CBSE Class 10 Quadratic Equations Sure Shot Questions Set 15

Short Answer Type Question

Question. Solve by completing the square: \( x^2 – 2\sqrt{5}x + 1 = 0 \)
Answer: \( \sqrt{5} + 2, \sqrt{5} – 2 \)

Question. Solve by completing the square: \( 4x^2 + x – 5 = 0 \)
Answer: \( 1, – \frac{5}{4} \)

Question. Solve by completing the square: \( 9x^2 + x + 15 = 0 \)
Answer: no real root.

Question. Solve by completing the square: \( x^2 – 5x + 7 = 0 \)
Answer: no real root.

Question. Solve by completing the square: \( x^2 + 4x – 9 = 0 \)
Answer: \( –2 \pm \sqrt{13} \)

Question. Solve by completing the square: \( 2x^2 – 5x + 3 = 0 \)
Answer: \( 1, \frac{3}{2} \)

Question. Solve by completing the square: \( 5x^2 – 6x – 2 = 0 \)
Answer: \( \frac{3 \pm \sqrt{19}}{5} \)

Question. Solve using quadratic formula: \( x^2 – 2\sqrt{2}x – 6 = 0 \)
Answer: \( 3\sqrt{2}, – \sqrt{2} \)

Question. Solve using quadratic formula: \( \sqrt{6}x^2 – 4x – 2\sqrt{6} = 0 \)
Answer: \( \sqrt{6}, – \frac{2}{\sqrt{6}} \)

Question. Solve using quadratic formula: \( \sqrt{3}x^2 + 11x + 6\sqrt{3} = 0 \)
Answer: \( – 2\sqrt{3}, – \frac{3}{\sqrt{3}} \)

Question. Solve using quadratic formula: \( 16x^2 – 1 = 0 \)
Answer: \( \frac{1}{4}, – \frac{1}{4} \)

Question. Solve using quadratic formula: \( 5x^2 – x – 4 = 0 \)
Answer: \( 1, – \frac{4}{5} \)

Question. Solve using quadratic formula: \( 4x^2 – 7x + 3 = 0 \)
Answer: \( 1, \frac{3}{4} \)

Question. Solve using quadratic formula: \( x^2 = 3x \)
Answer: \( 0, 3 \)

Question. Solve using quadratic formula: \( 3x^2 – 5x = 0 \)
Answer: \( 0, \frac{5}{3} \)

Question. Determine the set of values of p for which \( px^2 + 4x + 1 = 0 \) has real roots.
Answer: \( p \leq 4 \)

Question. Determine the set of values of p for which \( 2x^2 + 3x + p = 0 \) has real roots.
Answer: \( p \leq \frac{9}{8} \)

Question. Determine the set of values of p for which \( 2x^2 + px + 3 = 0 \) has real roots.
Answer: \( p^2 \geq 24 \)

Question. Determine the set of values of p for which \( 3x^2 – 2px – 5 = 0 \) has real roots.
Answer: \( p^2 + 15 \geq 0 \)

Question. Determine the set of values of p for which \( 2px^2 – 6x – 3 = 0 \) has real roots.
Answer: \( p \geq – \frac{3}{2} \)

Question. Determine whether \( x^2 + 6x + 6 = 0 \) has real roots and if so find the roots.
Answer: yes, \( – 3 \pm \sqrt{3} \)

Question. Determine whether \( x^2 – 3x + 4 = 0 \) has real roots and if so find the roots.
Answer: no

Question. Determine whether \( 4x^2 + x – 3 = 0 \) has real roots and if so find the roots.
Answer: yes, \( –1, \frac{3}{4} \)

Question. Determine whether \( 9x^2 + 30x + 25 = 0 \) has real roots and if so find the roots.
Answer: yes, \( – \frac{5}{3}, – \frac{5}{3} \)

Question. Determine whether \( 4x^2 – 12x + 9 = 0 \) has real roots and if so find the roots.
Answer: yes, \( \frac{3}{2}, \frac{3}{2} \)

Question. Determine whether \( 3x^2 – 3x + 1 = 0 \) has real roots and if so find the roots.
Answer: no

Question. Determine whether \( 3x^2 – 3x – 1 = 0 \) has real roots and if so find the roots.
Answer: yes, \( \frac{3 \pm \sqrt{21}}{6} \)

Question. Determine whether \( 4x^2 + 5\sqrt{3}x + 3 = 0 \) has real roots and if so find the roots.
Answer: yes, \( – \sqrt{3}, – \frac{\sqrt{3}}{4} \)

Question. Determine whether \( 5x^2 – 2\sqrt{5}x – 3 = 0 \) has real roots and if so find the roots.
Answer: yes, \( \frac{3\sqrt{5}}{5}, – \frac{\sqrt{5}}{5} \)

Question. Find the quadratic equation whose roots are 5 and – 5.
Answer: \( x^2 – 25 = 0 \)

Question. Find the quadratic equation whose roots are 8 and 3.
Answer: \( x^2 – 11x + 24 = 0 \)

Question. Find the quadratic equation whose roots are – 8 and – 3.
Answer: \( x^2 + 11x + 24 = 0 \)

Question. Find the quadratic equation whose roots are – 8 and 3.
Answer: \( x^2 + 5x – 24 = 0 \)

Question. Find the quadratic equation whose roots are \( \sqrt{3} \) and \( 5\sqrt{3} \).
Answer: \( x^2 – 6\sqrt{3}x + 15 = 0 \)

Question. Find the quadratic equation whose roots are \( 2\sqrt{2} \) and \( – 3\sqrt{2} \).
Answer: \( x^2 + \sqrt{2}x – 12 = 0 \)

Question. Find the quadratic equation whose roots are \( – 3\sqrt{5} \) and \( – 4\sqrt{5} \).
Answer: \( x^2 + 7\sqrt{5}x + 60 = 0 \)

Question. Find the quadratic equation whose roots are \( 1 + \sqrt{2} \) and \( 1 – \sqrt{2} \).
Answer: \( x^2 – 2x – 1 = 0 \)

Question. Find the quadratic equation whose roots are \( 4 – \sqrt{5} \) and \( 4 + \sqrt{5} \).
Answer: \( x^2 – 8x + 11 = 0 \)

Question. Find the quadratic equation whose roots are \( 7 + \sqrt{7} \) and \( 7 – \sqrt{7} \).
Answer: \( x^2 – 14x + 42 = 0 \)

Question. Find the quadratic equation whose roots are \( \frac{3 + \sqrt{2}}{3} \) and \( \frac{3 – \sqrt{2}}{3} \).
Answer: \( 9x^2 – 18x + 7 = 0 \)

Question. Find the quadratic equation whose roots are \( \frac{4 – \sqrt{5}}{2} \) and \( \frac{4 + \sqrt{5}}{2} \).
Answer: \( 4x^2 – 16x + 11 = 0 \)

Long Answer Type Question

Question. Find the value of ‘m’ so that the roots of the equation \( (4 – m)x^2 + (2m + 4)x + (8m + 1) = 0 \) may be equal.
Answer: 0, 3

Question. For the quadratic equation \( ax^2 + 7x + c = 0 \); the sum of roots is –1 and the product of roots is 1 ; find the values of ‘a’ and ‘c’.
Answer: a = 7, c = 7

Question. For the quadratic equation \( ax^2 – 3x – b = 0 \); the sum of roots is 6 and the product of roots is – 8; find the values of ‘a’ and ‘b’.
Answer: a = 1/2, b = 4

Question. Find the value of p ; if one root of quadratic equation \( 3x^2 – px – 6 = 0 \) is 3. Also, find the second (other) roots of the equation.
Answer: p = 7, \( – \frac{2}{3} \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x – 4 = 0 \); find the value of : \( \alpha^2 + \beta^2 \)
Answer: \( \frac{41}{4} \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x – 4 = 0 \); find the value of : \( \alpha^2 + \beta^2 – 3\alpha – 3\beta \)
Answer: \( \frac{71}{4} \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x – 4 = 0 \); find the value of : \( \alpha^2 + \beta^2 – 4\alpha\beta \)
Answer: \( \frac{73}{4} \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x – 4 = 0 \); find the value of : \( \alpha^3 + \beta^3 \)
Answer: \( – \frac{245}{8} \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x – 4 = 0 \); find the value of : \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \)
Answer: \( – \frac{41}{8} \)

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 – 6x + 1 = 0 \); find the value of : \( \alpha^2 + \beta^2 \)
Answer: 34

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 – 6x + 1 = 0 \); find the value of : \( \alpha^4 + \beta^4 \)
Answer: 1154

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 – 6x + 1 = 0 \); find the value of : \( \alpha^3 + \beta^3 \)
Answer: 198

Question. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 – 6x + 1 = 0 \); find the value of : \( \alpha^2 + \beta^2 – 2\alpha\beta \)
Answer: 32

Question. Find the value(s) of p so that \( 2x^2 – 7x + p = 0 \) has equal roots :
Answer: \( 6\frac{1}{8} \)

Question. Find the value(s) of p so that \( 6x^2 + 12x – p = 0 \) has equal roots :
Answer: –6

Question. Find the value(s) of p so that \( px^2 + 4x + p = 0 \) has equal roots :
Answer: \( \pm 2 \)

Question. Find the value(s) of p so that \( 2px^2 – 20x + (13p – 1) = 0 \) has equal roots :
Answer: \( 2, – \frac{25}{13} \)

Question. Find the value(s) of p so that \( 3px^2 + 18x + p = 0 \) has equal roots :
Answer: \( \pm 3\sqrt{3} \)