CBSE Class 10 Quadratic Equations Sure Shot Questions Set 14

EXERCISE # 1

A. Very Short Answer Type Question

Question. Which of the following are quadratic polynomials
(i) \( 5x^2 – 8x + 12 \)
(ii) \( 3 + 4x – 7x^2 \)
(iii) \( 8x^2 – 15 \)
(iv) \( 8x – 15 \)
(v) \( 8x^3 – 3x \)
(vi) \( x^2 – \sqrt{5}x + 2\sqrt{3} \)
(vii) \( \sqrt{3}x^2 – 10x – 5\sqrt{3} \)
(viii) \( 7 – \sqrt{5}x – \sqrt{3}x^3 \)
(ix) \( \sqrt{15}x^2 – \sqrt{5}x + 7 \)
Answer: (i), (ii), (iii), (vi), (vii), (ix)

Question. Find the value of each given polynomial at the given value of its variable :
(i) \( 5x^2 – 7x + 2 \) at \( x = 3 \)
(ii) \( x^2 + 15x – 4 \) at \( x = – 1 \)
(iii) \( 2y^2 – y + 2 \) at \( y = – 2 \)
(iv) \( 3y + 8 – 2y^2 \) at \( y = – 3 \)
(v) \( \sqrt{2}x^2 + 3x + 1 \) at \( x = \sqrt{2} \)
(vi) \( x^3 – 3x^2 + 5x + 2 \) at \( x = – 4 \)
(vii) \( 5\sqrt{2}x^3 + 2x^2 – \sqrt{2}x + 1 \) at \( x = 2\sqrt{2} \)
Answer: (i) 26 (ii) –18 (iii) 12 (iv) –19 (v) \( 5\sqrt{2} + 1 \) (vi) –130 (vii) 173

Question. Find the value of constant ‘m’ if \( x = – 2 \) is a zero of quadratic polynomial \( 4x^2 – 3mx + 5 \).
Answer: \( – \frac{7}{2} \)

Question. Find the value of constant ‘m’ if \( y = – 5 \) is a zero of quadratic polynomial \( 7 + 4(m + 2)y – y^2 \).
Answer: \( – \frac{29}{10} \)

Question. Which of the following are quadratic equations:
(i) \( x^2 – 9x + 5 = 0 \)
(ii) \( x^2 – \frac{3}{x} = 2 \)
Answer: (i)

Question. Which of the following are quadratic equations:
(i) \( x – \frac{3}{x} = 2x^2 \)
(ii) \( 15x^2 + 27x – 33 = 0 \)
Answer: (ii)

Question. Which of the following are quadratic equations:
(i) \( \sqrt{3}x^2 + 8x = 3\sqrt{2} \)
(ii) \( \frac{7}{8}x^2 – \frac{3}{5}x + \frac{5}{7} = 0 \)
Answer: (i), (ii)

Question. Determine whether \( x = – \frac{2}{\sqrt{3}} \) and \( x = – 3\sqrt{3} \) are solutions of given equation or not :
\( \sqrt{3}x^2 + 11x + 6\sqrt{3} = 0 \)
Answer: yes

Question. Determine if \( x = 5 \) is a root of equation given below or not :
\( \sqrt{2x^2 + 4x - 5} - \sqrt{x^2 - 4x + 4} = \sqrt{1 - 12x + 3x^2} \)
Answer: no

Question. Find the value of ‘m’ for which \( x = 2 \) is a solution of the equation: \( (2m + 1)x^2 + 2x – 3 = 0 \)
Answer: \( – \frac{5}{8} \)

Question. Find the value of ‘m’ for which \( 2x – 1 = 0 \) is a solution of the equation: \( 3x^2 + 2mx – 3 = 0 \)
Answer: \( \frac{9}{4} \)

Question. Find the value of ‘m’ for which \( x + a = 0 \) is a solution of the equation: \( x^2 + 2ax – m = 0 \)
Answer: \( – a^2 \)

Question. Determine whether \( x = \frac{1}{2} \) and \( x = \frac{3}{2} \) are solutions of the equation \( 2x^2 – 5x + 3 = 0 \) or not.
Answer: no [only \( x = 3/2 \) is a solution of the given equation]

Question. Solve the following quadratic equation: \( x^2 + 5x + 6 = 0 \)
Answer: \( –3, –2 \)

Question. Solve the following quadratic equation: \( x^2 – 8x – 33 = 0 \)
Answer: \( 11, – 3 \)

Question. Solve the following quadratic equation: \( x^2 + 4x – 32 = 0 \)
Answer: \( –8, 4 \)

Question. Solve the following quadratic equation: \( x^2 + 5x – 6 = 0 \)
Answer: \( –6, 1 \)

Question. Solve the following quadratic equation: \( x^2 – 5x – 6 = 0 \)
Answer: \( 6, – 1 \)

Question. Solve the following quadratic equation: \( x^2 – 5x + 6 = 0 \)
Answer: \( 3, 2 \)

Question. Solve the following quadratic equation: \( 5x^2 – 2ax – 3a^2 = 0 \)
Answer: \( a, – \frac{3a}{5} \)

Question. Solve the following quadratic equation: \( x^2 + 8x = 0 \)
Answer: \( 0, –8 \)

Question. Solve the following quadratic equation: \( 3x^2 + 2ax – a^2 = 0 \)
Answer: \( \frac{a}{3}, – a \)

Question. Solve the following quadratic equation: \( 4x^2 – 25x – 21 = 0 \)
Answer: \( 7, – \frac{3}{4} \)

Question. Solve the following quadratic equation: \( 10x^2 – 7x – 12 = 0 \)
Answer: \( \frac{3}{2}, – \frac{4}{5} \)

Question. Solve the following quadratic equation: \( 8x^2 – 2x – 3 = 0 \)
Answer: \( \frac{3}{4}, – \frac{1}{2} \)

Question. Solve the following quadratic equation: \( 3x^2 – 7x – 6 = 0 \)
Answer: \( 3, – \frac{2}{3} \)

Question. Solve the following quadratic equation: \( x(4x – 7) = 0 \)
Answer: \( 0, \frac{7}{4} \)

Question. Solve the following quadratic equation: \( x(x + 1) + (x + 2)(x + 3) = 26 \)
Answer: \( 2, – 5 \)

Question. Solve the following quadratic equation: \( x(x – 1) + (x – 2)(x – 3) = 42 \)
Answer: \( 6, –3 \)

Question. Solve the following quadratic equation: \( 4x^2 = 25 \)
Answer: \( \frac{5}{2}, – \frac{5}{2} \)

Question. Without solving, examine the nature of roots of the equations :
(i) \( 3x^2 + 2x – 1 = 0 \)
(ii) \( 4x^2 + 3x – 1 = 0 \)
(iii) \( 6x^2 – 5x – 6 = 0 \)
(iv) \( x^2 – 6x + 9 = 0 \)
(v) \( 2x^2 – 5x + 5 = 0 \)
(vi) \( 3x^2 + 7x + 3 = 0 \)
(vii) \( 4x^2 – 4x + 1 = 0 \)
(viii) \( 5x^2 – 8x + 2 = 0 \)
(ix) \( x^2 + px – q^2 = 0 \)
Answer: (i) Rational (real) and unequal. (ii) Rational (real) and unequal. (iii) Rational and unequal. (iv) Real and equal. (v) Imaginary (vi) Irrational and unequal. (vii) Real and equal (viii) Irrational and unequal. (ix) Irrational and unequal.

Question. Find the discriminant of the following quadratic equations :
(i) \( x^2 – 3x + 1 = 0 \)
(ii) \( 4x^2 + 3x – 2 = 0 \)
(iii) \( x^2 – x + 1 = 0 \)
(iv) \( 9x^2 – px + 2 = 0 \)
(v) \( ax^2 – 3x – 5 = 0 \)
(vi) \( 4x^2 – 5x + c = 0 \)
(vii) \( \sqrt{2}x^2 + 5\sqrt{3}x – 2\sqrt{2} = 0 \)
(viii) \( 3\sqrt{5}x^2 – 8x + 2\sqrt{5} = 0 \)
Answer: (i) 5 (ii) 41 (iii) – 3 (iv) \( p^2 – 72 \) (v) \( 9 + 20a \) (vi) \( 25 – 16c \) (vii) 91 (viii) –56

Question. Find the sum and the product of the roots of the following equations :
(i) \( 2x^2 – 7x + 4 = 0 \)
(ii) \( 3x^2 + 4\sqrt{2}x + 9 = 0 \)
(iii) \( 2x^2 + 5\sqrt{3}x – 3 = 0 \)
(iv) \( x^2 – 2\sqrt{5}x – 15 = 0 \)
(v) \( 5x^2 – 10x + 3\sqrt{5} = 0 \)
Answer: (i) \( \frac{7}{2}, 2 \) (ii) \( – \frac{4\sqrt{2}}{3}, 3 \) (iii) \( – \frac{5\sqrt{3}}{2}, – \frac{3}{2} \) (iv) \( 2\sqrt{5}, – 15 \) (v) \( 2, \frac{3\sqrt{5}}{5} \)