If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and $${{37} \over 4}$$

respectively, then (a $$-$$ b)² is equal to :

Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$. Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$ is equal to :

The value of the definite integral

$$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $$ is equal to :

Let C be the set of all complex numbers. Let

$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} $$

$${S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} $$ and

$${S_3} = \{ z \in C||z - \overline z | \ge 8\} $$.

Then the number of elements in $${S_1} \cap {S_2} \cap {S_3}$$ is equal to :

If the area of the bounded region
$$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$$ is ,
$$\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $$, then the value of $${(\alpha + \beta - 2\lambda )^2}$$ is equal to :

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity $${1 \over 3}$$ and the distance of the nearer focus from this directrix is $${8 \over {\sqrt {53} }}$$, then the equation of the other directrix can be :

If the coefficients of x⁷ in $${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$$ and x^$$-$$7 in $${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$$, b $$\ne$$ 0, are equal, then the value of b is equal to :

If $$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equal to :

Let $$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$$. If A^$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to :

Let $$f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$$ be defined as $$f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr b & , & {x = 0} \cr {{e^{\cot 4x/\cot 2x}}} & , & {0 < x < {\pi \over 4}} \cr } } \right.$$

If f is continuous at x = 0, then the value of 6a + b² is equal to :

Let y = y(x) be solution of the differential equation

$${\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y$$, with y(0) = 0.

If $$y\left( { - {2 \over 3}{{\log }_e}2} \right) = \alpha {\log _e}2$$, then the value of $$\alpha$$ is equal to :

Let f : R $$\to$$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $$\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}$$ is equal to :

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to :

Let $$\alpha$$, $$\beta$$ be two roots of the

equation x² + (20)^1/4x + (5)^1/2 = 0. Then $$\alpha$$⁸ + $$\beta$$⁸ is equal to

The probability that a randomly selected 2-digit number belongs to the set {n $$\in$$ N : (2ⁿ $$-$$ 2) is a multiple of 3} is equal to :

Let $$A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\} $$, $$B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\} $$ and $$C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\} $$.

Then the minimum value of |r| such that $$A \cup B \subseteq C$$ is equal to

For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations :

x + y $$-$$ z = 2, x + 2y + $$\alpha$$z = 1, 2x $$-$$ y + z = $$\beta$$. If the system has infinite solutions, then $$\alpha$$ + $$\beta$$ is equal to ______________.

Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b $$ and $$\overrightarrow c = \widehat j - \widehat k$$ be three vectors such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$ and $$\overrightarrow a \,.\,\overrightarrow b = 1$$. If the length of projection vector of the vector $$\overrightarrow b $$ on the vector $$\overrightarrow a \times \overrightarrow c $$ is l, then the value of 3l² is equal to _____________.

If $${\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)$$ are in an arithmetic progression, then the value of x is equal to _____________.

Let the domain of the function

$$f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \right)$$ be (a, b). Then the value of the integral $$\int\limits_a^b {{{{{\sin }^3}x} \over {({{\sin }^3}x + {{\sin }^3}(a + b - x)}}} dx$$ is equal to _____________.

Let $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|,x \in [0,\pi ]$$. Then the maximum value of f(x) is equal to ______________.

Let $$F:[3,5] \to R$$ be a twice differentiable function on (3, 5) such that

$$F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt} $$. If $$F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}$$, then $$\alpha$$ + $$\beta$$ is equal to _______________.

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $$\to$$ S
such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m . n $$\in$$ S is equal to _____________.

If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0$$, with y(0) = 0, then $$5y'\left( {{\pi \over 2}} \right)$$ is equal to ______________.

Let $$f:[0,3] \to R$$ be defined by $$f(x) = \min \{ x - [x],1 + [x] - x\} $$ where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x $$\in$$ [0, 3] where f i discontinuous, and Q denote the set containing all x $$\in$$ (0, 3) where f is not differentiable. Then the sum of number of elements in P and Q is equal to ______________.