List of top Mathematics Questions

Evaluate the integral: $$ I = \int \frac{2}{1 + x + x^2} \,dx. $$ 

If the system of equations has a unique solution, find the values of \( a \) and \( b \).
If \( P \) and \( Q \) are two \( 3 \times 3 \) matrices such that \( |PQ| = 1 \) and \( |P| = 9 \), then the determinant of adjoint of the matrix \( P . Adj \ 3Q \) is:
If \( \tanh x = \text{sech } y = \frac{3}{5} \) and \( e^{x+y} \) is an integer, then \( e^{x+y} \) is:
If the eccentricity of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is \( \sec \alpha \), then the area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is:
If \( e_1 \) and \( e_2 \) are respectively the eccentricities of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and its conjugate hyperbola, then the line \( \frac{x}{2e_1} + \frac{y}{2e_2} = 1 \) touches the circle having center at the origin, then its radius is:
If set A has 5 elements, and set B has 7 elements, then the number of one-one functions that can be defined from A to B is
The domain of the real-valued function \( f(x) = \sqrt{9 - \sqrt{x^2 - 144}} \) is
If \( x - 4 = 0 \) is the radical axis of two orthogonal circles out of which one is \( x^2 + y^2 = 36 \), then the centre of the other circle is:
The variance of the first 10 natural numbers which are multiples of 3 is:
Evaluate the integral: \[ \frac{3}{25} \int_{0}^{25\pi} \sqrt{|\cos x - \cos^3 x|} \, dx. \]
If \[ y = \log \left( x - \sqrt{x^2 - 1} \right), \] then \[ (x^2 - 1)y'' + xy' + e^y + \sqrt{x^2 - 1} = \] evaluates to:
The maximum interval in which the slopes of the tangents drawn to the curve \[ y = x^4 + 5x^3 + 9x^2 + 6x + 2 \] increase is:

If

\[ A = \{ P(\alpha, \beta) \mid \text{the tangent drawn at P to the curve } y^3 - 3xy + 2 = 0 \text{ is a horizontal line} \} \]

and

\[ B = \{ Q(a, b) \mid \text{the tangent drawn at Q to the curve } y^3 - 3xy + 2 = 0 \text{ is a vertical line} \} \]

then \( n(A) + n(B) = \)

The sum of the maximum and minimum values of the function \[ f(x) = \frac{x^2 - x + 1}{x^2 + x + 1} \] is:
The trigonometric equation \( \sin^{-1}x = 2\sin^{-1}a \) has a solution. Find the valid range for \( a \).

Given the vectors:

\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]

\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]

where

\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]

\[ \mathbf{a} \cdot \mathbf{x} = 3 \]

Find:

\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]

If \[ \frac{1}{x^4 + x^2 + 1} = \frac{Ax + B}{x^2 + ax + 1} + \frac{Cx + D}{x^2 - ax + 1} \] then \( A + B - C + D = ? \)
If \( m \) is the slope and \( P(\beta, \beta) \) is the midpoint of a chord of contact of the circle \[ x^2 + y^2 = 125, \] then the number of values of \( \beta \) such that \( \beta \) and \( m \) are integers is: \

Given below are two statements, one is labelled as Assertion (A) and the other one labelled as Reason (R).
Assertion (A): \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty = \sqrt{4} \] Reason (R): \[ |x| <1, \quad (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \] 

If the coefficient of \( x^r \) in the expansion of \( (1 + x + x^2)^{100} \) is \( a_r \), and \( S = \sum\limits_{r=0}^{300} a_r \), then

\[ \sum\limits_{r=0}^{300} r a_r = \]

The equations \[ 7x + y - 24 = 0 \quad \text{and} \quad x + 7y - 24 = 0 \] represent the equal sides of an isosceles triangle. If the third side passes through \( (-1,1) \), then a possible equation for the third side is: \
If the extremities of the latus recta having positive ordinate of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b) $$ lie on the parabola $$ x^2 + 2ay - 4 = 0, $$ then the points $(a, b)$ lie on the curve:
If the tangent drawn at a point $P(t)$ on the hyperbola $$ x^2 - y^2 = c^2 $$ cuts the X-axis at $T$ and the normal drawn at the same point $P$ cuts the Y-axis at $N$, then the equation of the locus of the midpoint of $TN$ is:
If \( X \sim B(6, p) \) is a binomial variate and \[ \frac{P(X=4)}{P(X=2)} = \frac{1}{9}, \] then the value of \( p \) is: