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In a $△ABC$ with usual notation $CF$ is the internal angle bisector of $\angle C$ and $F$ lies on side $AB$, then the length of $CF$ is equal to

A tangent is drawn to a circle of radius r from an external point P .If length of tangent from a point to the circle is twice the minimum distance between circle and the point, what is the length of the tangent?

Assuming the plane $4x-3y+7z=0$ to be horizontal, the equation of the line of greatest slope through the point $(2,1,1)$ in the plane $2x+y-5z=0$ is

The value of $\underset{x\to 1}{lim}\left[sinsi{n}^{-1}x\right]$ is

Let A = {1, 2, 3,....., 14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where $x,yϵA$} Write down its domain, codomain and range.

The equation of locus of a point where distance from the y-axis is equal to its distance from the point A(2,1,-1) is-

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.

The range of $f\left(x\right)=\frac{e^x}{1+\left[x\right]},x\ge 0$ is

Two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are pependicular to each other. Another unit vector $\overrightarrow{c}$ is inclined at an angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+z(\overrightarrow{a}\times \overrightarrow{b})$, then

Let $S_n=\sum _{k=1}^n\frac{k}{(k-1{)}^{\frac{4}{3}}+(k^2-1{)}^{\frac{2}{3}}+(k+1{)}^{\frac{4}{3}}}$ then which of the following is/are true?

If $|x-{\log }_{\sqrt{3}/2}\left(\frac{64}{27}\right)|=1$, then the value of $x$ is/are

Let $f\left(x\right)=\frac{\mathrm{kx}}{x+1},x\ne -1.$If $f\left(f\right(x\left)\right)=x$ for all $x$ $\ne$

$-1$, then the value of $k$ is

If $A$ is a proper subset of $B$, then $n\left(A\right)$$n\left(B\right)$.

The range of $f\left(x\right)=\cos \left(\sqrt{x}\right)$ is

$\underset{x\to 0}{lim}\frac{\sqrt{1-cos2x}}{\sqrt{2}x}$ is

(JEE 2002)

In a single throw of two dice, determine the probability of not getting same number on the two dice.

A line passes through P(1, 2) such that its intercept between the axes is bisected at P. Find the equation of line.

For the diagram shown below, $A\cap B=$

If $\sum _{r=1}^n{T}_r=\frac{n}{8}(n+1)(n+2)(n+3)$ and $\sum _{r=1}^n\frac{1}{T_r}=\frac{n^2+3n}{4\sum _{k=1}^{p}k}$, then $p$ is equal to

For the given distinct values $x_1,x_2,x_3,.....x_n$ occurring with frequencies $f_1,f_2,f_3,...f_n$ respectively. The mean deviation about mean where $\overline{x}$ is the mean and N is total number of observations would be

If $({\alpha }^2,\alpha -2)$ be a point interior to the regions of the parabola $y^2=2x$ bounded by the chord joining the points $(2,2)$ and $(8,-4),$ then $\alpha$ belongs to the interval

Let $f\left(x\right)=x^5+a{x}^3+bx.$ The remainder when $f\left(x\right)$ is divided by $x+1$ is $-3.$ Then the remainder when $f\left(x\right)$ is divided by $x^2-1,$ is

Find the equation of line parallel to the y- axis and drawn through the point of intersection of $x-7y=-5and3x+y-7=0.$

If the points $(\sqrt{3}\text{sin}\theta ,\sqrt{4}\text{cos}\theta )$ where $\theta \in R$, lies outside the hyperbola $\frac{x^2}{4}-\frac{y^2}{5}=1$, then the value of $\theta$ is:

The point $(-2m,m+1)$ is an interior point of the smaller region bounded by the circle $x^2+y^2=4$ and the parabola $y^2=4x$, then $m$ lies in the interval

If $z$ is a complex number satisfying $\arg (z+a)=\frac{\pi }{6}$ and $\arg (z-a)=\frac{2\pi }{3},a\in R^{+}$, then

$In\Delta ABC,(a-b{)}^{2}co{s}^2\frac{C}{2}+(a+b{)}^{2}si{n}^2\frac{C}{2}\text{is equal to}$

The mean of 5 numbers is 18. If one number is excluded, their mean becomes 16. Then the excluded number is

A ray of light coming from the point $(2,2\sqrt{3})$ is incident at an angle ${30}^{\circ }$ on the line $x=1$ at the point $A.$ The ray gets reflected on the line $x=1$ and meets $X-$axis at the point $B.$ Then, the line $AB$ passes through the point:

If an integer $q$ is chosen at random in interval $-10\le q\le 10,$ then the probability that the roots of the equation $x^2+qx+\frac{3q}{4}+1=0$ are real, is

The equations to the common tangents to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ are

Intersecting the y-axis at a distance of 2 units above the origin and making an angle of ${30}^{\circ }$ with positive direction of the x-axis.

$\underset{x\to 0}{lim}\frac{2^x-1}{(1+x{)}^{\frac{1}{2}}-1}$

If $\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\cdots \text{upto}\infty =\frac{{\pi }^4}{90},$ then the value of $\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\cdots \text{upto}\infty$ is

Match $\text{List I}$ with $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Let}z,\omega ,\alpha \text{be complex numbers such that}& \text{(P)}& 0\\ & |z|=|\omega |=4\text{and}\alpha =\frac{z-\overline{\omega }}{16+z\overline{\omega }}.\text{Then Re}\left(\alpha \right)& & \\ & \text{is equal to}& & \\ \text{(B)}& \text{If}x=p+iq\text{is a complex number such that}& \text{(Q)}& 3\\ & x^2=3+4i\text{and}{x}^3=2+11i,\text{where}i=\sqrt{-1},& & \\ & \text{then}p+q\text{is equal to}& & \\ \text{(C)}& \text{Number of complex number(s)}z\text{satisfying the}& \text{(R)}& 4\\ & \text{equation}\overline{z}=i{z}^2,\text{where}i=\sqrt{-1},\text{is equal to}& & \\ \text{(D)}& \text{If}z\in C\text{satisfies}|z+2-i|=5\text{, then the}& \text{(S)}& 5\\ & \text{maximum value of}\frac{|3z+9-7i|}{4}\text{is equal to}& & \end{array}$



Which of the following is the only CORRECT combination?

If α and β be the roots of the equation $2x^2+2(a+b)x+a^2+b^2=0$, then the equation whose roots are $(\alpha +\beta {)}^2$ and $(\alpha -\beta {)}^2)$ is

If $F\left(x\right)={\left(f\left(\frac{x}{2}\right)\right)}^2+{\left(g\left(\frac{x}{2}\right)\right)}^2$ where $f^{\prime }\left(x\right)=-f\left(x\right)$ and $g\left(x\right)=f^{\prime }\left(x\right)$ and given that $F\left(5\right)=5$, then $F\left(10\right)$ is equal to

Find the equation of the circle orthogonal to the circles $x^2+y^2+3x-5y+6=0and4x^2+4y2-28x+29=0$ and whose center lies on the line 3x + 4y + 1 = 0.

If arg$⟮\frac{z-i}{z+i}⟯$ =$\frac{\pi }{4}$, then z represents a point on

Let $f:N\to R$ be a function satisfying the following conditions:

$f\left(1\right)=1$ and $f\left(1\right)+2f\left(2\right)+\dots +nf\left(n\right)=n(n+1)f\left(n\right)$ for $n\ge 2$.

If $f\left(1003\right)=\frac{1}{K}$, then $K$ equals

The distance between the orthocentre and circumcentre of a triangle whose vertices are $P(3,0),Q(0,0)$ and $R(\frac{3}{2},\frac{-3\sqrt{3}}{2})$ is