List of practice Questions

Evaluate: \[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \, dx \]
Find the domain of $\sin^{-1} \sqrt{x - 1}$.
Calculate the area of the region bounded by the curve \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] and the x-axis using integration.
Simplify $\sin^{-1} \left( \frac{x}{\sqrt{1 + x^2}} \right)$.
The principal value of $\sin^{-1} \left( \sin \left( -\frac{10\pi}{3} \right) \right)$ is :
Evaluate: \[ \sqrt{\sin^4 x + 4\cos^2 x} - \sqrt{\cos^4 x + 4\sin^2 x} = ? \]
Evaluate \( \sin 210^\circ \cdot \cos 240^\circ \cdot \tan 150^\circ \):
The value of \( \cos \left( \frac{\pi}{6} + \cot^{-1}(-\sqrt{3}) \right) \) is:
In a $\triangle ABC$, $\frac{2(r_1 + r_3)}{a c (1 + \cos B)} =$
In a triangle ABC, if A, B, and C are in arithmetic progression, $r_3 = r_1 r_2$, and $c = 10$, then $a^2 + b^2 + c^2 =$
In a triangle ABC, if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$, then $\cos A : \cos B : \cos C =$
The number of solutions of \( \tan^{-1} 1 + \frac{1}{2} \cos^{-1} x^2 - \tan^{-1}\left(\frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}}\right) = 0 \) is
The general solution satisfying both the equations \(\sin x = -\frac{3}{5}\) and \(\cos x = -\frac{4}{5}\) is:
If \(\cos\theta = -\frac{3}{5}\) and \(\theta\) does not lie in second quadrant, then \(\tan\frac{\theta}{2} =\)
If \( \sqrt{3} \cos \theta + \sin \theta > 0 \), then the range of \( \theta \) is:
\( \csc 48^\circ + \csc 96^\circ + \csc 192^\circ + \csc 384^\circ = \)
In a triangle ABC, if \((r_1 - r_3)(r_1 - r_2) - 2r_2r_3 = 0\), then \(a^2 - b^2 =\)
In triangle $ ABC $, if $ a = 13 $, $ b = 8 $, $ c = 7 $, then $ \cos(B+C) = $
In triangle $ ABC $, if $ a = 5,\ b = 4,\ \cos(A - B) = \frac{31}{32} $, then $ c = ? $
In triangle $ ABC $, if $ C = 120^\circ $, $ c = \sqrt{19} $, and $ b = 3 $, then $ a = ? $
From the top of a tower 50 meters high, the angle of depression to a point on the ground is 30°. What is the distance from the base of the tower to that point on the ground? (Assume the ground is horizontal)
If $\dfrac{2\tan30^\circ}{1+\tan^2 30^\circ} = \dfrac{2\tan30^\circ}{\sqrt{1 - \tan^2 30^\circ}}$, then $x : y =$
A peacock sitting on the top of a tree of height 10 m observes a snake moving on the ground. If the snake is $10\sqrt{3}$ m away from the base of the tree, then angle of depression of the snake from the eye of the peacock is
In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is
Verify that \(\sin 2A = \frac{2 \tan A}{1 + \tan^2 A}\) for \(A = 30^\circ\).