List of top Mathematics Questions

A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, -1, 1), then the projection of $\vec{OP}$ on this plane is of length :

The following system of linear equations
2x + 3y + 2z = 9
3x + 2y + 2z = 9
x - y + 4z = 8

If for the matrix, A = \( A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix} \), and \( A A^T = I_2 \), then the value of \( \alpha^4 + \beta^4 \) is :

The number of elements in the set \[ \left\{ A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\} \text{ and } (I - A)^3 = I - A^3 \right\}, \] where \( I \) is the \( 2 \times 2 \) identity matrix, is _________.

If lim$_{x\to 0} \frac{ax - (e^{4x}-1)}{ax(e^{4x}-1)}$ exists and is equal to b, then the value of a$-$2b is ________ .

If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k)^6$ is equal to ________ .

A line 'l' passing through origin is perpendicular to the lines
$l_1: \vec{r} = (3+t)\hat{i} + (-1+2t)\hat{j} + (4+2t)\hat{k}$
$l_2: \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}$
If the co-ordinates of the point in the first octant on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of 'l' and '$l_1$' are (a, b, c), then 18(a+b+c) is equal to ________ .

The value of $\int_{-2}^{2} |3x^2-3x-6| \,dx$ is ________ .

The total number of two digit numbers 'n', such that $3^n + 7^n$ is a multiple of 10, is ________ .

If the remainder when x is divided by 4 is 3, then the remainder when $(2020+x)^{2022}$ is divided by 8 is ________ .

cosec[2cot$^{-1}$(5) + cos$^{-1}$($\frac{4}{5}$)] is equal to :

The contrapositive of the statement "If you will work, you will earn money" is:

If $\alpha, \beta \in R$ are such that 1$-$2i (here $i^2$=$-$1) is a root of z$^2$+$\alpha$z+$\beta$=0, then ($\alpha-\beta$) is equal to:

The shortest distance between the line $x-y=1$ and the curve $x^2 = 2y$ is:

The integral $\int \frac{e^{3\log_e{2x}} + 5e^{2\log_e{2x}}}{e^{4\log_e{x}} + 5e^{3\log_e{x}} - 7e^{2\log_e{x}}} \,dx$, x>0, is equal to: (where c is a constant of integration)

$\lim_{n\to\infty} [\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \dots + \frac{n}{(2n-1)^2}]$ is equal to :

Let $\alpha$ and $\beta$ be the roots of $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \ge 1$, then the value of $\frac{a_{10} - 2a_8}{3a_9}$ is:

If I$_n$ = $\int_{\pi/4}^{\pi/2} \cot^n x \,dx$, then :

Let A be a 3$\times$3 matrix with det(A)=4. Let R$_i$ denote the i$^{th}$ row of A. If a matrix B is obtained by performing the operation R$_2$ $\rightarrow$ 2R$_2$+5R$_3$ on 2A, then det(B) is equal to:

A function f(x) is given by f(x) = $\frac{5^x}{5^x + 5}$, then the sum of the series $f(\frac{1}{20}) + f(\frac{2}{20}) + \dots + f(\frac{39}{20})$ is equal to:

Suppose the line $\frac{x - 2}{\alpha} = \frac{y - 2}{-5} = \frac{z + 2}{2}$ lies on the plane $x + 3y - 2z + \beta = 0$. Then $(\alpha + \beta)$ is equal to _________.

Let B be the centre of the circle \( x^2 + y^2 - 2x + 4y + 1 = 0 \). Let the tangents at two points P and Q on the circle intersect at the point \( A(3, 1) \). Then \( 8 \cdot \frac{\text{area } \Delta APQ}{\text{area } \Delta BPQ} \) is equal to _________.

If the line \( y = mx \) bisects the area enclosed by the lines \( x = 0, y = 0, x = \frac{3}{2} \) and the curve \( y = 1 + 4x - x^2 \), then 12 m is equal to _________.

If \( S = \frac{7}{5} + \frac{9}{5^2} + \frac{13}{5^3} + \frac{19}{5^4} + ... \), then 160 S is equal to _________.