List of top Mathematics Questions

Let n be an odd natural number such that the variance of 1, 2, 3, 4, ..., n is 14. Then n is equal to _________.

The number of distinct real roots of the equation $3x^4 + 4x^3 - 12x^2 + 4 = 0$ is _________.

If $(\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a; 0<x<1, a \neq 0$, then the value of $2x^2 - 1$ is :

The statement $(p \land (p \to q) \land (q \to r)) \to r$ is :

The distance of the point $(1, -2, 3)$ from the plane $x - y + z = 5$ measured parallel to a line, whose direction ratios are $2, 3, -6$ is :

Let us consider a curve, $y = f(x)$ passing through the point $(-2, 2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x) + x f'(x) = x^2$. Then :

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is :

If $y = y(x)$ is an implicit function of $x$ such that $\log_e (x + y) = 4xy$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is equal to _________.

The area of the region $S = \{(x, y) : 3x^2 \leq 4y \leq 6x + 24\}$ is _________.

The sum of all integral values of $k$ ($k \neq 0$) for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is _________.

Let \( a, b \in \mathbb{R}, b \neq 0 \). Define a function \[ f(x) = \begin{cases} a \sin \frac{\pi}{2}(x - 1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases} \] If \( f \) is continuous at \( x = 0 \), then \( 10 - ab \) is equal to ________.

The sum of solutions of the equation \(\frac{\cos x}{1 + \sin x} = |\tan 2x|\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) - \left\{\frac{\pi}{4}, -\frac{\pi}{4}\right\}\) is :

Let \(\theta \in \left(0, \frac{\pi}{2}\right)\). If the system of linear equations
\((1 + \cos^2\theta)x + \sin^2\theta y + 4\sin3\theta z = 0\)
\(\cos^2\theta x + (1 + \sin^2\theta)y + 4\sin3\theta z = 0\)
\(\cos^2\theta x + \sin^2\theta y + (1 + 4\sin3\theta)z = 0\)
has a non-trivial solution, then the value of \(\theta\) is :

If \( A = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ -\frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ i & 1 \end{bmatrix} \), \( i = \sqrt{-1} \), and \( Q = A^T B A \), then the inverse of the matrix \( A Q^{2021} A^T \) is equal to :

Let the line $L$ be the projection of the line \[ \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2} \] in the plane $x - 2y - z = 3$. If $d$ is the distance of the point $(0, 0, 6)$ from $L$, then $d^2$ is equal to _________.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left( \frac{4}{\pi} + 1 \right)k$ is equal to _________.

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If \(\alpha\) and \(\sqrt{\beta}\) are the mean and standard deviation respectively for correct data, then \((\alpha, \beta)\) is :

A plane \(P\) contains the line \(x + 2y + 3z + 1 = 0 = x - y - z - 6\), and is perpendicular to the plane \(-2x + y + z + 8 = 0\). Then which of the following points lies on \(P\) ?

If the truth value of the Boolean expression \(((p \vee q) \wedge (q \to r) \wedge (\sim r)) \to (p \wedge q)\) is false, then the truth values of the statements \(p, q, r\) respectively can be :

Let \(ABC\) be a triangle with \(A(-3, 1)\) and \(\angle ACB = \theta, 0<\theta \leq \frac{\pi}{2}\). If the equation of the median through \(B\) is \(2x + y - 3 = 0\) and the equation of angle bisector of \(C\) is \(7x - 4y - 1 = 0\), then \(\tan \theta\) is equal to :

Let \(f(x) = \cos\left(2\tan^{-1} \sin \left(\cot^{-1} \sqrt{\frac{1-x}{x}}\right)\right)\), \(0<x<1\). Then :

The value of \(\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}\) is :

If the sum of an infinite GP \(a, ar, ar^2, ar^3, \dots\) is 15 and the sum of the squares of its each term is 150, then the sum of \(ar^2, ar^4, ar^6, \dots\) is :

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :