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There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, is a random order till both the faulty machines are identified. Then the probability that only two tests are needed

The number of integer(s) $x$ satisfying the inequation $\frac{x(x^2+2x+2)({\log }_{2}x-3)}{(x+3{)}^2(x^2-x-6)}\le 0,$ is

The domain of $f\left(x\right)=\sqrt{x-\sqrt{1-x^2}}$ is

Let $y=y\left(x\right)$ be a solution curve of the differential equation $(y+1){\tan }^{2}xdx+\tan xdy+ydx=0,x\in (0,\frac{\pi }{2}).$ If $\underset{x\to 0^{+}}{lim}xy\left(x\right)=1,$ then the value of $y\left(\frac{\pi }{4}\right)\text{is}$

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $p_k=(\alpha {)}^k+(\beta {)}^k,k\ge 1$, then which one of the following statements is not true ?

If the slope of the first line is half the slope of the second line and the tangent of the angle between them is $\frac{1}{4}$, then the slope of the first line can be

For the quadratic equation, $a{x}^2+bx+c=0$ which has non-zero roots $\alpha \&\beta$, the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ is

Let $I_n={\int }_0^{\infty }{e}^{-x}(sinx{)}^{n}dx,nϵN,n>1$ then $\frac{I_{2008}}{I_{2006}}$ equals

If $\alpha ,\beta$ be the roots of $x^2–x+3=0$, then the value of $({\omega }^2\alpha +\omega \beta )(\omega \alpha +{\omega }^2\beta )$ is, ($\omega$ is complex cube root of unity)

If $|a|<1,|b|<1$ then the sum of the series $1+(1+a)b+(1+a+a^2)b^2+(1+a+a^2+a^3)b^3+$ ………… to infinite terms is

The Boolean expression $\sim (p\vee q)\vee (\sim p\wedge q)$ is equivalent to

If α and β are the roots of the equation a $x^2$ + bx + c. find the equation whose roots are - α and - β.

The number of values of $x$ satisfying the equation $(x^2+7x+11{)}^{(x^2-4x-21)}=1$ is

The area bounded by the curve $5\left\{y\right\}=3\left[x\right]-2,$ $1\le y\le 2$ $($where $\{.\}$ and $[.]$ represent the fractional part and greatest integer function respectively$)$ is

The most recycled quantity of garbage is .

If the graph of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is





Then, the possible value(s) of $a$ can be

If $\alpha ={30}^{\circ }$ and $\beta ={60}^{\circ }$, then the value of $\frac{\sin \alpha +\sec 2\alpha +\tan (\alpha +{15}^{\circ })}{\tan \beta +\cot (\frac{\beta }{2}+{15}^{\circ })+\tan \alpha }$ is

If $\alpha$ + $\beta$ - $\gamma$ = $\pi$, then $si{n}^2\alpha$ + $si{n}^2\beta$ - $si{n}^2\gamma$ =

If the function $f\left(x\right)=\{\begin{array}{cc}1+x^{a-3},& x<0\\ \cos x,& x\ge 0\end{array}$ is differentiable, then the least integral value of $a$ is

The value of $co{s}^{-1}x+co{s}^{-1}(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^2})$ where $(\frac{1}{2}\le x\le 1)$ is equal to

Question 4 (iii)

Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.

(iii) -1.2, -3.2, -5.2, -7.2 …

Let
R be a relation from N
to N
defined by R = {(a,
b):
a,
b
∈
N
and a
= b²}.
Are the following true?

(i) (a,
a)
∈
R, for all
a ∈
N (ii) (a,
b)
∈
R, implies (b,
a)
∈
R

(iii) (a,
b)
∈
R, (b,
c)
∈
R implies (a,
c)
∈
R.

Justify your answer
in each case.

The number of solutions of $3sec\theta -5=4tan\theta$ in $[0,4\pi ]$ must be

If $b+ic=(1+a)z$ and $a^2+b^2+c^2=1,$ where $a,b,c\in R,$ then $\frac{1+iz}{1-iz}=\frac{a+ib}{k}$ where $k$ is equal to

What is the adjoint of the matrix $\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 1\\ 2& 3& 4\end{array}\right]$?

Find the principal and general solutions of the equation

If m is the A.M. of two distinct real numbers I and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals: