The shortest distance between the line x $$-$$ y = 1 and the curve x² = 2y is :

A function f(x) is given by $$f(x) = {{{5^x}} \over {{5^x} + 5}}$$, then the sum of the series $$f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} \right)$$ is equal to :

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $$\times$$ B. Then :

The integral $$\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx$$, x > 0, is equal to : (where c is a constant of integration)

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :

Let $$\alpha$$ and $$\beta$$ be the roots of x² $$-$$ 6x $$-$$ 2 = 0. If a_n = $$\alpha$$ⁿ $$-$$ $$\beta$$ⁿ for n $$ \ge $$ 1, then the value of $${{{a_{10}} - 2{a_8}} \over {3{a_9}}}$$ is :

If 0 < x, y < $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to :

cosec$$\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]$$ is equal to :

Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let R_i denote the i^th row of A. If a matrix B is obtained by performing the operation R₂ $$ \to $$ 2R₂ + 5R₃ on 2A, then det(B) is equal to :

If $$\alpha$$, $$\beta$$ $$\in$$ R are such that 1 $$-$$ 2i (here i² = $$-$$1) is a root of z² + $$\alpha$$z + $$\beta$$ = 0, then ($$\alpha$$ $$-$$ $$\beta$$) is equal to :

The minimum value of $$f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$$, where a, $$x \in R$$ and a > 0, is equal to :

If $${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx} $$, then :

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

If for the matrix, $$A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]$$, $$A{A^T} = {I_2}$$, then the value of $${\alpha ^4} + {\beta ^4}$$ is :

If the curve x² + 2y² = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

The following system of linear equations

2x + 3y + 2z = 9

3x + 2y + 2z = 9

x $$-$$ y + 4z = 8

A hyperbola passes through the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :

If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)²⁰²² is divided by 8 is __________.

The total number of two digit numbers 'n', such that 3ⁿ + 7ⁿ is a multiple of 10, is __________.

The value of $$\int\limits_{ - 2}^2 {|3{x^2} - 3x - 6|dx} $$ is ___________.

A line 'l' passing through origin is perpendicular to the lines

$${l_1}:\overrightarrow r = (3 + t)\widehat i + ( - 1 + 2t)\widehat j + (4 + 2t)\widehat k$$

$${l_2}:\overrightarrow r = (3 + 2s)\widehat i + (3 + 2s)\widehat j + (2 + s)\widehat k$$

If the co-ordinates of the point in the first octant on 'l₂‘ at a distance of $$\sqrt {17} $$ from the point of intersection of 'l' and 'l₁' are (a, b, c) then 18(a + b + c) is equal to ___________.

A function f is defined on [$$-$$3, 3] as

$$f(x) = \left\{ {\matrix{ {\min \{ |x|,2 - {x^2}\} ,} & { - 2 \le x \le 2} \cr {[|x|],} & {2 < |x| \le 3} \cr } } \right.$$ where [x] denotes the greatest integer $$ \le $$ x. The number of points, where f is not differentiable in ($$-$$3, 3) is ___________.

Let $$\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$$ and $$\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ is $$8\sqrt 3 $$ square units, then $$\overrightarrow a $$ . $$\overrightarrow b $$ is equal to __________.

If $$\mathop {\lim }\limits_{x \to 0} {{ax - ({e^{4x}} - 1)} \over {ax({e^{4x}} - 1)}}$$ exists and is equal to b, then the value of a $$-$$ 2b is __________.

If the curve, y = y(x) represented by the solution of the differential equation (2xy² $$-$$ y)dx + xdy = 0, passes through the intersection of the lines, 2x $$-$$ 3y = 1 and 3x + 2y = 8, then |y(1)| is equal to _________.