List of top Mathematics Questions

Let \( a_1 = b_1 = 0 \), and for each \( n \geq 2 \), let \( a_n \) and \( b_n \) be real numbers given by \[ a_n = \sum_{m=2}^{n} (-1)^{m} m (\log(m))^m \] \[ b_n = \sum_{m=2}^{n} \frac{1}{(\log(m))^m}. \] Then which one of the following is TRUE about the sequences \( \{a_n\} \) and \( \{b_n\} \)?

The eigenvalue(s) of the matrix 

The function \( f(x) = x e^{-x^2} \) has a minimum at
Let \( \mathbf{a} = 4i - 2j + 6k \) and \( \mathbf{b} = 7i + j - 12k \). If \( \mathbf{a} \times \mathbf{b} = \alpha i + \beta j + \gamma k \), then the value of \( \alpha + \beta + \gamma \) equals .............

Let \( \mathbb{N} \) be the set of natural numbers and \( f : \mathbb{N} \to \mathbb{N} \) be defined by 

Let \( f^n(x) \) denote the \( n \)-fold composition of \( f(x) \). What is the smallest integer \( n \) such that \( f^n(13) = 1 \)? 
 

The value of \( \int_0^\pi x \sin x \, dx \) is .......
Let XYZ be an equilateral triangle and let P, Q, R be the midpoints of YZ, XZ, and XY, respectively.
Let \( r = \frac{\text{Area of } \Delta PQR}{\text{Area of } \Delta XYZ} \).
Let \( U = \{1, 2, \dots, 15\} \). Let \( P \subseteq U \) consist of all prime numbers, \( Q \subseteq U \) consist of all even numbers and \( R \subseteq U \) consist of all multiples of 3. Let \( T = P - Q \). Then, which of the following is/are CORRECT?
Let \( f(x) = (x - 1)(x - 2)(x - 3)(x - 4) \) and let \( \alpha = f\left( \frac{3}{2} \right) \), \( \beta = f\left( \frac{5}{2} \right) \) and \( \gamma = f\left( \frac{7}{2} \right) \). Which of the following is/are CORRECT?
Simplify \( \frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} \).
Let \( a = \frac{\sqrt{5}+1}{2} \) and \( b = \frac{\sqrt{5}-1}{2} \). Then, \( \lim_{n \to \infty} \frac{a^n + b^n}{a^n - b^n} \) is ________
In how many ways can one write the elements 1, 2, 3, 4 in a sequence \( x_1, x_2, x_3, x_4 \) with \( x_i \neq i \) for all \( i \)?
Let \( U = \{1, 2, 3, 4, 5\} \). A subset \( S \) is chosen uniformly at random from the non-empty subsets of \( U \). What is the probability that \( S \) does NOT have two consecutive elements?
Which one of the points P = \(\left(\frac{3}{2}, \frac{1}{2}\right)\), Q = \(\left(\frac{1}{2}, \frac{3}{2}\right)\), R = \(\left(\frac{3}{2}, \frac{11}{2}\right)\) and S = \(\left(\frac{11}{2}, \frac{3}{2}\right)\) lies ABOVE the parabola \(y = 2x^2\) and INSIDE the circle \(x^2 + y^2 = 4\)?
The number of 4 digit numbers without repetition that can be formed using the digits 1, 2, 3, 4, 5, 6, 7 in which each number has two odd digits and two even digits is
If the coordinates at one end of a diameter of the circle $x^2 + y^2 - 8x - 4y + c = 0$ are $(-3, 2)$, then the coordinates at the other end are
The radius of a right circular cylinder increases at the rate of $0.1\, cm/min$, and the height decreases at the rate of $0.2\, cm/min$. The rate of change of the volume of the cylinder, in $cm^3/min$, when the radius is $2\, cm$ and the height is $3\, cm$ is
The equation $y^2 + 3 = 2(2x + y)$ represents a parabola with the vertex at
A circle has radius 3 and its centre lies on the line $y = x -1$ . The equation of the circle, if it passes through $(7, 3)$, is
If $\left(\vec{a}\times\vec{b}\right)^{2}+\left(\vec{a}.\vec{b}\right)^{2}=676$ and $\left|\vec{b}\right|=2,$ then $\left|\vec{a}\right|$ is equal to
The system of linear equations : $x + y + z = 0, 2x + y - z = 0, 3x + 2y = 0$ has :
The area under the curve $y = | cos\, x - sin \,x |, 0 \le x\le\frac{\pi}{2}$, and above x-axis is :
The mean and the median of the following ten numbers in increasing order $10, 22, 26, 29, 34, x 42, 67, 70,\, y$ are $42$ and $35$ respectively, then $\frac{y}{x}$ is equal to :
If vector equation of the line$\frac{x-2}{2}=\frac{2y-5}{-3}=z+1,$ is $\vec{r}=\left(2\hat{i}+\frac{5}{2} \hat{j}-\hat{k}\right)+\lambda\left(2\hat{i}-\frac{3}{2} \hat{j}+p\hat{k}\right)$ then $p$ is equal to
If $x = 3 \,tan \,t $ and $y = 3 \,sec \,t$, then the value of $\frac{d^2 y}{dx^2}$ at $t = \frac{\pi}{4}$, is :