Embibe Experts Solutions for Chapter: Solved Mathematics Paper-2, Exercise 1: Exercise

Find $\int \left(1+x-\frac{x^3}{4}\right)\mathrm{d}x.$

Show that $\left(a+1\right)$ is a factor of $\left|\begin{array}{ccc}a+1& 2& 3\\ 1& a+1& 3\\ 3& -6& a+1\end{array}\right|$

Solve the differential equation $\left(1+x^2\right)\frac{dy}{dx}+2xy-4x^2=0$, subject to the initial condition $y\left(0\right)=0$.

Solve the following differential equation:

$\frac{dy}{dx}+\frac{y}{x}=x^2,y\left(1\right)=1$

In a certain college, $4\%$ of boys and $1\%$ of girls are taller than $1.75$ metres. Furthermore, $60\%$ of the students in the college are girls. A student selected at random from the college is found to be taller than $1.75$ metres. Find the probability that the selected student is a girl.

If a fair coin is tossed $10$ times, find the probability of getting exactly six heads.

Using product rule, show that the following four points with given position vectors are coplanar: $4\hat{i}+8\hat{j}+12\hat{k}, 2\hat{i}+4\hat{j}+6\hat{k}, 3\hat{i}+5\hat{j}+4\hat{k}$ and $5\hat{i}+8\hat{j}+5\hat{k}$.

Let $X$ denote the number of vowels in a name a student selected from the following group of $15$ students at random. Sauchintya, Debapriya, Sanchari, Madhurima, Priyadarshini, Kalpak, Dishan, Rik, Ashmit, Aparajita, Rishav, Ritwik, Sayani, Dipanwita, Saranya. Write the probability distribution of $X$ and then find the expectation of $X$.