Paper Generator

Show that $f:R\rightarrow R$ defined as $f(x)=\frac{x}{\sqrt{1+x^{2}}}$ is one-one but not onto.

A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.

A die is rolled. Consider events: $A=\{1,2,5\}$, $B=\{3,5\}$, $C=\{2,3,4,5\}$ and hence find: (i) $P(A|C)$ and $P(C|A)$ (ii) $P(A\cap B|C)$ and $P(A\cup B|C)$.

In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that: (i) exactly one surgery is successful, (ii) at most two surgeries are successful.

Mother, Father and Son line up at random for a family picture. Let events E: Son on one end and F: Father in the middle. Find $P(E/F)$.

The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.

Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y) \text{ is divisible by } n, x, y\in Z\}$. Determine if R is an equivalence relation.

Let $A=\mathbb{R}-\{3\}$ and $B=\mathbb{R}-\{1\}$. A function $f:A\rightarrow B$ is defined by $f(x)=(\frac{x-2}{x-3})$. Find whether f is one-one and onto.

Check whether $f:R-\{3\} \rightarrow R$ defined as $f(x)=\frac{x-2}{x-3}$ is onto or not.

A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red?

If $3P(A)=P(B)=\frac{3}{5}$ and $P(A|B)=\frac{2}{3}$ then $P(A\cup B)$ is:

For two events A and B such that $P(A) \ne 0$ and $P(B) \ne 1$, $P(A^{\prime}/B^{\prime})=$

If E and F are two independent events such that $P(E)=\frac{3}{10}$, $P(E\cup F)=\frac{1}{2}$ then $P(E|F)-P(F|E)$ is equal to:

The corner points of the feasible region determined by the system of linear constraints are (0,0), (0, 40), (20, 40), (60, 20) and (60, 0). If the objective function of an LPP is $Z=4x+3y$, then the maximum value is :

Direction ratios of lines $l_{1}$ and $l_{2}$ are <12,-3, 9> and <4, q,-p> respectively. The values of p and q for which $l_{1}$ and $l_{2}$ are parallel are respectively:

Direction cosines of the line given by equations $\frac{2x-1}{4}=\frac{1-y}{3}=\frac{-z}{6}$ are

If points $(2, 3)$, $(0, 4)$ and $(p, 2)$ are collinear, then the value of p is:

If $(3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0}$ then the values of p and q are :

If $(\vec{a}+\vec{b}).(\vec{a}-\vec{b})=198$ and $|\vec{a}|=10|\vec{b}|$, then :