Title Quadratic Equations.PDF ✅

Goyal's ICSE Mathematics Question Bank with MTP for Class 10 25 B. Short Answer Type Questions [3 Marks] 1. Find the value of k for which the following equation has equal roots. x2 + 4kx + (k2 – k + 2) = 0 Sol. Given: x2 + 4kx + (k2 – k + 2) = 0 For equal roots, D = 0 i.e.; b2 – 4ac = 0 ⇒ (4k)2 – 4 (k2 – k + 2) = 0 ⇒ 16k2 – 4 (k2 – k + 2) = 0 ⇒ 4k2 – k2 + k – 2 = 0 ⇒ 3k2 + k – 2 = 0 ⇒ (k + 1) (3k – 2) = 0 ⇒ k = –1 or k = 2 3 . 2. Find the values of p for which the equation px2 – 5x + p = 0 has real and equal roots. Sol. Given px2 – 5x + p = 0 ∴ Roots are equal and real, then b2 – 4ac = 0 ⇒ (–5)2 – 4 × p × p = 0 ⇒ 4p2 = 25 ⇒ p = ± 5 2 . 3. For what values of m the equation 2x2 + mx + 2 = 0 has real roots? Sol. Given: 2x2 + mx + 2 = 0 For real roots, b2 – 4ac ≥ 0 ⇒ (m)2 – 4 × 2 × 2 ≥ 0 ⇒ (m)2 – 16 ≥ 0 ⇒ (m + 4)(m – 4) ≥ 0. m ≥ 4 or m ≤ – 4. 4. Find the values of p for which the equation px2 + 2x + 1 = 0 has distinct real roots. Sol. Given : px2 + 2x + 1 = 0 For distinct real roots b2 – 4ac > 0 ⇒ (2)2 – 4 × p × 1 > 0 ⇒ 4p < 4 ⇒ p < 1 5. Show that the equation 2(p2 + q2) x2 + 2(p + q)x + 1 = 0 has no real roots when p ≠ q. Sol. Given : 2 (p2 + q2) x2 + 2 (p + q)x + 1 = 0 D = b2 – 4ac = [2(p + q)]2 – 4 × 2(p2 + q2) × 1 = 4(p2 + q2 + 2pq) – 8(p2 + q2) = – 4 (p2 + q2 – 2pq) = – 4 (p – q)2 So for p ≠ q, D < 0, i.e., it has no real roots. Proved. 6. Solve the quadratic equation x2 – 4x – 8 = 0 for x. Give your answer correct to three significant figures. Sol. Given : x2 – 4x – 8 = 0 Using quadratic formula, x = 4 1 8 ( ) 4 4 ( ) 2 4 4 3 2 2 = 2 (1 ± 3 ) = 2 (1 + 1.732) or 2 (1 – 1.732) = 2 × 2.732 or 2 (– 0.732) ⇒ x = 5.46 or – 1.46. 7. Solve 3x2 + 11x + 10 = 0, when x∈I by factorisation. Sol. Given : 3x2 + 11x + 10 = 0 ⇒ (x + 2) (3x + 5) = 0 ⇒ (x + 2) = 0 or (3x + 5) = 0