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The total number of natural numbers of six digits that can be made with digits 1, 2, 3, 4, if all digits are to appear in the same number at least once, is

Adjoint of matrix $A=\left[\begin{array}{cc}4& 5\\ 6& 7\end{array}\right]$ is



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If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

Let $S_1$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$.............${}^n{C}_n$ and $S_2$ = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ ..............+ $(-1{)}^n$ ${}^n{C}_n$

Find the value of $\frac{S_1}{S_1+S_2}$ is ___.

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

The number of unordered pairs $(A,B)$ of subsets of the sets $S=\{1,2,3,4,5,6\}$ such that $A\cap B=\varphi$ and $A\cup B=S$ is

The number of permutations of 'n' different objects taken 'r' at a time is given by

If $y+b=m_1(x+a),y+b=m_2(x+a)$ are two tangents to the parabola $y^2=4ax,$ then

The period of the function $5+3\tan (\pi x+\frac{\pi }{4})$ is

Consider the hyperbola xy = 4 and a line 2x + y = 4. O is the centre of the hyperbola. Tangent at any point P on the hyperbola intersects the coordinate axes at A and B.

Locus of circumcentre of $\Delta OAB$ is ___

A box $B_1$ contains $1$ white ball, $3$ red balls, and $2$ black balls. Another box $B_2$ contains $2$ white balls, $3$ red balls, and $4$ black balls. A third box $B_3$ contains $3$ white balls, $4$ red balls, and $5$ black balls.

If $2$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $2$ balls are drawn from box $B_2$ is

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL are arranged in a dictionary. Then the position of the word SMALL is : -

Find the number of solutions of $z^2+|z^2|=0.$

If the abscissae and ordinates of two points P and Q are roots of the equations $x^2+2ax-b^2=0$
and $x^2+2px-q^2=0$ respectively,
then write the equation of the circle with PQ as diameter.

If the P^th, q^th , and r^th tems of an A.P. are in G.P., then common ratio of the G.P. is

If $A$ is a non-singular matrix of order $5$ and $|adjA|=256$, then the value of $|A|$ is

In the equation of the line parallel to the line 2x - 3y=1 And passing through the middle point of the line segment joining the Points (1,3) and (1,-7) is

If $a=lo{g}_x\left(yz\right),b=lo{g}_y\left(zx\right),c=lo{g}_z\left(xy\right)$ where $x,y,z$ are positive reals not equal to unity then $abc-a-b-c$ is equal to -

The graph shows the relation between the time taken by an ant to travel from one place to another place.





The domain of function shown in the graph is

If x and y
are connected parametrically by the equation, without eliminating the
parameter, find.

The value of $\underset{x\to 0}{lim}\frac{x\ln (1+7x)}{1-\cos 3x}$ is

Integrate the function.
$\int xco{s}^{-1}xdx.$

Let $f\left(x\right)={\sin }^{-1}\left(2x\sqrt{1-x^2}\right).$ If $\frac{1}{\sqrt{2}}<x<1,$ then $f^{\prime }\left(x\right)=$

Show
that the matrix
is
symmetric or skew symmetric according as A
is symmetric or skew symmetric.