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An electrical system has open-closed switches S₁, S₂ and S₃ as shown in Figure.
The switches operate independently of one another and the current will flow from A and B either if $S_1$ is closed or if both $S_2$ and $S_3$ are closed. If $P\left(S_1\right)=P\left(S_2\right)=P\left(S_3\right)=\frac{1}{2}$, then find the probability that the circuit will work

Match the domain of the following functions:

If $lo{g}^2,log(2^x-1)$ and $log(2^x+3)$ are in A.P., write the value of x.

If R is a relation from set A = {11, 12, 13} to set B = {8, 10, 12} defined by y = x - 3, then write $R^{-1}$.

Given AB = 0. Which of the following options is correct

Let a, b, c be three real numbers such that a < b < c. Let f(x) be continuous $\forall x\in [a,c]$ and differentiable $\forall x\in (a,c)$. If $f^{\prime \prime }\left(x\right)>0\forall x\in (a,c)$ then

If $x^2$ + 2ax + 10 - 3a > 0 for each $xϵR$, then

If $\alpha and\beta$ are the roots of the equation $x^2-x+1=0then{\alpha }^{2009}+{\beta }^{2009}=$

If a function $y=f\left(x\right)$ is such that $f^{\prime }\left(x\right)$ is continuous function and satisfies $\left(f\right(x){)}^2=K+{\int }_0^x\left(\right(f\left(t\right){)}^2+\left(f^{\prime }\right(t\left){)}^2\right)dt,KϵR^{+}$, then

If a, b, c are in A.P. b, c, d are in G.P. and $\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P., prove that a, c, e are in G.p.

$cos{40}^{\circ }+cos{80}^{\circ }+cos{160}^{\circ }+cos{240}^{\circ }=$

Two dice are thrown. Find the odds in favour getting the sum

(i) 4

(ii) 5

(iii) What are the odd against getthing the sum 6?

Area bounded by curve xy=c, x-axis and ordinates x=1 and x=4, is

The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 192. The common ratio of the original G.P. is

Find the ratio of the coefficients of $x^n$ and $x^q$ in the expansion of $(1+x{)}^{p+q}.$

If a and b are two arbitrary constants, then the straight line (a-2b)x + (a+3b)y + 3a+4b = 0 will pass through

If x + y + z = xyz. Then

$\frac{3x-x^3}{1-3x^2}$ + $\frac{3y-y^3}{1-3y^2}$ + $\frac{3z-z^3}{1-3z^3}$ =

The angle of elevation of the top of a vertical tower from a point A, due east of it is ${45}^{\circ }$. The angle of elevation of the top of the same tower from a point B, due south of A is ${30}^{\circ }$. If the distance between A and B is $54\sqrt{2}$ m, then the height of the tower (in metres), is

If $\sin x+\cos x=\sqrt{2}\cos x$, then cosx - sin x is equal to

Find the Cartesian equation of the following plane:
$r.(2\hat{i}+3\hat{j}-4\hat{k})=1$

If A={(x,y):y=e^x,x belongs to R} and B={(x,y):y=e^-x,x belongs to R}, then write A intersection B.

Which of the following are true for the function $f\left(x\right)={\log }_e(3x^2-4x+5)$ (where $D_f,R_f$ represents domain and range of function $f\left(x\right)$ reapectively)

Find the integrals of the functions.
$\int \frac{si{n}^{3}x+co{s}^{3}x}{si{n}^{2}xco{s}^{2}x}dx.$

Letters of the word RANDOM are arranged in all possible ways and those words are arranged as in the dictionary. What is the position of the word RANDOM in this list?

Let $f\left(x\right)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7$. The value of $\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{h^3+3h}$ is equal to

If $f\left(x\right)=\{\begin{array}{c}K{x}^2\text{if}x\le 2\\ 3\text{if}x>2\end{array}$ is continuous at $x=2,$ then the value of $K$ is

Given cot=7/8 then evaluate (1+sin)(1-sin)/(1+cos)(1-cos)

And (1+sin)/cos

${lim}_{x\to 3}\frac{x^2-x-6}{x^3-3x^2+x-3}$

There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3 : 1. Find the value of n.

A man is employed to count Rs. 10710. He counts at the rate of Rs. 180 per minute for half an hour. After this he counts at the rate of Rs. 180 per minute for half an hour. After this he counts at the rate of Rs. 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.

In throwing a pair of dice, find the probability of getting a total of 8.

If (1 + i) (1 + 2i) (1 + 3i) ...... (1+ni) = a + ib, then 2.5.10.17 ...... ($1+n^2$) =

Let A = {1, 2, 3} and B = {3, 4}. Find $A\times B$ and show it graphically.

$Ina\Delta ABC,ifa=5,b=6andC={60}^{\circ },\text{show that its area is}\frac{15\sqrt{3}}{2}sq.units.$

Answer the following by appropriately matching the lists based on the information in Column I and Column II​​​​​​
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a}.y=f\left(x\right)\text{is given by}& \\ x=t^5-5t^3-20t+7& \\ \text{and}y=4t^3-3t^2-18t+3.& \\ \text{Then}-5\times \frac{dy}{dx}\text{at}t=1& \text{p.}0\\ \text{b}.\text{Let}P\left(x\right)\text{be a polynomial of degree}4,& \\ \text{with}P\left(2\right)=-1,P^{\prime }\left(2\right)=0,& \\ P^{\prime \prime }\left(2\right)=2,P^{\prime \prime \prime }\left(2\right)=-12\text{and}& \\ P^{iv}\left(2\right)=24,\text{then}{P}^{\prime \prime }\left(3\right)\text{is}& \text{q.}-2\\ \text{c}.y=\frac{1}{x},\text{then}\frac{\frac{dy}{\sqrt{1+y^4}}}{\frac{dx}{\sqrt{1+x^4}}}& r.2\\ \text{d}.f\left(\frac{2x+3y}{5}\right)=\frac{2f\left(x\right)+3f\left(y\right)}{5}\text{and}& \\ f^{\prime }\left(0\right)=p\text{and}f\left(0\right)=q.\text{Then},f^{\prime \prime }\left(0\right)\text{is}& \text{s.}-1\end{array}$