List of practice Questions

For what values of $a \in \mathbb{Z}$, the quadratic expression $(x + a)(x + 1991) + 1$ can be factorised as $(x + b)(x + c)$, where $b, c \in \mathbb{Z}$?
If $A + B + C = \pi$, $\cos B = \cos A \cos C$, then $\tan A \tan C =$
In a triangle $ABC$, $2(bc \cos A + ac \cos B + ab \cos C) =$
Evaluate: $$ \int \frac{e^{\tan^{-1}x}}{1+x^2} \left[\left(\sec^{-1}(\sqrt{1+x^2})\right)^2 + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)\right] dx $$
The straight line touching the circle \(x^2+y^2-2x-3=0\) and remaining normal to the circle \(x^2+y^2-4y-6=0\) is
The number of real values of \( m \) such that the equation \[ x^2 + (2m + 1)x + m = 0 \] has equal roots is:
The number of positive real roots of the equation \[ 3^{x+1} + 3^{-x+1} = 10 \] is:
In a triangle ABC, $r_1 \cot \dfrac{A}{2} + r_2 \cot \dfrac{B}{2} + r_3 \cot \dfrac{C}{2} = $ ?
If the solution of \( \frac{dy}{dx} = y \log_e 0.5 = 0 \), \( y(0) = 1 \), and \( y(x) \to k \) as \( x \to \infty \), then \( k = \)
Let \( z \) and \( w \) be two complex numbers such that \( \bar{z} + i\bar{w} = 0 \) and \( \text{Arg}(zw) = \pi \). Then \( \text{Arg}(z) = \)
Simplify: \( \sin(x + y) \sec x \sec y = \)
For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), if the length of the transverse axis is 8 and the distance between the foci is \( 2\sqrt{41} \), then the length of its latus rectum is:
Evaluate \( \int \frac{\cos(\sec^2 x) - 2022}{\cos^{2022} x} dx = f(x) + C \Rightarrow f\left( \frac{\pi}{4} \right) = \)
If the points of intersection of the coordinate axes and $|x + y| = 2$ form a rhombus, then its area is
Find the length of the intercept made by the line $ x + 1 = 0 $ between the lines: $$ 3x + 2y = 5 \quad \text{and} \quad 3x + 2y = 3 $$
The values of \(\theta\), for which \( \frac{3+2i\sin\theta}{1-2i\sin\theta} \) is real are
The number of solutions of the equations \[ x + y + z = 1,\quad x^2 + y^2 + z^2 = 1,\quad x^3 + y^3 + z^3 = 1 \] is:
Evaluate the limit: \[ \lim_{x \to \infty} \frac{A + e^{mx}}{x + Ae^{mx}} \]
If $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$, then the value of $x$ is:
Assertion (A): \( I_n = \int \cot^n x \, dx \), then \( I_6 + I_4 = -\frac{\cot^5 x}{5} \)
Reason (R): \( \int \cot^n x \, dx = \frac{-\cot^{n-1} x}{n - 1} - \int \cot^{n - 2} x \, dx \)
What is the sum of the first \( n \) terms of the series, whose \( k \)-term is \( k! \cdot k \)?
The range of the data set: \( 35, 12, 21, 24, 15, 7, 16, 12, 30, 32, 13, 17 \) is:
An ellipse has lengths of major and minor axes as 6 and 2 respectively. If the center is at \( (5,6) \) and the major axis lies along the line \( x - y + 1 = 0 \), then the equation of the ellipse is:
Which of the following functions is differentiable at \( x = 0 \)?
If \( \int e^{\sqrt{x}} / \sqrt{x} (x + \sqrt{x}) dx = e^{\sqrt{x}}[Ax + B\sqrt{x} + C] + K \), then \( A + B + C = \):