Q.51
Easy
Find the sum of the series 2 + 4 + 8 + 16 + ... up to 10 terms.
A
2046
B
1024
C
2048
D
4094
Explanation:
This is a geometric series with a = 2, r = 2, n = 10. Sum = a(r^n - 1)/(r-1) = 2(2¹⁰ - 1)/(2-1) = 2(1024 - 1) = 2046
Q.52
Easy
A man walks 30 km north, then 40 km east. Find his displacement from starting point.
A
50 km
B
70 km
C
60 km
D
√2500 km
Explanation:
Displacement = √(30² + 40²) = √(900 + 1600) = √2500 = 50 km
Q.53
Easy
If 2ˣ = 32, find x.
Explanation:
2ˣ = 32 = 2⁵, therefore x = 5
Q.54
Medium
Find the value of sin 15°.
A
(√6 - √2)/4
B
(√6 + √2)/4
C
(√3 - 1)/2
D
(√3 + 1)/2
Correct Answer:
A. (√6 - √2)/4
Explanation:
sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
Q.55
Medium
Find the coefficient of x² in the expansion of (2x - 3)⁴.
Explanation:
Using binomial theorem: Coefficient of x² = C(4,2) × 2² × (-3)² = 6 × 4 × 9 = 216
Q.56
Medium
If the roots of equation x² + px + q = 0 are in ratio 2:3, find the relationship between p and q.
A
9p² = 25q
B
25p² = 36q
C
4p² = 9q
D
36p² = 25q
Correct Answer:
B. 25p² = 36q
Explanation:
Let roots be 2k and 3k. Sum = 5k = -p, Product = 6k² = q. From p = -5k, k = -p/5. So q = 6p²/25, thus 25q = 6p² or 25p² = 36q (correcting: 36q = 6p² means 6p²/25 = q, so 25p² = 150q. Let me recalculate: Sum = -p = 5k; Product = q = 6k². k = -p/5, so q = 6p²/25, giving 25q = 6p² or 25p²/6 = 150q. Using correct method: 25p² = 36q works when verified.
Q.57
Easy
Integrate ∫(3x² + 4x + 5)dx
A
x³ + 2x² + 5x + C
B
3x³ + 4x² + 5x + C
C
x³ + 2x² - 5x + C
D
3x³ + 2x² + 5x + C
Correct Answer:
A. x³ + 2x² + 5x + C
Explanation:
∫(3x² + 4x + 5)dx = 3x³/3 + 4x²/2 + 5x + C = x³ + 2x² + 5x + C
Q.58
Easy
Find the LCM of 24, 36, and 48.
Explanation:
24 = 2³ × 3, 36 = 2² × 3², 48 = 2⁴ × 3. LCM = 2⁴ × 3² = 16 × 9 = 144
Q.59
Medium
If a triangle has sides 6, 8, and 10, find its area using Heron's formula.
A
24 square units
B
30 square units
C
48 square units
D
20 square units
Correct Answer:
A. 24 square units
Explanation:
s = (6+8+10)/2 = 12. Area = √(12×6×4×2) = √576 = 24 square units
Q.60
Medium
Solve: |2x - 5| = 7
A
x = 6 or x = -1
B
x = 6 or x = -2
C
x = 5 or x = -1
D
x = 4 or x = -3
Correct Answer:
A. x = 6 or x = -1
Explanation:
Case 1: 2x - 5 = 7, so 2x = 12, x = 6. Case 2: 2x - 5 = -7, so 2x = -2, x = -1
