List of top Mathematics Questions

Let the foot of perpendicular from a point $P(1, 2, -1)$ to the straight line $L: \frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the plane $x + y + 2z = 0$ which meets $L$ at point $Q$. If $\alpha$ is the acute angle between the lines $PN$ and $PQ$, then $\cos \alpha$ is equal to ________.

A spherical gas balloon of radius 16 meter subtends an angle $60^{\circ}$ at the eye of the observer A while the angle of elevation of its center from the eye of A is $75^{\circ}$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :

Let $S_n$ be the sum of the first n terms of an arithmetic progression. If $S_{3n} = 3S_{2n}$, then the value of $\frac{S_{4n}}{S_{2n}}$ is :

Let \(\vec{p} = 2\hat{i} + 3\hat{j} + \hat{k}\) and \(\vec{q} = \hat{i} + 2\hat{j} + \hat{k}\) be two vectors. If a vector \(\vec{r} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}\) is perpendicular to each of the vectors \((\vec{p} + \vec{q})\) and \((\vec{p} - \vec{q})\), and \(|\vec{r}| = \sqrt{3}\), then \(|\alpha| + |\beta| + |\gamma|\) is equal to __________.

Let \(M = \left\{ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \{ \pm 3, \pm 2, \pm 1, 0 \} \right\}\). Define \(f: M \to \mathbb{Z}\) as \(f(A) = \det(A)\), for all \(A \in M\), where \(\mathbb{Z}\) is the set of all integers. Then the number of \(A \in M\) such that \(f(A) = 15\) is equal to __________.

Let \(S = \left\{ n \in \mathbb{N} : \begin{pmatrix} 0 & i \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \forall a, b, c, d \in \mathbb{R} \right\}\), where \(i = \sqrt{-1}\). Then the number of 2-digit numbers in the set S is __________.

The values of a and b, for which the system of equations $2x + 3y + 6z = 8$ $x + 2y + az = 5$ $3x + 5y + 9z = b$ has no solution, are :

If \(\alpha, \beta\) are roots of the equation \(x^2 + 5(\sqrt{2})x + 10 = 0, \alpha>\beta\) and \(P_n = \alpha^n - \beta^n\) for each positive integer n, then the value of \(\frac{P_{17}P_{20} + 5\sqrt{2} P_{17}P_{19}}{P_{18}P_{19} + 5\sqrt{2} P_{18}^2}\) is equal to __________.

If the value of \(\left( 1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + ....... \text{upto } \infty \right)^{\log_{0.25} \left( \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + ....... \text{upto } \infty \right)}\) is \(l\), then \(l^2\) is equal to __________.

The value of the definite integral $\int_{\pi/24}^{5\pi/24} \frac{dx}{1 + \sqrt[3]{\tan 2x}}$ is :

The sum of all values of $x$ in $[0, 2\pi]$, for which $\sin x + \sin 2x + \sin 3x + \sin 4x = 0$, is equal to :

The Boolean expression $(p \implies q) \wedge (q \implies \sim p)$ is equivalent to :

The number of real roots of the equation $e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^x + 1 = 0$ is :

Let $f(x) = 3 \sin^4 x + 10 \sin^3 x + 6 \sin^2 x - 3, x \in [-\frac{\pi}{6}, \frac{\pi}{2}]$. Then, $f$ is :

If \( |x + 3| + x>1 \), then \( x \in \):

The distance of the point \( (-5, -5, -10) \) from the point of intersection of the line \[ r = -2i - j + 2k + \lambda(3i + 4j + 2k) \] and the plane \( r \cdot (i - j + k) = 5 \) is:

\[ \int_{\log \sqrt{n}}^{\log \sqrt{r}} 2x \sec^2\left( \frac{1}{3} \cdot 2x \right) \, dx \] is equal to:

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000 if the rate of growth of bacteria is proportional to the number present?

What is the angle between the two straight lines \[ y = (2 - \sqrt{3})x + 5 \quad \text{and} \quad y = (2 + \sqrt{3})x - 7? \]

If the angle \( \theta \) between the line \[ \frac{x + 1}{2} = \frac{z - 2}{2} = \frac{y + 4}{\sqrt{n}} \] and the plane \( 2x - y + z + 4 = 0 \) is such that \( \sin \theta = \frac{1}{3} \), then the value of \( n \) is:

The value of \( \tan^{-1} (1) + \tan^{-1} (0) + \tan^{-1} (2) + \tan^{-1} (3) \) is equal to:

Area bounded by the curve \( y = \log x \) and the coordinate axes is:

The angle of intersection of the curve \( y = x^2 \), \( dy = 7 - x^2 \) at \( (1, 1) \) is:

The angle formed by the positive Y-axis and the tangent to \( y = x^2 + 4x - 17 \) at \( (2, -3) \) is: