Paper Generator

Obtain the value of $\Delta=\begin{vmatrix}1+x&1&1\\1&1+y&1\\1&1&1+z\end{vmatrix}$ in terms of x, y, and z. Further, if $\Delta=0$ and x, y, z are non-zero real numbers, prove that $x^{-1}+y^{-1}+z^{-1}=-1$

If $P=\begin{bmatrix}1&-1&0\\2&3&4\\0&1&2\end{bmatrix}$ and $Q=\begin{bmatrix}2&2&-4\\-4&2&-4\\2&-1&5\end{bmatrix}$, find (QP) and hence solve the following system of equations using matrices: $x-y=3$, $2x+3y+4z=17$, $y+2z=7$

Find the domain of $g(x)=\cos^{-1}(x^{2}-1)$. Hence, find the value of x for which $g(x)=\frac{\pi}{3}$. Also, write the range of $\cos^{-1}x$ other than its principal branch.

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.

Solve the following Linear Programming Problem graphically: Maximize $Z=\frac{2x}{5}+\frac{3y}{10}$ subject to constraints $2x+y\le 1000$, $x+y\le 800$, $x, y \ge 0$.

Solve the following Linear Programming Problem graphically: Maximise $Z=200x+120y$ subject to the constraints $x+y\le 300$, $3x+y\le 600$, $x-y\ge -100$, $x,y\ge 0$

Solve the following linear programming problem graphically: Maximize $Z=10500x+9000y$ Subject to constraints $x+y \le 50$, $2x+y \le 80$, $x, y \ge 0$

Find the shortest distance between the lines $\vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k}$ and $\vec{r}=(1+2\mu)\hat{i}+(2-5\mu)\hat{k}+(4\mu-1)\hat{j}$.

Find a point on the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z-3}{2}$ at a distance of $\sqrt{2}$ units from the point (1, 2, 3).

If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then prove that $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}\le 9$.

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $5\hat{i}-2\hat{j}$, $5\hat{i}+8\hat{j}$ and $a\hat{i}-52\hat{j}$ respectively, find the value of 'a'.

Solve the differential equation $(x+2y^{3})dy=y~dx$.

Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2 e^{-y}-1$ given that $y=0$ when $x=0$.

Find the general solution of the differential equation $2x^{2}\frac{dy}{dx}=y^{2}+2xy.$