List of top Mathematics Questions

Let \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \), \( \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k} \), and \( \vec{c} = \hat{i} - 2\hat{j} + 3\hat{k} \) be three vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \) and \( \vec{r} . \vec{c} = 18 \), then the magnitude of the orthogonal projection of \( 4\hat{i} + 3\hat{j} - \hat{k} \) on \( \vec{r} \) is:

If the position vectors of A, B, C, D are \( \vec{A} = \hat{i} + 2\hat{j} + 2\hat{k}, \vec{B} = 2\hat{i} - \hat{j}, \vec{C} = \hat{i} + \hat{j} + 3\hat{k}, \vec{D} = 4\hat{j} + 5\hat{k} \), then the quadrilateral ABCD is a:

If \( \sum\limits_{i=1}^{9} (x_i - 5) = 9 \) and \( \sum\limits_{i=1}^{9} (x_i - 5)^2 = 45 \), then the standard deviation of the nine observations \( x_1, x_2, \ldots, x_9 \) is

Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is \( \frac{1}{4} \) and the probability that the second student gets qualified in the same exam is \( \frac{2}{5} \), then the probability that at least one of them gets qualified in that exam is

The equation \[ \cos^{-1}(1 - x) - 2 \cos^{-1} x = \frac{\pi}{2} \] has:

In \( \triangle ABC \), if A, B, C are in arithmetic progression, then \[ \sqrt{a^2 - ac + c^2} . \cos\left(\frac{A - C}{2}\right) =\ ? \]

If \( x \) is a positive real number and the first negative term in the expansion of \[ (1 + x)^{27/5} \text{ is } t_k, \text{ then } k =\ ? \]

Evaluate the following expression: \[ \frac{1}{81^n} - \binom{2n}{1} . \frac{10}{81^n} + \binom{2n}{2} . \frac{10^2}{81^n} - .s + \frac{10^{2n}}{81^n} = ? \]

If \( \alpha, \beta \) are the roots of \( x^2 - 5x - 68 = 0 \) and \( \gamma, \delta \) are the roots of \( x^2 - 5\alpha x - 6\beta = 0 \), then \( \alpha + \beta + \gamma + \delta = \) ?

If \( \omega_1 \) and \( \omega_2 \) are two non-zero complex numbers and \( a, b \) are non-zero real numbers such that \[ |a\omega_1 + b\omega_2| = |a\omega_1 - b\omega_2|, \] then \( \dfrac{\omega_1}{\omega_2} \) is:

If \( f : \mathbb{R} \to A \), defined by \( f(x) = \cos x + \sqrt{3}\sin x - 1 \), is an onto function, then \( A = \)

The general solution of the differential equation \((1 + \sin^2 x) \, \frac{dy}{dx} + \sin 2x = 0\) is?

Evaluate \[ \int \frac{1}{x^4 + 1} \, dx = ? \]

Evaluate \[ \int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ? \]

Evaluate \[ \int \sin^3 x \cos^2 x \, dx = ? \]

Evaluate \[ \int (\log 2x)^3 \, dx = ? \]

If \(y = \sin^{-1} \left(\frac{2x}{1 + x^2}\right)\) and \(\left(\frac{d^2 y}{dx^2}\right)_{x=2} = k\), then find \(25k\).

If the displacement \(S\) of a particle travelling along a straight line in \(t\) seconds is given by \[ S = 2t^3 + 2t^2 - 2t - 3, \] then the time taken (in seconds) by the particle to change its direction is?

If \(f(x) = x^{\sec^{-1} x}\), then find \(f'(2)\).

If the tangent to the curve \(xy + ax + by = 0\) at \((1,1)\) makes an angle \(\tan^{-1} 2\) with X-axis, then find \(\frac{ab}{a+b}\).

Evaluate \[ \lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ? \]

If the image of the point \(A(1,1,1)\) with respect to the plane \(4x + 2y + 4z + 1 = 0\) is \(B(\alpha, \beta, \gamma)\), then find \(\alpha + \beta + \gamma\).

If a tangent having slope \(\frac{1}{3}\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b)\) is normal to the circle \((x+1)^2 + (y+1)^2 = 1\), then \(a^2\) lies in the interval?