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If the tangent at point $(1,1)$ on $y^2=x(2-x{)}^2$ meets the curve again at point $P,$ then $P$ is

If the sum of the coefficients in the expansion of $(a+b{)}^n$ is 4096, then the greatest coefficient in the expansion is

If $\int \frac{dx}{x+x^7}=p\left(x\right),$ then $\int \frac{x^6}{x+x^7}dx$ is equal to

(where $C$ is constant of integration)

A cup of coffee cools from ${90}^{\circ }\text{C}$ to ${80}^{\circ }\text{C}$ in $t$ minutes, when the room temperature is ${20}^{\circ }\text{C}$. The time taken by a similar cup of coffee to cool from ${80}^{\circ }\text{C}$ to ${60}^{\circ }\text{C}$ at a room temperature same at ${20}^{\circ }\text{C}$ is -

Let a function $f$ is a strictly increasing and $f^{\prime \prime }\left(x\right)<0$, also $a,b$ and $c$ are three distinct real numbers in the domain of inverse of $f\left(x\right).$ If $A=\frac{f^{-1}\left(a\right)+f^{-1}\left(b\right)+f^{-1}\left(c\right)}{3}$ and $B=f^{-1}\left(\frac{a+b+c}{3}\right),$ then which of the following is correct

From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that $\angle QPR$ is a right angle, then the possible value(s) of
$\lambda$ is (are)

In how many ways can a club consisting of $20$ people choose a president, a secretary, and a treasurer?

If $-4\le |x|\le 2$, then $x$ belongs to

Let $f$ be a function defined by $f\left(x\right)=\frac{x-5}{x-3};x\ne 3$, $f^k\left(x\right)$ denote the composition of $f$ with itself taken $k$ times i.e $f^3\left(x\right)=f\left(f\right(f\left(x\right)\left)\right)$

Then

If $\underset{x\to \infty }{lim}(\sqrt{x^2-x+1}-ax-b)=0,$ then for $k\ge 2,$ $\underset{n\to \infty }{lim}{\sec }^{2n}(k!\pi b)$ is equal to

Equation that represents the given graph is

Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum is of length 12

The rank of the word $SUCCESS$, if all possible permutations of the word $SUCCESS$ are arranged in dictionary order is

Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a facter of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then value of $P\left(1\right)$ is -

Probability that A speaks truth is 4/5, A coin is tossed, A reports that a head appears. the probability that actually there was head is

(a) $\frac{4}{5}$

(b)$\frac{1}{2}$

(c)$\frac{1}{5}$

(d) $\frac{2}{5}$

The maximum value of $sin(x+\frac{\pi }{6})+cos(x+\frac{\pi }{6})$ in the

interval $(0,\frac{\pi }{2})$ is attained at

If the extremities of the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $(\alpha ,\beta )$, then the distance between the point $P(1,1)$ and $(\alpha ,\beta )$, when $\alpha >0$ is/are

Let ${(1-x+x^4)}^{10}=a_0+a_{1}x+a_2{x}^2+.....+a_{40}{x}^{40}$, then the correct option(s) is/are

For given binary operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative.
(ii)On Q, define $a\ast b=ab+1$

If $co{t}^{-1}(4+\frac{2}{4})+co{t}^{-1}(4+\frac{6}{4})+co{t}^{-1}(4+\frac{12}{4})+.....\infty =ta{n}^{-1}\left(\frac{a}{b}\right),$ where a and b are relatively prime, then

State whether the following statement are true or false. Justify
(ii) If $\ast$ is a commutative binary operation on N, then $a\ast (b\ast c)=(c\ast b)\ast a.$

If α,β are the roots of the quadratic equation $x^2$ – (a – 2)x –(a+1) =0, where 'a' is a variable then the least value of ${\alpha }^2$+${\beta }^2$

What can be the maximum number of points of a team after the second round?

The term independent of $x(x>0,x\ne 1)$ in the expansion of ${[\frac{(x+1)}{(x^{2/3}-x^{1/3}+1)}-\frac{(x-1)}{(x-\sqrt{x})}]}^{10}is$

Let $f\left(x\right)=si{n}^{-1}\left(2x\sqrt{1-x^2}\right)$, then

The sum of the series $S=1+2\left(\frac{10}{11}\right)+3{\left(\frac{10}{11}\right)}^2+\cdots \text{upto}\infty$ is equal to

Which of the following is an empty relation on AxB where A= {0, 2, 5}, B = {1, 4, 7}

$co{s}^1\left(\frac{y}{b}\right)=2log\left(\frac{x}{2}\right),x>0\Rightarrow x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=$

Let $p:2$ is a prime number
$q:\cos {30}^{\circ }=\frac{1}{2}$
$r:{\sec }^{2}x+{\tan }^{2}x=1$
$s:\sqrt{7}$ is an irrational number
$u:{\pi }^2$ is greater than $10$

The statement which are all false is

If x and y
are connected parametrically by the equation, without eliminating the
parameter, find.

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ where $y_1,y_2<0,$ be the end points of the latus rectum of the ellipse $x^2+4y^2=4.$ Then equation(s) of the parabola with latus rectum $PQ$ is/are