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Let $L=\underset{y\to 0}{lim}{\int }_0^{\pi }\frac{tan\left(ysinx\right)}{y}dx$ and $l=\underset{k\to \infty }{lim}{\int }_0^k{(1-\frac{x}{k})}^k{e}^{\frac{x}{3}}dx$, then

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$Ina\Delta ABC,ifb=\sqrt{3},c=1and\angle A={30}^{\circ },finda$

${lim}_{x\to 0}\frac{3sin2x+2x}{3x+2tan3x}$

The relation between time taken in min (x) and distance covered in km (y) is given below. Which of the following holds true for the condition that represents the distance covered is 12 km throughout?

If $a+b+c=1$, where $a,b,c$ are positive real numbers, then the minimum value of $\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}$ is

Let $F\left(x\right)=\int e^{si{n}^{-1}x}(1-\frac{x}{\sqrt{1-x^2}})dx,andF\left(0\right)=1,ifF\left(\frac{1}{2}\right)=\frac{k{\sqrt{3e}}^{\frac{x}{6}}}{\pi }$, then k =

Let $y=y\left(x\right)$ be the solution of the differential equation, $\frac{2+\text{sin}x}{y+1}.\frac{dy}{dx}=-\text{cos}x,y>0,y\left(0\right)=1.$

If $y\left(\pi \right)=a,$ and $\frac{dy}{dx}$ at $x=\pi$ is $b$, then the ordered pair $(a,b)$ is equal to:

A curve $y=f\left(x\right)$ is passing through $(0,0).$ If the slope of the curve at any point $(x,y)$ is equal to $(x+xy),$ then the number of solution(s) of the equation $f\left(x\right)=1,$ is

The normal a point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$ then

If two lines are intersecting at $(4,3)$ and angle between them is ${45}^{\circ }$. If the slope of one line is $2$, then the equation(s) of other line is/are

Equation of a line which is parallel to the line common to the pair of lines given by $6x^2-xy-12y^2-0$ and $15x^2+14xy-8y^2-0$ and at a distance 7 from it is

In $\Delta$ ABC having vertices $A(acos{\theta }_1,asin{\theta }_1),B(acos{\theta }_2,asin{\theta }_2)$ and $C(acos{\theta }_3,asin{\theta }_3)$ is equilateral, then which of the followings is/are true?

If a ray of light along the line $y=4$ gets reflected from a parabolic mirror whose equation is $(y-2{)}^2=4(x+1)$, then equation of reflected ray will be

Which of the following statements is / are true?
1.Set of two vectors $\overrightarrow{a},\overrightarrow{b}$ is linearly dependent if and only if either any of $\overrightarrow{a}\text{and}\overrightarrow{b}$ is zero or they are parallel
2.$\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are linearly dependent $\iff \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are coplanar.

Find the
particular solution of the differential equation

,
given that y = 1 when x = 0

The perpendicular distance from origin to the obtuse angular bisector between the lines $\frac{x-1}{-1}=\frac{y+1}{2}=\frac{z-1}{1}$ and $\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-1}{2}$ is:

The order
of the differential equation

is

(A) 2 (B) 1 (C) 0 (D) not
defined

A paper contains $10$ questions with total marks of $40$ and solving each question is mandatory. In the paper, there are $3$ types of questions i.e., $2$ marks, $4$ marks and $6$ marks questions. If someone attempts the paper and got $38$ marks, then the probability that question paper has only $3$ question of $6$ marks is (there is no negative marking)

If the local maximum of f(x) = sin2x - xϵ(0,π) is at x = a, then find the value of 36 $\frac{a}{\pi }$

A fair die is thrown until a score of less than five points is obtained. The probability of obtaining less than three points on the last throw is

The solution lof $\frac{dy}{dx}-xtan(y-x)=1$

Rearrange the following sentences (A), (B), (C), (D), (E) and (F) to make a meaningful paragraph and answer the questions which follow:

1. 16

The equation of the line passing through the point $(c,d)$ and parallel to the line $ax+by+c=0$, is

If a straight line passing through the point $P(-3,4)$ is such that its intercepted portion between the coordinate axes is bisected at $P$, the its equation is :