BITSAT 2024

Consider the function $ f(x)=\frac{|x-1|}{x^{2}} $, then $ f(x) $ is

The maximum area of rectangle inscribed in a circle of diameter $ R $ is

The area enclosed by the curves $ y=x^{3} $ and $ y=\sqrt{x} $ is

The line $ y=m x $ bisects the area unclosed by lines $ x=0, y=0 $ and $ x=\frac{3}{2} $ and the curve $ y=1+4 x-x^{2} $. Then, the value of $ m $ is

The coefficient of $ x^{2} $ term in the binomial expansion of $ \left(\frac{1}{3} x^{\frac{1}{2}}+x^{\frac{-1}{4}}\right)^{10} $ is

The locus of the point of intersection of the lines $ x=a\left(\frac{1-t^{2}}{1+t^{2}}\right) $ and $ y=\frac{2 a t}{1+t^{2}} $ represent $ (t $ being a parameter)

The locus of the mid-point of a chord of the circle $ x^{2}+y^{2}=4 $, which subtends a right angle at the origin is

The points represented by the complex number $ 1+i,-2+3 i, \frac{5}{3} i $ on the argand plane are

The modulus of the complex number $ z $ such that $ |z+3-i|=1 $ and $ \arg (z)=\pi $ is equal to

$ \int_{0}^{\infty} \frac{d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)} $ is

The value of definite integral $ \int_{0}^{\pi / 2} \log (\tan x) d x $ is .

The solution of the differential equation
$ (x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^{2} $ is

If $ x \sqrt{1+y}+y \sqrt{1+x}=0 $, then $ \frac{d y}{d x}= $

Let $ [x] $ denote the greatest integer $ \leq x $. If $ f(x)=[x] $ and $ g(x)=|x| $, then the value of $ f\left(g\left(\frac{8}{5}\right)\right)-g\left(f\left(-\frac{8}{5}\right)\right) $ is

The foci of hyperbola $ 4 x^{2}-9 y^{2}-1=0 $ are

If $ y=\tan ^{-1}\left(\frac{\sqrt{x}-x}{1+x^{\frac{3}{2}}}\right) $, then $ y^{\prime}(1) $ is equal to

Let $ f $ be the function defined by$ f(x)=\left\{\begin{array}{cc} \frac{x^{2}-1}{x^{2}-2|x-1|-1}, & x \neq 1 \\ \frac{1}{2}, & x=1 \end{array}\right. $

If the number of available constraints is 3 and the number of parameters to be optimise is 4 , then

The Boolean expression $ \sim(p \vee q) \vee(\sim p \wedge q) $ is equivalent to

If $ A=\frac{1}{3}\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right] $ is an orthogonal matrix, then

If matrix $ A=\left[\begin{array}{ccc}3 & -2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1\end{array}\right] $ and $ A^{-1}=\frac{1}{k} \operatorname{adj}(A) $,

Suppose $ p, q, r \neq 0 $ and system of equation $ (p+a) x+b y+c z=0 $, $ a x+(q+b) y+c z=0 $, $ a x+b y+(r+c) z=0 $, has a non-trivial solution, then the value of $ \frac{a}{p}+\frac{b}{q}+\frac{c}{r} $ is

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions

A person invites a party of 10 friends at dinner and place so that 4 are on one round table and 6 on the other round table. The number of ways in which he can arrange the guests is

In a binomial distribution, the mean is 4 and variance is 3 . Then, its mode is

The probability of getting 10 in a single throw of three fair dice is

Let $ A, B $ and $ C $ are the angles of a triangle and $ \tan \frac{A}{2}=\frac{1}{3}, \tan \frac{B}{2}=\frac{2}{3} $. Then, $ \tan \frac{C}{2} $ is equal to

$ A B C $ is a triangular park with $ A B=A C=100 \mathrm{~m} $. A TV tower stands at the mid-point of $ B C $. The angles of elevation of the top of the tower at $ A $, $ B, C $ are $ 45^{\circ}, 60^{\circ}, 60^{\circ} $ respectively. The ...

Number of real solution of $ \sqrt{5-\log _{2}|x|} $ $ =3-\log _{2}|x| $ is equal to

Roots of the equation $ x^{2}+b x-c=0(b, c > 0) $ are

If $ a > 0, b > 0, c > 0 $ and $ a, b, c $ are distinct, then $ (a+b)(b+c)(c+a) $ is greater than

The coefficient of $ x^{n} $ in the expansion of $ \frac{e^{7 x}+e^{x}}{e^{3 x}} $ is

There are four numbers of which the first three are in GP and the last three are in AP, whose common difference is 6 . If the first and the last numbers are equal, then two other numbers are

If $ \sum\limits_{k=1}^{n} k(k+1)(k-1)=p n^{4}+q n^{3}+t n^{2}+s n $, where $ p, q, t $ and $ s $ are constants, then the value of $ s $ is equal to

In a statistical investigation of 1003 families of Calcutta, it was found that 63 families has neither a radio nor a TV, 794 families has a radio and 187 has TV. The number of families in that group having both a radio and a TV is

If $ a, c, b $ are in GP, then the area of the triangle formed by the lines $ a x+b y+c=0 $ with the coordinates axes is equal to

Number of solutions of equations $ \sin 9 \theta=\sin \theta $ in the interval $ [0,2 \pi] $ is

From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be $ 30^{\circ} $ and $ 45^{\circ} $. The height of tower is

Let $ \mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=x \hat{\mathbf{i}}+\hat{\mathbf{j}}+(1-x) \hat{\mathbf{k}} $ and $ \mathbf{c}=y \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(1+x-y) \hat{\mathbf{k}} $. Then, $ [\mathbf{a} \mathbf{b} \m...

The magnitude of projection of line joining ( 3,4 , $ 5) $ and $ (4,6,3) $ on the line joining $ (-1,2,4) $ and $ (1,0,5) $ is