CBSE Class 10 Quadratic Equations Sure Shot Questions Set Q

SECTION-A(VSA)

Question. Which one of the following is not a quadratic equation?
(a) \( (x + 2)^2 = 2(x + 3) \)
(b) \( x^2 + 3x = (-1)(1 - 3x)^2 \)
(c) \( (x + 2)(x - 1) = x^2 - 2x - 3 \)
(d) \( x^3 - x^2 + 2x + 1 = (x + 1)^3 \)
Answer: (c) \( (x + 2)(x - 1) = x^2 - 2x - 3 \)

Question. Which of the following equations has 2 as a root?
(a) \( x^2 - 4x + 5 = 0 \)
(b) \( x^2 + 3x - 12 = 0 \)
(c) \( 2x^2 - 7x + 6 = 0 \)
(d) \( 3x^2 - 6x - 2 = 0 \)
Answer: (c) \( 2x^2 - 7x + 6 = 0 \)

Question. If \( \frac{1}{2} \) is a root of the quadratic equation \( x^2 + kx - \frac{5}{4} = 0 \), then value of k is
(a) 2
(b) -2
(c) \( \frac{1}{4} \)
(d) \( \frac{1}{2} \)
Answer: (a) 2

Question. Which of the following equations has the sum of its roots as 3?
(a) \( 2x^2 - 3x + 6 = 0 \)
(b) \( -x^2 + 3x - 3 = 0 \)
(c) \( \sqrt{2}x^2 - \frac{3}{\sqrt{2}}x + 1 = 0 \)
(d) \( 3x^2 - 3x + 3 = 0 \)
Answer: (b) \( -x^2 + 3x - 3 = 0 \)

Question. Values of k for which the quadratic equation \( 2x^2 - kx + k = 0 \) has equal roots is
(a) 0 only
(b) 4
(c) 8 only
(d) 0, 8
Answer: (d) 0, 8

Question. Equation of \( (x + 1)^2 - x^2 = 0 \) has number of real roots equal to:
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. The roots of \( 100x^2 - 20x + 1 = 0 \) is:
(a) 1/20 and 1/20
(b) 1/10 and 1/20
(c) 1/10 and 1/10
(d) None of the options
Answer: (c) 1/10 and 1/10

Question. The sum of two numbers is 27 and product is 182. The numbers are:
(a) 12 and 13
(b) 13 and 14
(c) 12 and 15
(d) 13 and 24
Answer: (b) 13 and 14

Question. If \( \frac{1}{2} \) is a root of the quadratic equation \( x^2 - mx - 5/4 = 0 \), then value of m is:
(a) 2
(b) -2
(c) -3
(d) 3
Answer: (b) -2

Question. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, the other two sides of the triangle are equal to:
(a) Base=10cm and Altitude=5cm
(b) Base=12cm and Altitude=5cm
(c) Base=14cm and Altitude=10cm
(d) Base=12cm and Altitude=10cm
Answer: (b) Base=12cm and Altitude=5cm

Question. The roots of quadratic equation \( 2x^2 + x + 4 = 0 \) are:
(a) Positive and negative
(b) Both Positive
(c) Both Negative
(d) No real roots
Answer: (d) No real roots

Question. The sum of the reciprocals of Rehman’s ages 3 years ago and 5 years from now is 1/3. The present age of Rehman is:
(a) 7
(b) 10
(c) 5
(d) 6
Answer: (a) 7

Question. If one root of equation \( 4x^2 - 2x + k - 4 = 0 \) is reciprocal of the other. The value of k is:
(a) -8
(b) 8
(c) -4
(d) 4
Answer: (b) 8

Question. Which of the following equations has 2 as a root?
(a) \( x^2 - 4x + 5 = 0 \)
(b) \( x^2 + 3x - 12 = 0 \)
(c) \( 2x^2 - 7x + 6 = 0 \)
(d) \( 3x^2 - 6x - 2 = 0 \)
Answer: (c) \( 2x^2 - 7x + 6 = 0 \)

Question. The quadratic formula to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by
(a) \( \frac{-b \pm \sqrt{b^2 - ac}}{2a} \)
(b) \( \frac{-b \pm \sqrt{b^2 - 2ac}}{a} \)
(c) \( \frac{-b \pm \sqrt{b^2 - 4ac}}{4a} \)
(d) \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Answer: (d) \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

ASSERTION AND REASON

Directions:
(a) If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
(b) If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
(c) If Assertion is correct but Reason is incorrect.
(d) If Assertion is incorrect but Reason is correct.

Question. Assertion: If one root of the quadratic equation \( 6x^2 - x - k = 0 \) is 2/3, then the value of k is 2.
Reason: The quadratic equation \( ax^2 + bx + c = 0, a \neq 0 \) has almost two roots.
(a) A
(b) B
(c) C
(d) D
Answer: (b) Both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.

Question. Assertion: \( (2x - 1)^2 - 4x^2 + 5 = 0 \) is not a quadratic equation.
Reason: An equation of the form \( ax^2 + bx + c = 0, a \neq 0 \), where \( a, b, c \in R \) is called a quadratic equation.
(a) A
(b) B
(c) C
(d) D
Answer: (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question. Assertion: The roots of the quadratic equation \( x^2 + 2x + 2 = 0 \) are imaginary
Reason: If discriminant \( D = b^2 - 4ac < 0 \) then the roots of quadratic equation \( ax^2 + bx + c = 0 \) are imaginary.
(a) A
(b) B
(c) C
(d) D
Answer: (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question. Assertion: \( 3x^2 - 6x + 3 = 0 \) has repeated roots.
Reason: The quadratic equation \( ax^2 + bx + c = 0 \) have repeated roots if discriminant D > 0
(a) A
(b) B
(c) C
(d) D
Answer: (c) Assertion is correct but Reason is incorrect.

Question. Assertion: \( x^2 + 4x + 5 \) has two real zeroes.
Reason: A quadratic polynomial can have at the most two zeroes.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion is incorrect but Reason is correct.

Question. Assertion: \( y^3 + 3y \) 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) Assertion is correct but Reason is incorrect.

Question. Assertion: The graph \( y = f(x) \) is shown in figure, for the polynomial f (x). The number of zeros of f(x) is 3.
Reason: The number of zero of the polynomial f(x) is the number of points of which f(x) cuts or touches the axes.
(a) A
(b) B
(c) C
(d) D
Answer: (c) Assertion is correct but Reason is incorrect.

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) Both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.

Question. Assertion: \( x^2 + 11x + 30 \) has no real zeroes.
Reason: A quadratic polynomial can have at the most two zeroes.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion is incorrect but Reason is correct.

Question. Assertion: If the sum of the zeroes of the quadratic polynomial \( x^2 - 2kx + 8 \) is 2, then value of k is 1.
Reason: Sum of zeroes of a quadratic polynomial \( ax^2 + bx + c \) is (-b)/a .
(a) A
(b) B
(c) C
(d) D
Answer: (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question. Assertion: A quadratic polynomial, sum of whose zeroes is 6 and their product is 8 is \( x^2 - 14x + 48 \).
Reason: If \( \alpha \) and \( \beta \) be the zeroes of the polynomial f(x), then polynomial is given by \( f(x) = x^2 - (\alpha + \beta)x + \alpha\beta \).
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion is incorrect but Reason is correct.

Question. Assertion: \( P(x) = 3x^3 - 2x^2 + 4x^4 + x - 2 \) is a polynomial of degree 3.
Reason: The highest power of x in the polynomial P(x) is the degree of the polynomial.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion is incorrect but Reason is correct.

Question. Assertion: If the sum and product of zeroes of a quadratic polynomial are 3 and -2 respectively, then quadratic polynomial is \( x^2 - 3x - 2 \).
Reason: If S is the sum of the zeroes and P is the product of the zeroes of a quadratic polynomial, then the corresponding quadratic polynomial is \( x^2 - Sx + P \).
(a) A
(b) B
(c) C
(d) D
Answer: (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

Question. Assertion: If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( x^2 + 2x - 15 \), then \( 1/\alpha + 1/\beta \) is 2/15.
Reason: If \( \alpha \) and \( \beta \) are the zeroes of a quadratic polynomial \( ax^2 + bx + c \), then \( \alpha + \beta \) is (-b)/a and \( \alpha\beta = c/a \)
(a) A
(b) B
(c) C
(d) D
Answer: (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.