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equals

A. − cot
(ex^x) + C

B. tan (xe^x)
+ C

C. tan (e^x)
+ C

D. cot (e^x)
+ C

Equation of hyperbola whose axes are parallel to coordinate axis with $e=\sqrt{2}$ and having distance between the foci as $1$ unit is :

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& M_r\text{is defined as}{M}_r=\left[\begin{array}{cc}(r-1)& \frac{1}{r}\\ 1& \frac{1}{(r-1{)}^2}\end{array}\right]\text{. Then}\underset{n\to \infty }{lim}{(det{M}_2+det{M}_3+\cdots +det{M}_n)}^{\ln n}\text{equals}& \text{(P)}& 16\\ \text{(B)}& \text{If}\left|\begin{array}{ccc}1& \cos \alpha & \cos \beta \\ \cos \alpha & 1& \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0& \cos \alpha & \cos \beta \\ \cos \alpha & 0& \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|,\text{then}{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma \text{equals}& \text{(Q)}& 4\\ \text{(C)}& \text{If}A=\left[\begin{array}{cc}3& 1\\ -1& 1\end{array}\right]\text{and a matrix}C\text{is defined as}C=\left(BA{B}^{-1}\right)\left(B^{-1}{A}^{\text{T}}B\right),\text{where}|B|\ne 0,& \text{(R)}& 2\\ & \text{then}detC\text{is a square of natural number equal to}& & \\ \text{(D)}& \text{If}A=\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]\text{and}{A}^4=-\lambda I,\text{then}\lambda \text{equals}& \text{(S)}& 1\end{array}$



Which of the following is a CORRECT combination?

Equation of curve which passes through point $(1,1)$ and satisfies the differential equation $3x{y}^{2}dy=(x^2+2y^3)dx$ is

A pair of tangents are drawn from origin to the circle $x^2+y^2-8x+12=0$. Another tangent is drawn to this circle such that this circle will become the incircle of the triangle which is formed by these three tangents. What will be the area of the triangle formed?

[Hint:
Put sin x = t]

In a $△ABC$, if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A\sin B\sin C=\frac{3+\sqrt{3}}{8}$, then



The value of $\tan A\tan B+\tan B\tan C+\tan C\tan A$ is

If in a $\Delta$ ABC, perimeter is 12 units, b = 3 units, c = 5 units then find $sin\left(\frac{A}{2}\right)$.

Let P(6, 3) be a point on hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$
If the normal at the point P intersect the x - axis at (9, 0), then the eccentricity of the hyperbola is

If $y_1\left(x\right)$ is a solution of the differential equation $\frac{dy}{dx}-f\left(x\right)y=0$, then a solution of the differential equation $\frac{dy}{dx}-f\left(x\right)y=r\left(x\right)$ is

A common tangent of the two circles is

Which of the following statements hold true for a quadrilateral ABCD?

The linear operation $L\left(x\right)$ is defined by the cross product $L\left(x\right)=v\times X$ , where $b=[010{]}^T$ and $X=[x_1{x}_2{x}_3{]}^T$ are three dimensional vectors. The $3\times 3$ matrix $M$ of this operation satisfies

$L\left(x\right)=M\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$

then the eigen values of $M$ are

In a $△ABC,AB=AC,P$ and $Q$ are points on $AC$ and $AB$ respectively such that $CB=BP=PQ=QA.$ If $\angle AQP=\theta ,$then ${\tan }^2\theta$ is a root of the equation

Let $a_1,a_2,a_3\cdots ,a_{21}$ be an $AP$ such that $\sum _{n=1}^{20}\frac{1}{a_n{a}_{n+1}}=\frac{4}{9}$ If the sum of this $AP$ is $189$, then $a_6{a}_{16}$ is equal to:

If the plane $3x-4y+5z=0$ is parallel to $\frac{2x-1}{2}=\frac{1-y}{3}=\frac{z-2}{a},$ then the value of $a$ is:

Tan ÷1-cot +cot ÷1-tan =1+sec cosec in this We can sin and cos instead of tan could u explain

If $2a+3b+6c=0,a,b,cϵR$, then the quadratic equation $a{x}^2+bx+c=0$ has

Without repetition of the numbers, four digit numbers are formed with the numbers 0, 2, 3, 5. The probability of such a number divisible by 5 is

Consider a parabola $y=\frac{x^2}{4}$ and the point $F(0,1)$. Let $A_1(x_1,y_1),A_2(x_2,y_2),A_3(x_3,y_3),...,A_k(x_k,y_k)$ are $\text{'}n\text{'}$ points on parabola such as $x_k>0$ and $\angle OF{A}_k=\frac{k\pi }{2n},(k=1,2,3,...,n)$. Then the value of $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^{n}F{A}_k$ is

If direction cosines of a vector of magnitude $3$ are $\frac{2}{3},\frac{-1}{3},\frac{2}{3}$ and $a>0$, then vector is

For $x\in (-\pi ,\pi ),tanx>0$ for

Shortest distance between the lines $\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{1}$ is equal to

${lim}_{x\to 0}\frac{a^{mx}-b^{nx}}{sinkx}$

The real number x when added to its inverse gives the minimum value of the sum at x equal to

A) -2. B) 2. C) 1. D) -1.

Explain in Detail.

Using elementary transformations, Find the inverse of matrix $\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ if it exists.

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 36x² + 4y² = 144

If $(x+iy{)}^{\frac{1}{3}}$ = a + ib, then $\frac{x}{a}$ + $\frac{y}{b}$ =

If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t) then find $\frac{dy}{dx}att=\frac{\pi }{4}$.