CBSE Class 10 Mathematics Quadratic Equations Assignment Set 24

Previous Year Board Problems

Question 1. Solve for x: \( 4\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0 \).
Answer:
D = 25 + 96 = 121. \( x = \dfrac{-5 \pm 11}{8\sqrt{3}} \): x = √3/4, −2/√3 (i.e. −2√3/3).
In simple words: Apply the quadratic formula.

Teacher's Note:
1. 8√3 comes from 2a.

Question 2. For what value of k are the roots of \( kx(x - 2\sqrt{5}) + 10 = 0 \) equal?
Answer:
\( kx^2 - 2\sqrt{5}kx + 10 = 0 \). D = 20k² − 40k = 0, so k = 0 or 2. For k = 0 the equation is not quadratic: k = 2.
In simple words: Set D = 0.

Teacher's Note:
1. Reject k = 0 because the equation must be quadratic.

Question 3. The sum of the areas of two squares is 400 cm². If the difference of their perimeters is 16 cm, find the sides.
Answer:
Sides x and y (x > y): \( x^2 + y^2 = 400 \) and \( 4x - 4y = 16 \), so y = x − 4. Then \( x^2 + (x - 4)^2 = 400 \), \( x^2 - 4x - 192 = 0 \), \( (x - 16)(x + 12) = 0 \).
Sides: 16 cm and 12 cm.
In simple words: Perimeter of a square = 4 × side.

Teacher's Note:
1. Check: 256 + 144 = 400.

Question 4. Solve for x: \( \sqrt{3}x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0 \).
Answer:
D = 8 + 24 = 32. \( x = \dfrac{2\sqrt{2} \pm 4\sqrt{2}}{2\sqrt{3}} \): x = √6, −√6/3.
In simple words: Use the formula with a = √3.

Teacher's Note:
1. √32 = 4√2.

Question 5. If 2 is a root of 3x² + px − 8 = 0 and the equation 4x² − 2px + k = 0 has equal roots, find k.
Answer:
12 + 2p − 8 = 0, so p = −2. Then \( 4x^2 + 4x + k = 0 \) with D = 16 − 16k = 0: k = 1.
In simple words: First find p.

Teacher's Note:
1. Then use D = 0.

Question 6. The sum of the squares of two consecutive even numbers is 340. Find the numbers.
Answer:
Let them be n and n + 2. \( n^2 + (n + 2)^2 = 340 \), so \( n^2 + 2n - 168 = 0 \), \( (n - 12)(n + 14) = 0 \). The numbers are 12 and 14 (or −14 and −12 if negatives are allowed).
In simple words: Consecutive even numbers differ by 2.

Teacher's Note:
1. Check: 144 + 196 = 340.

Question 7. If the equation \( px^2 - 2\sqrt{5}px + 15 = 0 \) has two equal roots, find p.
Answer:
D = 20p² − 60p = 0, so p = 0 or 3. p ≠ 0, hence p = 3.
In simple words: D = 0.

Teacher's Note:
1. p = 0 would not be a quadratic.

Question 8. Solve for x: \( 4x^2 + 4bx - (a^2 - b^2) = 0 \).
Answer:
D = 16b² + 16(a² − b²) = 16a². \( x = \dfrac{-4b \pm 4a}{8} \): x = (a − b)/2, −(a + b)/2.
In simple words: Compute D first.

Teacher's Note:
1. Factorised: (2x − a + b)(2x + a + b).

Question 9. The diagonal of a rectangular field is 16 m more than the shorter side. If the longer side is 14 m more than the shorter side, find the sides.
Answer:
Shorter side s: \( s^2 + (s + 14)^2 = (s + 16)^2 \), so \( s^2 - 4s - 60 = 0 \), \( (s - 10)(s + 6) = 0 \). Sides 10 m and 24 m (diagonal 26 m).
In simple words: Use Pythagoras.

Teacher's Note:
1. Check: 100 + 576 = 676.

Question 10. A train travels 54 km at a certain average speed and then 63 km at a speed 6 km/h more. If the total time is 3 hours, find the first speed.
Answer:
\( \dfrac{54}{x} + \dfrac{63}{x + 6} = 3 \), so \( x^2 - 33x - 108 = 0 \), \( (x - 36)(x + 3) = 0 \): 36 km/h.
In simple words: Add the two times.

Teacher's Note:
1. Check: 1.5 + 1.5 = 3.

Question 11. If −5 is a root of \( 2x^2 + px - 15 = 0 \) and \( p(x^2 + x) + k = 0 \) has equal roots, find k.
Answer:
50 − 5p − 15 = 0, so p = 7. Then \( 7x^2 + 7x + k = 0 \), D = 49 − 28k = 0: k = 7/4.
In simple words: Find p, then D = 0.

Teacher's Note:
1. Check: 7x² + 7x + 7/4 = 7(x + 1/2)².

Question 12. A motor boat of speed 24 km/h in still water takes 1 hour more to go 32 km upstream than to return downstream. Find the speed of the stream.
Answer:
\( \dfrac{32}{24 - x} - \dfrac{32}{24 + x} = 1 \), so \( x^2 + 64x - 576 = 0 \), \( (x - 8)(x + 72) = 0 \): 8 km/h.
In simple words: Upstream speed 24 − x, downstream 24 + x.

Teacher's Note:
1. Check: 2 − 1 = 1.

Question 13. Find p if one root of px² − 14x + 8 = 0 is 6 times the other.
Answer:
Roots α, 6α: sum \( 7\alpha = \dfrac{14}{p} \), product \( 6\alpha^2 = \dfrac{8}{p} \). With \( \alpha = \dfrac{2}{p} \): \( \dfrac{24}{p^2} = \dfrac{8}{p} \), p = 3.
In simple words: Use sum and product of roots.

Teacher's Note:
1. Check: roots 2/3 and 4.

Question 14. If ad ≠ bc, prove that (a² + b²)x² + 2(ac + bd)x + (c² + d²) = 0 has no real roots.
Answer:
D = 4(ac + bd)² − 4(a² + b²)(c² + d²) = −4(ad − bc)². Since ad ≠ bc, D < 0, so there are no real roots.
In simple words: Expand D and use Lagrange's identity.

Teacher's Note:
1. (a²+b²)(c²+d²) = (ac+bd)² + (ad−bc)².

Question 15. Two taps together fill a tank in \( 3\tfrac{1}{13} \) hours. One tap takes 3 hours more than the other. In what time will each fill it?
Answer:
\( 3\tfrac{1}{13} = \tfrac{40}{13} \). \( \dfrac{1}{x} + \dfrac{1}{x + 3} = \dfrac{13}{40} \), so \( 13x^2 - 41x - 120 = 0 \), \( (x - 5)(13x + 24) = 0 \).
One tap takes 5 hours, the other 8 hours.
In simple words: Add the rates of filling.

Teacher's Note:
1. Check: 1/5 + 1/8 = 13/40.