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Let the circles \( C_1: |z| = r \) and \( C_2: |z - 3 - 4i| = 5, \ z \in \mathbb{C}, \) be such that \( C_2 \) lies within \( C_1 \). If \( z_1 \) moves on \( C_1, z_2 \) moves on \( C_2 \) and \( \min |z_1 - z_2| = 2 \), then \( \max |z_1 - z_2| \) is equal to:

If the system of equations \( x + 5y + 6z = 4 \), \( 2x + 3y + 4z = 7 \), \( x + 6y + az = b \) has infinitely many solutions, then the point \( (a,b) \) lies on the line:

Let \( a_1, a_2, a_3, \ldots \) be an A.P. and \( g_1 = a_1, g_2, g_3, \ldots \) be an increasing G.P. If \( a_1 = a_2 + g_2 = 1 \) and \( a_3 + g_3 = 4 \), then \( a_{10} + g_5 \) is equal to:

The sum \( \frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \ldots \) up to \( 8 \) terms, is:

If for then \( m \) equals:

Let \( p_n \) denote the total number of triangles formed by joining the vertices of an \( n \)-side regular polygon. If \( p_{n+1} - p_n = 66 \), then the sum of all distinct prime divisors of \( n \) is:

A man throws a fair coin repeatedly. He gets \( 10 \) points for each head he throws and \( 5 \) points for each tail he throws. If the probability that he gets exactly \( 30 \) points is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to:

The mean and variance of \( n \) observations are \( 8 \) and \( 16 \), respectively. If the sum of the first \( (n-1) \) observations is \( 48 \) and the sum of squares of the first \( (n-1) \) observations is \( 496 \), then the value of \( n \) is:

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines \( x + (k-1)y + 3 = 0 \) and \( 2x + k^2 y - 4 = 0 \). If the line \( x - y + 2 = 0 \) intersects the circle at the points \( A \) and \( B \), then \( (AB)^2 \) is equal to:

Let \( O \) be the origin, and \( P \) and \( Q \) be two points on the rectangular hyperbola \( xy = 12 \) such that the mid-point of the line segment \( PQ \) is \( \left( \frac{1}{2}, -\frac{1}{2} \right) \). Then the area of the triangle \( OPQ \) equals:

Let the parabola \( y = x^2 + px + q \) passing through the point \( (1, -1) \) be such that the distance between its vertex and the \( x \)-axis is minimum. Then the value of \( p^2 + q^2 \) is:

Let \( P=\{\theta\in[0,4\pi]:\tan^2\theta=1\} \) and \( S=\{a\in\mathbb{Z}: 2(\cos^8\theta-\sin^8\theta)\sec 2\theta=a^2,\; \theta\in P\} \). Then \( n(S) \) is:

Let the vectors \( \vec{a}=-\hat{i}+\hat{j}+3\hat{k} \) and \( \vec{b}=\hat{i}+3\hat{j}+\hat{k} \). For some \( \lambda,\mu\in \mathbb{R} \), let \( \vec{c}=\lambda \vec{a}+\mu \vec{b} \). If \( \vec{c}\cdot(3\hat{i}-6\hat{j}+2\hat{k})=10 \) and \( \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=-2 \), then \( |\vec{c}|^2 \) is equal to:

Let the point \( A \) be the foot of perpendicular drawn from the point \( P(a,b,0) \) on the line \( \dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-\alpha}{3} \). If the midpoint of the line segment \( PA \) is \( \left(0,\dfrac{3}{4},-\dfrac{1}{4}\right) \), then the value of \( a^2+b^2+\alpha^2 \) is equal to:

Two adjacent sides of a parallelogram \( PQRS \) are given by \( \overrightarrow{PQ}=\hat{j}+\hat{k} \) and \( \overrightarrow{PS}=\hat{i}-\hat{j} \). If the side \( PS \) is rotated about the point \( P \) by an acute angle \( \alpha \) in the plane of the parallelogram so that it becomes perpendicular to the side \( PQ \), then \( \sin^{2}\!\left(\dfrac{5\alpha}{2}\right)-\sin^{2}\!\left(\dfrac{\alpha}{2}\right) \) is equal to:

The value of \( \displaystyle \int_{0}^{20\pi}\left(\sin^{4}x+\cos^{4}x\right)dx \) is equal to:

Let \( f(x) \) be a polynomial of degree \( 5 \), and have extrema at \( x=1 \) and \( x=-1 \). If \( \displaystyle \lim_{x\to0}\left(\dfrac{f(x)}{x^{3}}\right)=-5 \), then \( f(2)-f(-2) \) is equal to:

Let \( f(x)=\displaystyle \int \left(\dfrac{16x+24}{x^{2}+2x-15}\right)dx \). If \( f(4)=14\log_{e}(3) \) and \( f(7)=\log_{e}\!\left(2^{\alpha}\cdot3^{\beta}\right),\,\alpha,\beta\in\mathbb{N} \), then \( \alpha+\beta \) is equal to:

Let \( x = x(y) \) be the solution of the differential equation \( 2y^2 \frac{dx}{dy} - 2xy + x^2 = 0, \; y > 1, x(e) = e \). Then \( x(e^2) \) is equal to:

Let \( A = \{2, 3, 4, 5, 6\} \). Let \( R \) be a relation on the set \( A \times A \) given by \( (x,y)R(z,w) \) if and only if \( x \) divides \( z \) and \( y \leq w \). Then the number of elements in \( R \) is ________.

Consider the matrices and . If matrices \( P \) and \( Q \) are such that \( PA = B \) and \( AQ = B \), then the absolute value of the sum of the diagonal elements of \( 2(P+Q) \) is ______.

Let \( A \) be the point \((3, 0)\) and circles with variable diameter \( AB \) touch the circle \( x^2 + y^2 = 36 \) internally. Let the curve \( C \) be the locus of the point \( B \). If the eccentricity of \( C \) is \( e \), then \( 72e^2 \) is equal to __________.

If the area of the region bounded by \( 16x^2 - 9y^2 = 144 \) and \( 8x - 2y = 24 \) is \( A \), then \( 3(A + 6\log_e(3)) \) is equal to ________.

The number of points in the interval \( [2,4] \), at which the function \( f(x)=\left[x^{2}-x-\frac{1}{2}\right] \), where \( [.] \) denotes the greatest integer function, is _________.