List of practice Questions

$\int\frac{2x+\sin2x}{1+\cos2x}\, dx$ is equal to

Let $t_n$ denote the $n^{th}$ term in a binomial expansion. If $ \frac{t_{6}}{t_{5}}$ in the expansion of $(a+ b)^{n+4}$ and $ \frac{t_{5}}{t_{4}}$ in the expansion of $(a + b)^n$ are equal, then $n$ is

The value of $\sum\limits^{n}_{k=0}\left(i^{k}+i^{k+1}\right)$, where $i^2 = -1$, is equal to

If $f\left(x\right) = \int\limits^{sin\,x}_{2x}cos\left(t^{3}\right)dt$, then $f'{x}$ is equal to

If $A$ and $B$ are mutually exclusive events and if $ p(B)=\frac{1}{3},p(A\cup B)=\frac{13}{21}, $ then $P(A)$ is equal to

Out of $15$ persons $10$ can speak Hindi and $8$ can speak English. If two persons are chosen at random, then the probability that one person speaks Hindi only and the other speaks both Hindi and English is

$\int \frac{e^{x}}{x}\left(x\,log\,x+1\right)dx$ is equal to

If z =$\frac{2-i}{i}$ = , then Re(z$^2$) + lm(z$^2$) is equal to

If the equation $2x^2 - (a+3)x + 8 = 0$ has equal roots, then one of the values of $a$ is

The plane $ \overrightarrow{r}=s(\hat{i}+2\hat{j}-4\hat{k})+t(3\hat{i}+4\hat{j}-4\hat{k}) $ $ +(1-t)(2\hat{i}-7\hat{j}-3\hat{k}) $ is parallel to the line

If $ {{x}^{2}}+4ax+2>0 $ for all values of $ x, $ then $a$ lies in the interval

If $z = \cos\left(\frac{\pi}{3} \right) - i \sin \left(\frac{\pi }{3}\right),$ the $z^{2} - z +1 $ is equal to

If $z = i^9 + i^{19}$ , then $z$ is equal to

Let $S_{1}$ be a square of side $5\,cm$. Another square $S_{2}$ is drawn by joining the midpoints of the sides of $S_{1}$ Square $S_{3}$ is drawn by joining the midpoints of the sides of $S_{2}$ and so on. Then (area of $S_{1}$ + area of $S_{2}$ + area of $S_{3}$ $+\ldots+$ area of $S_{10}$) =

Let $x_{1},x_{2},\cdots,x_{n}$ be in an $A.P$. If $x_{1}+x_{4}+x_{9}+x_{11}+x_{20}+x_{22}+x_{27}+x_{30}=272, $ then $x_{1}+x_{2}+x_{3}+\cdots+x_{30}$ is equal to

If $ \alpha ,\beta ,\gamma $ are the cube roots of a negative number $p$, then for any three real numbers, $ x,y,z $ the value of $ \frac{x\alpha +y\beta +z\gamma }{x\beta +y\gamma +z\alpha } $ is

If one of the roots of the quadratic equation $ax^2 - bx + a = 0$ is $6$, then value of $\frac{ b}{ a}$ is equal to

If $ {{x}^{2}}-px+q=0 $ has the roots $ \alpha $ and $ \beta $ then the value of $ {{(\alpha -\beta )}^{2}} $ is equal to

If $a + 1, 2a + 1, 4a - 1$ are in arithmetic progression, then the value of $a$ is

Two tangent galvanometers $A$ and $B$ have coils of radii $8 \,cm$ and $16\, cm$ respectively and resistance $ 8\,\Omega $ , each. They are connected in parallel with a cell of emf $4\, V$ and negligible internal resistance. The deflections produced in the tangent galvanometers $A$ and $B$ are $ 30{}^\circ $ and $ 60{}^\circ $ respectively. If $A$ has $2$ turns, then $B$ must have

In the Wheatstone's network given, $ P=10\,\Omega , Q=20\,\Omega ,\,R=15\,\Omega , S=30\,\Omega , $ the current passing through the battery (of negligible internal resistance) is

A uniform heavy rod of length L and area of cross section 'A' is hanging from a fixed support. It young's modulus of the material of the rod is Y, the increase the length of rod is ( p is density of the mateiral of the rod)

A research satellite of mass 200 kg circles the earth in an orbit of average radius $\frac{3R}{2}$ where R is the radius of earth. Assuming the gravitatioaal pull on a mass of 1 kg on the earth's surface to be 10 N, the puII on the sateIlite will be

A bullet is fired from a gun. The force on the bullet is given by $ F=600-2\times {{10}^{5}}t, $ where F is in newton and t is in second. The force on the bullet becomes zero as soon as it leaves the barrel. What is the average impulse imparted to the bullet?

A fighter aircraft is looping in a vertical plane. The minimum velocity at the highest point is (Given, r = radius of the loop)