Let the foci of a hyperbola $$H$$ coincide with the foci of the ellipse $$E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$$ and the eccentricity of the hyperbola $$H$$ be the reciprocal of the eccentricity of the ellipse $$E$$. If the length of the transverse axis of $$H$$ is $$\alpha$$ and the length of its conjugate axis is $$\beta$$, then $$3 \alpha^2+2 \beta^2$$ is equal to

Let $$z$$ be a complex number such that the real part of $$\frac{z-2 i}{z+2 i}$$ is zero. Then, the maximum value of $$|z-(6+8 i)|$$ is equal to

Let the range of the function $$f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R}$$ be $$[a, b]$$. If $$\alpha$$ and $$\beta$$ ar respectively the A.M. and the G.M. of $$a$$ and $$b$$, then $$\frac{\alpha}{\beta}$$ is equal to

Let $$\int_\limits0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0$$. Then at $$x=2, y^{\prime \prime}+y+1$$ is equal to

Two vertices of a triangle $$\mathrm{ABC}$$ are $$\mathrm{A}(3,-1)$$ and $$\mathrm{B}(-2,3)$$, and its orthocentre is $$\mathrm{P}(1,1)$$. If the coordinates of the point $$\mathrm{C}$$ are $$(\alpha, \beta)$$ and the centre of the of the circle circumscribing the triangle $$\mathrm{PAB}$$ is $$(\mathrm{h}, \mathrm{k})$$, then the value of $$(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$$ equals

The integral $$\int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x$$ is equal to

Let $$\alpha, \beta ; \alpha>\beta$$, be the roots of the equation $$x^2-\sqrt{2} x-\sqrt{3}=0$$. Let $$\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N}$$. Then $$(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$$ is equal to

If $$\log _e y=3 \sin ^{-1} x$$, then $$(1-x^2) y^{\prime \prime}-x y^{\prime}$$ at $$x=\frac{1}{2}$$ is equal to

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $$i^{\text {th }}$$ roll than the number obtained in the $$(i-1)^{\text {th }}$$ roll, $$i=2,3$$, is equal to

$$\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$$ is equal to

The value of the integral $$\int_\limits{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$$ is

Between the following two statements:

Statement I : Let $$\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$$ and $$\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$$. Then the vector $$\vec{r}$$ satisfying $$\vec{a} \times \vec{r}=\vec{a} \times \vec{b}$$ and $$\vec{a} \cdot \vec{r}=0$$ is of magnitude $$\sqrt{10}$$.

Statement II : In a triangle $$A B C, \cos 2 A+\cos 2 B+\cos 2 C \geq-\frac{3}{2}$$.

If the variance of the frequency distribution

$$x$$	$$c$$	$$2c$$	$$3c$$	$$4c$$	$$5c$$	$$6c$$
$$f$$	2	1	1	1	1	1

is 160, then the value of $$c\in N$$ is

Let $$B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$$ and $$A$$ be a $$2 \times 2$$ matrix such that $$A B^{-1}=A^{-1}$$. If $$B C B^{-1}=A$$ and $$C^4+\alpha C^2+\beta I=O$$, then $$2 \beta-\alpha$$ is equal to

$$\lim _\limits{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$$ is equal to

Consider the line $$\mathrm{L}$$ passing through the points $$(1,2,3)$$ and $$(2,3,5)$$. The distance of the point $$\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$$ from the line $$\mathrm{L}$$ along the line $$\frac{3 x-11}{2}=\frac{3 y-11}{1}=\frac{3 z-19}{2}$$ is equal to

The area (in square units) of the region enclosed by the ellipse $$x^2+3 y^2=18$$ in the first quadrant below the line $$y=x$$ is

Let $$a, a r, a r^2$$, ............ be an infinite G.P. If $$\sum_\limits{n=0}^{\infty} a r^n=57$$ and $$\sum_\limits{n=0}^{\infty} a^3 r^{3 n}=9747$$, then $$a+18 r$$ is equal to

The sum of the coefficient of $$x^{2 / 3}$$ and $$x^{-2 / 5}$$ in the binomial expansion of $$\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9$$ is

Let $$\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}$$, where $$\alpha$$ and $$\beta$$ are integers and $$\alpha \beta=-6$$. Let the values of the ordered pair $$(\alpha, \beta)$$, for which the area of the parallelogram of diagonals $$\vec{a}+\vec{b}$$ and $$\vec{b}+\vec{c}$$ is $$\frac{\sqrt{21}}{2}$$, be $$\left(\alpha_1, \beta_1\right)$$ and $$\left(\alpha_2, \beta_2\right)$$. Then $$\alpha_1^2+\beta_1^2-\alpha_2 \beta_2$$ is equal to

Let $$A=\{(x, y): 2 x+3 y=23, x, y \in \mathbb{N}\}$$ and $$B=\{x:(x, y) \in A\}$$. Then the number of one-one functions from $$A$$ to $$B$$ is equal to _________.

Let $$A, B$$ and $$C$$ be three points on the parabola $$y^2=6 x$$ and let the line segment $$A B$$ meet the line $$L$$ through $$C$$ parallel to the $$x$$-axis at the point $$D$$. Let $$M$$ and $$N$$ respectively be the feet of the perpendiculars from $$A$$ and $$B$$ on $$L$$. Then $$\left(\frac{A M \cdot B N}{C D}\right)^2$$ is equal to __________.

Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\left[\begin{array}{l}x \\ y\end{array}\right]$$. Let the set of all $$m$$, for which the system of equations $$A X=B$$ has a negative solution (i.e., $$x<0$$ and $$y<0$$), be the interval $$(a, b)$$. Then $$8 \int_\limits a^b|A| d m$$ is equal to _________.

For a differentiable function $$f: \mathbb{R} \rightarrow \mathbb{R}$$, suppose $$f^{\prime}(x)=3 f(x)+\alpha$$, where $$\alpha \in \mathbb{R}, f(0)=1$$ and $$\lim _\limits{x \rightarrow-\infty} f(x)=7$$. Then $$9 f\left(-\log _e 3\right)$$ is equal to _________.

Consider the circle $$C: x^2+y^2=4$$ and the parabola $$P: y^2=8 x$$. If the set of all values of $$\alpha$$, for which three chords of the circle $$C$$ on three distinct lines passing through the point $$(\alpha, 0)$$ are bisected by the parabola $$P$$ is the interval $$(p, q)$$, then $$(2 q-p)^2$$ is equal to __________.

The square of the distance of the image of the point $$(6,1,5)$$ in the line $$\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$$, from the origin is __________.

Let the set of all values of $$p$$, for which $$f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$$ does not have any critical point, be the interval $$(a, b)$$. Then $$16 a b$$ is equal to _________.

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is __________.

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $$2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5}$$, is __________.

If $$\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots . .+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots \ldots+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}$$, then $$\alpha$$ is equal to ___________.