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List of top Mathematics Questions

Let $ f : [0,3] \to \mathbb{R} $ be defined by \[ f(x) = \begin{cases} 0, & 0 \leq x < 1, \\ e^{x^2} - e, & 1 \leq x < 2, \\ e^{x^2} + 1, & 2 \leq x \leq 3. \end{cases} \] Now, define $ F : [0, 3] \to \mathbb{R} $ by \[ F(0) = 0 \,\,\text{and}\,\, F(x) = \int_0^x f(t) \, dt, \text{ for } 0 < x \leq 3. \] Then

If $ x, y, z $ are real numbers such that $ 4x + 2y + z = 31 \,\,\text{and}\,\, 2x + 4y - z = 19, $ then the value of $ 9x + 7y + z $ is

Let \[ M = \begin{pmatrix} 1 & -1 & 1 \\ 1 & -1 & -1 \end{pmatrix}. \] If \[ V = \{ (x, y, 0) \in \mathbb{R}^3 : M \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \}, \] \[ W = \{ (x, y, z) \in \mathbb{R}^3 : M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \}, \] then

Let $ M $ be a $ 3 \times 3 $ non-zero, skew-symmetric real matrix. If $ I $ is the $ 3 \times 3 $ identity matrix, then

Let $ \{a_n\}_{n \geq 1} $ be a sequence of real numbers such that \[ a_n = \sum_{k=n+1}^{2n} \frac{1}{k}, n \geq 1. \] Then which of the following statement(s) is (are) true?

Let $ \sum_{n \geq 1} a_n $ be a convergent series of positive real numbers. Then which of the following statement(s) is (are) true?

Let $ f : \mathbb{R} \to \mathbb{R} $ be defined by \[ f(x) = \begin{cases} x^4(2 + \sin \frac{1}{x}), & x \neq 0, \\ 0, & x = 0. \end{cases} \] Then which of the following statement(s) is (are) true?

Let $ \{a_n\}_{n \geq 1} $ be a sequence of real numbers such that \[ a_n = \frac{1 + 3 + 5 + \dots + (2n - 1)}{n!}, n \geq 1. \] Then $ \sum_{n \geq 1} a_n $ converges to ................

Let \[ S = \{(x, y) \in \mathbb{R}^2 : x \geq 0, \sqrt{4 - (x - 2)^2} \leq y \leq \sqrt{9 - (x - 3)^2} \}. \] Then the area of $ S $ equals .............

Let \[ S = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \leq 1\}. \] Then the area of $ S $ equals ...............

Let \[ J = \frac{1}{\pi} \int_0^1 t^{-\frac{1}{2}} (1 - t)^{\frac{3}{2}} \, dt. \] Then the value of $ J $ equals ..............

Let $ M = \sum_{i=1}^{4} X_i X_i^T $, where \[ X_1^T = [1 \ -1 \ 1 \ 0], X_2^T = [1 \ 1 \ 0 \ 1], X_3^T = [1 \ 3 \ 1 \ 0] \, \text{and} \, X_4^T = [1 \ 1 \ 1 \ 0]. \] Then the rank of $ M $ equals ...............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function such that $ f^\prime $ is continuous on $ \mathbb{R} $ with $ f^\prime(3) = 18 $. Define \[ g_n(x) = n \left( f \left( x + \frac{5}{n} \right) - f \left( x - \frac{2}{n} \right) \right). \] Then $ \lim_{n \to \infty} g_n(3) $ equals ...............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function with $ \lim_{x \to \infty} f(x) = \infty $ and $\lim_{x \to \infty} f'(x) = 2. $ Then \[ \lim_{x \to \infty} \left( 1 + \frac{f(x)}{x^2} \right)^x \] equals ..............

The value of \[ \int_0^\frac{\pi}{2} \left( \int_0^x e^{\sin y} \sin x \, dy \right) dx \] equals .............

Let $ \mathbb{C}^* = \mathbb{C} \setminus \{0\} $ denote the group of non-zero complex numbers under multiplication. Suppose \[ Y_n = \{ z \in \mathbb{C} \mid z^n = 1 \}, n \in \mathbb{N}. \] Which of the following is (are) subgroup(s) of $ \mathbb{C}^* $?

Let $ P $ and $ Q $ be two non-empty disjoint subsets of $ \mathbb{R} $. Which of the following is (are) FALSE?

Suppose $ f, g, h $ are permutations of the set $ \{ \alpha, \beta, \gamma, \delta \} $, where
f interchanges $ \alpha $ and $ \beta $ but fixes $ \gamma $ and $ \delta $,
g interchanges $ \beta $ and $ \gamma $ but fixes $ \alpha $ and $ \delta $,
h interchanges $ \gamma $ and $ \delta $ but fixes $ \alpha $ and $ \beta $.
Which of the following permutations interchange(s) $ \alpha $ and $ \delta $ but fix(es) $ \beta $ and $ \gamma $?

Consider the group $ \mathbb{Z}^2 = \{ (a, b) | a, b \in \mathbb{Z} \} $ under component-wise addition. Then which of the following is a subgroup of $ \mathbb{Z}^2 $?

Let $ G $ be a group satisfying the property that $ f: G \to \mathbb{Z}_{221} $ is a homomorphism implies $ f(g) = 0 $, $ \forall g \in G $. Then a possible group $ G $ is

Suppose that $ f, g : \mathbb{R} \to \mathbb{R} $ are differentiable functions such that $ f $ is strictly increasing and $ g $ is strictly decreasing. Define $ p(x) = f(g(x)) $ and $ q(x) = g(f(x)) $, $ \forall x \in \mathbb{R} $. Then, for $ t > 0 $, the sign of $ \int_0^t p'(x) (q'(x)-3) \, dx $ is

Let $ f : \mathbb{R}^3 \to \mathbb{R} $ be a scalar field, $ \vec{v} : \mathbb{R}^3 \to \mathbb{R}^3 $ be a vector field and let $ \vec{a} \in \mathbb{R}^3 $ be a constant vector. If $ \vec{r} $ represents the position vector $ x\hat{i} + y\hat{j} + z\hat{k} $, then which one of the following is FALSE?

The equation of the circle which passes through the point $(4, 5)$ and has its centre at $(2, 2)$ is

In a binomial distribution, the mean is $4$ and variance is $3$. Then its mode is :

The value of $\displaystyle\lim_{n \to\infty} \frac{1+2+3+...n}{n^{2}+100}$ is equal to :