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

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 \(2\sqrt{3}\)?

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

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

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:

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

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

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

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 \(\mathbf{e}_2 = (1, 1, 1)\).

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?

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

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?

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

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

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

Find the derivative of \[ y = (1 - x)^m (1 + x)^n \text{ at } x = 0, \text{ where } m, n>0 \]

Angles A, B, and C of a triangle \(\Delta ABC\) are in AP and \( b:c = \sqrt{3} : \sqrt{2} \), then angle \( \angle A \) is given by

The straight line \( r = (i - j + k) + \lambda (2i + j - k) \) and the plane \( r \cdot (2i + j - k) = 4 \) are

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

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 \(x>0\)
\[ \log_3\left( \frac{3}{x} \right) + \log_3 x = 1 \]

A real solution of the equation \[ \cosh x - 5 \sinh x - 5 = 0 \text{ is} \]