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Sketch the curve $\{(x, y): 100x^{2}+25y^{2}=2500\}$ and find the area of the region enclosed by it, using integration.

Using integration, find the area of the region enclosed by the curve $y=|x-6|$, the x-axis, and between $x=4$ and $x=8$.

Evaluate: $\int_{0}^{1}\frac{x~\tan^{-1}x}{(1+x^{2})^{3/2}}dx$

Find: $\int\frac{x}{(x-1)(x^{2}+4)}dx$

A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.

Find the sub intervals in which $f(x)=\cot^{-1}(\sin x+\cos x)$, $x \in (0,\pi)$ is increasing and decreasing.

Find the differential of $x^{\cot x}+\frac{2x^{2}-3}{2x^{2}-x+2}$ with respect to x.

If $y\sqrt{x^{2}+1}=\log\sqrt{x^{2}+1}-x$, show that $(x^{2}+1)\frac{dy}{dx}+xy+1=0$.

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.

Given that $P=\begin{bmatrix}2&-1\\3&4\end{bmatrix}$, $Q=\begin{bmatrix}5&2\\7&4\end{bmatrix}$ and $R=\begin{bmatrix}2&5\\3&8\end{bmatrix}$, find a matrix S such that PQ - RS is a null matrix.

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

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 $P(E/F)$.