COMEDK 2026 Afternoon Shift

The function $f(x)=e^{a x}+e^{-a x}, x \in \mathbb{R}$ and $a<0$, is strictly decreasing for all values of ' $x$ ', where

The absolute maximum and minimum values of the function $f(x)=\sin x+\sqrt{3} \cos x$ in $[0, \pi]$ are

An open hemispherical storage tank has radius 13 m . Oil flows into the tank such that the depth ' $\boldsymbol{h}$ ' of oil in the tank changes at the rate of $3 \mathrm{~m} / \mathrm{hr}$. When the depth $\boldsymbol{h}=1 \mathrm{~m}$, th...

A movie screen on a wall is $\mathbf{2 0}$ feet high and $\mathbf{1 0}$ feet above the floor. What is the maximum viewing angle $\boldsymbol{\theta}$ (in radians) that can be achieved by positioning yourself at the optimal distance from the...

If $f(x)=x^3+\frac{3}{2} x^2+3 x+3$, then $f(x)$ is

Find the area bounded by the curve $y=|2-x|$, the $x$-axis, and the lines $x=0$ and $x=5$

The area enclosed by the curve $y=-x^2$ and the line $x+y+2=0$ is

If the coefficients of $x^2$ and $x^3$ in the expansion of $(3+k x)^9$ are equal, then the value of ' $\boldsymbol{k}$ ' is

The conjugate of the multiplicative inverse of the complex number $\boldsymbol{z}=\frac{\mathbf{1}+\mathbf{7} \boldsymbol{i}}{\mathbf{3}+\boldsymbol{i}}$ is:

\(\int_0^{\frac{\pi}{2}} \frac{3 \sin x+4 \cos x}{\sin x+\cos x} d x=\)

\(\text { The particular solution of the differential equation }(x-y)(d x+d y)=(d x-d y) \text { when } y=-1 \text { and } x=0 \text { is }\)

The function $\boldsymbol{x}+\boldsymbol{y}=\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}} \boldsymbol{y}$ is the solution of which of the following differential equations?

\(\text { The degree of the differential equation } \sqrt{1+\left(\frac{d y}{d x}\right)^{1 / 3}}=\frac{d^2 y}{d x^2} \text { is: }\)

Let the population of a species of birds surviving at a time ' $\boldsymbol{t}$ ' be governed by the differential equation $\frac{d p}{d t}-p=-100$. If $p(0)=50$, then $p\left(-\log _e 2\right)$ is equal to

\(\text { The second derivative of } \sin 3 \boldsymbol{x} \boldsymbol{\operatorname { c o s }} \mathbf{5 x} \text { is: }\)

\(\text { If } y=\tan ^{-1}\left(\frac{\sqrt{1+x^3}+\sqrt{1-x^3}}{\sqrt{1+x^3}-\sqrt{1-x^3}}\right) \text { then } \frac{\boldsymbol{d} \boldsymbol{y}}{\boldsymbol{d x}}=\)

If $\log y=\log (\sin x)-x^2$, then $\frac{d^2 y}{d x^2}+\mathbf{4} x \frac{d y}{d x}+\mathbf{4} x^2 y=$

If the two ends of the major axis of an ellipse are $(5,0)$ and $(-5,0)$ and one focus lies on the line $3 x-5 y-9=0$, then its equation is

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{x}{x^2+1} \quad \forall x \in \mathbb{R}$ is

The foci of a hyperbola are the same as those of the ellipse with equation $9 x^2+16 y^2=144$.
If the length of the transverse axis of this hyperbola is $2 \cos \alpha$, then its equation is:

\(\int \frac{d x}{x \sqrt{x^2+4}}=\)

\(\int \tan ^{-1}\left(\sqrt{\frac{1-\sin x}{1+\sin x}}\right) d x=\)

\(\int\left(\sin ^6 x+\cos ^6 x+3 \sin ^2 x \cos ^2 x\right) d x=\)

\(\int \frac{\log x}{(1+x)^2} d x\)

\(\text { The domain of the function } f(x)=\sin ^{-1}(\sqrt{x-1})\)

If $X=\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]+\cos ^{-1}\left[\cos \left(\frac{7 \pi}{6}\right)\right]$ and $Y=\sin ^{-1}\left[\sin \left(\frac{11 \pi}{6}\right)\right]+\tan ^{-1}\left[\tan \left(\frac{4 \pi}{3}\...

\(\mathop {\lim }\limits_{x \to {\pi \over 2}}\left(\frac{1-\sin x}{\cos x}\right) \text { is equal to }\)

If $\mathop {\lim }\limits_{x \to 0}\left(\frac{p \sin 2 x+1-\cos 2 x}{x+\tan x}\right)=1$ then the value of ' $p$ ' is

The function $\boldsymbol{f}(\boldsymbol{x})=|\boldsymbol{x}|+|\boldsymbol{x}-\mathbf{1}|$ is:

The feasible region represented by the constraints:
$$ \begin{aligned} & x+2 y \leq 120 \\ & x+y \geq 60 \\ & x-2 y \geq 0 \\ & x \geq 0 \text { and } y \geq 0 \end{aligned} $$

Matrix $A=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & -2 & 2 \\ 1 & 0 & -1\end{array}\right]$,
Given $\boldsymbol{M}_{\mathbf{2 2}}$ and $\boldsymbol{A}_{\mathbf{3 2}}$ are the minor and cofactor of the adjoint matrix of $\boldsymbol{A}$ respe...

If the matrix $M=\left[\begin{array}{ccc}x+5 & a & -4 \\ -2 & 0 & b \\ c & 6 & y+1\end{array}\right]$ is a skew symmetric matrix, the value of the expression $\boldsymbol{a} \boldsymbol{b}+\boldsymbol{c}^{\mathbf{2}}-\boldsymbol{x} \boldsym...

Let A be a square matrix of order $3 \times 3$. If $|A|=-4$, then the value of $\left|\frac{A^{-1}}{-2}\right|$ is:

Let $A=\left[a_{i j}\right]$ be a square matrix of order $3 \times 3$, where the elements are defined as $a_{i j}=\left\{\begin{array}{ll}i-2 j & \text { if } i=j \\ 0 & \text { if } i> j \\ 1 & \text { if } i < j\end{array} \quad\right.$ t...

Given $A=\left[\begin{array}{lll}x & 1 & -2\end{array}\right]$ and $B=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ If $\boldsymbol{A} \boldsymbol{B} \boldsymbol{A}^{\boldsymbol{t}}=[-\mathbf{2 0}]$ then the...

A coach needs to select a $\mathbf{4}$-player starting lineup from a pool of $\mathbf{1 0}$ players:

5 guards
3 forwards
2 centres

Find the number of different selections if the 4-player starting lineup must include:

At least 1 guard
At ...

\(\text { If }{ }^{\mathrm{n}} C_{13},{ }^{\mathrm{n}} C_{14} \text { and }{ }^{\mathrm{n}} C_{15} \text { are in arithmetic progression, then the positive integer value of ' } \mathbf{n} \text { ' can be }\)

A company is migrating its database, and two software engineers, Ishaan and Kavya, take turns running a data-sync script that has a constant success rate of $\frac{3}{8}$ per attempt.
If Ishaan initiates the first attempt and they persist u...

Vishnu has two jars of marbles, Jar A and Jar B.

Jar A contains 3 yellow marbles and 2 green marbles.
Jar B contains 4 yellow marbles and 3 green marbles.

Vishnu flips a fair coin.

If it lands heads, he picks two marbles at random withou...

If $P(A \cup B)=0.85, P(B)=0.50$ and $P(A \cap B)=0.30$. Then $P\left(A \cap B^{\prime}\right)=$

Cards are numbered from 12 to 51 . Two cards are drawn one after the other without replacement. Find the probability that one card is a multiple of $\mathbf{6}$ and the other card is a multiple of $\mathbf{8}$.

A coffee roaster has $\mathbf{1 2}$ rare coffee beans with intensity scores ranked from $\mathbf{1}$ (mildest) to $\mathbf{1 2}$ (strongest).
You choose 7 beans at random and line them up from mildest to strongest:
$$ C_1< C_2< C_3< C_4< C_...

Every term of a geometric progression is positive, and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is:

If $\boldsymbol{k}$ is the arithmetic mean of two given quantities and $\boldsymbol{p}, \boldsymbol{q}$ are the geometric means between the same two quantities, then $\boldsymbol{p}^{\mathbf{3}}+\boldsymbol{q}^{\mathbf{3}}$ is:

Given the sets $A=\{1,2,3\} ; B=\{2,3,5\}$ and $C=\{4,5,6\}$ identify which of the following statement is incorrect.

Let $A$ and $B$ be two subsets of $\xi=\{\mathbf{1}, \mathbf{2}, \mathbf{3},-------, \mathbf{4 4}, \mathbf{4 5}\}$ such that
$A=\{x: x$ is divisible by 3 and 4$\}$
$B=\{x: x$ is a perfect square number $\}$
Then $n(B-A)$ equals

\(\text { The range of the relation } R=\left\{(x, y): y=x+\frac{6}{x} \text {; where } x, y \in \mathbb{N} \text { and } x<6\right\} \text { is: }\)

A student needs to buy notebooks $(n)$ for a semester. Double the number of notebooks plus 5 must strictly exceed 15 , but the number of notebooks plus 10 must be no more than 22 . What is the range of notebooks they can buy?

The variance of a set of 20 observations is 16 . If 7 is added to each observation, and then $\mathbf{5}$ is subtracted from each resulting observation, what will be the new standard deviation?

Let point Q be the image of point $P(2,-1)$ in the line $3 x+5=4 y$.
Find the area of the circle that has the segment PQ as the diameter.