2 + 2x + a = 0\) has two distinct real roots, then
(a) \(–1 < a < 1\)
(b) \(a < –1\)
(c) \(a > 1\)
(d) None of these
Answer: A

Question. Which of the following equations has two distinct real roots?
(a) \(2x^2 - 3\sqrt{2}x + \frac{9}{4} = 0\)
(b) \(x^2 + x - 5 = 0\)
(c) \(x^2 + 3x + 2\sqrt{2} = 0\)
(d) \(5x^2 - 3x + 1 = 0\)
Answer: B

Question. The necessary condition for \(ax^2 + bx + c = 0\) to be quadratic is
(a) \(a \neq 0\)
(b) \(a = 0\)
(c) \(c \neq 0\)
(d) None of these
Answer: A

Question. Find the positive value of \(k\) for which quadratic equations \(x^2 + kx + 64 = 0\) and \(x^2 - 8x + k = 0\) will have real roots.
(a) 16
(b) –16
(c) 12
(d) –12
Answer: A

Question. Find the roots of the quadratic equation \(3\sqrt{2}x^2 - 5x - \sqrt{2} = 0\).
(a) \(\frac{9}{4}, \frac{3}{2}\)
(b) \(\frac{2}{3}, \sqrt{2}\)
(c) \(\frac{-\sqrt{2}}{6}, \sqrt{2}\)
(d) \(\pm \frac{\sqrt{2}}{3}\)
Answer: C

Question. Which of the following equations has no real roots?
(a) \(x^2 = 10x - 2\)
(b) \(x^2 - 12x = 16\)
(c) \(7x^2 - 1 = -8x\)
(d) \(2x^2 + 5x + 5 = 0\)
Answer: D

Question. If \(x = k\) be a solution of the quadratic equation \(x^2 + 4x + 3 = 0\), then \(k = –1\) and
(a) 2
(b) – 3
(c) 3
(d) – 2
Answer: B

Question. Which of the following is not a quadratic equation?
(a) \((x + 1)(x + 3) - x + 7 = 0\)
(b) \(x^2 + 2x + \frac{1}{x} = 0\)
(c) \(2y(3y + 7) = y^2 + 3\)
(d) None of these
Answer: B

Question. In the Maths test two representatives, while solving a quadratic equation, committed the following mistakes:
(i) One of them made a mistake in the constant term and got the roots as 5 and 9.
(ii) Another one committed an error in the coefficient of \(x\) and got the roots as 12 and 4.
But in the meantime, they realised that they are wrong and they managed to get it right jointly. Find the correct quadratic equation.
(a) \(x^2 + 4x + 14 = 0\)
(b) \(2x^2 + 7x - 24 = 0\)
(c) \(x^2 - 14x + 48 = 0\)
(d) \(3x^2 - 17x + 52 = 0\)
Answer: C

Question. The integral value of \(k\) for which the equation \((k - 12) x^2 + 2 (k - 12) x + 2 = 0\) possesses no real solutions, is
(a) 12
(b) 13
(c) 14
(d) All of the above
Answer: B

Question. The roots of the equation \(x^2 + x - p(p + 1)= 0\), where \(p\) is a constant, are
(a) \(p, p + 2\)
(b) \(-p, p - 1\)
(c) \(p, - (p + 1)\)
(d) \(-p, - (p + 1)\)
Answer: C

Question. The value(s) of \(k\) for which the quadratic equation \(2x^2 + kx + 2 = 0\) has equal roots, is
(a) 4
(b) \(\pm 4\)
(c) –4
(d) 0
Answer: B

Case Based MCQs

Case I : Read the following passage and answer the questions from 39 to 43.
Formation of Quadratic Equation
Quadratic equations started around 3000 B.C. with the Babylonians. They were one of the world’s first civilisation, and came up with some great ideas like agriculture, irrigation and writing. There were many reasons why Babylonians needed to solve quadratic equations. For example to know what amount of crop you can grow on the square field.
Now represent the following situations in the form of quadratic equation.

Question. The sum of squares of two consecutive integers is 650.
(a) \(x^2 + 2x - 650 = 0\)
(b) \(2x^2 + 2x - 649 = 0\)
(c) \(x^2 - 2x - 650 = 0\)
(d) \(2x^2 + 6x - 550 = 0\)
Answer: B

Question. The sum of two numbers is 15 and the sum of their reciprocals is 3/10.
(a) \(x^2 + 10x - 150 = 0\)
(b) \(15x^2 - x + 150 = 0\)
(c) \(x^2 - 15x + 50 = 0\)
(d) \(3x^2 - 10x + 15 = 0\)
Answer: C

Question. Two numbers differ by 3 and their product is 504.
(a) \(3x^2 - 504 = 0\)
(b) \(x^2 - 504x + 3 = 0\)
(c) \(504x^2 + 3 = x\)
(d) \(x^2 + 3x - 504 = 0\)
Answer: D

Question. A natural number whose square diminished by 84 is thrice of 8 more of given number.
(a) \(x^2 + 8x - 84 = 0\)
(b) \(3x^2 - 84x + 3 = 0\)
(c) \(x^2 - 3x - 108 = 0\)
(d) \(x^2 - 11x + 60 = 0\)
Answer: C

Question. A natural number when increased by 12, equals 160 times its reciprocal.
(a) \(x^2 - 12x + 160 = 0\)
(b) \(x^2 - 160x + 12 = 0\)
(c) \(12x^2 - x - 160 = 0\)
(d) \(x^2 + 12x - 160 = 0\)
Answer: D

Assertion & Reasoning Based MCQs

Directions In these questions, a statement of Assertion is followed by a statement of Reason is given.
Choose the correct answer out of the following choices :
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion : \(2x^2 - 4x + 3 = 0\) is a quadratic equation.
Reason : All polynomials of degree \(n\), when \(n\) is a whole number can be treated as quadratic equation.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: C

Question. Assertion : \(3y^2 + 17y - 30 = 0\) have distinct roots.
Reason : The quadratic equation \(ax^2 + bx + c = 0\) have distinct roots (real roots) if \(D > 0\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: A

Question. Assertion : \(9x^2 - 3x - 20 = 0 \Rightarrow (3x - 5) (3x + 4) = 0\) If the roots are calculated by splitting the middle term.
Reason : To factorise \(ax^2 + bx + c = 0\), we write it in the form \(ax^2 + b_1x + b_2x + c = 0\) such that \(b_1 + b_2 = b\) and \(b_1b_2 = ac\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: A

Question. Assertion : The value of \(k\) for which the equation \(kx^2 - 12x + 4 = 0\) has equal roots, is 9.
Reason : The equation \(ax^2 + bx + c = 0, (a \neq 0)\) has equal roots, if \((b^2 - 4ac) > 0\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: C

Question. Assertion : Both the roots of the equation \(x^2 - x + 1 = 0\) are real.
Reason : The roots of the equation \(ax^2 + bx + c = 0\) are real if and only if \(b^2 - 4ac \geq 0\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: D

Question. Assertion : \(2\sqrt{2}\) is a root of the quadratic equation \(x^2 - 4\sqrt{2}x + 8 = 0\).
Reason : The root of a quadratic equation satisfies it.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: A

Question. Assertion : \(\frac{1}{(x-1)(x-2)} + \frac{1}{(x-2)(x-3)} = \frac{2}{3} (x \neq 1,2,3)\) is a quadratic equation.
Reason : An equation of the form \(ax^2 + bx + c = 0\), where \(a, b, c \in R\) is a quadratic equation.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: C