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The function defined by $f\left(x\right)=\{\begin{array}{cc}kx+1,& ifx\le 5\\ 3x-5,& ifx>5\end{array}$ is continuous at x = 5, the k =

If $A$ and $B$ are two events such that $P\left(A\right)=\frac{1}{2}$ and $P\left(B\right)=\frac{2}{3}$, then which of the following is/are correct?

A straight line $L$ through the point $(3,-2)$ is inclined at an angle of ${60}^{\circ }$ to the line $\sqrt{3}x+y=1.$ If $L$ also intersects the $x-$ axis, then the equation of $L$ is

Let E denotes the event "India scores less than 300” in a cricket match between India and South Africa. How many elements are there in the sample space which will favor the event E?

Find the ratio in which the line segment joining A(1, -5) B(-4, 5) is divided by the x-axis

The incircle touches side $BC$ of triangle $ABC$ at $D$ and $ID$ is produced to $H$ so that $DH=s,$ where $s$ and $I$ are the semi-perimeter and incentre of triangle $ABC$ respectively. If $HBIC$ is cyclic, then $\cot \left(\frac{A}{4}\right)$ is equal to

Find the graph of $|y|=e^{-|x|}-1$

From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Find the value of 1 + $\omega$ + ${\omega }^2$ + ${\omega }^3$

Find the equation of the straight line passing through the origin and bisecting the portion of the line ax + by + c = 0 intercepted between the coordinate axes.

If $\int \frac{x^3}{x^4+3x^2+2}dx=\ln \left|\frac{x^2+2}{\sqrt{f\left(x\right)}}\right|+C,$ where $C$ is constant of integration, then the value of $f\left(7\right)$ is

The value of a cos $\theta$ + b sin $\theta$ lies between

Let $n_1$ and $n_2$ represent respectively the number of possible ordered and unordered triplets $(a,b,c)$ such that $abc=144,(a,b,c\in N)$, then

If the product of $n$ positive numbers is unity, then the sum of these numbers can not be less than

If the tangents are drawn at the points of intersection of the curves $3x^2+y^2-4y=0$ and $3x^2-y-2=0,$ then the slope of tangents to the first curve is/are

If ${\log }_{0.3}(x-1)<{\log }_{0.09}(x-1)$, then $x$ lies in the interval

Let $f:(-\infty ,+1]\to R,g:[-1,\infty )\to R$ be such that $f\left(x\right)=\sqrt{1-x}$ and $g\left(x\right)=\sqrt{1+x}$, then $f\left(x\right)+\frac{1}{g\left(x\right)}$ exist if $x\in$

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

The magnitude of $-\sqrt{4}$ is

The principal solution(s) for $\cos x=-\frac{1}{\sqrt{2}}$ is/are

The value of $\underset{1}{\overset{2}{\int }}\frac{dx}{(x+1)(x+2)}$ is:

Check whether the relation R in R defined by $R=\left\{\right(a,b):a\le b^3\}$ is reflexive, symmetric or transitive

Do ATC and AVC curves intersect? Explain.

Equations of the line(s) which makes an angle of ${45}^{\circ }$ with $y$ axis and passing through the point $(2,3)$ is

The slope intercept form of the line $3x+7y+8=0$ is

A body is projected horizontally from the top of a building. It strikes the ground after a time $t$ with its velocity vector making an angle $\theta$ with the horizontal. The speed with which the body is projected is

Let $\overrightarrow{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}.$ If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}\cdot \overrightarrow{c}=|\overrightarrow{c}|,|\overrightarrow{c}-\overrightarrow{a}|=2\sqrt{2}$ and the angle between $\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{c}$ is ${30}^{\circ }$, then the value of $|(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}|$ is