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For a statistical data \( x_1, x_2, \ldots, x_{10} \) of \( 10 \) values, a student obtained the mean as \( 5.5 \) and \( \sum_{i=1}^{10} x_i^2 = 371 \). He later found that he had noted two values in the data incorrectly as \( 4 \) and \( 5 \), instead of the correct values \( 6 \) and \( 8 \), respectively. The variance of the corrected data is:

Let \( S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots \) upto \( n \) terms. If the sum of the first six terms of an A.P. with first term \( -p \) and common difference \( p \) is \( \sqrt{2026\, S_{2025}} \), then the absolute difference between \( 20^{\text{th}} \) and \( 15^{\text{th}} \) terms of the A.P. is:

If \( \alpha \) and \( \beta \) are the roots of the equation \( 2z^2 - 3z - 2i = 0 \), where \( i = \sqrt{-1} \), then \( 16\,. \mathrm{Re}\!\left( \frac{\alpha^{19}+\beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot \mathrm{Im}\!\left( \frac{\alpha^{19}+\beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \) is equal to:

For some \( n \neq 10 \), let the coefficients of the \( 5^{\text{th}} \), \( 6^{\text{th}} \) and \( 7^{\text{th}} \) terms in the binomial expansion of \( (1 + x)^{n+4} \) be in A.P. Then the largest coefficient in the expansion of \( (1 + x)^{n+4} \) is:

\( A \) and \( B \) alternately throw a pair of dice. \( A \) wins if he throws a sum of \( 5 \) before \( B \) throws a sum of \( 8 \), and \( B \) wins if he throws a sum of \( 8 \) before \( A \) throws a sum of \( 5 \). The probability, that \( A \) wins if \( A \) makes the first throw, is:

Let the product of the focal distances of the point \( (\sqrt{3}, \tfrac{1}{2}) \) on the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( (a > b) \), be \( \tfrac{7}{4} \). Then the absolute difference of the eccentricities of two such ellipses is:

If the system of equations
\( 2x - y + z = 4 \)
\( 5x + \lambda y + 3z = 12 \)
\( 100x - 47y + \mu z = 212 \),
has infinitely many solutions, then \( \mu - 2\lambda \) is equal to:

\( \displaystyle \lim_{x\to 0} cosec x \left( \sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4} \right) \) is:

Let circle \( C \) be the image of \( x^2 + y^2 - 2x + 4y - 4 = 0 \) in the line \( 2x - 3y + 5 = 0 \) and \( A \) be the point on \( C \) such that \( OA \) is parallel to \( x \)-axis and \( A \) lies on the right hand side of the centre \( O \) of \( C \). If \( B\,(\alpha,\beta) \), with \( \beta < 4 \), lies on \( C \) such that the length of the arc \( AB \) is \( \left(\tfrac{1}{6}\right)^{\text{th}} \) of the perimeter of \( C \), then \( \beta - \sqrt{3}\alpha \) is equal to:

The product of all the rational roots of the equation \( (x^2 - 9x + 11)^2 - (x - 4)(x - 5) = 3 \), is equal to:

Let \( f(x) = \dfrac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32} \). Then the value of \( 8\!\left( f\!\left(\tfrac{1}{15}\right) + f\!\left(\tfrac{2}{15}\right) + \cdots + f\!\left(\tfrac{59}{15}\right) \right) \) is equal to:

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to:

Let the line passing through the points \( (-1, 2, 1) \) and parallel to the line \( \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4} \) intersect the line \( \frac{x+2}{3} = \frac{y-3}{2} = \frac{z-4}{1} \) at the point \( P \). Then the distance of \( P \) from the point \( Q (4, -5, 1) \) is:

Let in a \( \triangle ABC \), the length of the side \( AC \) be \( 6 \), the vertex \( B \) be \( (1, 2, 3) \) and the vertices \( A, C \) lie on the line \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} \). Then the area (in sq. units) of \( \triangle ABC \) is:

Let \( y = y(x) \) be the solution of the differential equation \( \left( xy - 5x^{2}\sqrt{1 + x^{2}} \right) dx + \left( 1 + x^{2} \right) dy = 0, \ y(0) = 0 \). Then \( y(\sqrt{3}) \) is equal to:

Consider the region \( R = \left\{ (x, y) : x \le y \le 9 - \frac{11}{3} x^{2}, \ x \ge 0 \right\} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is:

If \( I(m, n) = \int_{0}^{1} x^{m-1} (1 - x)^{n-1} \, dx, \ m, n > 0 \), then \( I(9, 14) + I(10, 13) \) is:

Let the lines \( 3x - 4y - \alpha = 0 \), \( 8x - 11y - 33 = 0 \) and \( 2x - 3y + \lambda = 0 \) be concurrent. If the image of the point \( (1, 2) \) in the line \( 2x - 3y + \lambda = 0 \) is \( \left( \frac{57}{13}, \frac{-40}{13} \right) \), then \( |\alpha \lambda| \) is equal to:

The area of the region \( \{(x,y): x^{2}+4x+2 \le y \le |x+2|\} \) is equal to:

Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^{T} A X = 0 \) for all non-zero \( 3 \times 1 \) matrices  , and \( \det(\operatorname{adj}(2(A+I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} \), \( \alpha, \beta, \gamma \in \mathbb{N} \), then \( \alpha^{2} + \beta^{2} + \gamma^{2} \) is ________.

If for some \( \alpha, \beta : \alpha \le \beta \), \( \alpha + \beta = 8 \) and \( \sec^{2}\!\left(\tan^{-1}\! \alpha \right) + cosec^{2}\!\left(\cot^{-1}\! \beta \right) = 36 \), then \( \alpha^{2} + \beta \) is ______.

Let \( f \) be a differentiable function such that \( 2(x+2)^{2} f(x) - 3(x+2)^{2} = 10 \int_{0}^{x} (t+2) f(t) \, dt, \ x \ge 0 \). Then \( f(2) \) is equal to ________.

The number of \( 3 \)-digit numbers, that are divisible by \( 2 \) and \( 3 \), but not divisible by \( 4 \) and \( 9 \), is _______.

Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y) \), \( x \in S \), \( y \in A \), such that \( x \) divides \( y \), is ________.