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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 $P(E/F)$.

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 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 $2x^{2}\frac{dy}{dx}=y^{2}+2xy.$

Find the general solution of the differential equation: $y\log y\frac{dx}{dy}+x=\frac{2}{y}$

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

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.

Let two rods placed on the ground be represented by vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

Determine the values of x for which $f(x)=\frac{x-3}{x+1}$, $x\ne -1$ is an increasing function.