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Length of the line segment joining the points -1-i and 2+3i is

Let $f\left(x\right)$ be a continuous and differentiable function for all $x\in (0,\frac{\pi }{2})$, such that $\left(f\right(x){)}^2={\int }_0^{x}f\left(t\right)\cdot \frac{2{\sec }^{2}t}{4+\tan t}dt,f\left(x\right)\ne 0,f\left(\frac{\pi }{3}\right)=\ln (4+\sqrt{3}).$ Then which of the following is/are correct

The value of $\left|\begin{array}{ccc}{a}^2+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right|,a\ne 1$ is

20 cards are numbered from 1 to 20. One card is drawn at random. What is the probability that the number on the cards is:
(i) a multiple of 4?

(ii) not a multiple of 4?

(iii) odd?

(iv) greater than 12?

(v) divisible by 5?

(vi) not a multiple of 6?

Find the value of sin$\left(2ta{n}^{-1}\frac{1}{3}\right)+cos\left(ta{n}^{-1}2\sqrt{2}\right).$

If $\theta \in (0,2\pi )$ and $2\cos \theta =\sqrt{3}\cos {10}^{\circ }-\sin {10}^{\circ }$, then $\theta$ can be

The equation of the locus of the point of intersection of two normals to the parabola $y^2=4ax$ which are perpendicular to each other is

In the table, trigonometric ratios and the intervals of $\theta$ are given. Match the intervals with the ratios which are positive in those intervals.

Form point $P(8,27)$, tangent $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$.If the angle subtended by $QR$ at origin is $\varphi$, then $\tan \varphi =$

A bag contains 7 black and 4 white balls two balls are drawn at a time from the bag. The probability at least one white ball is selected is

A car has a velocity $\overrightarrow{v}=80km/hr$ towards east. Another car on the road has a velocity = . $\overrightarrow{-v}$The speed and the direction of vecthe second car is

a) 80 km/hr towards south

b) 80 km/hr towards west

c) Direction cannot be determined from the given information

d) Speed of the second car cannot be determined from the given information.

Evaluate the Given limit:

The value of $a$ and $b$ respectively so that the function

$f\left(x\right)=\{\begin{array}{cc}x+a\sqrt{2}\sin x,& 0\le x<\frac{\pi }{4}\\ 2x\cot x+b,& \frac{\pi }{4}\le x\le \frac{\pi }{2}\\ a\cos 2x-b\sin x,& \frac{\pi }{2}<x\le \pi \end{array}$

is continuous for $x\in [0,\pi ]$ is

Let $R=\left\{\right(x,y):x,yϵZ,y=2x-4\}$. If (a, -2) and $4,b^2)ϵR,$ then write the values of a and b.

Let $I=\int \frac{{\sin }^{2}x+\sin x}{1+\sin x+\cos x}dx$, $J=\int \frac{{\cos }^{2}x+\cos x}{1+\sin x+\cos x}dx$ and $c$ is the constant of integration.



$$\begin{array}{cc}\text{Function}& \text{Integral}\\ \begin{array}{c}\text{(a) I}\end{array}& \text{(p)}\frac{1}{2}(x-\sin x-\cos x)+c\\ \begin{array}{c}\text{(b) J}\end{array}& \text{(q)}\frac{1}{2}(x+\sin x+\cos x)+c\\ \begin{array}{c}\text{(c) I + J}\end{array}& \text{(r)}x+c\\ \begin{array}{c}\text{(d) I - J}\end{array}& \text{(s)}c-\cos x-\sin x\\ \begin{array}{}\end{array}& \text{(t)}c+\cos x+\sin x\\ \begin{array}{}\end{array}& \text{(u)}-\frac{1}{2}(x+\sin x+\cos x+c)\end{array}$$



Which of the following is the only CORRECT combination?

Let $y=y\left(x\right)$ be the solution of the differential equation $e^x\sqrt{1-y^2}dx+\left(\frac{y}{x}\right)dy=0,$ $y\left(1\right)=-1.$ Then the value of $\left(y\right(3){)}^2$ is equal to

If $\alpha$ is the inclination of a tangent to the parabola $y^2=4ax$ then the distance between the tangent and a parallel normal is

If $f\left(x\right)=1+\frac{1}{x}\underset{1}{\overset{x}{\int }}f\left(t\right)dt$, then the value of $f\left(e^{-1}\right)$ is

If a,b,c $ϵ$ R and the equations a$x^2+bx+c=0and{x}^3+3x^2+3x+2=0$ have two roots in common, then

The value of $\underset{-1}{\overset{1}{\int }}{x}^2{e}^{\left[x^3\right]}dx,$ where $\left[t\right]$ denotes the greatest integer $\le t$ ,is :

If $A$ and $B$ are two independent events such that $P\left(A\right)>0.5,$ $P\left(B\right)>0.5,$ $P(A\cap \overline{B})=\frac{3}{25},$ $P(\overline{A}\cap B)=\frac{8}{25},$ then the value of $P(A\cap B)$ is

An ellipse, with foci at $(0,2)$ and $(0,–2)$ and minor axis of length $4$ units, passes through which of the following points?

If $lo{g}_{10}x=ythenlo{g}_{1000}{x}^2$ is equal to

The product of the real roots of the equation $(x-1{)}^4+(x-5{)}^4=82$ is

Evaluate $\int (1+\ln x)dx$

(where $C$ is constant of integration)

If $\frac{1}{lo{g}_3\pi }+\frac{1}{lo{g}_4\pi }>x$, then x be

If $3\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ x+w& 3\end{array}\right]$, find the values of x, y, z and w.

If from the vertex of the parabola $y^2=4ax$ pair of chords be drawn perpendicular to each other and with these chords as adjacent sides a rectangle is completed then the locus of the vertex of the farther angle of the rectangle is the parabola

Determine if f(x) defined by f(x) =$\{\begin{array}{c}{x}^{2}sin\frac{1}{x},ifx\ne 0\\ o,ifx=0\end{array}$ is a continuous function ?

If tan A = - $\frac{1}{2}$ and tan B = - $\frac{1}{3}$, then A + B =

[IIT 1967; MNR 1987; MP PET 1989]