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Given, A={2,3,4}, B ={2,5,6,7}. Construct an example of each of the following

(i) an injective mapping from A to B.

(ii) a mapping from A to B which is not injective.

(iii) a mapping from B to A.

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE ?

A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2).$ If $O$ is the origin and $P$ is $(2,-1,1),$ then the projection of $\overrightarrow{OP}$ on this plane is of length :

A stroboscopic system is used for measuring the speed of a rotating shaft. The shaft has one target mark on it. The maximum strobe rate at which synchronism is achieved is $r_1$ flashes per minute. The next lower flash rate at which synchronism is achieved is $r_2$ flashes per minute. The speed of the shaft in rpm is

The last two digits of the number $3^{400}$ is

Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :

(i) $4,-2,1,-\frac{1}{2},.......$

(ii) $\frac{-2}{3}=-6=-54,....$

(iii) $a,\frac{3a^2}{4},\frac{9a^3}{16},......$

(iv) $\frac{1}{2},\frac{1}{3},\frac{2}{9},\frac{4}{27},.......$

Distinguish between unity of command and unity of direction.

The equation of the plane which bisects the line joining the points (-1, 2, 3) and (3, -5, 6) at right angle, is

For a function $f\left(x\right)=\sqrt{0.5-\cos x}\forall x\in [3\pi ,4\pi ],$ to be defined $x\in$

Point of inflection of the function $f\left(x\right)=x^3$ is

∫e^x² dx = ?

Let $g_i:[\frac{\pi }{8},\frac{3\pi }{8}]\to R,i=1,2$ and $f:[\frac{\pi }{8},\frac{3\pi }{8}]\to R$ be functions such that $g_1\left(x\right)=1,g_2\left(x\right)=|4x-\pi |$ and $f\left(x\right)={\sin }^{2}x,$ for all $x\in [\frac{\pi }{8},\frac{3\pi }{8}].$

If $S_i=\underset{\pi /8}{\overset{3\pi /8}{\int }}f\left(x\right)\cdot g_i\left(x\right)dx,i=1,2,$ then the value of $\frac{16S_1}{\pi }$ is

${lim}_{x\to \frac{\pi }{4}}\frac{1-tanx}{x-\frac{\pi }{4}}$

Evaluate $\int \frac{3x+4}{x^2-8x+15}dx$

(where $C$ is constant of integration)

If both roots of the equation $x^2+ax+2=0$ lie in the interval $(0,3),$ then the range of values of $a$ is

Evaluate sin(3 sin

The sum ${\sum }_{i=0}^m(\frac{10}{i})(\frac{20}{m-i}),where(\frac{p}{q})=0ifp>q,$ is maximum when m is equal to

Let $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ be two $3\times 3$ real matrices such that $b_{ij}=(3{)}^{(i+j-2)}{a}_{ji}$, where $i,j=1,2,3$. If the determinant of $B$ is $81$, then the determinant of $A$ is :

On the circle with centre O, points A, B are such that OA = AB. A point C is located on the tangent at B to the circle such that A and C are on the opposite sides of the line OB and AB = BC. The line segment AC intersects the circle again at F. Then the ratio $\angle$BOF: $\angle$BOC is equal to:

If x < 2, then write the value of $\frac{d}{dx}\left(\sqrt{x^2-4x+4}\right)$.

Which term in the A.P 7,3, 1.................. is -73?

The equation of a line which is parallel to the angle bisector between the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{0}$ and $\frac{x-1}{0}=\frac{y-4}{-1}=\frac{z-5}{-1}$ and passing through the point $(1,-2,3)$ is

If $a,b,c$ are three consecutive positive integers and $\log (1+ac)=2k,$ then the value of $k$ is

The solution set of inequality ${\log }_5(x-3)+\frac{1}{2}{\log }_{5}3<\frac{1}{2}{\log }_5(2x^2-6x+7)$ is

Suppose that p,q and r are three non-coplanar vectors in $R^3$. Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$, $y$ and $z$ respectively, then the value of $2x+y+z$ is ___

The number of ways in which $13$ non-distinguishable books can be distributed among $7$ students so that every student get at least one book and at least one student gets $4$ books but not more, is

​​​​​​​(correct answer + 1, wrong answer - 0.25)

If $\sin \theta +\sin \varphi =a$ and $\cos \theta +\cos \varphi =b$, $(a\ne b,a\ne 0,b\ne 0)$ then -

Find points at which the tangent to the curve $y=x^3-3x^2-9x+7$ is parallel to the x-axis.

Find the value of the trigonometric function

Sum of first $n$ terms of the sequence $5,7,11,17,25,\dots$ is equal to

In the correct grammar above, what is the length of the derivation (number of steps starting from S) to generate the string $a^l{b}^m\text{with}l\ne m?$