NCERT 10th Maths - May 11, 05:12 – 3 Practice Problems & Printable PDF

1. If α and β are roots of x² - 5x + 6 = 0, find the equation whose roots are α + 2 and β + 2

A. x² - 9x + 16 = 0 B. x² - 9x + 20 = 0 C. x² - 4x + 6 = 0 D. x² - 11x + 30 = 0

Solution

Correct: B

Given α + β = 5 and αβ = 6. New roots: α + 2 and β + 2. Sum = (α + 2) + (β + 2) = α + β + 4 = 5 + 4 = 9. Product = (α + 2)(β + 2) = αβ + 2α + 2β + 4 = 6 + 4 + 10 = 20. The equation becomes x² - (sum)x + product = 0 ⇒ x² - 9x + 20 = 0.

2. Find the value of k for which the pair of equations x + 2y - 3 = 0 and 5x + ky + 7 = 0 represents parallel lines

A. k = -10 B. k = 10 C. k = 5 D. k = -5

Solution

Correct: B

For parallel lines, the coefficients must be proportional: 1/5 = 2/k ≠ -3/7. Solving 1/5 = 2/k ⇒ k = 10. However, 1/5 ≠ -3/7, which is true. Thus, k = 10 satisfies the condition.

3. The angles of elevation of the top of a tower from two points on the same straight path are 30° and 60°. If the points are 800 m apart, find the height of the tower

A. 200√3 m B. 400√3 m C. 600 m D. 800√3 m

Solution

Correct: B

Let h be the height. From the first point: h/x = tan30° ⇒ x = h√3. From the second point: h/(x - 800) = tan60° ⇒ x - 800 = h/√3. Substitute x = h√3 into the second equation: h√3 - 800 = h/√3 ⇒ Multiply by √3: 3h - 800√3 = h ⇒ 2h = 800√3 ⇒ h = 400√3 m.