CBSE Class 10 Quadratic Equations Sure Shot Questions Set 06

QUADRATIC EQUATIONS

• An equation involving single variable with a term having highest degree 2 of variable is called quadratic equation.

• In general form, \( ax^2 + bx + c = 0, a \neq 0 \) is a quadratic equation in variable \( x \).

How to Check Whether a Given Equation is Quadratic or Not

• To check whether a given equation is quadratic or not, first write the given equation in its simplest form and then compare the equation with the standard form of a quadratic equation, i.e., \( ax^2 + bx + c = 0, a \neq 0 \).
• If the given equation follows the form of quadratic equation \( (ax^2 + bx + c = 0) \), then it is a quadratic equation otherwise not.

SOLUTION OF A QUADRATIC EQUATION

• The zeroes of the quadratic polynomial or the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is called the solution of the quadratic equation. Solution of a Quadratic Equation can be found by using following methods :

(i) By Factorisation Method

• To find the solution of a quadratic equation by factorisation method, first represent the given equation as a product of two linear factors by splitting the middle term or by using identities and then equate each of the factor equal to zero to get the desired roots.

(ii) By Quadratic Formula

• Quadratic Formula is used to find the solutions or roots of a quadratic equation of the form \( ax^2 + bx + c = 0, a \neq 0 \).
• Thus, for a quadratic equation \( ax^2 + bx + c = 0 \), we have \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( b^2 - 4ac \geq 0 \).
• Roots of the quadratic equation is given by, \( \alpha = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \) or \( \beta = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \).

NATURE OF ROOTS OF QUADRATIC EQUATION

• By quadratic formula, the roots of the quadratic equation are given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, a \neq 0 \).
• \( b^2 - 4ac \) is called discriminant of the quadratic equation and denoted by \( D \).
• The following cases arise :
  • (i) If \( D = b^2 - 4ac > 0 \), then the roots of the equation are real and distinct.
  • (ii) If \( D = b^2 - 4ac = 0 \), then roots of the equation are equal and real.
  • (iii) If \( D = b^2 - 4ac < 0 \), then there does not exist any real root.
  • (iv) If \( D = b^2 - 4ac > 0 \), and perfect square, then the roots are real, rational and unequal.
  • (v) If \( D = b^2 - 4ac > 0 \) and not a perfect square, then the roots are real, irrational and unequal.

Note :

• (i) \( x^2 - a^2 > 0 \Rightarrow x < -a \) or \( x > a \)
• (ii) \( x^2 - a^2 \geq 0 \Rightarrow x \leq -a \) or \( x \geq a \)
• (iii) \( x^2 - a^2 < 0 \Rightarrow -a < x < a \)
• (iv) \( x^2 - a^2 \leq 0 \Rightarrow -a \leq x \leq a \)

Multiple Choice Questions 

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) \( \frac{-5 + \sqrt{5}}{2}, \frac{-5 - \sqrt{5}}{2} \)
(b) \( \frac{5 + \sqrt{5}}{2}, \frac{5 - \sqrt{5}}{2} \)
(c) \( \frac{-3 + \sqrt{5}}{2}, \frac{-3 - \sqrt{5}}{2} \)
(d) \( \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2} \)
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) \( \frac{\sqrt{2}}{3}, \sqrt{2} \)
(c) \( \frac{-\sqrt{2}}{6}, \sqrt{2} \)
(d) \( \pm\sqrt{\frac{2}{3}} \)
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)

Question. The roots of the quadratic equation \( 2x^2 - x - 6 = 0 \) 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) \( 2x^2 - 3x + 6 = 0 \)
(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)

Question. If \( (x + 4)(x - 4) = 9 \), then the values of \( x \) are
(a) \( \pm 5 \)
(b) \( \pm \frac{1}{5} \)
(c) \( -\frac{1}{3}, \frac{1}{5} \)
(d) \( \pm 4 \)
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) \( x^2 + x - 20 = 0 \)
(b) \( x^2 - x - 20 = 0 \)
(c) \( x^2 + x + 20 = 0 \)
(d) \( x^2 - x + 20 = 0 \)
Answer: (a)

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

Question. If \( \frac{1}{3} \) is a root of the equation \( x^2 + kx - \frac{5}{9} = 0 \), then find the value of \( k \).
(a) \( \frac{3}{4} \)
(b) \( \frac{4}{3} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{3}{2} \)
Answer: (b)

Question. If –2 is a root of the quadratic equation \( 3x^2 + px - 8 = 0 \) and the quadratic equation \( 4x^2 - 2px + k = 0 \) 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 \( 2x^2 - 3x - 5 = 0 \) 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 \( \sqrt{3}x^2 + 10x + 7\sqrt{3} = 0 \)?
(a) \( -\sqrt{3} \)
(b) \( \sqrt{3} \)
(c) \( 7\sqrt{3} \)
(d) \( -7\sqrt{3} \)
Answer: (a)

Question. The discriminant of the equation \( x^2 + 9x - 13 = 0 \) is
(a) 157
(b) 141
(c) 133
(d) 129
Answer: (a)

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. \( x^2 - 3\sqrt{5}x + 10 = 0 \).
(a) \( -2\sqrt{5}, \sqrt{5} \)
(b) \( 2\sqrt{5}, \sqrt{5} \)
(c) \( -2\sqrt{5}, -\sqrt{5} \)
(d) \( 2\sqrt{5}, -\sqrt{5} \)
Answer: (b)

Question. Solve the following quadratic equation for \( x \) : \( 4\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0 \)
(a) \( \frac{\sqrt{3}}{4}, \frac{-2}{\sqrt{3}} \)
(b) \( \frac{-\sqrt{3}}{4}, \frac{-2}{\sqrt{3}} \)
(c) \( \frac{\sqrt{3}}{4}, \frac{2}{\sqrt{3}} \)
(d) \( \frac{-\sqrt{3}}{4}, \frac{2}{\sqrt{3}} \)
Answer: (a)

Question. If \( x = \sqrt{2 + \sqrt{2 + \sqrt{2 + ... \infty}}} \) and \( x \) is a natural number, then
(a) \( x^2 + x - 2 = 0 \)
(b) \( x^2 + 2x + 2 = 0 \)
(c) \( x^2 - x - 2 = 0 \)
(d) \( x^2 - x + 2 = 0 \)
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 \( t, x = \frac{2}{3} \) is a root of \( 7x^2 + tx - 3 = 0 \)?
(a) \( \frac{1}{6} \)
(b) \( -\frac{1}{6} \)
(c) \( \frac{1}{5} \)
(d) \( \frac{1}{8} \)
Answer: (a)

Question. The roots of the quadratic equation \( \frac{x^2 - 8}{x^2 + 20} = \frac{1}{2} \) are
(a) \( \pm 3 \)
(b) \( \pm 2 \)
(c) \( \pm 6 \)
(d) \( \pm 4 \)
Answer: (c)

Question. Find the roots of the following quadratic equation. \( 2\sqrt{3}x^2 - 5x + \sqrt{3} = 0 \).
(a) \( \frac{-\sqrt{3}}{2}, \frac{1}{\sqrt{3}} \)
(b) \( \frac{\sqrt{3}}{2}, \frac{-1}{\sqrt{3}} \)
(c) \( \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{3}} \)
(d) \( \frac{-\sqrt{3}}{2}, \frac{-1}{\sqrt{3}} \)
Answer: (c)

Question. The roots of the equation \( \sqrt{x^2 + 15} = 8 \) are
(a) \( x = 7 \)
(b) \( x = \pm 7 \)
(c) \( x = -7 \)
(d) \( x = 0 \)
Answer: (b)