List of practice Questions

$\lim_{x \to 0} \frac{\tan^{-1} \left( \frac{-x}{\sqrt{1 - x^2}} \right)}{\ln(1 - x)} =\ ?$

Find the component of the vector a = (-1, 2, 0) perpendicular to the plane of the vectors \(\mathbf{e}_1(1, 0, 1)\) and \(\mathbf{e}_2(1, 1, 1)\)

\(\lim_{x \to \infty} \frac{\ln x}{x^n}\) is equal to

There are 3 true coins and 1 false coin with 'head' on both sides. A coin is chosen at random and tossed 4 times. If 'head' occurs all the 4 times, then the probability that the false coin has been chosen and used is

Choose the most appropriate options.
The value of \({}^{40}C_0 + {}^{40}C_1 + {}^{40}C_2 + \ldots + {}^{40}C_{20}\) is

If \(y^{\frac{1}{m}} + x^{\frac{1}{m}} = 2x\) then

On the ellipse \(9x^2 + 25y^2 = 225\), find the point, the distance from which to the focus \(F_2\) is four times the distance to the focus \(F_1\).

Find the points of intersection of the given surface $\frac{x^2}{81} + \frac{y^2}{36} + \frac{z^2}{4} = 1$ and the straight line  $\frac{x - 3}{3} = \frac{y - 4}{-6} = \frac{z + 2}{4}$

Let \( a = \cos \theta_1 + i \sin \theta_1 \), \( b = \cos \theta_2 + i \sin \theta_2 \), \( c = \cos \theta_3 + i \sin \theta_3 \) and \( a + b + c = 0 \), then \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = ? \]

Find the real solution of the system of equations: \[ x^4 + y^4 - x^2 y^2 = 13,\quad x^2 - y^2 + 2xy = 1 \] Satisfying condition: \( xy \geq 0 \)

Which of the following complex numbers is conjugate to its square?

If \(i = \sqrt{-1}\), then \[ \lim_{n \to \infty} \frac{(n + 2i)(3 + 7in)}{(2 - i)(6n^2 + 1)} \] is equal to:

What is the shape of the figure given by the following equations?

What is the equation of the curve traced by point \(M\), if the sum of distances to \(A(-1, -1)\) and \(B(1, 1)\) is constant and equals \(2\sqrt{3}\)?

If \(y = \sec(\tan^{-1} x)\), then \(y\) at \(x = 1\) is equal to term is the sum of two preceding terms. Then, the common ratio of the G.P. is:

Every term of G.P. is positive and also every term is the sum of two preceding terms. Then, the common ratio of the G.P. is

Choose the most appropriate options.
\[ \int \frac{x^2 - 2}{x \sqrt{x^2 - 1}}\,dx \text{ equal to} \]

A die is thrown twice and the sum of the numbers appearing is 6. Then, the conditional probability that the number 4 has appeared at least once is

If \(x = e^y + e^{-y} - x,\, x>0\) then \(\frac{dy}{dx}\) is

\[ \int_0^1 x e^{2x} dx \text{ is equal to} \]

For any two vectors \( \vec{u} \) and \( \vec{v} \), if \( |\vec{u} + \vec{v}| = |\vec{u} - \vec{v}| \) then the angle between them is equal to

For \(x>1\), how many roots/solutions of the following equation exist: \[ \log_2\left( \frac{2}{x} \right)\log^2 x + \log^2 x = 1 \]

Evaluate $ \int_0^1 x^5 \sqrt{1 - x^3} \, dx $

\[ \lim_{x \to \pi/4} \frac{(1 - \cos x)^2}{\tan^2 x - \sin^2 x} \text{ is equal to:} \]