List of top Mathematics Questions

The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
\( \int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \, dx \) is equal to:
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
Let \( f : \mathbb{R}_+ \to [-5, \infty) \) be defined as \( f(x) = 9x^2 + 6x - 5 \), where \( \mathbb{R}_+ \) is the set of all non-negative real numbers. Then, \( f \) is:
If \( \begin{vmatrix} -a & b & c
a & -b & c
a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:
If the sum of all the elements of a \( 3 \times 3 \) scalar matrix is 9, then the product of all its elements is:
If \( A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} \) and \( A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix} \), find the value of \((a + x) - (b + y)\).
Evaluate: \[ \int_{-2}^{2} \frac{x^3 + |x| + 1}{x^2 + 4|x| + 4} \, dx. \]
Find: \[ \int \frac{(3 \cos x - 2) \sin x}{5 - \sin^2 x - 4 \cos x} \, dx. \]
Using integration, find the area of the ellipse: \[ \frac{x^2}{16} + \frac{y^2}{4} = 1, \] included between the lines \(x = -2\) and \(x = 2\).
Solve the following differential equation: \[ (\tan^{-1}y - x) \, dy = (1 + y^2) \, dx. \]
Find the equation of a line \( l_2 \) which is the mirror image of the line \( l_1 \) with respect to line \( l : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \), given that line \( l_1 \) passes through the point \( P(1, 6, 3) \) and is parallel to line \( l \).
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
Find: \[ \int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx. \]
A relation \( R \) on set \( A = \{1, 2, 3, 4, 5\} \) is defined as \[ R = \{(x, y) : |x^2 - y^2| <8\}. \] Check whether the relation \( R \) is reflexive, symmetric, and transitive.
E and F are two independent events such that \( P(\overline{E}) = 0.6 \) and \( P(E \cup F) = 0.6 \). Find \( P(F) \) and \( P(\overline{E} \cup \overline{F}) \).
Evaluate: \[ \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx \]
If \( x^{1/3} + y^{1/3} = 1 \), find \( \frac{dy}{dx} \) at the point \( \left( \frac{1}{8}, \frac{1}{8} \right) \).
Find: \[ \int x \sqrt{1 + 2x} \, dx \]
A linear programming problem deals with the optimization of a/an:
If \( P(A|B) = P(A'|B) \), then which of the following statements is true?
If the direction cosines of a line are \( \sqrt{3}k, \sqrt{3}k, \sqrt{3}k \), then the value of \( k \) is:
If \( f(x) \) is an odd function, then: \[ \int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx \, \text{equals:} \]
Let \( f(x) \) be a continuous function on \([a, b]\) and differentiable on \((a, b)\). Then, this function \( f(x) \) is strictly increasing in \((a, b)\) if: