COMEDK 2025 Afternoon Shift

Let $A$ and $G$ denote the arithmetic mean and geometric mean of positive real numbers $5^x$ and $5^{1-x}$. Then the minimum value of the expression $5^x+5^{1-x}$ where $x \in R$ is

The curve $4 y=3 x^4-2 x^2$ attains ----------- at the points $x=-\frac{1}{\sqrt{3}}$ and $x=\frac{1}{\sqrt{3}}$

In the interval $(0,1)$ the function $f(x)=x^2-x+1$ is

$x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ represents the equation of a curve. If $\theta$ changes at a constant rate $k$ then the rate of change of the slope of the tangent to the curve at $\theta=\frac{\pi}{3}$ is

The least area of a circle circumscribing any right-angle triangle of area $\frac{9}{\pi}$ sq units is

Quadrilateral PQRS is inscribed inside a rectangle of dimensions $10 \mathrm{~cm} \times 8 \mathrm{~cm}$. The value of ' $x$ ', if the area of the quadrilateral is minimum is

The area of the region enclosed by the lines $2 x+y=10, y=1, y=5$ and the $y$-axis is

Evaluate the value of $(1.02)^8$ using binomial theorem up to two decimal places.

The complex number $\frac{1+7 i}{(2-i)^2}$ lies in

$\int\limits_0^{\frac{\pi}{2}} \log \left(\frac{5+4 \sin x}{5+4 \cos x}\right) d x=$

Solution of the differential equation $y \frac{d y}{d x}+x=0$ represents a family of

The degree of the differential equation $\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{4}}=\left(\frac{d^2 y}{d x^2}\right)^{\frac{1}{3}}$

Integrating factor of the differential equation $\frac{d y}{d x}+y=\frac{x^3+y}{x}$ is

Which of the following transformations reduce the differential equation $\frac{d z}{d x}+\frac{z}{x} \log z=\frac{z}{x^2}(\log z)^2$ into the form $\frac{d u}{d x}+P(x) u=Q(x)$

If $ 2 y=\left[\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right]^2 \forall x \in\left(0, \frac{\pi}{2}\right)$ then $\frac{d y}{d x}$ is equal to :

If $y=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$,
then $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}=$

Differentiate $\log _a x$ with respect to $a^x$

The radius of the circle passes through the foci of a conic $\frac{x^2}{16}+\frac{y^2}{9}=1$ and has its centre at $(0,3)$, then the diameter of the circle is ---

The area of the region bounded by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

If a function $f:[2, \infty) \rightarrow R$ defined by $f(x)=x^2-4 x+5$, then the range of $f$ is

The length of the latus rectum of a conic $49 y^2-16 x^2=784$ is

$\int \frac{d x}{x \sqrt{4 x^2-9}}=$

$\int \frac{x}{(x-1)(x-2)^2} d x=a \log \left|\frac{x-1}{x-2}\right|+\frac{b}{(x-2)}+c$ then

$\int \frac{\sin x+\cos x}{\sqrt{1+2 \sin x \cos x}} d x=\varphi(x)+C$ Then $\varphi(x)=$

$\int \frac{e^{\tan ^{-1} x}}{\left(1+x^2\right)}\left(1+x+x^2\right) d x=$

Domain of the function $f(x)=\sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}}$ for real valued of $x$ is

The value of $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ is

$\lim _\limits{\theta \rightarrow \frac{\pi}{2}} \frac{1-\sin \theta}{\left(\frac{\pi}{2}-\theta\right) \cos \theta}$ is equal to :

$\lim _\limits{x \rightarrow 1} \frac{(\sqrt{x}-1)(2 x-3)}{2 x^2+x-3}$ is

The function $f(x)=\left\{\begin{array}{l}\frac{|x|}{x}, \text { if } x \neq 0 \\ 0, \text { if } x=0\end{array}\right.$ is discontinuous at

The corner points of the feasible region determined by the system of linear constraints are $(0,3),(1,1)$ and $(3,0)$, If objective function is $Z=p x+q y, p, q>0$ then the condition on $p$ and $q$ so that the minimum of $Z$ occurs at $(3,0...

If $A=\frac{1}{\pi}\left[\begin{array}{cc}\sin ^{-1} \frac{1}{2} & \tan ^{-1} \frac{x}{\pi} \\ \sin ^{-1} \frac{x}{\pi} & \cot ^{-1} \sqrt{3}\end{array}\right] \quad B=\frac{1}{\pi}\left[\begin{array}{cc}-\cos ^{-1} \frac{1}{2} & \tan ^{-1}...

If $A(\operatorname{adj} A)=\left[\begin{array}{lll}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{array}\right]$, then the value of $|A|+|\operatorname{adj} A|$ is equal to :

The cofactor of the element $a_{21}$ in the expansion of $\Delta=\left|\begin{array}{ccc}1 & 4 & 4 \\ -3 & 5 & 9 \\ 2 & 1 & 2\end{array}\right|$ is

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ₹ 500 .
The cost of 1 kg onion, 2 kg wheat and 3 kg rice is ₹ 300 .
The cost of 6 kg onion, 2 kg wheat and 3 kg rice is ₹ 575 .
The above situation can be represented in matrix form as $\m...

If $A=\left[\begin{array}{ccc}0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0\end{array}\right]$, then $A+2 A^T=$

If ${ }^{n+2} C_8:{ }^{n-2} P_4=57: 16$, then ' $n$ ' is

On each working day of a school there are six periods. The number of ways in which five subjects are arranged if each subject is allotted at least one period and no period remains vacant is

If for two events $A$ and $B, P(A-B)=\frac{1}{5}$ and $P(A)=\frac{3}{5}$ then $P(B / A)=$

In a kabaddi league, two matches are being played between Jaipur and Delhi. It is assumed that the outcomes of the two games are independent. The probability of Jaipur winning, drawing and losing the game against Delhi are $\frac{1}{2}, \fr...

A bag contains $(n+1)$ coins. It is known that one of these coins has a head on both sides, whereas the other coins are fair. One of these coins is selected at random and tossed. If the probability that the toss results in heads is $\frac{7...

Three bags contain a number of red and white balls are as follows.
Bag I: 3 red balls
Bag II: 2 red balls and 1 white ball
Bag III: 3 White balls
The probability that bag $i$ will be chosen and a ball is selected from it is $\frac{i}{6}, i=...

Five persons entered the lift cabin on the ground floor of an eight-floor apartment. Suppose that each of them independently and with equal probability, can leave the cabin at any floor beginning with the first floor, then the probability o...

The terms of an infinitely decreasing geometric progression in which all the terms are positive, the first term is $\mathbf{4}$, and the difference between third and fifth term is $\frac{32}{81}$, then which of the following is not true

For real numbers $x$ and $y, x R y \Leftrightarrow x-y+\sqrt{2}$ is an irrational number. Then the relation R is:

If $A=\{1,2,4\} \quad B=\{2,4,5\} \quad C=\{2,5\}$ then $(A-B) \cap(B-C)=$

Let $A=\{x: x=4 n+1, n \in Z, 0 \leq n<4\}$
$$\begin{aligned} & B=\{x: x=15 n+4, n \in N, n \leq 3\} \\ & C=\{x: x \text { is a prime number }, x \in A \cup B\} \end{aligned}$$
Then the cardinal number of set C is

The inequality representing the following graph is

If the mean of $4,7,2,8,6$ and $k$ is 7 . Then the mean deviation from the mean of these observations is

The point on the line $x+y=4$ that lie at a unit distance from the line $4 x+3 y=10$ is