CBSE Class 10 Mathematics Quadratic Equations MCQs Set 08

Question. \( (x^2 + 1)^2 - x^2 = 0 \) has
(a) four real roots
(b) two real roots
(c) no real roots
(d) one real root
Answer: C
Given equation is, \( (x^2 + 1)^2 - x^2 = 0 \)
\( x^4 + 1 + 2x^2 - x^2 = 0 \) [\( (a + b)^2 = a^2 + b^2 + 2ab \)]
\( x^4 + x^2 + 1 = 0 \)
Let, \( x^2 = y \)
\( y^2 + y + 1 = 0 \)
On comparing with \( ay^2 + by + c = 0 \), we get \( a = 1, b = 1 \) and \( c = 1 \)
Discriminant, \( D = b^2 - 4ac \)
\( = (1)^2 - 4(1)(1) \)
\( = 1 - 4 = -3 \)
Since, \( D < 0 \), \( y^2 + y + 1 = 0 \) i.e., \( x^4 + x^2 + 1 = 0 \) or \( (x^2 + 1)^2 - x^2 = 0 \) has no real roots.

Question. The equation \( 2x^2 + 2(p + 1)x + p = 0 \), where \( p \) is real, always has roots that are
(a) Equal
(b) Equal in magnitude but opposite in sign
(c) Irrational
(d) Real
Answer: D
The discrimination of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \).
Here, \( a = 2, b = 2(p + 1) \) and \( c = p \)
\( b^2 - 4ac = [2(p + 1)]^2 - 4(2)(p) \)
\( = 4(p + 1)^2 - 8p \)
\( = 4[(p + 1)^2 - 2p] \)
\( = 4[p^2 + 2p + 1 - 2p] \)
\( = 4(p^2 + 1) \)
For any real value of \( p \), \( 4(p^2 + 1) \) will always be positive as \( p^2 \) cannot be negative for real \( p \).
Hence, the discriminant \( b^2 - 4ac \) will always be positive. When the discriminant is greater than '0' or is positive, then the roots of a quadratic equation will be real.

Question. Out of a certain number of saras birds, one-fourth the number are moving about lotus plants, \( \frac{1}{9} \text{th} \) are coupled with \( \frac{1}{4} \text{th} \) as well as 7 times the square root of the number move on a hill, 56 birds remain in vakula tree. What is the total number of birds?
(a) 576
(b) 567
(c) 556
(d) 557
Answer: A
Let the total number of birds be \( x \). Then, number of birds moving about lotus plants \( = \frac{x}{4} \) and number of birds moving on a hill \( = \frac{x}{9} + \frac{x}{4} + 7\sqrt{x} \).
Given, number of birds in vakula tree \( = 56 \)
According to the given condition,
\( \frac{x}{4} + \left( \frac{x}{9} + \frac{x}{4} + 7\sqrt{x} \right) + 56 = x \)
\( x - \frac{x}{4} - \frac{x}{9} - \frac{x}{4} - 7\sqrt{x} - 56 = 0 \)
\( \frac{36x - 9x - 4x - 9x}{36} - 7\sqrt{x} - 56 = 0 \)
\( \frac{14x}{36} - 7\sqrt{x} - 56 = 0 \)
\( \frac{7x}{18} - 7\sqrt{x} - 56 = 0 \)
\( \frac{x}{18} - \sqrt{x} - 8 = 0 \) [dividing both sides by 7]
\( x - 18\sqrt{x} - 144 = 0 \)
Put \( \sqrt{x} = y \), then above equation becomes
\( y^2 - 18y - 144 = 0 \)
\( y^2 - 24y + 6y - 144 = 0 \)
\( y(y - 24) + 6(y - 24) = 0 \)
\( (y - 24)(y + 6) = 0 \)
\( \Rightarrow y = 24 \) or \( -6 \)
But \( y \neq -6 \) as \( \sqrt{x} = y \)
\( y = 24 \)
\( \Rightarrow \sqrt{x} = 24 \Rightarrow x = 576 \)
Hence, total number of birds is 576.

Question. If \(\sqrt{x + 10} - \frac{6}{\sqrt{x + 10}} = 5\), then extraneous root of this equation is
(a) 26
(b) -9
(c) -26
(d) 9
Answer: B
Given, \(\sqrt{x + 10} - \frac{6}{\sqrt{x + 10}} = 5\) ...(1)
\(\frac{x + 10 - 6}{\sqrt{x + 10}} = 5\)
\(x + 4 = 5\sqrt{x + 10}\)
On squaring both sides, we get
\(x^2 + 16 + 8x = 25(x + 10)\)
\(x^2 + 8x - 25x - 250 + 16 = 0\)
\(x^2 - 17x - 234 = 0\)
\(x^2 - 26x + 9x - 234 = 0\)
\(x(x - 26) + 9(x - 26) = 0\)
\((x + 9)(x - 26) = 0\)
\(x + 9 = 0\) or \(x - 26 = 0\)
\(x = 26\) or \(-9\)
On putting \(x = -9\) in Eq. (1), we get
\(\sqrt{-9 + 10} - \frac{6}{\sqrt{-9 + 10}} = 5\)
\(1 - \frac{6}{1} = 5 \Rightarrow -5 = 5\)
Which is not true.
Hence, extraneous root of given equation is -9.

Question. If \(\sin \alpha\) and \(\cos \alpha\) are the roots of the equation \(ax^2 + bx + c = 0\), then \(b^2\) is
(a) \(c^2 + 2ac\)
(b) \(a^2 + ac\)
(c) \(a^2 + 2ac\)
(d) \(c^2 + ac\)
Answer: C
Given equation is, \(ax^2 + bx + c = 0\)
Since, \(\sin \alpha\) and \(\cos \alpha\) are the roots of the equation.
Sum of the roots, \(\sin \alpha + \cos \alpha = \frac{-b}{a}\) ...(1)
and product of the roots, \(\sin \alpha \cdot \cos \alpha = \frac{c}{a}\) ...(2)
On squaring both sides of Eq. (1), we get
\((\sin \alpha + \cos \alpha)^2 = \left(\frac{-b}{a}\right)^2\)
\(\sin^2 \alpha + \cos^2 \alpha + 2 \sin \alpha \cos \alpha = \frac{b^2}{a^2}\)
\(1 + 2 \sin \alpha \cos \alpha = \frac{b^2}{a^2}\) [\(\sin^2 \theta + \cos^2 \theta = 1\)]
\(2 \sin \alpha \cos \alpha = \frac{b^2}{a^2} - 1\)
\(2 \times \left(\frac{c}{a}\right) = \frac{b^2 - a^2}{a^2}\) [From Eq. 2]
\(2ac = b^2 - a^2\)
\(b^2 = a^2 + 2ac\)
Hence proved.

Question. Draw the graph of \(y = x^2 + x - 12\). If \(y = 0\), then area of the triangle formed by joining the intersection point of curve.
(a) 12 sq. units
(b) 24 sq. units
(c) 42 sq. units
(d) 48 sq. units
Answer: C

Question. Plot the roots of the equations \(x^2 - 4x + 3 = 0\) and \(2y^2 - 7y + 3 = 0\) and find the area of the smallest triangle formed by joining these points and origin.
(a) 0.5 sq units
(b) 0.05 sq units
(c) 0.15 sq. units
(d) 0.25 sq. units
Answer: D

Question. The condition for one root of the quadratic equation \(ax^2 + bx + c = 0\) to be twice the other, is
(a) \(b^2 = 4ac\)
(b) \(2b^2 = 9ac\)
(c) \(c^2 = 4a + b^2\)
(d) \(c^2 = 9a - b^2\)
Answer: B
\(\alpha + 2\alpha = -\frac{b}{a}\)
and \(\alpha \times 2\alpha = \frac{c}{a}\)
\(3\alpha = -\frac{b}{a}\)
\(\alpha = -\frac{b}{3a}\)
and \(2\alpha^2 = \frac{c}{a}\)
\(2\left(-\frac{b}{3a}\right)^2 = \frac{c}{a}\)
\(\frac{2b^2}{9a^2} = \frac{c}{a}\)
\(2ab^2 - 9a^2c = 0\)
\(a(2b^2 - 9ac) = 0\)
Since, \(a \neq 0\)
\(2b^2 = 9ac\)
Hence, the required condition is \(2b^2 = 9ac\).

Question. If \(x^2 + y^2 = 25\), \(xy = 12\), then \(x =\)
(a) \(\{3, 4\}\)
(b) \(\{3, -3\}\)
(c) \(\{3, 4, -3, -4\}\)
(d) \(\{3, -3\}\)
Answer: C
\(x^2 + y^2 = 25\)
\(xy = 12 \Rightarrow y = \frac{12}{x}\)
\(x^2 + \left(\frac{12}{x}\right)^2 = 25\)
\(x^4 + 144 - 25x^2 = 0\)
\((x^2 - 16)(x^2 - 9) = 0\)
Hence, \(x^2 = 16\) and \(x^2 = 9\)
\(x = \pm 4\) and \(x = \pm 3\)

Question. If \(x = \sqrt{7 + 4\sqrt{3}}\), then \(x + \frac{1}{x} =\)
(a) 4
(b) 6
(c) 3
(d) 2
Answer: A
We have \(x = \sqrt{7 + 4\sqrt{3}}\)
\(\frac{1}{x} = \frac{1}{\sqrt{7 + 4\sqrt{3}}} = \frac{\sqrt{7 - 4\sqrt{3}}}{\sqrt{7 + 4\sqrt{3}} \cdot \sqrt{7 - 4\sqrt{3}}}\)
\(= \sqrt{7 - 4\sqrt{3}}\)
\(x + \frac{1}{x} = \sqrt{7 + 4\sqrt{3}} + \sqrt{7 - 4\sqrt{3}}\)
\(= (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4\)

Question. If the roots of the equation \(px^2 + 2qx + r = 0\) and \(qx^2 - 2\sqrt{pr}x + q = 0\) be real, then
(a) \(p = q\)
(b) \(q^2 = pr\)
(c) \(p^2 = qr\)
(d) \(r^2 = pq\)
Answer: B
Equation \(px^2 + 2qx + r = 0\) has real roots, so
\((2q)^2 - 4pr \geq 0 \Rightarrow 4q^2 - 4pr \geq 0 \Rightarrow q^2 \geq pr\) ...(i)
and from second \(qx^2 - 2\sqrt{pr}x + q = 0\), for real roots:
\((-2\sqrt{pr})^2 - 4(q)(q) \geq 0 \Rightarrow 4pr - 4q^2 \geq 0 \Rightarrow pr \geq q^2\) ...(ii)
From (i) and (ii), we get result \(q^2 = pr\)

Question. If the ratio of the roots of the equation \(x^2 + bx + c = 0\) is the same as that of \(x^2 + qx + r = 0\), then
(a) \(r^2b = qc^2\)
(b) \(r^2c = qb^2\)
(c) \(c^2r = q^2b\)
(d) \(b^2r = q^2c\)
Answer: D
Let 1, 2 be the roots of equation (i) and \(x^2 + qx + r = 0\), let 4, 8 be the roots.
For \(x^2 + 3x + 2 = 0\), \(b = 3, c = 2\)
For \(x^2 - 6x + 8 = 0\), \(q = -6, r = 8\)
Putting these values in the options, we find that option (d) is satisfied: \(b^2r = 3^2 \times 8 = 72\) and \(q^2c = (-6)^2 \times 2 = 72\).

FILL IN THE BLANK

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

Question. A polynomial of degree 2 is called the .......... polynomial.
Answer: quadratic

Question. If \(a, b\) are the roots of \(x^2 + x + 1 = 0\), then \(a^2 + b^2 = ..........\)
Answer: -1

Question. If \(\alpha, \beta\) are the roots of \(x^2 + bx + c = 0\) and \(\alpha + h, \beta + h\) are the roots of \(x^2 + qx + r = 0\), then \(h = .........\)
Answer: \(\frac{1}{2}(b - q)\)

Question. A quadratic equation cannot have more than ......... roots.
Answer: two

Question. Let \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers, \(a \neq 0\), be a quadratic equation, then this equation has no real roots if and only if .........
Answer: \(b^2 < 4ac\)

Question. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, the other two sides are ..........
Answer: 5 cm, 12 cm

QUADRATIC EQUATIONS

Question. If \( \alpha, \beta \) are roots of the equation \( ax^2 + bx + c = 0 \), then the quadratic equation whose roots are \( a\alpha + b \) and \( a\beta + b \) is .........
Answer: \( x^2 - bx + ca = 0 \)

Question. If \( r, s \) are roots of \( ax^2 + bx + c = 0 \), then \( \frac{1}{r^2} + \frac{1}{s^2} \) is .........
Answer: \( \frac{b^2 - 2ac}{c^2} \)

Question. The quadratic equation whose roots are the sum and difference of the squares of roots of the equation \( x^2 - 3x + 2 = 0 \) is ..........
Answer: \( x^2 - 8x + 15 = 0 \)

Question. The equation \( x^2 + x - 5 = 0 \) then, product of its two roots is ..........
Answer: -5

TRUE/FALSE

Question. Sum of the reciprocals of the roots of the equation \( x^2 + px + q = 0 \) is \( 1/p \).
Answer: False

Question. If the coefficient of \( x^2 \) and the constant term have the same sign and if the coefficient of \( x \) term is zero, then the quadratic equation has no real roots.
Answer: True

Question. The nature of roots of equation \( x^2 + 2x\sqrt{3} + 3 = 0 \) are real and equal.
Answer: True

Question. Every quadratic equation has at least one real root.
Answer: False

Question. For the expression \( ax^2 + 7x + 2 \) to be quadratic, the possible values of \( a \) are non-zero real numbers.
Answer: True

Question. Every quadratic equation has exactly one root.
Answer: False

Question. A quadratic equation cannot be solved by the method of completing the square.
Answer: False

Question. If the value of discriminant is equal to zero, then the equation has real and distinct roots.
Answer: False

Question. \( 0.2 \) is a root of the equation \( x^2 - 0.4 = 0 \).
Answer: False

Question. A quadratic equation has its degree at least two.
Answer: False

Question. \( (x^2 + 3x + 1) = (x - 2)^2 \) is not a quadratic equation.
Answer: True

Question. If the coefficient of \( x^2 \) and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
Answer: True

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 : 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) 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 : If roots of the equation \( x^2 - bx + c = 0 \) are two consecutive integers, then \( b^2 - 4c = 1 \).
Reason : If \( a, b, c \) are odd integer then the roots of the equation \( 4abcx^2 + (b^2 - 4ac)x - b = 0 \) are real and distinct.
(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 equation \( 9x^2 + 3kx + 4 = 0 \) has equal roots for \( k = \pm 4 \).
Reason : If discriminant ‘\( D \)’ of a quadratic equation is equal to zero then the roots of equation are real and equal.
(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 : A quadratic equation \( ax^2 + bx + c = 0 \), has two distinct real roots, if \( b^2 - 4ac > 0 \).
Reason : A quadratic equation can never be solved by using method of completing the squares.
(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 : Sum and product of roots of \( 2x^2 - 3x + 5 = 0 \) are \( 3/2 \) and \( 5/2 \) respectively.
Reason : If \( \alpha \) and \( \beta \) are the roots of \( ax^2 + bx + c = 0, a \neq 0 \), then sum of roots \( \alpha + \beta = -b/a \) and product of roots \( \alpha\beta = c/a \).
(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