Section A — MCQ (Single Correct)
Question 1
The value of $\tan\dfrac{\pi}{16} \cdot \tan\dfrac{3\pi}{16} \cdot \tan\dfrac{5\pi}{16} \cdot \tan\dfrac{7\pi}{16}$ is:
A
$1$
B
$\sqrt{2}$
C
$0$
D
$-1$
Question 2
If $\alpha$, $\beta$, $\gamma$ are roots of $\sin\theta = k$ in $[0, 2\pi]$, then $\sin\alpha + \sin\beta + \sin\gamma$:
A
$= 3k$
B
$= 2k$ (when 2 roots only)
C
$= 0$
D
Depends on $k$
Question 3
The number of values of $\theta \in (0, \pi)$ satisfying $\sin 5\theta = 5\sin\theta$ is:
A
$0$
B
$1$
C
$2$
D
infinitely many
Question 4
If $\tan^{-1}(\cot\theta) + \cot^{-1}(\tan\theta) = \dfrac{\pi}{6}$, then $\theta$ equals:
A
$\dfrac{\pi}{6}$
B
$\dfrac{\pi}{3}$
C
$\dfrac{5\pi}{12}$
D
$\dfrac{2\pi}{3}$
Question 5
In a triangle, if $\cos A + 2\cos B + \cos C = 2$, then the sides are in:
A
Arithmetic Progression
B
Geometric Progression
C
Harmonic Progression
D
No general relation
Question 6
If the circumradius of a triangle with sides $a, b, c$ is $R$, then $a\cot A + b\cot B + c\cot C$ equals:
A
$\dfrac{abc}{2R^2}$
B
$2(R + r)$
C
$a + b + c$
D
$\dfrac{a + b + c}{2R} \cdot 2R$, i.e., $a+b+c$
Question 7
The value of the expression $\sin^4\dfrac{\pi}{8} + \sin^4\dfrac{3\pi}{8} + \sin^4\dfrac{5\pi}{8} + \sin^4\dfrac{7\pi}{8}$ is:
A
$1$
B
$\dfrac{3}{2}$
C
$\dfrac{1}{2}$
D
$2$
Question 8
If $A + B + C = \pi$ and $\tan A \tan B = 2$, then $\dfrac{\cos(A-B)}{\cos(A+B)}$ equals:
A
$3$
B
$-3$
C
$1$
D
$-1$
Question 9
If $\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \dfrac{3\pi}{2}$, then $x^{100} + y^{100} + z^{100} - \dfrac{9}{x^{101} + y^{101} + z^{101}}$ equals:
A
$0$
B
$1$
C
$2$
D
$3$
Question 10
In a right-angled triangle with hypotenuse $\sqrt{2}$ and one acute angle $\theta$, the area is maximum when $\theta$ equals:
A
$\dfrac{\pi}{4}$
B
$\dfrac{\pi}{3}$
C
$\dfrac{\pi}{6}$
D
$\dfrac{\pi}{8}$
Section B — Integer Type
Question 11 — Integer answer
Let $\alpha, \beta$ be the roots of $\sin x = \dfrac{1}{2}$ in $[0, 2\pi]$. Find $\alpha + \beta$ in units of $\pi$, i.e., if $\alpha + \beta = k\pi$, find $k$.
Question 12 — Integer answer
The value of $\dfrac{\sin 47° + \sin 61° - \sin 11° - \sin 25°}{\cos 7°}$ is.
Question 13 — Integer answer
In triangle $ABC$, $a = 13$, $b = 14$, $c = 15$. Find $r_1$ (the radius of the escribed circle opposite to $A$).
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): For any triangle, $r_1 + r_2 + r_3 = 4R + r$.Reason (R): This follows from the relations between $r_1, r_2, r_3, r, R$ via half-angle formulas of $A, B, C$.
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.
Question 15 — Assertion / Reason
Assertion (A): $\cos^{-1}(\cos(2\pi/3)) = 2\pi/3$.Reason (R): $\cos^{-1}(\cos x) = x$ when $x \in [0, \pi]$.
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.