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If $n$ is the number of real solutions of the equation $min(e^{-|x|},1-e^{-|x|})=\frac{1}{4}$ and $L=\underset{x\to 0^{-}}{lim}(\frac{e^{2x}-1}{x}+\frac{e^{3x}-1}{x}+\frac{e^{4x}-1}{x}+\cdots \text{upto}n\text{terms}),$ then the value of $L$ is

If $A,B$ and $C$ are three subsets of a non-empty set $X$, then $(A^{\prime }\cap B^{\prime }\cap C)\cup (B\cap C)\cup (A\cap C)$ is equal to

The total number of points of non-differentiability of $f\left(x\right)=\text{min}\{|\sin x|,|\cos x|,\frac{1}{4}\}$ in $(0,2\pi )$ is -

Classify the following as scalar and vector quantities:
(i) Distance

Find the solution set in ${\log }_2(x-1)+{\log }_2(x-2)>1$

If one end point of the focal chord of the parabola $y^2=4ax$ is $(1,2)$, then second end point lies on

Is th function defined by f(x) = $\{\begin{array}{c}x+5,ifx\le 1\\ x-5,ifx>1\end{array}$ a contionuous functions ?

The maximum value of 3cos $\theta$ - 4 sin$\theta$ is

The equation whose roots are the values of $r$ satisfying the equation ${}^{69}{C}_{3r-1}-{}^{69}{C}_{r^2}={}^{69}{C}_{r^2-1}-{}^{69}{C}_{3r}$ is

Arrange the following limits in the ascending order of their values

$\left(1\right)\underset{x\to \infty }{lim}{\left(\frac{1+x}{2+x}\right)}^{x+2}$

$\left(2\right)\underset{x\to 0}{lim}{(1+2x)}^{3/x}$

$\left(3\right)\underset{\theta \to 0}{lim}\left(\frac{\sin \theta }{2\theta }\right)$

$\left(4\right)\underset{x\to 0}{lim}\frac{\ln (1+x)}{x}$

If $\pi$ < 2$\theta$ < $\frac{3\pi }{2}$ then $\sqrt{2+\sqrt{2+2cos4\theta }}$ is equal to

A point equidistant from the line $4x+3y+10=0,5x-12y+26=0$ and $7x+24y-50=0$ is

Consider the circle $|z-5-5i|=2$ in the complex number plane $(x,y)$ with $z=x+iy$. The minimum distance from the origin to the circle is

Three lines $L_1:\overrightarrow{r}=\lambda \hat{i},\lambda \in R,$$L_2:\overrightarrow{r}=\hat{k}+\mu \hat{j},\mu \in R,$ and $L_3:\overrightarrow{r}=\hat{i}+\hat{j}+\nu \hat{k},\nu \in R$ are given.

For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P,Q$ and $R$ are collinear ?

Q42. What will come in place of (?) in the following number series?

If the determinant

$\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$ and $x,y,z,p\in R^{+}$ then

The values of $m$ such that exactly one root of $x^2+2(m-3)x+9=0$ lies between $1$ and $3$, is

Consider the equation of curve $y=x^2-3x+3$ and $x\ne 3.$

Then number of critical point(s) for given curve is



[2 marks]

Find the equation of the circle passing through the point of intersection of the lines x + 3y = 0 and 2x - 7y = 0 and whose centre is the point of intersection of the lines x +y +1 = 0 and x - 2y + 4 = 0.

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempts 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?

If $\alpha$ and $\beta$ are the roots of the equation $x^2+5x-7=0$. Then a equation with roots

$\frac{1}{\alpha }and\frac{1}{\beta }$

is .