List of top Mathematics Questions

For $a, b > 0$, let $ f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0, \\ \frac{x}{3}, & x = 0, \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a x \sqrt{x}}}, & x > 0 \end{cases} $ be a continuous function at $x = 0$. Then $\frac{b}{a}$ is equal to: 

Let A= {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15} Let R be a relation on A × B define by (a, b)R(c, d) if and only if 3ad – 7bc is an even integer. Then the relation R is

Let \[ \int_{\log_e a}^{4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}. \] Then \(e^\alpha\) and \(e^{-\alpha}\) are the roots of the equation:

If the function \(f(x) = 2x^3 - 9ax^2 + 12a^2x + 1, \, a>0\) has a local maximum at \(x = \alpha\) and a local minimum at \(x = \alpha^2\), then \(\alpha\) and \(\alpha^2\) are the roots of the equation:

If the line segment joining the points \((5, 2)\) and \((2, a)\) subtends an angle \(\frac{\pi}{4}\) at the origin, then the absolute value of the product of all possible values of \(a\) is:

If the value of \[ \frac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ} = \frac{a\sqrt{5} - b}{c}, \] where \(a, b, c\) are natural numbers and \(\text{gcd}(a, c) = 1\), then \(a + b + c\) is equal to:

If the system of equations \(x + 4y - z = \lambda\), \(7x + 9y + \mu z = -3\), \(5x + y + 2z = -1\) has infinitely many solutions, then \((2\mu + 3\lambda)\) is equal to:

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \[ \frac{70}{3} \] and the product of the third and fifth terms is 49. Then the sum of the \(4^\text{th}, 6^\text{th}\), and \(8^\text{th}\) terms is:

If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

Let \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}\), and \(\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}\) be three vectors. Let \(\vec{r}\) be a unit vector along \(\vec{b} + \vec{c}\). If \(\vec{r} \cdot \vec{a} = 3\), then \(3\lambda\) is equal to:

If the image of the point \((-4, 5)\) in the line \(x + 2y = 2\) lies on the circle \((x + 4)^2 + (y - 3)^2 = r^2\), then \(r\) is equal to:

Let \( A = \{2, 3, 6, 7\} \) and \( B = \{4, 5, 6, 8\} \). Let \( R \) be a relation defined on \( A \times B \) by \((a_1, b_1) R (a_2, b_2)\) if and only if \(a_1 + a_2 = b_1 + b_2\). Then the number of elements in \( R \) is __________.

Let the centre of a circle, passing through the point \((0, 0)\), \((1, 0)\) and touching the circle \(x^2 + y^2 = 9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k)\), \(4(h^2 + k^2)\) is equal to __________.

Let A be a non-singular matrix of order 3. If \[ \text{det}\left(3 \text{adj}(2 \text{adj}((\text{det} A) A))\right) = 3^{-13} \cdot 2^{-10} \] and \[ \text{det}\left(3 \text{adj}(2 A)\right) = 2^m \cdot 3^n, \] then \( |3m + 2n| \) is equal to __________.

Let \( f: (0, \pi) \to \mathbb{R} \) be a function given by
\[ f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\tan 8x / \tan 7x}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ \left(1 + |\cot x|\right)^{b^{\lfloor \tan x \rfloor}}, & \frac{\pi}{2} < x < \pi \end{cases} \]
Where \( a, b \in \mathbb{Z} \). If \( f \) is continuous at \( x = \frac{\pi}{2} \), then \( a^2 + b^2 \) is equal to __________.

The remainder when \( 4^{28^{2024}} \) is divided by 21 is __________.

Let \(\lim_{n \to \infty} \left( \frac{n}{\sqrt{n^4 + 1}} - \frac{2n}{\left(n^2 + 1\right)\sqrt{n^4 + 1}} + \frac{n}{\sqrt{n^4 + 16}} - \frac{8n}{\left(n^2 + 4\right)\sqrt{n^4 + 16}} + \ldots + \frac{n}{\sqrt{n^4 + n^4}} - \frac{2n \cdot n^2}{\left(n^2 + n^2\right)\sqrt{n^4 + n^4}} \right)\)  be \(\frac{\pi}{k},\) using only the principal values of the inverse trigonometric functions. Then \(k^2\) is equal to ______.

Let the set of all positive values of \( \lambda \), for which the point of local minimum of the function \((1 + x (\lambda^2 - x^2)) \frac{x^2 + x + 2}{x^2 + 5x + 6} < 0\) be \((\alpha, \beta)\).
Then \( \alpha^2 + \beta^2 \) is equal to ________.

Let \( f(x) = x^2 + 9 \), \( g(x) = \frac{x}{x-9} \), and \[ a = f \circ g(10), \, b = g \circ f(3). \]
If \( e \) and \( l \) denote the eccentricity and the length of the latus rectum of the ellipse \[ \frac{x^2}{a} + \frac{y^2}{b} = 1, \] then \( 8e^2 + l^2 \) is equal to:

Let $\alpha, \beta$ be the roots of the equation $x^2 + 2\sqrt{2}x - 1 = 0$. The quadratic equation, whose roots are $\alpha^4 + \beta^4$ and $\frac{1}{10} \left( \alpha^6 + \beta^6 \right)$, is:

The frequency distribution of the age of students in a class of 40 students is given below:

\(Age\)	15	16	17	18	19	20
No. of Students	5	8	5	12	x	y

If the mean deviation about the median is 1.25, then \(4x + 5y\) is equal to:

The shortest distance between the line:
\[ \frac{x-3}{4} = \frac{y+7}{-11} = \frac{z-1}{5} \] and \[ \frac{x-5}{3} = \frac{y-9}{-6} = \frac{z+2}{1} \] is:

Let a circle passing through (2, 0) have its centre at the point \( (h, k) \). Let \( (x_c, y_c) \) be the point of intersection of the lines \( 3x + 5y = 1 \) and \( (2 + c)x + 5c^2y = 1 \). If \( h = \lim_{c \to 1} x_c \) and \( k = \lim_{c \to 1} y_c \), then the equation of the circle is:

Let \( f(x) = ax^3 + bx^2 + ex + 41 \) be such that \( f(1) = 40 \), \( f'(1) = 2 \) and \( f''(1) = 4 \). Then \( a^2 + b^2 + c^2 \) is equal to:

If the sum of the series $$ \frac{1}{1 \cdot (1 + d)} + \frac{1}{(1 + d)(1 + 2d)} + \cdots + \frac{1}{(1 + 9d)(1 + 10d)} $$ is equal to 5, then \(50d\) is equal to: