Section A — MCQ (Single Correct)
Question 1
If $\sin\theta = \dfrac{12}{13}$ and $\theta$ is acute, then $\cos\theta + \tan\theta$ equals:
A
$\dfrac{25}{65}$
B
$\dfrac{181}{65}$
C
$\dfrac{169}{65}$
D
$\dfrac{157}{65}$
Question 2
The value of $\tan 9° - \tan 27° - \tan 63° + \tan 81°$ is:
A
$0$
B
$2$
C
$4$
D
$-4$
Question 3
The general solution of $\cos 2\theta = -\dfrac{1}{2}$ is:
A
$\theta = n\pi \pm \dfrac{\pi}{3}$
B
$\theta = \dfrac{n\pi}{2} \pm \dfrac{\pi}{3}$
C
$\theta = n\pi \pm \dfrac{2\pi}{3}$
D
$\theta = n\pi \pm \dfrac{\pi}{6}$
Question 4
The value of $\sin^{-1}\dfrac{4}{5} + \sin^{-1}\dfrac{3}{5}$ is:
A
$\dfrac{\pi}{2}$
B
$\dfrac{\pi}{4}$
C
$\sin^{-1}\dfrac{7}{5}$
D
$\dfrac{\pi}{3}$
Question 5
In a triangle, if $a = 5$, $b = 12$, $c = 13$, then the radius of the circumscribed circle is:
A
$6.5$
B
$13$
C
$\dfrac{13}{2}$
D
$5$
Question 6
If $\tan A + \tan B + \tan C = \tan A \tan B \tan C$ in a triangle, then $A + B + C$ equals:
A
$\pi$
B
$\dfrac{\pi}{2}$
C
$2\pi$
D
Any value
Question 7
The number of solutions of $2\sin^2\theta - 5\sin\theta + 2 = 0$ in $[0, 2\pi]$ is:
A
$0$
B
$1$
C
$2$
D
$4$
Question 8
If $\cos x = \dfrac{1}{3}$, $0 < x < \pi$, then $\tan(x/2)$ equals:
A
$\sqrt{2}$
B
$\dfrac{1}{\sqrt{2}}$
C
$\sqrt{3}$
D
$\sqrt{6}$
Question 9
The angle of elevation of a cloud above a lake from a point $20$ m above the surface is $30°$. The angle of depression of its reflection in the lake from the same point is $60°$. The height of the cloud above the lake is:
A
$20$ m
B
$40$ m
C
$60$ m
D
$80$ m
Question 10
If $A + B + C = \pi$ in a triangle, then $\cos A + \cos B + \cos C$ equals:
A
$1 - 4\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2}$
B
$1 + 4\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2}$
C
$4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\cos\dfrac{C}{2}$
D
$4\sin A\sin B\sin C$
Section B — Integer Type
Question 11 — Integer answer
If $\sin\theta + \sin^2\theta = 1$, then the value of $\cos^2\theta + \cos^4\theta$ equals which integer?
Question 12 — Integer answer
In a triangle with sides $a, b, c$ and area $\Delta$, if $a = 6$, $b = 8$, $c = 10$, find $\Delta$.
Question 13 — Integer answer
The number of solutions of $\sin x \cdot \sin 2x \cdot \sin 3x = 0$ in $[0, \pi]$ is.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): $\sin^{-1}(\sin 3) = 3$.Reason (R): For all $x \in \mathbb{R}$, $\sin^{-1}(\sin x) = x$.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: A is false ($.
Question 15 — Assertion / Reason
Assertion (A): In any triangle, $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$.Reason (R): The sine rule is derived using the fact that the diameter of the circumscribed circle equals $\dfrac{a}{\sin A}$.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: Both A and R are true, and R is the correct explanation.