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For any two sets, prove that :

(i) $A\cup (A\cap B)=A$

(ii) $A\cap (A\cup B)=A$

Find the area bounded by the curves $y=-2x^2\text{and}y=2x^2-1$

If $A=\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right],$ then $A^{-1}+(A-aI)(A-cI)=$

If $x^2+px+1$ is a factor of 2 $co{s}^2\theta x^3+2x+sin2\theta$, then

Let f: R$\to$ Rbe defined by $f\left(x\right)=\frac{1}{x}\forall xϵR,$ then f is.....

If the standard deviation of $0,1,2,\mathrm{3...9}$ is $K$, then the standard deviation of $10,11,12,\mathrm{13...19}$ is

Let A = R
− {3} and B = R − {1}. Consider the function f:
A → B defined by

.
Is f one-one and onto? Justify your answer.

Inverse of thefollowing matrix using elementary Row transformations would be

$=\left|\begin{array}{ccc}1& 3& 1\\ 0& 1& 1\\ 3& 6& 1\end{array}\right|$

Question 4

The lengths of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:

$\begin{array}{cc}Length\left(inmm\right)& Numberofleaves{f}_i\\ 118-126& 3\\ 127-135& 5\\ 136-144& 9\\ 145-153& 12\\ 154-162& 5\\ 163-171& 4\\ 172-180& 2\end{array}$

Find the median length of the leaves.

In a race between Achilles and tortoise, people assigned probability to Achilles winning and tortoise winning. These probability pairs are listed below. How many of these pairs satisfy the axiomatic approach, assuming only two possible results are tortoise wins and Achilles wins.

If the range of $f\left(x\right)=({\sin }^{-1}x{)}^3+({\cos }^{-1}x{)}^3;x\in [-1,1]$ and $g\left(x\right)=({\tan }^{-1}x{)}^3+({\cot }^{-1}x{)}^3;x\in [-1,1]$ are $[a,b]$ and $[c,d]$ respectively, then

The point of inflection for $y=f\left(x\right)=x{e}^x$ is:

The solution of the diffeential equaion $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$ is

${lim}_{x\to 0}xlo{g}_{e}x$ will be equal to _____

An examination paper containing 12 questions consists of two parts, A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to attempt 8 questions, selecting at least 3 from each part. In how many ways can the candidate select the questions?

The lowest integer which is greater than ${(1+\frac{1}{{10}^{100}})}^{{10}^{100}}$ is

Let $f\left(x\right)=\{\begin{array}{cc}(x-1)sin\frac{1}{x-1},& ifx\ne 1\\ 0,& ifx=1\end{array}$ Then, which one of the following is true?

If the function $f\left(x\right)=\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

If the equation $(cose{c}^2\theta -4)x^2+(cot\theta +\sqrt{3})x+co{s}^2\frac{3\pi }{2}=0$ holds true for all real x, then the general value of $\theta$ can given by $\left(nϵZ\right)$

If $2a+2b+c=0,$ then the equation of the straight line $ax+by+c=0$ which is farthest from $(1,1)$ is

The equation of the ellipse having its centre at the point $(2,-3)$, one focus at $(3,-3)$ and one vertex at $(4,-3)$ is

If the major axis is "n” times the minor axis of the ellipse, then its eccentricity is

The value of
is

A. 0

B. 2

C. π

D. 1

Consider the binary opeartion $\stackrel{\Lambda }{}$ on the set {1,2,3,4,5} defined by a $\stackrel{\Lambda }{}$ b =min {a,b}. Write the multiplication table of the operation.

The equation of the straight line passing through the intersection of the lines $x-y-1=0$ and $2x-3y+1=0$ and parallel to $3x+4y-14=0$ is