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Let \( \vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{b} \). If \( \vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4 \), then \( \left|\vec{a} + \vec{c}\right|^2 \) is equal to:

If \( x^2 + x + 1 = 0 \), then the value of \( \left(x + \frac{1}{x}\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^3 + \frac{1}{x^3}\right)^4 + \cdots + \left(x^{25} + \frac{1}{x^{25}}\right)^4 \) is:

The value of \( \ cosec 10^\circ - \sqrt{3}\sec 10^\circ \) is equal to:

Let \( a_1, a_2, a_3, \ldots \) be a G.P. of increasing positive terms such that \( a_2 \cdot a_3 \cdot a_4 = 64 \) and \( a_1 + a_3 + a_5 = \frac{813}{7} \). Then \( a_3 + a_5 + a_7 \) is equal to:

If the domain of the function \( f(x) = \cos^{-1}\!\left(\frac{2x - 5}{11 - 3x}\right) + \sin^{-1}\!\left(2x^2 - 3x + 1\right) \) is the interval \( [\alpha, \beta] \), then \( \alpha + 2\beta \) is equal to:

Let a point \( A \) lie between the parallel lines \( L_1 \) and \( L_2 \) such that its distances from \( L_1 \) and \( L_2 \) are \( 6 \) and \( 3 \) units, respectively. Then the area (in sq. units) of the equilateral triangle \( ABC \), where the points \( B \) and \( C \) lie on the lines \( L_1 \) and \( L_2 \), respectively, is:

Let \( O \) be the vertex of the parabola \( x^2 = 4y \) and \( Q \) be any point on it. Let the locus of the point \( P \), which divides the line segment \( OQ \) internally in the ratio \( 2:3 \) be the conic \( C \). Then the equation of the chord of \( C \), which is bisected at the point \( (1, 2) \) is:

The number of relations, defined on the set \( \{a,b,c,d\} \), which are both reflexive and symmetric, is equal to:

Let \( y = y(x) \) be the solution curve of the differential equation \( (1 + x^2)\,dy + \left(y - \tan^{-1}x\right)dx = 0,\, y(0) = 1 \). Then the value of \( y(1) \) is:

If the coefficient of \( x \) in the expansion of \( (a x^2 + b x + c)(1 - 2x)^{26} \) is \( -56 \) and the coefficients of \( x^2 \) and \( x^3 \) are both zero, then \( a + b + c \) is equal to:

The sum of all the roots of the equation \( (x - 1)^2 - 5|x - 1| + 6 = 0 \), is:

Let the foci of a hyperbola coincide with the foci of the ellipse \( \frac{x^2}{36} + \frac{y^2}{16} = 1 \). If the eccentricity of the hyperbola is \( 5 \), then the length of its latus rectum is:

The number of strictly increasing functions \( f \) from the set \( \{1, 2, 3, 4, 5, 6\} \) to the set \( \{1, 2, 3, \ldots, 9\} \) such that \( f(i) \neq i \) for \( 1 \le i \le 6 \), is equal to:

The area of the region, inside the ellipse \( x^2 + 4y^2 = 4 \) and outside the region bounded by the curves \( y = |x| - 1 \) and \( y = 1 - |x| \), is:

Let \( \vec{c} \) and \( \vec{d} \) be vectors such that \( |\vec{c} + \vec{d}| = \sqrt{29} \) and \( \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d} \). If \( \lambda_1, \lambda_2 \) \( (\lambda_1 > \lambda_2) \) are the possible values of \( (\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) \), then the equation \( K^2 x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right) y^2 - 8x + 12y + \lambda_2 = 0 \) represents a circle, for \( K \) equal to:

Let \( PQ \) and \( MN \) be two straight lines touching the circle \( x^2 + y^2 - 4x - 6y - 3 = 0 \) at the points \( A \) and \( B \) respectively.
Let \( O \) be the centre of the circle and \( \angle AOB = \frac{\pi}{3} \). Then the locus of the point of intersection of the lines \( PQ \) and \( MN \) is:

Let the mean and variance of \( 7 \) observations \( 2, 4, 10, x, 12, 14, y \), \( x > y \), be \( 8 \) and \( 16 \) respectively. Two numbers are chosen from \( \{1, 2, 3, x - 4, y, 5\} \) one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than \( 4 \), is:

Let \( f : \mathbb{R} \to (0,\infty) \) be a twice differentiable function such that \( f(3)=18 \), \( f'(3)=0 \) and \( f''(3)=4 \). Then 

The value of   is equal to:

Let \( S=\{(m,n): m,n\in \{1,2,3,\ldots,50\}\} \). If the number of elements \( (m,n) \) is \( S \) such that \( 6^{m}+9^{n} \) is a multiple of \( 5 \) is \( p \) and the number of elements \( (m,n) \) in \( S \) such that \( m+n \) is a square of a prime number is \( q \), then \( p+q \) is equal to _________.

\( 6 \displaystyle \int_{0}^{\pi} \left| (sin 3x + \sin 2x + \sin x) \right| dx \) is equal to __________.

 For some \( \alpha,\beta \in \mathbb{R} \), let   be such    that \( A^{2}-4A+2I=B^{2}-3B+I=O \). Then \( \left( \det\!\left( \operatorname{adj}\!\left(A^{3}-B^{3}\right) \right) \right)^{2} \) is equal to ________.

Let \( f:\mathbb{R}\to \mathbb{R} \) be a twice differentiable function such that the quadratic equation \( f(x)m^{2}-2f'(x)m+f''(x)=0 \) in \( m \) has two equal roots for every \( x\in\mathbb{R} \). If \( f(0)=1 \), \( f'(0)=2 \), and \( (\alpha,\beta] \) is the largest interval in which the function \( f\left(\log_{e}x - x\right) \) is increasing, then \( \alpha+\beta \) is equal to ________.

Let \( a_1 = 1 \) and for \( n \ge 1 \), \( a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \). Then \( \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \) is equal to _________.