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If $f:R\to R$ is defined by $f\left(x\right)=\frac{sin\left[x\right]\pi +tan\left[x\right]\pi }{1+\left[x^2\right]}$, then the range of f(x) (where [x] denotes integral part of x)

The equation of the plane parallel to the lines $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (2\hat{i}+\hat{j}+4\hat{k})$ and $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and is passing through the point $(0,1,-1)$

If three points $A,B$ and $C$ lie on a line and $A\equiv (3,4),$ $B\equiv (7,7)$ and $AC=10$, then the coordinates of the point $C$ can be

If the lines $\frac{x}{1}=\frac{y}{2}=\frac{z}{3},\frac{x-1}{3}=\frac{y-2}{-1}=\frac{z-3}{4}$ and $\frac{x-a}{3}=\frac{y-1}{2}=\frac{z-2}{b}$ are concurrent, then the value of $b-2a$ is equal to

The number of common solution(s) of $y=\sin x$ and $y=(2x-\pi {)}^2$ is

The equation of the line parallel to the line joining $(4,2)$ and $(2,4)$ and whose $y$-intercept is $4$ units along positive $y$- axis is

Let $f\left(x\right)$ be an invertible function such that $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ for all $x\in R,$ then which of the following is/are correct ?
(where $x_1,x_2,\cdots ,x_n$ are different points)

In a $\Delta ABC$ of A $-ta{n}^{-1}$ and B $=ta{n}^{-1}3$ then C is equal to

If the function $f\left(x\right)=\{\begin{array}{cc}\frac{1-\cos 2px}{x^2},& x<0\\ q^2,& x=0\\ \frac{1-\sqrt{1-x}}{x},& x>0\end{array}$ is continuous at $x=0,$ then the value(s) of $2(p+q^2)$ can be

The average value of $\sin 2°,\sin 4°,\sin 6°,...,\sin 180°\mathrm{is}$

If the coefficients of $r^{th},(r+1{)}^{th}$ and $(r+2{)}^{th}$ terms in the binomial expansion of $(1+y{)}^m$are in $A.P.$, then $m$ and $r$ will satisfy the equation

The value of $x$ for $\sin x+\sqrt{3}\cos x\ge 1$ in the interval $-\pi <x\le \pi$ is

$Ify=e^{aco{s}^{-1}x},-1\le x\le 1showthat(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-a^{2}y=0$

The interval for $x$ which satisfies the equation $2{\tan }^{-1}2x={\sin }^{-1}\frac{4x}{1+4x^2}$ is

Solution set for the inequality $\frac{5}{4}{\sin }^{2}x+{\sin }^{2}x\cdot {\cos }^{2}x>\cos 2x$ is

What needs to be added to the sum of $\frac{5}{3}{x}^3-\frac{7}{4}{x}^2+\frac{11}{4}$ and $\frac{3}{4}{x}^3-\frac{5}{4}{x}^2+\frac{9}{4}$ to get $3x^3-\frac{8}{7}{x}^2+\frac{50}{9}$?

The equation of the circle passing through the foci of the ellipse
$\frac{x^2}{16}+\frac{y^2}{9}=1$ and having centre at (0,3) is

Find the principal and general solutions of the equation

The infinite series $f\left(x\right)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+....+\infty$ converges to

The term independent of $x$ in expansion of ${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$ is :

If $y=1+t^4$ and $x=3t^3+t$ then what is $\frac{dy}{dx}$

A pipe can fill a tank in 6 hours , due to a leak in the tank it gets filled in7 hours. When the tank is full how much time will it take to empty the tank?

The
general solution of the differential equation

A.

B.

C.

D.

The value of X such that b is the inverse of the matrix A, where

If $p^{th},q^{th}$ and $r^{th}$ terms of an A.P. and G.P. are both a, b and c respectively, show that $a^{b-c}{b}^{c-a}{c}^{a-b}=1$.

Determine $P\left(\frac{E}{F}\right)$
A coin is tossed three times
E: head on third toss F: head on first two tosses

E: Atleast two heads F : atmost two heads

E: Atmost two tails F : atleast one tail

If roots of the equation $x^3+3p{x}^2+3qx+r=0,p,q,r\ne 0$ are in H.P., then which of the following is correct?

The vertices of triangle $OBC$ are $O(0,0),B(-2,-5),C(-5,-2)$. The equation of the line parallel to $BC$, intersecting the sides $OB$ and $OC$ and whose perpendicular distance from the origin is $\frac{1}{4}$ is

Evalutate: ${\int }_2^4\frac{x}{x^2+1}dx$.

Let $\left[x\right]$ denote the greatest integer less than or equal to $x.$ If the domain of the function $\frac{1}{[x{]}^2-7[x]+12}$ is $R-[a,b),$ then the value of $a+b$ is

Numbers are selected at random, one at a time, form the two digit numbers 00,01,02, ........, 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find the probability that the event E occurs at least 3 times.

If $I={\int }_0^{1}cos\left(2Co{t}^{-1}\sqrt{\frac{1-x}{1+x}}\right)dxthen$

A coin is tossed and a dice is rolled. The probability that the coin shows the head and the dice shows 6 is

[MP PET 1994; Pb. CET 2001]