Q2
What is the fundamental period of the function f(x) = tan(3x)?
π
2π
π/3
2π/3
The fundamental period of tan(x) is π. For a function f(x) = tan(bx), the period is π/|b|. Here b = 3, so the period is π/3.
Q4
A pizza of radius 21 cm has a sector of angle 60°. Its area is:
231 cm²
462 cm²
154 cm²
77 cm²
Area=(60/360)×π×21²=231 cm² using π=22/7.
Q6
What is the exact value of sin 75°?
(√3 − 1)/(2√2)
(√3 + 1)/(2√2)
√3/2
1/√2
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (1/√2)×(√3/2) + (1/√2)×(1/2) = (√3 + 1)/(2√2).
Q9
A wheel of radius 14 cm rotates through π/2 radians. Distance travelled by a point on rim is:
7π cm
14π cm
21π cm
28π cm
Arc length = rθ =14×π/2=7π.
Q10
If sin θ + cosec θ = 2, then what is the value of sin²θ + cosec²θ?
1
2
4
0
Squaring both sides of sin θ + cosec θ = 2 gives sin²θ + cosec²θ + 2(sin θ × cosec θ) = 4. Since sin θ × cosec θ = 1, we get sin²θ + cosec²θ + 2 = 4, so sin²θ + cosec²θ = 2.
Q11
Find the value of sin(31π/3).
1/2
−1/2
√3/2
−√3/2
sin(31π/3) = sin(10π + π/3). Since the sine function has a period of 2π, this equals sin(π/3) = √3/2.
Q13
What is the range of the function y = sec²x + cosec²x?
(0, ∞)
[1, ∞)
[2, ∞)
[4, ∞)
sec²x + cosec²x = 1/cos²x + 1/sin²x = (sin²x + cos²x) / (sin²x cos²x) = 1 / (1/4 sin²2x) = 4 / sin²2x. Since the maximum value of sin²2x is 1, the minimum value of the expression is 4. Range is [4, ∞).
Q14
Find the number of real solutions to the equation sin x = x.
0
1
2
Infinite
The graph of y = sin x and y = x intersect only at the origin (0,0) because for x > 0, sin x < x, and for x < 0, sin x > x. Thus, there is exactly 1 solution: x = 0.
Q15
A bus wheel of radius 35 cm rotates through 4 radians. Arc length covered by a point on the tyre is:
70 cm
105 cm
140 cm
175 cm
Arc length = rθ = 35×4 =140 cm.