List of top Mathematics Questions

The length of the subnormal at any point on the curve \( y = \left(\frac{x}{2024}\right)^k \) is constant if the value of \( k \) is:
The number of ways in which 17 apples can be distributed among four guests such that each guest gets at least 3 apples is:
In \(\triangle ABC\), if \(4r_1 = 5r_2 = 6r_3\), then \(\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} =\)
Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} \, dx \):
For any real value of \(x\), if \( \frac{11x^{2}+12x+6}{x^{2}+4x+2} \not\in (a, b) \), then the value for \(x\) for which $$ \frac{11x^{2}+12x+6}{x^{2}+4x+2} = b - a + 3 $$ is: 
Two natural numbers are chosen at random from 1 to 100 and are multiplied. If \(A\) is the event that the product is an even number and \(B\) is the event that the product is divisible by 4, then \(P(A \cap \bar B) = \)
If the points having the position vectors \( \mathbf{r}_1 = -i + 4j - 4k\), \( \mathbf{r}_2 = 3i + 2j - 5k\), \( \mathbf{r}_3 = -3i + 8j - 5k\) and \( -3i + 2j + \lambda k\) are coplanar, then \(\lambda = \)
If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are three vectors such that \( |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = \sqrt{3} \) and \( ( \mathbf{a} + \mathbf{b} - \mathbf{c} )^2 + ( \mathbf{b} + \mathbf{c} - \mathbf{a} )^2 + ( \mathbf{c} + \mathbf{a} - \mathbf{b} )^2 = 36\), then \(|2\mathbf{a} - 3\mathbf{b} + 2\mathbf{c}|^2 =\)
Evaluate the integral \( \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx \):
The radical center of the circles \(x^2 + y^2 + 2x + 3y + 1 = 0\), \(x^2 + y^2 - x + y + 3 = 0\), and \(x^2 + y^2 - 3x + 2y + 5 = 0\) is:
If \(\sec \theta + \tan \theta = \frac{1}{3}\), then the quadrant in which \(2\theta\) lies is:
Determine the values of \(a\) and \(b\) for which the function \(f(x)\) defined as: \[ f(x) = \begin{cases} 1 + |\sin x|^{\left(\frac{a}{|\sin x|}\right)} & \text{if } \frac{-\pi}{6} < x < 0, \\ b & \text{if } x = 0, \\ e^{\left(\frac{\tan 2x}{\tan 3x}\right)} & \text{if } 0 < x < \frac{\pi}{6} \end{cases} \] is continuous at \(x = 0\).
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be defined by \( f(x + y) = f(x) + 12y \), for all \( x, y \in \mathbb{R} \). If \( f(1) = 6 \), then the value of \( \sum_{r=1}^n f(r) \) is:
Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{4\sqrt{2} - (\cos x + \sin x)^5}{1 - \sin 2x} = \]
If the coefficients of \(x^5\) and \(x^6\) are equal in the expansion of \( \left( a + \frac{x}{5} \right)^{65} \), then the coefficient of \(x^2\) in the expansion of \( \left( a + \frac{x}{5} \right)^4 \) is:
If \( L_1 \) and \( L_2 \) are two lines which pass through origin and have direction ratios \( (3, 1, -5) \) and \( (2, 3, -1) \) respectively, then the equation of the plane containing \( L_1 \) and \( L_2 \) is:
If \(f(x) = \begin{cases} 2x + 3, & \text{if } x \leq 1 \\ 2ax + bx, & \text{if } x > 1 \end{cases}\) is differentiable \(\forall x \in \mathbb{R}\), then \(f(2) =\)
Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x} - a - \log(1+x)}{\sin x} = 0, { then find } a. \]
Equation of a tangent line of the parabola \(y^2 = 8x\), which passes through the point \((1, 3)\) is:
The angle between the line with the direction ratios \( (2, 5, 1) \) and the plane \( 8x + 2y - z = 4 \) is given by
If \([x]\) is the greatest integer function, then evaluate the integral \( \int_{0}^{5} [x] \, dx \):
A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is
Evaluate the integral \( \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx \):
If \( a_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}} \), then \( \sum_{r=0}^{n} r \binom{n}{r} = \):
The coefficient of \( x^5 \) in the expansion of \( \left( 2x^3 - \frac{1}{3x^2} \right)^5 \) is: