List of top Mathematics Questions

Let the set \( C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\} \).Then\[\sum_{(x, y) \in C} (x + y)\]is equal to ______.

Remainder when \( 64^{32^{32}} \) is divided by 9 is equal to ______.

Let\[f(x) = \sqrt{\lim_{r \to x} \left( \frac{2r^2 \left[ (f(r))^2 - f(x) f(r) \right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right)}\]be differentiable in \( (-\infty, 0) \cup (0, \infty) \) and \( f(1) = 1 \). Then the value of \( ea \), such that \( f(a) = 0 \), is equal to ______.

Let \( O \) be the origin, and \( M \) and \( N \) be the points on the lines \[\frac{x - 5}{4} = \frac{y - 4}{1} = \frac{z - 5}{3} \]and\[\frac{x + 8}{12} = \frac{y + 2}{5} = \frac{z + 11}{9} \]respectively, such that \( MN \) is the shortest distance between the given lines. Then \( OM \cdot ON \) is equal to ______.

Let the area of the region \( \{(x, y): 0 \leq x \leq 3, 0 \leq y \leq \min\{x^2 + 2, 2x + 2\}\} \) be \( A \). Then \( 12A \) is equal to ______.

If\[\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1 - \sin 2x} \, dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3},\]where \( \alpha \), \( \beta \), and \( \gamma \) are rational numbers, then \( 3\alpha + 4\beta - \gamma \) is equal to ______.

Let \( P(\alpha, \beta) \) be a point on the parabola \( y^2 = 4x \). If \( P \) also lies on the chord of the parabola \( x^2 = 8y \) whose midpoint is \( \left( 1, \frac{5}{4} \right) \), then \( (\alpha - 28)(\beta - 8) \) is equal to ______.

Let for any three distinct consecutive terms \( a, b, c \) of an A.P., the lines \( ax + by + c = 0 \) be concurrent at the point \( P \) and \( Q (\alpha, \beta) \) be a point such that the system of equations \[x + y + z = 6,\]\[2x + 5y + \alpha z = \beta,\]\[x + 2y + 3z = 4,\]has infinitely many solutions. Then \( (PQ)^2 \) is equal to ______.

If \( R \) is the smallest equivalence relation on the set \( \{1, 2, 3, 4\} \) such that \( \{(1,2), (1,3)\} \subseteq R \), then the number of elements in \( R \) is ______.

Let \( y = \log_e \left( \frac{1 - x^2}{1 + x^2} \right), -1 < x < 1 \). Then at \( x = \frac{1}{2} \), the value of \( 225(y' - y'') \) is equal to

Let \( x = \frac{m}{n} \) ( \( m, n \) are co-prime natural numbers) be a solution of the equation \( \cos \left( 2 \sin^{-1} x \right) = \frac{1}{9} \) and let \( \alpha, \beta (\alpha > \beta) \) be the roots of the equation \( mx^2 - nx - m + n = 0 \). Then the point \( (\alpha, \beta) \) lies on the line

Let \( A \) be the point of intersection of the lines \( 3x + 2y = 14 \), \( 5x - y = 6 \) and \( B \) be the point of intersection of the lines \( 4x + 3y = 8 \), \( 6x + y = 5 \). The distance of the point \( P(5, -2) \) from the line \( AB \) is

If each term of a geometric progression \( a_1, a_2, a_3, \dots \) with \( a_1 = \frac{1}{8} \) and \( a_2 \neq a_1 \), is the arithmetic mean of the next two terms and \( S_n = a_1 + a_2 + \dots + a_n \), then \( S_{20} - S_{18} \) is equal to

If \( \sin\left(\frac{y}{x}\right) = \log_e |x| + \frac{\alpha}{2} \) is the solution of the differential equation \[x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x\]and \( y(1) = \frac{\pi}{3} \), then \( \alpha^2 \) is equal to

The distance of the point \( (2, 3) \) from the line \( 2x - 3y + 28 = 0 \), measured parallel to the line \( \sqrt{3}x - y + 1 = 0 \), is equal to

If\[\int \frac{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}{\sqrt{\sin^{\frac{1}{3}} x \cos^{\frac{1}{3}} x \sin(x - \theta)}} \, dx = A \sqrt{\cos \theta \tan x - \sin \theta} + B \sqrt{\cos \theta \cot x + \sin(x - \theta)} + C,\]where \( C \) is the integration constant, then \( AB \) is equal to ______.

If \( \log_e a, \log_e b, \log_e c \) are in an A.P. and \( \log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a \) are also in an A.P., then \( a : b : c \) is equal to

Let \( \overrightarrow{OA} = \vec{a}, \overrightarrow{OB} = 12\vec{a} + 4\vec{b} \) and \( \overrightarrow{OC} = \vec{b} \), where \( O \) is the origin. If \( S \) is the parallelogram with adjacent sides \( OA \) and \( OC \), then\[\frac{\text{area of the quadrilateral OABC}}{\text{area of } S}\]is equal to ___.

The sum of the solutions \( x \in \mathbb{R} \) of the equation\[\frac{3 \cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - x^2 + 6\]is

The function \( f(x) = 2x + 3(x)^{\frac{2}{3}}, x \in \mathbb{R} \), has

If the mean and variance of five observations are \( \frac{24}{5} \) and \( \frac{194}{25} \) respectively and the mean of first four observations is \( \frac{7}{2} \), then the variance of the first four observations is equal to

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \), and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is

A line with direction ratios \( 2, 1, 2 \) meets the lines \(x = y + 2 = z\) and \(x + 2 = 2y = 2z\) respectively at the points \( P \) and \( Q \). If the length of the perpendicular from the point \( (1, 2, 12) \) to the line \( PQ \) is \( l \), then \( l^2 \) is

If the mean and variance of the data \( 65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60 \) where \( \alpha > \beta \) are \( 56 \) and \( 66.2 \) respectively, then \( \alpha^2 + \beta^2 \) is equal to

If the solution curve \( y = y \, x \) of the differential equation \((1 + y^2) \left(1 + \log_e x\right) dx + x \, dy = 0, \quad x > 0\) passes through the point \( (1, 1) \) and\[y(e) = \frac{\alpha - \tan\left(\frac{3}{2}\right)}{\beta + \tan\left(\frac{3}{2}\right)},\]then \( \alpha + 2\beta \) is