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If $y=\sqrt{\sin x+y}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

$\underset{x\to \infty }{lim}(\sqrt{(x^2+8x+3)}-\sqrt{(x^2+4x+3)})=$

If z = a + ib where a and b are less than zero. Conjugate of z lies on which quadrant in Argand plane.

1. 0

The domain and range of the function ${\text{cosec}}^{-1}\sqrt{{\log }_{\left(\frac{3-4\sec x}{1-2\sec x}\right)}2}$ are respectively

Prove by the principle of mathematical induction that $1\times 1!+2\times 2!+3\times 3!+...+n\times n!=(n+1)!-1$ for all natural numbers n.

For three events A, B and C, if P (exactly one of A or B occurs) = P(exactly one of B or C occurs) = P (exactly one of C or A occurs) $=\frac{1}{4}$ and P (all the three events occur simultaneosuly) $=\frac{1}{16},$ then the probability that atleast one of the events occurs is ?

Identify the function based on the description.

1.It is periodic with period $2\pi$.

2.Domain of the function is R and the range is [-1, 1]

3.F(x) decreases strictly from 1 to -1 as x increases from 0 to $\pi$. [For eg. If $x_2>x_1,F\left(x_1\right)>f\left(x_2\right),xϵ[0,\pi ]$

4.F(x) increases strictly from -1 to 1 as x increases from $\pi to2\pi$. (foreg. If $x_2>x_1,f\left(x_2\right)>f\left(x_1\right),xϵ[\pi ,2\pi ]$

From given functions, which of the following(s) is a point function

Out of all the patients in a hospital $89\%$ are found to be suffering from heart ailment and $98\%$ are suffering from lungs infection. If $K\%$ of them are suffering from both ailments, then $K$ can not belong to the set:

The area of the region described by $A=\left\{\right(x,y):x^2+y^2\le 1\text{and}{y}^2\le 1-x\}$ is :

The coefficient of the middle term in the binomial expansion of $(1+\alpha x{)}^4$ and $(1-\alpha x{)}^6$ is same if $\alpha$ equals .

Let $Z$ be the set of all integers.

$A=\left\{\right(x,y)\in Z\times Z:(x-2{)}^2+y^2\le 4\}$,

$B=\left\{\right(x,y)\in Z\times Z:x^2+y^2\le 4\}$ and

$C=\left\{\right(x,y)\in Z\times Z:(x-2{)}^2+(y^2-2{)}^2\le 4\}$

If the total number of relations from $A\cap B$ to $A\cap C$ is $2^p$, then the value of $p$ is

2 cos A = x + $\frac{1}{x}$ , 2cosB = y + $\frac{1}{y}$ . Find 2 cos(A-B)

The number of permutations that can be made out of the letters of the word “EQUATION” which start with a consonant and end with a consonant is

The sum of the series 1 + 2x + 3 $x^2$ + 4 $x^3$ + ..........upto n

terms is

The point equidistant from the points (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) is

If $a\in R\&b\in R$, then the equation $x^2-abx-a^2=0$ has ________.

If tan

$\frac{A}{2}$ =

$\frac{3}{2}$, then

$\frac{1+cosA}{1-cosA}$ =

The centre of regular polygon of $n$ sides is located at $z=0$ and one of its vertices is $z_1$. If $z_2$ is vertex adjacent to $z_1$, then $z_2=$

If the following functions have both domain and co-domain as $[-1,1]$, then select those which are not bijective?

If the solution set of $|x-k|<2$ is a subset of the solution set of the inequality $\frac{2x-1}{x+2}<1,$ then the number of possible integral value(s) of $k$ is

If $A$ = $\{x:x^2-5x+6$ = 0$\}$, B={2,4}, C={4,5}, then A$\times$(B$\cap$C) is

If p and q are simple statements, $p\iff \sim q$ is true when

Prove $(1+\frac{1}{1})(1+\frac{1}{2})(1+\frac{1}{3})\cdots (1+\frac{1}{n})=(n+1).$

If $\frac{(x+1{)}^2}{x^3+x}=\frac{A}{x}+\frac{Bx+C}{x^2+1}$, then $cose{c}^{-1}\left(\frac{1}{A}\right)+co{t}^{-1}\left(\frac{1}{B}\right)+se{c}^{-1}C=$

A = {1,2,3} and B = {a,b,c} . Which of the following is a function from A to B?

The centre and radius of the circle $x^2+y^2+2gx+2yf+c=0$ are

Find the value of $C_0{C}_3+C_1{C}_4+.....C_{n-3}{C}_n$, when $C_r={}^n{C}_r$

The total number of $4$ letter words that can be formed from the string $"AABBBBCC"$ is

If $u_1$ and $u_2$ are the units selected in two systems of unit of any physical quantity and $n_1$ and $n_2$ are their numerical values then

Find the number of discontinuities of the given function between x = 0 and x =2.

The value of ${\sin }^{2}A+{\sin }^2(A+B)-2\sin A\cos B\sin (A+B)$ when $B={45}^{\circ }$ is

Words are formed using all letters of the word 'JEEADVANCED'.

Let $a$ denotes the number of words in which all the vowels are together.

Let $b$ denotes the number of words in which vowels as well as consonants are separated.

Let $c$ denotes the number of words which begin and end with vowels.

In a exam there are $30$ true/false questions. If a student guesses all the $30$ questions, then the probability that he/she gets atleast $15$ correct, is

A body takes $T$ minutes to cool from ${62}^{\circ }C$ to ${61}^{\circ }C$ when the surrounding temperature is ${30}^{\circ }C$. The time taken by the body to cool form ${46}^{\circ }C$ to ${45.5}^{\circ }C$ is

Find the equations of the lines which cut-off intercepts on the axes whose sum and product are 1 and - 6 respectively.

Let $a,b\in R.$ If the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20,b,-a-9),$ then $|a+b|$ is equal to

Let the first term $a$ of an infinite G.P. is the value of $x$, where the function $f\left(x\right)=7+2x{\log }_{e}25-5^{x-1}-5^{2-x}$ has the greatest value and the common ratio $r$ is equal to $\underset{x\to 0}{lim}\underset{0}{\overset{x}{\int }}\frac{t^2}{x^2\tan (\pi +x)}dt$. Also, let $S$ be the sum of infinite terms of G.P.



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& a& \text{(P)}& 4\\ \text{(B)}& \frac{1}{r}& \text{(Q)}& 3\\ \text{(C)}& S& \text{(R)}& 2\\ \text{(D)}& a-rS& \text{(S)}& 1\\ & & \text{(T)}& 5\end{array}$



Which of the following is the only CORRECT combination?

Find the coefficient of $x^{17}$ in the expansion $(x+x^2+x^3+..............x^6{)}^6(1+x+x^2+x^3+.........)$

The equation $\overline{b}$z+$\overline{z}$b where b is a non-zero complex constant and c is real, represents

Let two distinct numbers $a$ and $b$ are selected from the set $\{1,2,3,\dots ,9,10\}.$ Then the probability that the last digit of the number $a^b$ will be $6,$ is

The locus of the midpoint of the portion between the axes of

$xcos\alpha +ysin\alpha =p,$ where p is a constant, is

If ${\sin }^{2}x+\sin x-1=0$, then the value of ${\cos }^{12}x+3{\cos }^{10}x+3{\cos }^{8}x+{\cos }^{6}x$ is

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3+2x^2+3x+1=0$. Find the equation whose roots are $\frac{1}{{\alpha }^3},\frac{1}{{\beta }^3},\frac{1}{{\gamma }^3}$

The median of the variables $x+4,x-\frac{7}{2},x-\frac{5}{2},x-3,x-2,x+\frac{1}{2}$$x-\frac{1}{2},x+5(x>0)$, is