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Let $f\left(x\right)=\{\begin{array}{cc}\ln (x^2+1)-e^{-x}+1,& x<0\\ \alpha ,& x\ge 0\end{array}.$

If $f\left(x\right)$ has local minimum at $x=0,$ then

The function, f(x) = 2x + 1 is

Given, A={2,3,4}, B ={2,5,6,7}. Construct an example of each of the following
(ii) a mapping from A to B which is not injective.

Prove that:

$\left(i\right)tan{225}^{\circ }cot{405}^{\circ }+tan{765}^{\circ }cot{675}^{\circ }=0$

$\left(ii\right)sin\frac{8\pi }{3}cos\frac{23\pi }{6}+cos\frac{13\pi }{3}sin\frac{35\pi }{6}=\frac{1}{2}$

$\left(iii\right)cos{24}^{\circ }+cos{55}^{\circ }+cos{125}^{\circ }+cos{204}^{\circ }+cos{300}^{\circ }=\frac{1}{2}$

$\left(iv\right)tan(-{225}^{\circ })cot(-{405}^{\circ })-tan(-{765}^{\circ })cot\left({675}^{\circ }\right)=0$

$\left(v\right)cos{570}^{\circ }sin{510}^{\circ }+sin(-{330}^{\circ })cos(-{390}^{\circ })=0$

$\left(vi\right)tan\frac{11\pi }{3}-2sin\frac{4\pi }{6}-\frac{3}{4}cose{c}^2\frac{\pi }{4}+4co{s}^2\frac{17\pi }{6}=\frac{3-4\sqrt{3}}{2}$

$\left(vii\right)3sin\frac{\pi }{6}sec\frac{\pi }{3}-4sin\frac{5\pi }{5}cot\frac{\pi }{4}=1$

The area (in sq. units) bounded by curves $y=\text{cosec}x,$ $y=\sec x,$ $y=\cos x$ and $y=\sin x$ on interval $(0,\frac{\pi }{2})$ is equal to

A person goes in for an examination in which there are four papers with a maximum of 20 marks from each paper. The number of ways in which one can get 40 marks is_______.

Identify the function based on the description.

1.It is periodic with period $2\pi$.

2.Domain of the function is R and the range is [-1, 1]

3.F(x) decreases strictly from 1 to -1 as x increases from 0 to $\pi$. [For eg. If $x_2>x_1,F\left(x_1\right)>f\left(x_2\right),xϵ[0,\pi ]$

4.F(x) increases strictly from -1 to 1 as x increases from $\pi to2\pi$. (foreg. If $x_2>x_1,f\left(x_2\right)>f\left(x_1\right),xϵ[\pi ,2\pi ]$

In triangle $ABC,a^2+c^2=2002b^2$, then $\frac{cotA+cotC}{cotB}$ is equal to

If the sum of a certain number of terms of the A.P 25, 22, 19, …. is 116. Find the last term.

In the matrix$\left[\begin{array}{cccc}2& 5& 19& -7\\ 35& -2& \frac{5}{2}& 12,\\ \sqrt{3}& 1& -5& 17\end{array}\right]$,write
(ii)The number of elements,

The vertex of the quadratic expression $y=3x^2+2x+5$ is given as:

Verify Lagrange's mean value theorem for

f(x): (4x-1)^-1 in the interval [1,4]

The value of [(cosA/cotA) + sinA] is:

If 7 divides ${32}^{{32}^{32}}$ then the remainder is ___

Differentiate the
functions with respect to x.

Value of the determinant $\left|\begin{array}{ccc}1& 3& 5\\ 3& 4& 5\\ 0& 2& 3\end{array}\right|is$___

Of the 10 prizes 5 prizes are of category Platinum 3 of gold and 2 of silver and they are placed in an enclosure for an olympiad contest. The prizes are awarded by allowing winners to select randomly from the prizes remaining. When the 8th participant goes to collect the prize what the probability that last 3 prizes are 1 of platinum 1 of gold and 1 of silver?

In answering a question on a multiple choice test, a student either
knows the answer or guesses. Let
be
the probability that he knows the answer and
be
the probability that he guesses. Assuming that a student who guesses
at the answer will be correct with probability
What is the probability that the student knows the answer given that
he answered it correctly?

The conjugate of complex no $\frac{1}{1+i}$ is ______.

Seventeen people are standing in a straight line facing South. What is Bhavna's position from the left end of the line? I. Sandeep is standing second to the left of Sheetal. Only five people stand between Sheetal and the one who is standing at the extreme right end of the line. Four people stand between Sandeep and Bhavna. II. Anita is standing fourth to the left of Sheetal. Less than three people are standing between Bhavna and Anita.