COMEDK 2025 Evening Shift

If the function $f(x)=\mu \sin x+\frac{1}{3} \sin 3 x$ has its derivative equal to zero at $x=\frac{\pi}{3}$, then the value of ' $\mu$ ' is

A man is moving away from a tower 41.6 m high at a rate of $2 \mathrm{~m} / \mathrm{s}$. If the eyelevel of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a di...

If a quadratic function in $x$ has the value 19 when $x=1$ and has a maximum value 20 when $x=2$, then the function is

If the length of the diagonal of a square is increasing at the rate of $0.1 \mathrm{~cm} / \mathrm{sec}$.

What is the rate of increase of its area when the side is $\frac{15}{\sqrt{2}} \mathrm{~cm}$ ?

Let $f(x)=x \sqrt{4 a x-x^2}, a>0$ then $f^{\prime}(x)$ at $x=2 a$ is :

The function $y=\frac{\log x}{x^3}$ is strictly increasing function for

If the area under the curve $y=\sqrt{a^2-x^2}$ included between the lines $x=0$ and $x=a$ is 4 sq units. Then the value of ' $a$ ' is

Area of the region bounded by the curve $y=\cos x$ between $x=-\frac{\pi}{2}$ and $x=\pi$ is ------------------

Given that $z$ is a real number and $z=\frac{\lambda+4 i}{1+\lambda i}$ where $\lambda \in R$, then the possible value of $\lambda$ is :

If $z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^5+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^5$, then

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\sqrt{\tan x}} d x=$

$\int_0^1 x(1-x)^{99} d x=$

$\int\limits_{-2}^2 \frac{|x-3|}{x-3} d x=$

Find the function ' $f$ ' which satisfies the equation $\frac{d f}{d x}=2 f$, given that $f(0)=e^3$

The number of solutions of $\frac{d y}{d x}=\frac{y+1}{x-1}$, when $y(1)=2$ is :

The solution of the differential equation: $x \cos y d y=\left(x e^x \log x+e^x\right) d x$ is

Solve the following differential equation $\cos ^2 x \frac{d y}{d x}+y=\tan x$, given that $y(0)=1$. Hence find $y\left(\frac{\pi}{4}\right)$

If $y=x+e^x$ then $\frac{d^2 x}{d y^2}=$

If $x=a\left[\left\{\cos t+\frac{1}{2} \log \left(\tan ^2 \frac{t}{2}\right)\right\}\right]$ and $y=a \sin t$ then $\frac{d y}{d x}=$

A function $f$ from the set of natural numbers to integers defined by
$$f(n)=\left\{\begin{array}{l}
\frac{n-1}{2}, \quad \text { when } n \text { is odd } \\
-\frac{n}{2}, \quad \text { when } n \text { is even }
\end{array} \quad\right. \...

If the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the foci of the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide, then the value of $b^2$ is

\(\int \frac{\sin 2 x}{(1+\sin x)(2+\sin x)} d x=a \log |1+\sin x|-b \log |2+\sin x|+c\)
then the value of $a$ and $b$ is ----------------

$\int\left(e^{x \log _e 6}\right) e^x d x=\phi(x)+c$ then $\phi(x)=$

The value of $\tan \left\{\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)-\frac{\pi}{2}\right\}$ is

$\sin ^{-1}(x-1)+\cos ^{-1}(x-3)+\tan ^{-1}\left(\frac{x}{2-x^2}\right)=\cos ^{-1} k+\pi$, then the value of ' $k$ ' is

The relationship between a and b for the continuous function
$f(x)=\left\{\begin{array}{ll}a x+1, & \text { if } x \leq 3 \\ b x+3, & \text { if } x>3\end{array}\right.$ at $x=3$ is

Evaluate: $\lim _\limits{x \rightarrow 0} \frac{\sqrt[3]{1+x}-\sqrt[3]{1-x}}{x}$

If $\lim\limits_{x \rightarrow 1} \frac{x^4-1}{x-1}=\lim\limits_{x \rightarrow k} \frac{x^3-k^3}{x^2-k^2}$, then the value of K is :

Given $Z=80 x+120 y$, subject to constraints are $x+3 y \leq 30 ; 3 x+4 y \leq 60 ; x \geq 0 ; y \geq 0$.
P is one of the corner points of the feasible region for the given Linear Programming Problem.
Then the coordinate of $P$ is

The solution for the following system of inequalities $3 x-7<5+x$ and $11-5 x \leq 1$ on a real number line is

If $A=\left[\begin{array}{ccc}4 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right]$ then $A^{-1}$ exists if :

Value of the determinant of a matrix $A$ of order $3 \times 3$ is 7 . Then the value of the determinant formed by the cofactors of matrix A is

If $X=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ and $Y=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$ then $B...

If $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ and $(A I)^2=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]$ where I is the identity matrix then

Kiran purchased 3 pencils, 2 notebooks and one pen for ₹41.
From the same shop Manasa purchased 2 pencils, one notebook and 2 pens for ₹ 29 , while Shreya purchased 3 pencils, 2 notebooks and 2 pens for ₹ 44.
The above situation can be repr...

The area of a triangle formed by the lines joining the vertex of the parabola $x^2=\lambda y$ to the ends of its latus rectum is 18 sq units then the value of $\lambda$ is

How many natural numbers are there between 100 and 1000 such that at least one of their digits is $6 ?$

The number of words that can be formed with the letters of the word 'DEFINITE' if two vowels are together and the other two are also together but separated from the first two is

Two numbers are selected at random from integers 1 to 9 .
If their sum is even, what is the probability that both the numbers are odd?

If A and B are two events such that $P(\bar{A})=0.3, P(B)=0.4, P(A \cap \bar{B})=0.5$, then find the value of $P(B / A \cup \bar{B})$

In a game, a man wins ₹ 1000 if he gets an even number greater than or equal to 4 on a fair dice and loses ₹ 200 for getting any other number on the dice. If he decides to throw the dice until he wins or maximum of three times, then his exp...

A pot contains 5 red and 2 green balls. A ball is drawn at random from this pot. If a drawn ball is green, then a red ball is added to the pot. If a drawn ball is red, then a green ball is added to the pot, while the original ball drawn is ...

An unbiased die is tossed twice. What is the probability of getting a 4,5 or 6 on the first toss and a $1,2,3$ or 4 on the second toss?

Given that n number of arithmetic means are inserted between two pairs of numbers $a, 2 b$ and $2 a, b$; where $a, b \in R$. If the $m^{\text {th }}$ means in the two cases are the same, then the ratio $a: b$ is equal to

A geometric progression consists of an even number of terms. If the sum of all the terms is five times the sum of the terms occupying the odd places, then the common ratio of the geometric progression is

If $P=\{5 m: m \in N\}$ and $Q=\left\{5^m: m \in N\right\}$, where $N$ is set of natural numbers, then

The relation $R=\{(1,1),(2,2),(3,3)\}$ on the set $\{1,2,3\}$ is

Two finite sets have $m$ and $n$ elements. The total number of proper subsets of the first set is 119 more than the total number of subsets of the second set. Find the value of $m-n$

If the standard deviation of $0,1,2,3 \cdots\cdots\cdots9$ is ' $k$ ' then the standard deviation of $10,11,12,13, \cdots\cdots\cdots\cdots19$ will be :

The angle between two lines is $45^{\circ}$ and slope of one line is $\frac{1}{4}$ then which is the possible value of the slope of the other line.