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$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& arg\left(\frac{z+1}{z-1}\right)=\frac{\pi }{4}& \text{(p)}& \text{Parabola}\\ \text{(b)}& z=\frac{3i-1}{2+it}\left(tϵR\right)& \text{(q)}& \text{Part of a circle}\\ \text{(c)}& argz=\frac{\pi }{4}& \text{(r)}& \text{Full circle}\\ \text{(d)}& z=t+i{t}^2\left(tϵR\right)& \text{(s)}& \text{Line}\end{array}$

Which of the following is correct?

The number of points at which the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and the circle $x^2+y^2-8x+15=0$ intersect is

The maximum value of ${\left(\frac{1}{x}\right)}^x,(x>0)$ is

If Rolle's theorem holds for the function $f\left(x\right)=\frac{2}{e^x+e^{-x}},x\in [-1,1]$ at the point $x=c$, then the value of $c$ is:

If $\tan A=\frac{1}{3}$ and $\sec B=\frac{13}{5}$ where $\pi <A<\frac{3\pi }{2}$, $\frac{3\pi }{2}<B<2\pi$, then $\cot (A+B)$ is equal to

Is the function defined by

a continuous function?

Calculate the time taken by the light to pass through a nucleus of diameter$1.3\times {10}^{-15}m$, if the speed of light is $3\times {10}^{8}m/s$.

Let F(x) = f(x) + $f\left(\frac{1}{x}\right)$, where f(x) = ${\int }_1^x\frac{logt}{1+t}dt$ . Then F(e) =

Differentiate $e^{ax}cos(bx+c).$

Ortho centre of the triangle formed by the vertices $(\sqrt{13},\sqrt{5})$, $(\sqrt{7},-\sqrt{11})$ and $(-\sqrt{18},0)$ is

The equation of the tangent to the parabola $y^2=16x$ inclined at an angle of ${60}^{\circ }$ to the positive $x-$axis is

The value of the integral $\underset{0}{\overset{\infty }{\int }}\frac{x\log x}{(1+x^2{)}^2}dx$ is

The distance between the line $\overline{r}=2\hat{i}-2\hat{j}-3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})$
and the plane $\overline{r}.(\hat{i}+5\hat{j}+\hat{k})=5$

Sir I don't understand calculus (differentiation) 😖😖😖please help me

If the normal to the curve y = f(x) at x = 0 be given by the equation 3x – y + 3 = 0, then the value of $\underset{x\to 0}{lim}{\left\{f\right(x^2)-5f(4x^2)+4f(7x^2\left)\right\}}^{-1}$ is

$\frac{c}{a+b}=\frac{1-tan\left(\frac{A}{2}\right)tan\left(\frac{B}{2}\right)}{1+tan\left(\frac{A}{2}\right)tan\left(\frac{B}{2}\right)}$

The value of $\lambda$ for which the equation$2x^2+7xy+3y^2+8x+14y+\lambda =0$ represents a pair of straight lines is:

The value of $\underset{-2021\pi }{\overset{2021\pi }{\int }}\left\{x^{2021}(1+\tan (\frac{\pi }{3}-x))(1+\tan (x-\frac{\pi }{12}))\right\}dx$ is

In a class of $120$ students numbered from $1$ to $120$, all even numbered students opt for Physics, those whose numbers are divisible by $5$, opt for Chemistry and those whose numbers are divisible by $7$, opt for Maths. How many opt for none of the three subjects?

If a rectangular hyperbola of latus rectum $4$ units passing through $(0,0)$ have $(2,0)$ as its one focus, then equation of locus of the other focus is

1. 66.3

A bird is sitting on the top of a vertical pole $20m$ high and its elevation from a point $O$ on the ground is ${45}^{\circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to ${30}^{\circ }$. Then the speed (in $m/s$) of the bird is

The solution(s) of the equation $9{\cos }^{12}x+{\cos }^{2}2x+1=6{\cos }^{6}x\cos 2x+6{\cos }^{6}x-2\cos 2x$ is/are $(n\in I)$.

Find the inverse of the given matrix
$\left[\begin{array}{cc}2& -2\\ 4& 3\end{array}\right]$

If $cosecA+secA=cosecB+secB,provethat:tanAtanB=cot\frac{A+B}{2}$

$c(acosB-bcosA)=a^2-b^2$

If $\frac{x^2}{36}-\frac{y^2}{k^2}=1$ is a hyperbola then which of the following statement can be true?

Locus of the centre of the circle which always passes through the fixed point $(a,0)$ and $(-a,0)$, where $a\ne 0$, is

Let $f:R\to R$ be a continuous function. Then $\underset{x\to \frac{\pi }{4}}{lim}\frac{\frac{\pi }{4}\underset{2}{\overset{{\sec }^{2}x}{\int }}f\left(x\right)dx}{x^2-\frac{{\pi }^2}{16}}$ is equal to