List of top Mathematics Questions

If a line makes angles \( \alpha, \beta, \gamma \) with the x-axis, y-axis, and z-axis respectively,
then prove that: \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]

Simplify:
\( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \)

Find the intervals in which the function
\( f(x) = x^4 - 4x^3 + 4x^2 + 15 \) is increasing .
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto
If \( |\vec{a}| = 7 \), \( |\vec{b}| = 24 \), \( |\vec{c}| = 25 \) and
\( \vec{a} + \vec{b} + \vec{c} = \vec{0} \),
find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).
Assertion (A): The principal value of $\cot^{-1}(\sqrt{3})$ is $\frac{\pi}{6}$.
Reason (R): Domain of $\cot^{-1} x$ is $\mathbb{R} - \{-1, 1\}$.
If $x = a \cos \theta + b \sin \theta$, $y = a \sin \theta - b \cos \theta$, then which one is true?
It is given that \[ X \begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}. \text{ Then matrix X is:} \]
If three non-zero vectors are \( \vec{a}, \vec{b} \) and \( \vec{c} \) such that
\( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \) and \( \vec{a} \times \vec{b} = \vec{a} \times \vec{c} \),
then show that \( \vec{b} = \vec{c} \).
The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function $Z = 2ax + by$ occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is:
corner points of the bounded
Assertion (A): Quadrilateral formed by vertices A(0, 0, 0), B(3, 4, 5), C(8, 8, 8) and D(5, 4, 3) is a rhombus.
Reason (R): ABCD is a rhombus if $AB = BC = CD = DA$, and $AC \ne BD$.
The point which lies in the half-plane $2x + y - 4 \le 0$ is:
If $(\cos x)^y = (\cos y)^x$, then $\frac{dy}{dx}$ is:
The difference of the order and the degree of the differential equation \[ \Big( \frac{d^2 y}{dx^2} \Big)^2 + \Big( \frac{dy}{dx} \Big)^3 + x^4 = 0 \text{ is:} \]
A family has 2 children and the elder child is a girl. The probability that both children are girls is:
If $\vec{a}$, $\vec{b}$ and $(\vec{a} + \vec{b})$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is:
The projection of vector $\hat{i}$ on the vector $\hat{i} + \hat{j} + 2\hat{k}$ is:
For which value of x, are the determinants \[ \begin{vmatrix} 2x & -3 \\ 5 & x \end{vmatrix} \text{and} \begin{vmatrix} 10 & 1 \\ -3 & 2 \end{vmatrix} \text{ equal?} \]
The value of the cofactor of the element of second row and third column in the matrix \[ \begin{bmatrix} 4 & 3 & 2 \\ 2 & -1 & 0 \\ 1 & 2 & 3 \end{bmatrix} \text{ is:} \]
The vector equation of a line which passes through the point (2, -4, 5) and is parallel to the line $\displaystyle \frac{x+3}{3} = \frac{4-y}{2} = \frac{z+8}{6}$ is:
The value of $\displaystyle \int_{0}^{\pi/6} \sin 3x \, dx$ is:
Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let $\vec{u}=\hat{i}-\hat{j}-2 \hat{k}, \vec{v}=2 \hat{i}+\hat{j}-\hat{k}, \vec{v} \cdot \vec{w}=2$ and $\vec{v} \times \vec{w}=\vec{u}+\lambda \vec{v}$.  Then $\vec{u} \cdot \vec{w}$ is equal to

If the domain of the function $f(x)=\frac{[x]}{1+x^2}$, where $[x]$ is greatest integer $\leq x$, is $[2,6)$, then its range is

Let y = y(x) be a solution of the differential equation (x cos x)dy +(xy sin x + y cos x – l)dx = 0, 0 < x < \(\frac{π}{2}\). If\(\frac{π}{3}y(\frac{π}{3})=\sqrt3\), then \(| \frac{π}{6 }y“( \frac{π}{6})+2 y'( \frac{π}{6})| \) is equal to______.