CBSE Class 10 Mathematics Quadratic Equations MCQs Set 07

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

Question. Each root of \( x^2 - bx + c = 0 \) is decreased by 2. The resulting equation is \( x^2 - 2x + 1 = 0 \), then
(a) \( b = 6, c = 9 \)
(b) \( b = 3, c = 5 \)
(c) \( b = 2, c = -1 \)
(d) \( b = -4, c = 3 \)
Answer: A
\( \alpha + \beta = b \)
\( \alpha \beta = c \)
According to the question
\( (\alpha + \beta - 4) = b - 4 \)
\( (\alpha - 2)(\beta - 2) = \alpha \beta - 2(\alpha + \beta) + 4 \)
\( = c - 2b + 4 \)
Now \( 2 = b - 4 \)
\( b = 6 \)
\( 1 = c - 2b + 4 \)
\( 1 = c - 2 \times 6 + 4 \)
\( 1 = c - 12 + 4 \)
\( c = 1 + 12 - 4 = 9 \)

Question. Value(s) of \( k \) for which the quadratic equation \( 2x^2 - kx + k = 0 \) has equal roots is/are
(a) 0
(b) 4
(c) 8
(d) 0, 8
Answer: D
Given equation is, \( 2x^2 - kx + k = 0 \)
On comparing with \( ax^2 + bx + c = 0 \), we get \( a = 2, b = -k \) and \( c = k \)
For equal roots, the discriminant must be zero.
\( D = b^2 - 4ac = 0 \)
\( (-k)^2 - 4(2)k = 0 \)
\( k^2 - 8k = 0 \)
\( k(k - 8) = 0 \)
\( k = 0, 8 \)
Hence, the required values of \( k \) are 0 and 8.

Question. If the equation \( (m^2 + n^2)x^2 - 2(mp + nq)x + p^2 + q^2 = 0 \) has equal roots, then
(a) \( mp = nq \)
(b) \( mq = np \)
(c) \( mn = pq \)
(d) \( mq = \sqrt{np} \)
Answer: B
\( b^2 = 4ac' \)
\( 4(mp + nq)^2 = 4(m^2 + n^2)(p^2 + q^2) \)
\( m^2q^2 + n^2p^2 - 2mnpq = 0 \)
\( (mq - np)^2 = 0 \)
\( mq - np = 0 \)
\( mq = np \)

Question. Which constant must be added and subtracted to solve the quadratic equation \( 9x^2 + \frac{3}{4}x - \sqrt{2} = 0 \) by the method of completing the square?
(a) \( \frac{1}{8} \)
(b) \( \frac{1}{64} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{9}{64} \)
Answer: B
Given equation is \( 9x^2 + \frac{3}{4}x - \sqrt{2} = 0 \)
\( (3x)^2 + \frac{1}{4}(3x) - \sqrt{2} = 0 \)
On putting \( 3x = y \), We have, \( y^2 + \frac{1}{4}y - \sqrt{2} = 0 \)
\( y^2 + \frac{1}{4}y + \left(\frac{1}{8}\right)^2 - \left(\frac{1}{8}\right)^2 - \sqrt{2} = 0 \)
\( \left(y + \frac{1}{8}\right)^2 = \frac{1}{64} + \sqrt{2} \)
\( \left(y + \frac{1}{8}\right)^2 = \frac{1 + 64\sqrt{2}}{64} \)
Thus, \( \frac{1}{64} \) must be added and subtracted to solve the given equation.

Question. Any line is said to be a tangent to the curve, if it intersects the curve at one point. If the line \( y = kx - 3 \) is a tangent to the curve \( y = 2x^2 + 7 \), then the possible values of \( k \) is
(a) \( 4\sqrt{5} \)
(b) \( -4\sqrt{5} \)
(c) Both (a) and (b)
(d) None of these
Answer: C
Given equations of line and curve are \( y = kx - 3 \) and \( y = 2x^2 + 7 \)
Now, for point of intersection consider, \( 2x^2 + 7 = kx - 3 \)
\( 2x^2 - kx + 10 = 0 \)
On comparing with \( ax^2 + bx + c = 0 \), we get \( a = 2, b = -k \) and \( c = 10 \)
Since, the line is a tangent to the curve, so the discriminant \( D = 0 \).
i.e. \( b^2 - 4ac = 0 \)
\( (-k)^2 - 4 \times 2 \times 10 = 0 \Rightarrow k^2 = 80 \)
\( k = \pm 4\sqrt{5} \)

Question. The linear factors of the quadratic equation \( x^2 + kx + 1 = 0 \) are
(a) \( k \geq 2 \)
(b) \( k \leq 2 \)
(c) \( k \geq -2 \)
(d) \( k \geq 2 \) and \( k \leq -2 \)
Answer: D
We have, \( x^2 + kx + 1 = 0 \)
On comparing with \( ax^2 + bx + c = 0 \), we get \( a = 1, b = k \) and \( c = 1 \)
For linear factors, \( D \geq 0 \)
\( b^2 - 4ac \geq 0 \)
\( k^2 - 4 \times 1 \times 1 \geq 0 \)
\( k^2 - 2^2 \geq 0 \)
\( (k - 2)(k + 2) \geq 0 \)
\( k \geq 2 \) and \( k \leq -2 \)

Question. If the coefficient of \( x \) in the quadratic equation \( x^2 + px + q = 0 \) was taken as 17 in the place of 13 and its roots were found to be -2 and -15 then the roots of the original equation.
(a) 3, 10
(b) -3, -10
(c) -3, 10
(d) 3, -10
Answer: B
Given, \( x^2 + px + q = 0 \)
When we take the coefficient of \( x \) as 17, i.e. \( p = 17 \), then the roots are -2 and -15.
Thus, we can say -2 is a root of the equation
\( x^2 + 17x + q = 0 \)
\( (-2)^2 + 17 \times (-2) + q = 0 \)
\( 4 - 34 + q = 0 \)
\( q = 30 \)
Clearly, the new quadratic equation will be
\( x^2 + 13x + 30 = 0 \)
\( x^2 + 10x + 3x + 30 = 0 \)
\( x(x + 10) + 3(x + 10) = 0 \)
\( (x + 10)(x + 3) = 0 \)
\( x = -10 \) or \( x = -3 \)

Question. If one root of the quadratic equation \( ax^2 + bx + c = 0 \) is the reciprocal of the other, then
(a) \( b = c \)
(b) \( a = b \)
(c) \( ac = 1 \)
(d) \( a = c \)
Answer: D
If one root is \( \alpha \), then the other \( \frac{1}{\alpha} \)
\( \alpha \cdot \frac{1}{\alpha} = \text{product of roots} = \frac{c}{a} \)
\( 1 = \frac{c}{a} \)
\( a = c \)

Question. One of the two students, while solving a quadratic equation in \( x \), copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of \( x^2 \) correctly as -6 and 1 respectively. The correct roots are
(a) 3, -2
(b) -3, 2
(c) -6, 1
(d) 6, -1
Answer: D
Let \( \alpha, \beta \) be the roots of the equation.
Then, \( \alpha + \beta = 5 \)
and \( \alpha\beta = -6 \).
So, the equation is \( x^2 - 5x - 6 = 0 \)
The roots of the equation are 6 and -1.

Question. The quadratic equation \( 2x^2 - \sqrt{5}x + 1 = 0 \) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Answer: C
Given equation is, \( 2x^2 - \sqrt{5}x + 1 = 0 \)
On comparing with \( ax^2 + bx + c = 0 \), we get \( a = 2, b = -\sqrt{5} \) and \( c = 1 \)
Discriminant, \( D = b^2 - 4ac \)
\( = (-\sqrt{5})^2 - 4 \times (2) \times (1) \)
\( = 5 - 8 = -3 < 0 \)
Since, discriminant is negative, therefore quadratic equation \( 2x^2 - \sqrt{5}x + 1 = 0 \) has no real roots i.e., imaginary roots.

Question. The real roots of the equation \( x^{2/3} + x^{1/3} - 2 = 0 \) are
(a) 1, 8
(b) -1, -8
(c) -1, 8
(d) 1, -8
Answer: D
The given equation is \( x^{2/3} + x^{1/3} - 2 = 0 \)
Put \( x^{1/3} = y \), then \( y^2 + y - 2 = 0 \)
\( (y - 1)(y + 2) = 0 \)
\( y = 1 \) or \( y = -2 \)
\( x^{1/3} = 1 \Rightarrow x = (1)^3 = 1 \)
or \( x^{1/3} = -2 \Rightarrow x = (-2)^3 = -8 \)
Hence, the real roots of the given equations are 1, -8.

FILL IN THE BLANK

Question. If the product \(ac\) in the quadratic equation \(ax^2 + bx + c\) is negative, then the equation cannot have .......... roots.
Answer: Non-real

Question. The equation \(ax^2 + bx + c = 0, a \neq 0\) has no real roots, if ..........
Answer: \(b^2 < 4ac\)

Question. A real number \(\alpha\) is said to be .......... of the quadratic equation \(ax^2 + bx + c = 0\), if \(a\alpha^2 + b\alpha + c = 0\).
Answer: root

Question. The equation of the form \(ax^2 + bx = 0\) will always have .......... roots.
Answer: real

Question. If the discriminant of a quadratic equation is greater than zero, then its roots are .......... and ..........
Answer: real, distinct

Question. A quadratic equation in the variable \(x\) is of the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ..........\)
Answer: \(\neq 0\)

Question. The roots of a quadratic equation is same as the .......... of the corresponding quadratic polynomial.
Answer: zero

QUADRATIC EQUATIONS

Question. A quadratic equation \( ax^2 + bx + c = 0 \) has two distinct real roots, if \( b^2 - 4ac \) ..........
Answer: \( > 0 \)

Question. For any quadratic equation \( ax^2 + bx + c = 0, b^2 - 4ac \), is called the .......... of the equation.
Answer: discriminant

Question. The values of \( k \) for which the equation \( 2x^2 + kx + 8 = 0 \) will have real and equal roots are ..........
Answer: \( 7 \) and \( -9 \)

Question. A quadratic equation does not have any real roots if the value of its discriminant is .......... zero.
Answer: less than

TRUE/FALSE

Question. \( x^2 + x - 306 = 0 \) represent quadratic equation where product of two consecutive positive integer is \( 306 \).
Answer: True

Question. If we can factorise \( ax^2 + bx + c, a \neq 0 \), into a product of two linear factors, then the roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found by equating each factor to zero.
Answer: True

Question. The equation \( (x + 2)^2 = 0 \) has real roots.
Answer: True

Question. Every quadratic equation has at most two roots.
Answer: True

Question. \( (x - 2)(x + 1) = (x - 1)(x + 3) \) is a quadratic equation.
Answer: False

Question. Every quadratic equation has at least two roots.
Answer: False

Question. The roots of the equation \( (x - 3)^2 = 3 \) are \( 3 \pm \sqrt{3} \).
Answer: True

Question. The degree of a quadratic polynomial is atmost 2.
Answer: False

Question. A quadratic equation may have no real root.
Answer: True

Question. If sum of the roots is 2 and product is 5, then the quadratic equation is \( x^2 - 2x + 5 = 0 \).
Answer: True

Question. If \( 2 \) is a zero of the quadratic polynomial \( p(x) \) then \( 2 \) is a root of the quadratic equation \( p(x) = 0 \).
Answer: True

Question. If the product \( ac \) in the quadratic equation \( ax^2 + bx + c \) is negative, then the equation cannot have non-real roots.
Answer: Ture

MATCHING QUESTIONS

Question. Column-II give roots of quadratic equations given in Column-I.
Column-I
(A) \( 6x^2 + x - 12 = 0 \)
(B) \( 8x^2 + 16x + 10 = 202 \)
(C) \( x^2 - 45x + 324 = 0 \)
(D) \( 2x^2 - 5x - 3 = 0 \)
Column-II
(p) \( (-6, 4) \)
(q) \( (9, 36) \)
(r) \( (3, -1/2) \)
(s) \( (-3/2, 4/3) \)
Answer: (A) - s, (B) - p, (C) - q, (D) - r.

Question. Match the following equations in Column-I with their descriptions in Column-II.
Column-I
(A) \( (x - 3)(x + 4) + 1 = 0 \)
(B) \( (x + 2)^3 = 2x(x^2 - 1) \)
(C) \( (2x - 2)^2 = 4x^2 \)
(D) \( (2x^2 - 2)^2 = 3 \)
Column-II
(p) Forth degree polynomial
(q) Quadratic equation
(r) Non-quadratic equation
(s) Linear equation
Answer: (A) - q, (B) - r, (C) - s, (D) - p.

Question. Match the statements in Column-I with their properties in Column-II.
Column-I
(A) If \( \alpha, \beta \) are roots of \( ax^2 + bx + c = 0 \) then one of the equation \( ax^2 + bx(x - 1) + c(x - 1)^2 = 0 \)
(B) If the roots of \( ax^2 + b = 0 \) are real, then
(C) Roots of \( 4x^2 - 4x + 1 = 0 \)
(D) Roots of \( (x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0 \) are always
Column-II
(p) \( a < 0, b > 0 \)
(q) real and equal
(r) \( \frac{\beta}{1+\beta} \)
(s) \( a > 0, b < 0 \)
(t) real
(u) \( \frac{\alpha}{1+\alpha} \)
Answer: (A) - (r, u), (B) - (p, s), (C) - q, (D) - t.

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : \( 4x^2 - 12x + 9 = 0 \) has repeated roots.
Reason : The quadratic equation \( ax^2 + bx + c = 0 \) have repeated roots if discriminant \( D > 0 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: C

Question. Assertion : The equation \( (x^2 + 3x + 1) = (x - 2)^2 \) is a quadratic equation.
Reason : Any equation of the form \( ax^2 + bx + c = 0 \) where \( a \neq 0 \), is called a quadratic equation.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: D

Question. Assertion : \( (2x - 1)^2 - 4x^2 + 5 = 0 \) is not a quadratic equation.
Reason : \( x = 0, 3 \) are the roots of the equation \( 2x^2 - 6x = 0 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: B

Question. Assertion : The values of \( x \) are \( -a/2, a \) for a quadratic equation \( 2x^2 + ax - a^2 = 0 \).
Reason : For quadratic equation \( ax^2 + bx + c = 0 \), \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: D

Question. Assertion : The equation \( 8x^2 + 3kx + 2 = 0 \) has equal roots then the value of \( k \) is \( \pm 8/3 \).
Reason : The equation \( ax^2 + bx + c = 0 \) has equal roots if \( D = b^2 - 4ac = 0 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: A

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