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If $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 2& 3& 4\end{array}\right]B=\left[\begin{array}{cc}-1& -2\\ -1& 0\\ 2& -1\end{array}\right]$ then third element of second column of AB = _______

Show that $tan\left(\frac{1}{2}si{n}^{-1}\frac{3}{4}\right)=\frac{4-\sqrt{7}}{3}$ and justify why the other value $\frac{4+\sqrt{7}}{3}$ is ignored?

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a^2+b^2$
Show that none of the opeartion has identity.

TanX+Tan2X+Tan3X=TanX.Tan2X.Tan3x

Prove this

If $S$ is the solution set of the inequality ${\log }_5(x^2-2)<{\log }_5(\frac{3}{2}|x|-1),$ then which of the following intervals lie(s) in $S$?

The line $x+y=1$ touches the parabola $y^2-y+x=0$ at the point

If the radius of a circle is at least $7\text{cm}$, then the minimum area of the circle is : (in $\text{sq. cm}$) $(\text{use}\pi =\frac{22}{7})$

Explain vector product of vectors

How many three digits no. Are multiple of 5?without repetitions

Of digit

Show that $y=log(1+x)-\frac{2x}{1+x},x>-1$, is an increasing function of x throughout its domain

Show that the function given by $f\left(x\right)=\frac{logx}{x}$ has maximum at x = e.

The locus of the foot of perpendicular from the centre upon any normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

If $f\left(x\right)=x^n,n$ being a non-negative integer, then the values of $n$ for which $f^{\prime }(\alpha +\beta )=f^{\prime }\left(\alpha \right)+f^{\prime }\left(\beta \right)$ for all $\alpha ,\beta >0$ is

Problems which seek to maximise or, minimise profit or, cost form a general class of problems called ………

If the letters of the word 'VIRAT' are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word ‘VIRAT’ is:

The value of $\cot 1^{\circ }\cot 2^{\circ }\cot 3^{\circ }\cdots \cot {179}^{\circ }$ is

The locus of the middle points of chords of hyperbola $3x^2-2y^2+4x-6y=0$ parallel to $y=2x$ is :

Let $f\left(x\right)=\frac{\alpha x}{x+1},x\ne -1$. Then write the value of $\alpha$ satisfying $f\left(f\right(x\left)\right)=x$ for all $x\ne -1$.

In how many ways can $15$ identical blankets be distributed among $6$ persons such that everyone gets atleast one blanket and two particular persons get equal blankets and another three particular persons get equal blankets.

If $2\cos \alpha =x+\frac{1}{x},2\cos \beta =y+\frac{1}{y}$ then $\frac{x^{10}}{y^{12}}-\frac{y^{12}}{x^{10}}=$

A line drawn through the point P (-1, 2) meets the hyperbola $xy=c^2$ at the points A and B (points A and B lie on same side of P) . Q is a point on AB such that PA, PQ and PB are in H.P then locus of Q is

If $z_1$ and $z_2$ are two $n^{\text{th}}$ roots of unity, then $\arg \left(\frac{z_1}{z_2}\right)$ is a multiple of

Four persons can hit a target correctly with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{8}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is:

The equation of the plane which is parallel to xy-plane and cuts intercept of length 3 from the z-axis is

These are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins.
$\begin{array}{cccc}& Column-I& & Column-II\\ \left(A\right)& Thenumberofwaysinwhichallthese& \left(P\right)& 3\\ & coinscanbedistributedifallcoins& & \\ & areidenticalbutallpotsaredifferent& & \\ \left(B\right)& Thenumberofwaysinwhichall& \left(Q\right)& 23\\ & thesecoinscanbedistrbutedifallcoins& & \\ & aredifferentbutallpotsareidentical& & \\ \left(C\right)& Thenumberofwaysallthesecoinscan& \left(R\right)& 15\\ & bedistributedsuchthatnopotisemptyifall& & \\ & coinsaredifferentbutallpotsareidentical& & \\ \left(D\right)& Thenumberofwaysallthesecoinscanbe& \left(S\right)& 12\\ & distributedsuchthatnopotisemptyifallcoins& & \\ & areidenticalbutallpotsaredifferent& & \end{array}$

The least positive integer n which will reduce ${\left(\frac{i-1}{i+1}\right)}^n$to a real number , is

If A be square matrix of order n and if |A| = D and |adj A| = D' , then [RPET 2000]​

The anti-derivative of $\frac{cos5x+cos4x}{1-2cos3x}$ is

${lim}_{x\to 0}\frac{e^{3+x}-sinx-e^3}{x}$

If centroid of the tetrahedron OABC, where coordinates of $A,B,C$ are $(a,2,3),(1,b,3)$ and $(2,1,c)$ respectively be $(1,2,3),$ then find the distance of a point $(a,b,c)$ from the origin, where O is the origin.

If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

If the volume of parallelopiped formed by $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ is 4 units, the volume of tetrahedron formed by $2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c},\overrightarrow{b},\overrightarrow{c}is\frac{\lambda }{3}$. Then $\lambda$ equals ___

If the roots of the equation $a{x}^2+bx+c=0$ are in the ratio m:n, then

If $\alpha ,\beta ϵ(\frac{\pi }{2},\pi )$ and $\alpha <\beta$, then which one of the following is true?

Find the mean deviation from the mean and from median of the following distribution :

$\begin{array}{cccccc}\text{Marks}& 0-10& 10-20& 20-30& 30-40& 40-50\\ \text{No. of students}& 5& 8& 15& 16& 6\end{array}$

If $a^2(1-\sin \theta )+b^2(1+\sin \theta )=2ab\cos \theta$, then the value of $\tan \theta$ is

If x${}_1$, x${}_2$, . . . . x${}_{20}$ are in H.P and x${}_1$, 2, x${}_{20}$ are in G.P., then$\sum _{r=1}^{19}{x}_1{x}_{r+1}=$

The value of $\frac{1}{2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{2^5}+\frac{1}{3^6}+\cdots \cdots \infty$ is

The point of intersection of two tangents at the ends of the latus rectum to the parabola $(y+3{)}^2=8(x-2)$ is

The angle between the lines whose direction cosines are proportional to (1, 2, 1) and (2, -3, 6) is

If a and b are integers, then the roots of the equation $2a{x}^2+(2a+b)x+b=0,a\ne 0$ are:

The domain of the function $f\left(x\right)=\sqrt{x^{12}-x^3+x^4-x+1}$ is