CBSE Class 10 Mathematics Quadratic Equations MCQs Set 06

Question. The quadratic equation ax² – 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 - 3/x = 1 are
(a) 1/2, −1
(b) 3/2, 1
(c) 3/2, −1
(d) None of the options

Answer: C

Question. If roots of the quadratic equation 3ax2 + 2bx + c = 0 are in the ratio 2 : 3, then which of the following statements is true?
(a) 8ac = 25b
(b) 8ac = 9b²
(c) 8b² = 9ac
(d) 8b² = 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) x² + 2x – 2 = 0
(b) x² + x + 2 = 0
(c) x² – 2x + 2 = 0
(d) x² – x – 2 = 0

Answer: D

Question. The roots of the equation x² + 5x + 5 = 0 are
(a) −5 + √5/2 ,  − 5 - √5/2
(b) 5 - √5/2 , 5 + √5/2,
(c) -3 + √5/2 , - 3 - √5/2
(d) 3 - √5/2 , 3 + √5/2

Answer: A

Question. ax² + 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 ax² + 2x + a = 0 has two distinct real roots, then
(a) –1 < a < 1
(b) a < –1
(c) a > 1
(d) None of the options

Answer: A

Question. Which of the following equations has two distinct real roots?
(a) 2x² - 3√2x + 9/4 = 0
(b) x² + x – 5 = 0
(c) x² + 3x + 2√2 = 0
(d) 5x² – 3x + 1 = 0

Answer: B

Question. The necessary condition for ax² + bx + c = 0 to be quadratic is
(a) a ≠ 0
(b) a = 0
(c) c ≠ 0
(d) None of the options

Answer: A

Question. Find the positive value of k for which quadratic equations x² + kx + 64 = 0 and x² – 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√2x² − 5x − √2 = 0.
(a) 9/4 , 3/2
(b) 2/3 ,  √2
(c) − √2/6, √2
(d) ± √2/3

Answer: C

Question. Which of the following equations has no real roots?
(a) x² = 10x – 2
(b) x² – 12x = 16
(c) 7x² – 1 = –8x
(d) 2x² + 5x + 5 = 0

Answer: D

Question. If x = k be a solution of the quadratic equation x² + 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) x² + 2x + 1/x = 0
(c) 2y(3y + 7) = y² + 3
(d) None of the options

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) x² + 4x + 14 = 0
(b) 2x² + 7x – 24 = 0
(c) x² – 14x + 48 = 0
(d) 3x² – 17x + 52 = 0

Answer: C

Question. The integral value of k for which the equation (k – 12) x² + 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 x² + 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 2x² + kx + 2 = 0 has equal roots, is
(a) 4
(b) ±4
(c) –4
(d) 0

Answer: B

Question. The roots of the quadratic equation 2x² – x – 6 = 0 are
(a) –2, 3/2
(b) 2, – 3/2
(c) –2, –3/2
(d) 2, 3/2

Answer: B

Question. Which of the following equations has the sum of its roots as 3?
(a) 2x² – 3x + 6 = 0
(b) –x² + 3x – 3 = 0
(c) √2x² - (3/√2) x + 1 = 0
(d) 3x² – 3x + 3 = 0

Answer: B

Question. If (x + 4) (x – 4) = 9, then the values of x are
(a) ± 5
(b) ± 1/5
(c) − 1/3, 1/5
(d) ± 4

Answer: A

Question. The sum of the squares of two consecutive natural numbers is 41. Represent this situation in the form of a quadratic equation.
(a) x² + x – 20 = 0
(b) x² – x – 20 = 0
(c) x² + x + 20 = 0
(d) x² – x + 20 = 0

Answer: A

Question. The roots of the equation x² – 2x – (r² – 1) = 0 are
(a) 1 – r, –r – 1
(b) 1 – r, r + 1
(c) 1, r
(d) 1 – r, r

Answer: B

Question. If 1/3 is a root of the equation x² kx - 5/9 = 0, then find the value of k.
(a) 3/4
(b) 4/3
(c) 2/3
(d) 3/2

Answer: B

Question. If –2 is a root of the quadratic equation 3x² + px – 8 = 0 and the quadratic equation 4x² – 2px + k = 0 has equal roots, then find the value of k.
(a) –1
(b) 2
(c) –2
(d) 1

Answer: D

Question. The roots of the quadratic equation 2x² – 3x – 5 = 0 are
(a) both equal
(b) opposite integers
(c) rational and unequal
(d) not real

Answer: C

Question. Which of the following is a root of the quadratic equation √3x² + 10x + 7√3 = 0?
(a) − √3
(b) √3
(c) 7√3
(d) −7√3

Answer: A

Question. The discriminant of the equation x² + 9x – 13 = 0 is
(a) 157
(b) 141
(c) 133
(d) 129

Answer: C

Question. The number of real roots of the equation (x – 1)² + (x – 2)² + (x – 3)² = 0 is
(a) 2
(b) 1
(c) 0
(d) 3

Answer: C

Question. Find the roots of the quadratic equation. x² − 3√5x + 10 = 0 .
(a) −2√5, √5
(b) 2√5, √5
(c) −2√5, − √5
(d) 2√5, − √5

Answer: B

Question. Solve the following quadratic equation for x : 4√3x² + 5x − 2√3 = 0
(a) √3/4 , - 2√3 = 0
(b) − √3/4 , − 2/√3
(c) √3/4 , 2/√3
(d) − √3/4 , 2/√3

Answer: A

Question. If x = √2 + √2 + √2 + ... ∞ and x is a natural number, then
(a) x² + x – 2 = 0
(b) x² + 2x + 2 = 0
(c) x² – x – 2 = 0
(d) x² – x + 2 = 0

Answer: C

Question. The roots of the quadratic equation 5(x – 3)² = 20 are
(a) 1, – 5
(b) 1, 5
(c) –1, –5
(d) –1, 5

Answer: B

Question. For what value of t, x = 2/3 is a root of 7x² + tx – 3 = 0?
(a) 1/6
(b) − 1/6
(c) 1/5
(d) 1/8

Answer: B

Question. The roots of the quadratic equation x² - 8 / x² + 20 = 1/2 are
(a) ± 3
(b) ± 2
(c) ± 6
(d) ± 4

Answer: C

Question. Find the roots of the following quadratic equation. 2√3x² − 5x + √3 = 0.
(a) − √3/2 , 1/√3
(b) √3/2 , -1/√3
(c) √3/2, 1/√3
(d) − √3/2, −1/√3

Answer: C

Question. The roots of the equation √x² + 15 = 8 are
(a) x = 7
(b) x = ± 7
(c) x = – 7
(d) x = 0

Answer: B

Case Based MCQs

Case I : Read the following passage and answer the questions.

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) x² + 2x – 650 = 0
(b) 2x² +2x – 649 = 0
(c) x² – 2x – 650 = 0
(d) 2x² + 6x – 550 = 0

Answer: B

Question. The sum of two numbers is 15 and the sum of their reciprocals is 3/10.
(a) x² + 10x – 150 = 0
(b) 15x²– x + 150 = 0
(c) x² – 15x + 50 = 0
(d) 3x² – 10x + 15 = 0

Answer: C

Question. Two numbers differ by 3 and their product is 504.
(a) 3x² – 504 = 0
(b) x² – 504x + 3 = 0
(c) 504x² + 3 = x
(d) x² + 3x – 504 = 0

Answer: D

Question. A natural number whose square diminished by 84 is thrice of 8 more of given number.
(a) x² + 8x – 84 = 0
(b) 3x² – 84x + 3 = 0
(c) x² – 3x – 108 = 0
(d) x² –11x + 60 = 0

Answer: C

Question. A natural number when increased by 12, equals 160 times its reciprocal.
(a) x² – 12x + 160 = 0
(b) x² – 160x + 12 = 0
(c) 12x² – x – 160 = 0
(d) x² + 12x – 160 = 0

Answer: D

Case II : Read the following passage and answer the questions from.

Factorization Method

Amit is preparing for his upcoming semester exam. For this, he has to practice the chapter of Quadratic Equations. So he started with factorization method. Let two linear factors of
ax² + bx + c be (px + q) and (rx + s).
∴ ax² + bx + c = (px + q) (rx + s)
= prx² + (ps + qr)x + qs.
Now, help Amit in factorizing the following quadratic equations and find the roots.

Question. 6x² + x – 2 = 0
(a) 1, 6
(b) 1/2 , 2/3
(c) 1/3 , -1/2
(d) 3/2, −2

Answer: B

Question. 2x² + x – 300 = 0
(a) 30, 2/15
(b) 60, -2/5
(c) 12, -25/2
(d) None of the options

Answer: C

Question. x² – 8x + 16 = 0
(a) 3, 3
(b) 3, –3
(c) 4, –4
(d) 4, 4

Answer: D

Question. 6x² – 13x + 5 = 0
(a) 2, 3/5
(b) − 2, -5/3
(c) 1/2, -3/5
(d) 1/2, 5/3

Answer: D

Question. 100x² – 20x + 1 = 0
(a) 1/10 , 1/10
(b) –10, –10
(c) −10, 1/10
(d) −1/10 , -1/10

Answer: A

Case-III : Read the following passage and answer the questions.

Nature of Roots

A quadratic equation can be defined as an equation of degree 2. This means that the highest exponent of the polynomial in it is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Every quadratic equation has two roots depending on the nature of its discriminant, D = b² – 4ac.

Question. Which of the following quadratic equation have no real roots?
(a) –4x² + 7x – 4 = 0
(b) –4x² + 7x – 2 = 0
(c) –2x² + 5x – 2 = 0
(d) 3x² + 6x + 2 = 0

Answer: A

Question. Which of the following quadratic equation have rational roots?
(a) x² + x – 1 = 0
(b) x² – 5x + 6 = 0
(c) 4x² – 3x – 2 = 0
(d) 6x² – x + 11 = 0

Answer: B

Question. Which of the following quadratic equation have irrational roots?
(a) 3x² + 2x + 2 = 0
(b) 4x² – 7x + 3 = 0
(c) 6x² – 3x – 5 = 0
(d) 2x² + 3x – 2 = 0

Answer: C

Question. Which of the following quadratic equations have equal roots?
(a) x² – 3x + 4 = 0
(b) 2x² – 2x + 1 = 0
(c) 5x² – 10x + 1 = 0
(d) 9x² + 6x + 1 = 0

Answer: D

Question. Which of the following quadratic equations has two distinct real roots?
(a) x² + 3x + 1 = 0
(b) –x² + 3x – 3 = 0
(c) 4x² + 8x + 4 = 0
(d) 3x² + 6x + 4 = 0

Answer: A

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 : 2x² – 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.

Answer: C

Question. Assertion : 3y2 + 17y – 30 = 0 have distinct roots.
Reason : The quadratic equation ax² + bx + c = 0 have distinct roots (real roots) if D > 0.

Answer: A

Question. Assertion : 9x² – 3x – 20 = 0 ⇒ (3x – 5) (3x + 4) = 0 If the roots are calculated by splitting the middle term.
Reason : To factorise ax² + bx + c = 0, we write it in the form ax² + b1x + b2x + c = 0 such that b1 + b2 = b and b1b2 = ac.

Answer: A

Question. Assertion : The value of k for which the equation kx² – 12x + 4 = 0 has equal roots, is 9.
Reason : The equation ax² + bx + c = 0, (a ≠ 0) has equal roots, if (b2 – 4ac) > 0.

Answer: C

Question. Assertion : Both the roots of the equation x² – x + 1 = 0 are real.
Reason : The roots of the equation ax² + bx + c = 0 are real if and only if b2 – 4ac ≥ 0.

Answer: D

Question. Assertion : 2 2 is a root of the quadratic equation x² − 4 2x + 8 = 0.
Reason : The root of a quadratic equation satisfies it.

Answer: A

Question. Assertion : 1 / (x − 1)(x − 2) + 1/(x - 2)(x - 3) = 2/3 (x ≠ 1,2,3) is a quadratic equation.
Reason : An equation of the form ax² + bx + c = 0, where a, b, c ∈ R is a quadratic equation.

Answer: C