List of top Mathematics Questions asked in Guru Gobind Singh Indraprastha University Common Entrance Test

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

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 \(2\sqrt{3}\)?

On the sphere \((x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25\), compute the distance from the point \(M_0\) to the plane \(3x - 4z + 19 = 0\)

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

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

\[ \lim_{n \to \infty} \left[ 5 - \frac{1}{5} + \frac{1}{25} - \dots + (-1)^{n-1} \cdot \frac{1}{5^n} \right] \]

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

Find the angle between unit vectors \(\mathbf{e_1}\) and \(\mathbf{e_2}\) if vectors \[ \mathbf{a} = \mathbf{e_1} + 2\mathbf{e_2},\quad \mathbf{b} = 5\mathbf{e_1} - 4\mathbf{e_2} \] are mutually perpendicular.

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

\[ \lim_{x \to 0} \frac{a^x - 1}{x} \text{ is equal to} \]

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

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

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

\[ \frac{1}{2 \sin 10^\circ} - 2 \sin 70^\circ \text{ is equal to} \]

Given \(\varepsilon = \cos\left(\frac{2\pi k}{n}\right) + i \sin\left(\frac{2\pi k}{n}\right)\), find the value of \[ \prod_{k=0}^{n-1} \left( \varepsilon^2k - 2\varepsilon k \cos \theta + 1 \right) \]

\[ (1 + 2i)^6 \text{ is equal to:} \]

What is the number of ordered pairs of real numbers (a, b) such that \[ (a + bi)^{2002} = a - bi \]

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

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

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

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: \( xy \geq 0 \)

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

Choose the most appropriate option. Solve for \(x>0\)
\[ \log_3\left( \frac{3}{x} \right) + \log_3 x = 1 \]

At what point of the curve \( y^2 = 2x^3 \) is the tangent line perpendicular to the straight line \[ 4x - 3y + 2 = 0? \]