BITSAT 2025

The function $f(x)=\frac{x}{\sin x}$ is strictly increasing in the interval.

For what values of the parameter ' $a$ ' does the function $f(x)=x^3+3(a-7) x^2+3\left(a^2-9\right) x-1$ have a positive point of maximum.

The slope of the curve $2 y^2=a x^2+b$ at $(1,-1)$ is -1 . Then, the value of $b$ is

What is the area enclosed by the curves $y=x^4$ and $y=x^{\frac{1}{3}}$

If ${ }^n C_{n-r}+3 \cdot{ }^n C_{n-r+1}+3 \cdot{ }^n C_{n-r+2} +{ }^n C_{n-r+3}={ }^x C_r$, then the value of $x$ is

What is the coefficient of $x^{50}$ in $(1+x)^{41}\left(1-x+x^2\right)^{40}$.

The locus of the middle points of chords of the circle $x^2+y^2=25$ which are parallel to the line $x-2 y+3=0$ is

What is the set of values of a for which the point ( $2 a, a+1$ ) is an interior point of the larger segment of the circle $x^2+y^2-2 x-2 y-8=0$ made by the chord $x-y+1=0$.

If ' $a$ ' is a complex number such that $|a|=1$. Find the value of $a$, so that the equation $a z^2+z+1=0$ has one purely imaginary root.

Let $z$ be a complex number for which $\left|2 z \cos \theta+z^2\right|>1$, if $|z|

The value of integral $\int_a^b e^x d x$ as limit of sums is

What is the value of the definite integral $\int_0^\pi \log (\sin x) d x$ ?

The solution of the differential equation $\frac{d y}{d x}+x \sin 2 y=x^3 \cos ^2 y$ is

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at points where it is intersected by the line $l x+m y+n=0$. The point of intersection of tangents at these points is

A rectangle is inscribed in an ellipse with the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
What is the maximum area of the rectangle that can be inscribed in the ellipse?

If $f: X \rightarrow Y$ be a function defined by $f(x)=a \sin \left(x+\frac{\pi}{4}\right)+b \cos x+c$ and $f$ is bijective, then the set $X$ with $\theta=\tan ^{-1}\left(\frac{a+\sqrt{2} b}{a}\right)$ is

If the function $f: R \rightarrow R$ is defined by $f(x)=x^2+5 x+9$, then $f^{-1}(9)$ is equal to

The locus of the mid-point of the chords of the circle $x^2+y^2=16$ which are tangents to the hyperbola $9 x^2-16 y^2=144$ is $\left(x^2+y^2\right)^2=k x^2-l y^2$. Then, the sum of values of $k$ and $l$ is

The value of $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^3}$ is

Consider the function $g(x)$ defined as
$$ g(x)=\left\{\begin{array}{cc} \frac{x^2-4}{x^2-2|x-2|-4}, & x \neq 2 \\ \frac{3}{4}, & x=2 \end{array}\right. $$
Which of the following statements is true about the continuity of $g(x)$ ?

If $\mathop {\lim }\limits_{x \to \infty }\left\{\frac{x^2-1}{x+1}-a x-b\right\}=2$. The value of $a$ is

The number of real solution of $\sqrt{\left(7-\log _3|x|\right)}=4-\log _3|x|$ is equal to

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

If $A, B, C$ are the angles of a $\triangle A B C$, then

$$ \Delta=\left|\begin{array}{ccc} \sin 2 A & \sin C & \sin B \\ \sin C & \sin 2 B & \sin A \\ \sin B & \sin A & \sin 2 C \end{array}\right| \text { is equal to } $$

What is the probability of getting a sum of 9 in a single throw of three fair dice?

In a binomial distribution, the mean is 10 and the variance is 6 . Then, its median is

A four digit number is formed with digits $1,3,4$, 5 with no repetition. What is the probability that the number is divisible by 5 ?

The angles of a triangle are in the ratio $1: 2: 7$. The ratio of the greatest side to the least side is $(k+1):(k-1)$. The value of $k$ is

Let $A, B$ and $C$ be the angles of a triangle and $\tan \frac{A}{2}=\frac{2}{5}, \tan \frac{B}{2}=\frac{3}{7}$. Then, $\tan \frac{C}{2}$ is equal to

For what value of $a, 6$ lies between the roots of the equation $x^2+2(a-3) x+9=0$.

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

The coefficient of $x^n$ in the expansion of $\frac{1-a x-x^2}{e^x}$ is

For three numbers $a, b, c$ between 2 and 18 such that their sum is 25 , the numbers $2, a, b$ are in AP and the numbers $b, c, 18$ are in GP Then, the value of $a+b+c$ is

If $a, b, c, d$ be four positive unequal quantities and $s=a+b+c+d$, then $(s-a)(s-b)(s-c) (s-d)>k a b c d$. Then, value of $k$ is

If $p, q, r$ are in GP and the line $p x+q y+r=0$ forms a triangle with the coordinate axes, what is the area of the triangle if $p=2, q=4$ and $r=8$ ?

If the equation $2 h x y+2 g x+2 f y+c=0$ represents two straight lines, then the area of rectangle formed with the coordinates axes is

If four lines $a x \pm b y \pm c=0$ encloses a rhombus, then the area is

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

If $A+B=\frac{\pi}{4}$, then $(1+\tan A)(1+\tan B)$ is equal to

If $\mathbf{a , b , c}$ are vectors such that $|\mathbf{b}|=|\mathbf{c}|$ then $\{(\mathbf{a}+\mathbf{b}) \times(\mathbf{a}+\mathbf{c})\} \times(\mathbf{b} \times \mathbf{c}) \cdot(\mathbf{b}+\mathbf{c})$ is equal to

A pure resistive circuit element $X$, when connected to an AC supply of peak voltage 200 V , gives a peak current of 5 A . A second circuit element $Y$, when connected to the same AC supply also gives the same value of peak current but the ...

Consider a hydrogen atom with its electron in the $n$th orbit. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV , then the value of $n$ is ( $h c=1242 \mat...

The mass of proton is 1.0073 u and that of neutron is $1.0087 \mathrm{u}(\mathrm{u}=$ atomic mass unit). The binding energy of ${ }_2 \mathrm{He}^4$ is

A dielectric slab of dielectric constant $k=3$ is filling $\frac{3}{4}$ th space of the capacitor.

When capacitor is charged, then \% of total energy stored in dielectric is

Two wires of the same material (Young's modulus $=Y$ ) and same length $L$ but radii $R$ and $2 R$ respectively are joined end to end and a weight $w$ is suspended from the combination as shown in the figure. The elastic potential energy in...

A current $I=10 \sin (100 \pi t) A$ is passed in the first coil, which induces a maximum emf of $5 \pi \mathrm{~V}$ in second coil. The mutual inductance between the coil is

A parallel plate capacitor with plate area $A$ and separation between plates $d$ is charged with constant current $I$. Consider a plane surface of area $\frac{5 A}{6}$ parallel to the plates and drawn between the plates. The displacement cu...

A large block of ice 10 cm thick with a vertical hole drilled through it is floating in a lake. The minimum length of the rope required to scoop out a bucket full of water through the hole is (density of ice $=0.9 \mathrm{~g} / \mathrm{cm}^...

A water film is made between two straight parallel wires of length 10 cm each and at a distance of 0.5 cm from each other. If the distance between the wires is increased by 1 mm , then work done (in ergs) is [Given, surface tension of water...

A body is projected vertically upwards from the surface of Earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be