The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of its circumcentre is :

The foot of perpendicular of the point (2, 0, 5) on the line $${{x + 1} \over 2} = {{y - 1} \over 5} = {{z + 1} \over { - 1}}$$ is ($$\alpha,\beta,\gamma$$). Then, which of the following is NOT correct?

The number of functions

$$f:\{ 1,2,3,4\} \to \{ a \in Z|a| \le 8\} $$

satisfying $$f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\} $$ is

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $$N-2,\sqrt{3N},N+2$$ are in geometric progression be $$\frac{k}{48}$$. Then the value of k is :

If the function $$f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$$

is continuous at $$x = {\pi \over 2}$$, then $$9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$$ is equal to

Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) A$$^{13}$$ B$$^{26}$$ $$-$$ B$$^{26}$$ A$$^{13}$$ is symmetric

(S2) A$$^{26}$$ C$$^{13}$$ $$-$$ C$$^{13}$$ A$$^{26}$$ is symmetric

Then,

The shortest distance between the lines $$x+1=2y=-12z$$ and $$x=y+2=6z-6$$ is :

Let T and C respectively be the transverse and conjugate axes of the hyperbola $$16{x^2} - {y^2} + 64x + 4y + 44 = 0$$. Then the area of the region above the parabola $${x^2} = y + 4$$, below the transverse axis T and on the right of the conjugate axis C is :

Let the function $$f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6$$ have a maxima for some value of $$x < 0$$ and a minima for some value of $$x > 0$$. Then, the set of all values of p is

Let $$z$$ be a complex number such that $$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then $$z$$ lies on the circle of radius 2 and centre :

The integral $$16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}} $$ is equal to

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\} $$, for some $$m$$, such that the range of $$f$$ is [0, 2]. Then the value of $$m$$ is _________

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

Let $$y=y(t)$$ be a solution of the differential equation $${{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}$$ where, $$\alpha > 0,\beta > 0$$ and $$\gamma > 0$$. Then $$\mathop {\lim }\limits_{t \to \infty } y(t)$$

Let $$A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - i} \cr 0 & 1 \cr } } \right]$$, where $$i = \sqrt { - 1} $$. If $$\mathrm{M=A^T B A}$$, then the inverse of the matrix $$\mathrm{AM^{2023}A^T}$$ is

$$\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}} $$ is equal to :

Let $$f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N$$, and $$f(4) = 133,f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) - f(2))$$ is

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $$\frac{k}{10}%$$. Then the value of k is __________.

A triangle is formed by X-axis, Y-axis and the line $$3x+4y=60$$. Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____________.

The remainder when (2023)$$^{2023}$$ is divided by 35 is __________.

If $$\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)} $$, where m and n are coprime natural numbers, then $${m^2} + {n^2} - 5$$ is equal to _____________.

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is ____________

For the two positive numbers $$a,b,$$ if $$a,b$$ and $$\frac{1}{18}$$ are in a geometric progression, while $$\frac{1}{a},10$$ and $$\frac{1}{b}$$ are in an arithmetic progression, then $$16a+12b$$ is equal to _________.

If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line $${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}$$ is $$\alpha$$, then 28$$\alpha^2$$ is equal to ____________.

Let $$\alpha \in\mathbb{R}$$ and let $$\alpha,\beta$$ be the roots of the equation $${x^2} + {60^{{1 \over 4}}}x + a = 0$$. If $${\alpha ^4} + {\beta ^4} = - 30$$, then the product of all possible values of $$a$$ is ____________.