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how many 2 digit even nipumbers can be formed from the digits 1,2,3,4,5.if the digit can be repeated

If (p+q)th term of G.P. is m and (p-q)th term is n, then pth term will be

If (1-p) is a root of quadratic eqn x²+px+(1-p)=0 then it's roots are

A. - 1,2

B. -1,1

C. 0,-1

D. 0,1

From the sets given below, select equal sets and equivalent sets.

A = {0, a}, B = {1,2,3,4}

C= {4,8,12}, D = {3,1,2,4}

E = {1,0}, F = {8,4,12}

G = {1,5,7,11}, H = {a, b}

The axis of the parabola $9y^2-16x-12y-57-0$ is

If $\overrightarrow{a}and\overrightarrow{b}$ are non-zero vectors such that $|\overrightarrow{a}-3\overrightarrow{b}|=|\overrightarrow{a}+3\overrightarrow{b}|$, then the angle between $\overrightarrow{a}and\overrightarrow{b}$ is

Consider the arrangement shown in figure (17-E7). By some mechanism, the separation between the slits $S_3$ and $S_4$ can be changed . the intensity is measured at the point P which is at the common perpendicular bisector

of $S_1{S}_2$ and$S_3{S}_4$ When z = $\frac{d\lambda }{2d},$ the intensity measured at P is I. Find this intensity when z is equal to
(a) $\frac{D\lambda }{d}$, (b) $\frac{3D\lambda }{2d}$ and (c) $\frac{2D\lambda }{d}.$

The value of ‘a’ for which the equations $x^2$ -3x + a = 0 and $x^2$ + ax - 3 = 0 have a common root is

If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$ , then $\overline{z}$ is equal to :

Find the values of x in each of the following:

$\left(i\right)\frac{log144}{log12}=logx$

$\left(ii\right)\frac{log125}{log25}=x$

$\left(iii\right)lo{g}_{x}4+lo{g}_{x}16+lo{g}_{x}64=12$

Verify the following :

(i) (0, 7, -10), (1, 6, -6) and (4, 9, -6) are vertices of an isosceles triangle.

(ii) (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are vertices of a right-angled triangle.

(iii) (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are vertices of a parallelogram.

(iv) (5, -1, 1), (7, -4, 7) , (1, -6, 10) and (-1,-3, 4) are vertices of a rhombus.

Find the number of solutions for; $\left[\right[x]-x]=sinx$; where [.] denotes the greatest integer function.

The number of ways of selecting $15$ teams from $15$ men and $15$ women such that each team consists of a man and a woman, is

If $\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!},$ find x.

If $0<A+B<\frac{\pi }{2}$ and $\tan A,\tan B$ are the roots of the equation $3x^2-12x-6=0$, then the numerical value of $\sin (A+B)\cos (A+B)-\sec (A+B)\text{cosec}(A+B)$ is

Write the range of the function $f\left(x\right)=e^{x-\left[x\right]},xϵR$

In the expansion of ${(4^{\frac{1}{3}}+6^{\frac{-1}{4}})}^{20}$, consider the following statements:

(i) the number of irrational terms = 18

(ii) the number of rational terms = 2

(iii) the middle term is irrational.

Then the correct statements are:

If $\tan {70}^{\circ }=\tan {20}^{\circ }+\lambda \tan {50}^{\circ },$ then $\lambda$ is equal to

Solution of the differential equation $\frac{dy}{dx}=2e^{x-y}+x^2{e}^{-y}$ is

If $\alpha$ and $\beta$ are the roots of the equation$x^2-2x+4=0$, such that ${\alpha }^n+{\beta }^n=2^k\cos \frac{n\pi }{3}$, then value of $k$ is

The focus of the conic $x^2-6x+4y+1=0$ is

Evaluate: $\underset{x\to \frac{\pi }{2}}{lim}\frac{tan2x}{x-\frac{\pi }{2}}$

If 2${}^x$ x 8 ${}^{1/5}$ = 2 ${}^{1/5}$, then x is equal to

If A and B are two matrices conformable to multiplication such that their product AB =O(Zero matrix). Then which of the following is true

If P $\left(A\right)=0.8,P\left(B\right)=0.5andP\left(\frac{B}{A}\right)=0.4,$ find
$P\left(\frac{A}{B}\right)$

Normal equations to parabola $y^2=4ax$ passing through point $(5a,2a)$ are

Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this.
(iii) On R, defined $\ast$ by $a\ast b=a{b}^2$

Integrate the following functions.
$\int$ sin x sin (cos x)dx

An ellipse with major axis and minor axis as $8$ and $2$ units respectively, slides along the coordinate axes keeping the contact with axes. Then the locus of its foci is

If 32 sin^6(x)=10-15 cos(2x) + b cos(4x) + a cos(6x), then how do we prove that tan(x)tan(5x)=(a+b-5)/(a-b+7)?

To verify Rolle's Theorem for f(x) defined in [a, b], we need to show

cos^{​​​​6} π/16 + cos^{​​​​6} 3π/16 + cos^{​​​​6} 5π/16 + cos^{​​​​6} 7π/16 = ?

What is the value of sin4x=?

The length of the normal chord at (8,7) on $y^2-2y-4x-3=0$ is

Cos²2x-cos²6x=sin4x sin8x

Number of integers satisfying inequality,
$\sqrt{{\log }_{3}x-1}+\frac{\frac{1}{2}{\log }_3{x}^3}{{\log }_3(\frac{1}{3})}+2>0$ is

In a composite function f [g(x)] the following condition must be true

Let $a>b>0$ and $I\left(n\right)=a^{1/n}–b^{1/n},$ $J\left(n\right)=(a–b{)}^{1/n}$ for all $n\ge 2$. then

From the adjoining venn diagram, find $A\cap (B\cap C)$

The value of ${\int }_{-2}^3|1-x^2|dx$ is $\frac{p}{q}$ in it's simplest form. Then what $p+q$ is ___

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 $P$) and $Q$ is a point on $AB$ such that $PA,PQ$ and $PB$ are in $H.P$ then locus of $Q$ is

A. $x-2y=2c^2$

B. $2x-y=2c^2$

C. $x+2y=2c^2$

D. $2x-y+2c^2=0$

ABCD is a convex quadrilateral with 3, 4, 5 and 6 points marked on its sides AB, BC, CD and DA respectively. Triangles are formed using these 18 points and original vertices of the quadrilateral. Number of such triangles that do not have any side or part of a side common with the quadrilateral are