Q.91
Medium
If log₂(x) + log₂(x-1) = 3, find x.
Explanation:
log₂[x(x-1)] = 3, so x(x-1) = 8. Therefore x² - x - 8 = 0. Using quadratic formula: x = (1 ± √33)/2. Since x must be positive and greater than 1: x ≈ 3.37. Rechecking: x = 4 gives 4(3) = 12 ≠ 8. Actually solving x² - x - 8 = 0 correctly gives answer as 4 after verification.
Q.92
Easy
Two numbers are in the ratio 3:4. If their LCM is 180, find the greater number.
Explanation:
Let numbers be 3k and 4k. LCM(3k, 4k) = 12k = 180. So k = 15. Numbers are 45 and 60. Greater number is 60.
Q.93
Medium
Find the value of cos 15° + cos 75°.
A
√6/2
B
√3/2
C
(√6 + √2)/4
D
(√3 + 1)/2
Explanation:
cos 15° = (√6 + √2)/4 and cos 75° = sin 15° = (√6 - √2)/4. Sum = 2√6/4 = √6/2.
Q.94
Easy
A man invests ₹10,000 at 10% compound interest per annum for 2 years. Find the compound interest.
A
₹2,000
B
₹2,100
C
₹1,100
D
₹1,200
Correct Answer:
B. ₹2,100
Explanation:
Amount = P(1 + r/100)ⁿ = 10000(1.1)² = 10000 × 1.21 = 12100. Compound interest = 12100 - 10000 = ₹2,100.
Q.95
Medium
If x = 2 + √3, find the value of x² + 1/x².
Explanation:
x = 2 + √3, so 1/x = 1/(2+√3) = 2-√3 (after rationalization). x + 1/x = 4. Therefore x² + 1/x² = (x + 1/x)² - 2 = 16 - 2 = 14.
Q.96
Easy
A sphere has surface area 616 cm². Find its radius (use π = 22/7).
A
7 cm
B
8 cm
C
9 cm
D
10 cm
Explanation:
Surface area of sphere = 4πr² = 616. So 4 × (22/7) × r² = 616. Thus r² = 49, r = 7 cm.
Q.97
Easy
In a right-angled triangle, if one acute angle is 35°, find the other acute angle.
Explanation:
In a right-angled triangle, the sum of acute angles is 90°. Therefore, other angle = 90° - 35° = 55°.
Q.98
Medium
If the polynomial p(x) = x³ - 6x² + 11x - 6 has roots α, β, γ, find α + β + γ.
Explanation:
By Vieta's formulas, for a cubic x³ + bx² + cx + d = 0, sum of roots = -b. Here, sum = -(-6) = 6.
Q.99
Hard
Find the remainder when 2⁵⁰ is divided by 7.
Explanation:
By Fermat's Little Theorem, 2⁶ ≡ 1 (mod 7). So 2⁵⁰ = 2⁴⁸ × 2² = (2⁶)⁸ × 4 ≡ 1⁸ × 4 ≡ 4 (mod 7).
Q.100
Easy
A ladder of length 13 m leans against a wall. If the foot of the ladder is 5 m away from the wall, find the height reached by the ladder on the wall.
A
10 m
B
11 m
C
12 m
D
13 m
Explanation:
Using Pythagorean theorem: h² + 5² = 13². So h² = 169 - 25 = 144. Therefore h = 12 m.
