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The radii $r_1,r_2,r_3$ of the escribed circles of the triangle $ABC$ are in H.P. If the area of the triangle is $24{\text{cm}}^2$ and its perimeter is $24\text{cm}$, then the length of its largest side is

If $x{y}^2=4$ and $lo{g}_3\left(lo{g}_{2}x\right)+lo{g}_{1/3}\left(lo{g}_{1/2}y\right)=1$, then x equals

Consider any set of 201 observations $x_1,x_2,\cdots ,x_{200},x_{201}$. It is given that $x_1<x_2<\cdots <x_{200}<x_{201}$. Then, the mean deviation of this set of observations about a point k is minimum, when k equals

The perpendicular distance from (1,2) to the straight line 12x+5y=7 is

If $|Z_1+Z_2|$ = $\left|Z_1\right|+\left|Z_2\right|$, then find the value of arg $\left(\frac{Z_1}{Z_2}\right)$

A ray emitting from the point $(-3,0)$ is incident on the ellipse $16x^2+25y^2=400$ at point $P$ with an ordinate $4.$ If the equation of the reflected ray after first reflection is $4x+3y=k,$ then the value of $k$ is

Plot the graph of $|y|=ln\left(x\right)$

The number of values of $x\in [0,\pi ]$, that satisfies the equation ${\log }_{|\sin x|}(1+\cos x)=2$, is

$\text{Let f(x)=}{x}^{n}andnϵN.\text{The the value of n for which}{f}^{\prime }(a+b)=f^{\prime }\left(a\right)+f^{\prime }\left(b\right)\text{is valid for a, b}>0\text{is}$

For any sets A and B, prove that

$(A\times B)\cap (B\times A)=(A\cap B)\times (B\cap A)$.

The most probable radius (in pm) for finding the electron in $H{e}^{+}$ is

If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$, then $A^n=$

Let $P(x,y)$ is a point where $x$ satisfies $x^2-|x|-6=0$ and $y$ satisfies $y^2+|y|-6=0$, if $A(1,1)$, then $\sum (PA{)}^2$ is

Find the general solution of 2sin 3x -1 =0

If the complex number $x^2+y^2+100i$ and $29-x^2{y}^{2}i$ are conjugate to each other, Find the value of $x^3+y^3$ when x and y are positive real number.

The geometric mean of $6$ and $24$ is

The relation f is defined by f(x) = $\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 3\le x\le 10\end{array}$ and

The relation g is defined by g(x) = $\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 2\le x\le 10\end{array}$ Then,

If $co{s}^{-1}x>si{n}^{-1}x$ then

If $\tan \theta =\sqrt{3}$ and 'θ' lies in the third quadrant, then $\sin \theta +\cos \theta$ is

How many of following statements are true?

(1)Period of sin $\theta$ is $\pi$, because sin 0 and sin $\pi$ = 0

(2) Period of cos $\theta$ is $\pi$

(3) Period of tan $\theta$ is $\pi$

If $P=\frac{\sin {300}^{\circ }\cdot \tan {330}^{\circ }\cdot \sec {420}^{\circ }}{\tan {135}^{\circ }\cdot \sin {210}^{\circ }\cdot \sec {315}^{\circ }}$ and $Q=\frac{\sec {480}^{\circ }\cdot \text{cosec}{570}^{\circ }\cdot \tan {330}^{\circ }}{\sin {600}^{\circ }\cdot \cos {660}^{\circ }\cdot \cot {405}^{\circ }},$ then the value of $P$ and $Q$ are respectively

Let the digits of a three digit number are in G.P. If $400$ is subracted from the number and the digits of the new number are in A.P., then the last digit of the original number is

Suppose there are four roads from station A to station B and 3 roads between station B and station C. Find the number of ways in which one can drive from station A to station C via station B.

The root(s) of the equation $x^4=16$ is/are :

If α is the value of $xϵ[0,\pi ]$ satisfying 3 cos x + 3 sin x + sin 3x - cos 3x = 0, then find the value of $\frac{4\alpha }{\pi }$ ?

Circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $6$ in such a manner that their common chord is of maximum length and has slope equal to $\frac{1}{2}.$ Then the co-ordinates of the centre of the circle(s) $C_2$ is (are)

If $p=\frac{1}{2}si{n}^2\theta +\frac{1}{3}co{s}^2\theta$ , then

What is the equation of chord of contact of tangents drawn from P(10, 8) to the ellipse

$\frac{x^2}{25}+\frac{y^2}{16}=1$.

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$

The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

The number of ordered triplets of non-negative integers which are solutions of the equation x + y + z = 100 is

If x co-ordinates of a point P of line joining the points Q(2, 2, 1) and R(5, 2, -2) is 4, then the z-coordinates of P is

[RPET 2000]

A tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts the axes in M and N. Then the least length of MN is

$7^{th}$ term of the sequence $\sqrt{2},\sqrt{10},5\sqrt{2},.........$ is

Prove the following:

sin (n+1)x sin (n+2)x + cos (n+1)x cos (n+2)x = cos x

If $(3x{)}^{log3}=(4y{)}^{log4}$ and $(4{)}^{logx}=(3{)}^{logy}$, then x equals

If $x\in (\frac{-\pi }{2},\frac{\pi }{2}),$ then $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to

Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.



$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. The coefficient of two consecutive terms in the}& \text{p.}9\\ \text{expansion of}(1+x{)}^n\text{will be equal, then}n\text{can be}& \\ \text{b. If}{15}^n+{23}^n\text{is divisible by}19,\text{then}n\text{can be}& \text{q.}10\\ \text{c. If the coefficients of}{T}_r,T_{r+1},T_{r+2}\text{terms of}& \text{r.}11\\ (1+x{)}^{14}\text{are in A.P., then}r\text{is less than}& \end{array}$

To receive Grade 'A' in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita's marks in first four examinations are 87, 92,94 and 95, find minimum marks that Sunita must obtain in fifth examination to get Grade 'A' in the course.

If $e_1$ is the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ and $e_2$ is the eccentricity of the hyperbola passing through the foci of the ellipse and $e_1{e}_2=1$, then equation of the hyperbola is

If the roots of the equation $6x^2-7x+k=0,k>-3$ are rational, then possible integral value(s) of $k$ is/are