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The equation of a tangent to the hyperbola $4x^2-5y^2=20$ parallel to the line $x-y=2$ is :

What is the equation of the normal to the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ at the point $(5\sqrt{3},2\sqrt{2})$

The value of $\underset{0}{\overset{2\pi }{\int }}|\sin x|dx$ is

Which
of the following pairs of sets are disjoint

(i) {1,
2, 3, 4} and {x:
x is
a natural number and 4 ≤
x
≤
6}

(ii) {a,
e,
i,
o,
u}and
{c,
d,
e,
f}

(iii) {x:
x
is an even integer} and {x:
x
is an odd integer}

The scalar
product of the vectorwith
a unit vector along the sum of vectors
and
is
equal to one. Find the value of.

If three coprime numbers $x,y,z$ are such that the product of $x$ and $y$ is $551$ and $y$ and $z$ is $1073,$ then $x+y+z$ can be

The value of $\int {\tan }^{4}xdx$ is

(where $C$ is constant of integration)

A number N is stored in a 4-bit 2's complement representation as

If a, b, c are in A.P. and x. y, z are in G.P., then the value of $x^{b-c}{y}^{c-a}{z}^{a-b}$ is

The value of $\sin {765}^{\circ }+\text{cosec}(-{1110}^{\circ })-\sin {405}^{\circ }+\cot {585}^{\circ }$ is equal to

If $p,p^{\prime }$ denote the lengths of the perpendiculars from the focus and the centre of an ellipse whose semi major axis is of length $a$ units on a tangent at a point on the ellipse and $r$ denotes the focal distance of the point, then

If the area of the pentagon formed by the vertices $A(1,3),B(-2,5),C(-3,-1),D(0,-2)$ and $E=(2,t)$ is $\frac{45}{2}\text{sq. units}$, then possible value(s) of $t$ is/are

A function $f$ is continuous and differentiable for all $x>0,$ such that $f^2\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)\frac{\cos t}{2+\sin t}dt$ and $f\left(x\right)\ne 0,f\left(\pi \right)=\ln 2,$ then $f\left(x\right)$ is

The value of $\int \frac{dx}{5+4\cos x}$ is

(where $C$ is integration constant)

If $x,y,z$ are in G.P. and $x+y,y+z,z+x$ are in A.P., where $x\ne y\ne z$, then common ratio of the G.P. is

If the four points with position vectors $-2\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{j}+\hat{k},\hat{j}-\hat{k}$ and $\lambda \hat{j}+\hat{k}$ are coplanar, then $\lambda =$

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation,

$x^2+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

The maximum value of the expression $y=2(a-x)(x+\sqrt{x^2+b^2})$ is

If $|2x-3|+|x-1|=|x-2|$, then $x\in$

If $f(x-y),f\left(x\right)f\left(y\right)$ and $f(x+y)$ are in A.P. for all $x,y\in R$ and $f\left(0\right)\ne 0$, then

If in a parallelogram $ABDC$, the coordinates of $A,B$ and $C$ are respectively $(1,2),(3,4)$ and $(2,5)$, then the equation of the diagonal $AD$ is :

$cos\frac{\pi }{15}cos\frac{2\pi }{15}cos\frac{3\pi }{15}cos\frac{4\pi }{15}cos\frac{5\pi }{15}cos\frac{6\pi }{15}cos\frac{7\pi }{15}=\frac{1}{128}$

Let $|A|=|a_{ij}{|}_{3\times 3}\ne 0$. Each element $a_{ij}$ multiplied by $k^{i-j}$. Let |B| be the resulting determinant, where $k_1|A|+k_2|B|=0.Then{k}_1+k_2=$

Find
the middle terms in the expansions of

Which of the following function describe the graph shown in the below figure?

$\text{The latus-rectum of the hyperbola}$

$16x^2-9y^2=144is$

Which of the following statements are correct regarding the function f(x) = $\sqrt{x}$

If given $g\left(x\right)={\int }_2^{x^2}\frac{1}{ln(1+t^2)}dtthenfind{g}^{\prime }\left(\sqrt{2}\right)$ .

If $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=3,$ then which of the following is $\text{TRUE}?$

If $I_n=\underset{0}{\overset{\pi /4}{\int }}{\tan }^n\theta d\theta$, then for any positive integer $n$, the value of $I_{n-1}+I_{n+1}$ is

In a four dimensional space, where unit vectors along the axes are $\hat{i},\hat{j},\hat{k}$ and $\hat{l},$$\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3},\overrightarrow{a_4}$ are four non-zero vectors such that no vector can be expressed as a linear combination of others and $(\lambda -1)({\overrightarrow{a}}_1-{\overrightarrow{a}}_2)+\alpha ({\overrightarrow{a}}_2+{\overrightarrow{a}}_3)+\gamma ({\overrightarrow{a}}_3+{\overrightarrow{a}}_4-2{\overrightarrow{a}}_2)+{\overrightarrow{a}}_3+\delta {\overrightarrow{a}}_4=\overrightarrow{0}.$

Then which of the following is/are correct?