List of top Mathematics Questions

Consider the relations $R_1$ and $R_2$ defined as \[a R_1 b \iff a^2 + b^2 = 1 \quad \text{for all } a, b \in \mathbb{R},\]and \[(a, b) R_2 (c, d) \iff a + d = b + c \quad \text{for all } (a, b), (c, d) \in \mathbb{N} \times \mathbb{N}.\]Then:

If the domain of the function \[f(x) = \frac{\sqrt{x^2 - 25}}{(4 - x^2)} + \log_{10}(x^2 + 2x - 15)\]is $(-\infty, \alpha) \cup [\beta, \infty)$, then $\alpha^2 + \beta^3$ is equal to:

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$, then $S_{15} - S_{5}$ is equal to:

If \[\int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = a\pi + b\sqrt{3},\]where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to:

Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of \[\left( \frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}} \right)^{18}.\]Then \[\left(\frac{n}{m}\right)^{\frac{1}{3}}\]is:

Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \[ f(x) = \frac{4^x}{4^x + 2} \] and \[ M = \int_{f(a)}^{f(1 - a)} x \sin^4 \left( x (1 - x) \right) \, dx, \] \[ N = \int_{f(a)}^{f(1 - a)} \sin^4 \left( x (1 - x) \right) \, dx; \quad a \neq \frac{1}{2}. \] If \( \alpha M = \beta N \), \( \alpha, \beta \in \mathbb{N} \), then the least value of \( \alpha^2 + \beta^2 \) is equal to _____

Let \( A = \{1, 2, 3, 4\} \) and \( R = \{(1, 2), (2, 3), (1, 4)\} \) be a relation on \( A \).Let \( S \) be the equivalence relation on \( A \) such that \( R \subseteq S \) and the number of elements in \( S \) is \( n \). Then, the minimum value of \( n \) is _____

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \) and \( \vec{a} \cdot \vec{b} = 2 \).If \( \vec{c} = (2 \vec{a} \times \vec{b}) - 3 \vec{b} \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \alpha \), then \( 192 \sin^2 \alpha \) is equal to _____

Let $\alpha$ be a non-zero real number. Suppose $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(0) = 2$ and \[\lim_{x \to \infty} f(x) = 1.\]If $f'(x) = \alpha f(x) + 3$, for all $x \in \mathbb{R}$, then $f(-\log 2)$ is equal to ________ .

Let the foci and length of the latus rectum of an ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \quad \text{be } (\pm 5, 0) \text{ and } \sqrt{50}, \] respectively. Then, the square of the eccentricity of the hyperbola \[ \frac{x^2}{b^2} - \frac{y^2}{a^2 b^2} = 1 \] equals _____

Let $P$ be a point on the ellipse \[\frac{x^2}{9} + \frac{y^2}{4} = 1.\]Let the line passing through $P$ and parallel to the $y$-axis meet the circle \[x^2 + y^2 = 9\]at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR : RQ = 4 : 3$ as $P$ moves on the ellipse, is:

Let \( A = \{ 1, 2, 3, \dots, 20 \} \). Let \( R_1 \) and \( R_2 \) be two relations on \( A \) such that \(R_1 = \{(a, b) : b \text{ is divisible by } a\}\)
and  \(R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}\).Then, the number of elements in \( R_1 - R_2 \) is equal to \(\_\_\_\_.\)

Let the line of the shortest distance between the lines \(L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})\)and  \(L_2: \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k})\) intersect \(L_1\) and \(L_2\) at \(P\) and \(Q\), respectively.  If \((\alpha, \beta, \gamma)\) is the midpoint of the line segment \(PQ\), then \(2(\alpha + \beta + \gamma)\) is equal to \(\_\_\_\_\).

Let \( P = \{ z \in \mathbb{C} : |z + 2 - 3i| \leq 1 \} \) and \( Q = \{ z \in \mathbb{C} : z(1 + i) + \overline{z}(1 - i) \leq -8 \} \).
Let \( z \) in \( P \cap Q \) have \( |z - 3 + 2i| \) be maximum and minimum at \( z_1 \) and \( z_2 \), respectively.  
If \( |z_1|^2 + 2|z_2|^2 = \alpha + \beta \sqrt{2} \), where \( \alpha \) and \( \beta \) are integers, then \( \alpha + \beta \) equals ____.

Let the line \( L : \sqrt{2}x + y = \alpha \) pass through the point of intersection \( P \) (in the first quadrant) of the circle \( x^2 + y^2 = 3 \) and the parabola \( x^2 = 2y \). Let the line \( L \) touch two circles \( C_1 \) and \( C_2 \) of equal radius \( 2\sqrt{3} \). If the centers \( Q_1 \) and \( Q_2 \) of the circles \( C_1 \) and \( C_2 \) lie on the y-axis, then the square of the area of the triangle \( PQ_1Q_2 \) is equal to ____.

Let \(\{x\}\) denote the fractional part of \(x\), and \(f(x) = \frac{\cos^{-1}(1 - \{x\}^2) \sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}, \quad x \neq 0\).If \(L\) and \(R\) respectively denote the left-hand limit and the right-hand limit of \(f(x)\) at \(x = 0\), then  \(\frac{32}{\pi^2} \left(L^2 + R^2\right)\) is equal to \(\_\_\_\_\_\_\_\_\).

If the coefficient of \(x^{30}\) in the expansion of \(\left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8, \, x \neq 0\) is \(\alpha\), then \(|\alpha|\) equals ____.

The value of \[\int_{0}^{1} \left(2x^3 - 3x^2 - x + 1\right)^{\frac{1}{3}} \, dx\]is equal to:

In the expansion of \[ (1 + x)(1 - x^2) \left( 1 + \frac{3}{x} + \frac{3}{x^2} + \frac{1}{x^3} \right)^5, \quad x \neq 0, \]the sum of the coefficients of \( x^3 \) and \( x^{-13} \) is equal to ____

The number of solutions of the equation \[4 \sin^2 x - 4 \cos^3 x + 9 - 4 \cos x = 0, \, x \in [-2\pi, 2\pi]\]is:

Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x = y, z = 1 and x = –y, z = –1 respectively. If ∠QPR is a right angle, then 12a² is equal to _____

The total number of words (with or without meaning) that can be formed out of the letters of the word ‘DISTRIBUTION’ taken four at a time, is equal to _____

Let 3, 7, 11, 15, ...., 403 and 2, 5, 8, 11, . . ., 404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to _________.

Let\(f(x) = |2x^2 + 5|x - 3|, x \in \mathbb{R}\). If \(m\) and \(n\) denote the number of points were \(f\)is not continuous and not differentiable respectively, then\(m + n\)is equal to:

Let \( S = (-1, \infty) \) and \( f : S \rightarrow \mathbb{R} \) be defined as \[ f(x) = \int_{-1}^{x} (e^t - 1)^{11} (2t - 1)^5 (t - 2)^7 (t - 3)^{12} (2t - 10)^{61} \, dt \] Let \( p = \) Sum of squares of the values of \( x \), where \( f(x) \) attains local maxima on \( S \). And \( q = \) Sum of the values of \( x \), where \( f(x) \) attains local minima on \( S \). Then, the value of \( p^2 + 2q \) is ______