COMEDK 2024 Afternoon Shift

\(\text { If } f(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x} \text { is decreasing for all } x \text {, then }\)

If \((x-a)^2+(y-b)^2=c^2\), where \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are some constants, \(c>0\) then \(\frac{\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2 y}{d x^2}}\) is independent of

\(\text { The function } y=\tan x-x \text { is }\)

If \(f(x)=\log x+b x^2+a x, x \neq 0\) has extreme values (or turning points) at \(x=-1\) and \(x=2\) then the values of \(\mathrm{a}\) and \(\mathrm{b}\) are

The dimensions of the largest rectangle of side \(x\) and \(y\) that can be inscribed in the right angled triangle of sides \(\mathrm{a}\) and \(\mathrm{b}\) is

The area of the region (in sq units) bounded by the curve \(y=\sqrt{16-x^2} \text { and } x \text {-axis is }\)

\(\text { The area of the region (in square units) bounded by the line } y+3=x ; x=1 \text { and } x=5 \text { is }\)

If the sum of the coefficients of the first three terms in the expansion of \(\left(x-\frac{a}{x^2}\right)^{12}, x \neq 0\) is 559. Find the value of '\(a\)' if '\(a\)' belongs to positive integers

\(\text { The value of } \frac{i^{1004}+i^{1006}+i^{1008}+i^{1010}+i^{1012}}{i^{510}+i^{508}+i^{506}+i^{504}+i^{502}} \text { is }\)

\(\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\cos x-\cos ^3 x} d x \text { is equal to }\)

\(\int_\limits0^{\frac{\pi}{2}} \frac{\cos x}{1+\cos x+\sin x} d x=\)

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

\(\text { The general solution of } \frac{d y}{d x}=\sin ^{-1} x \text { is }\)

\(\text { The general solution of the differential equation } \frac{d y}{d x}=\frac{x y}{x^2+y^2} \text { is }\)

Degree of the differential equation \(\log \left(\frac{d y}{d x}\right)^{\frac{1}{2}}=5 x+4 y\) is

\(\text { If } x^2+y^2=t+\frac{1}{t} \text { and } x^4+y^4=t^2+\frac{1}{t^2} \text { then } \frac{d y}{d x}=\)

The points on the ellipse \(16 x^2+9 y^2=400\) at which the ordinate decreases at the same rate at which the abscissa increases are

If an ellipse has an equation in the standard form and it passes through the points \(\left(\frac{5}{2}, \frac{\sqrt{6}}{4}\right)\) and \(\left(-2, \frac{\sqrt{15}}{5}\right)\) then the length of its latus rectum is

\(\text { Which of the following function is injective? }\)

If the distance between the foci and the distance between the two directrixes are in the ratio \(3: 2\) for a hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), then a : b is

\(\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x \text { is equal to }\)

\(\int \frac{f^{\prime}(x)}{f(x) \log (f(x))} d x \text { is equal to }\)

\(\int \log x(\log x+2) d x \text { equals to }\)

\(\text { If } \alpha=\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right) \text { and } \beta=\tan ^{-1}\left(-\tan \frac{2 \pi}{3}\right) \text { then }\)

Evaluate :

\(\operatorname{cosec}^{-1}\left(-\frac{2 \sqrt{3}}{3}\right)+\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)+\sec ^{-1} 2+\cos ^{-1}\left(-\frac{1}{2}\right)-\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)\)

\(\text { If } y=\sin ^{-1}(\sqrt{\sin x}) \text {, then } \frac{d y}{d x} \text { equals }\)

\(\lim _\limits{x \rightarrow 0} \frac{a^x-b^x}{c^x-d^x}=\)

\(\text { The value of } \lim _\limits{x \rightarrow 0} \frac{\sin (a+x)-\sin (a-x)}{x} \text { is }\)

\(\text { The number of points of discontinuity of the rational function } f(x)=\frac{x^2-3 x+2}{4 x-x^3}\)

\(\text { The maximum value of } Z=3 x+4 y \text { for the given constraints } x+2 y \leq 76,2 x+y \leq 104, x \geq 0, y \geq 0 \text { is }\)

$$ \text { If } A=\left[\begin{array}{cc} 1 & -2 \\ 4 & 5 \end{array}\right] \text { and } f(t)=t^2-3 t+7 \text { then } f(A)+\left[\begin{array}{cc} 3 & 6 \\ -12 & -9 \end{array}\right] \text { is } $$

If $$A=\left[\begin{array}{lll}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{array}\right] \quad B^{-1}=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$$ then \((A B)^{-1}\) is equal to

$$ \left|\begin{array}{ccc} \cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \end{array}\right| $$ is independent of

\(\text { If A }(\operatorname{adj} A)=5 I \text {, where I is the identity matrix of order } 3 \text {, then }|\operatorname{adj} A|=\)

\(\text { A square matrix } P \text { satisfies } P^2=I-P \text { where } I \text { is identity matrix. If } P^n=5 I-8 P \text {, then } n \text { is equal to }\)

The number of four digit numbers strictly greater than 4321 formed using the digits \(0,1,2,3,4,5\) with repetition of digit is

A student has 3 library cards and 8 books of his interest in the library. Out of these 8 books he does not want to borrow Chemistry part 2 unless he can borrow Chemistry part 1 also. In how many ways can he choose the three books to be borr...

There are some baskets. The chances of picking a loaded basket and choosing a red coloured one is 0.2 . For every 100 tries to pick one basket, 60 times a basket is either loaded or red in colour. What is the probability of choosing an empt...

A and B are two independent events. The probability of their simultaneous occurrence is \(\frac{1}{8}\) and the probability that neither of them occurs is \(\frac{3}{8}\). Then their individual probabilities are

P and Q are considering to apply for a job. The probability that P applies for the job is \(\frac{1}{4}\). The probability that \(\mathrm{P}\) applies for the job given that \(\mathrm{Q}\) applies for the job is \(\frac{1}{2}\), and the pro...

A determinant of the second order is made with elements 0 and 1 . What is the probability that the determinant made is non-negative?

The probability of inviting three friends on 5 consecutive days, exactly one friend a day and no friend is invited on more than two days is

The inequality \(4 x-3 \geq \frac{10 x-1}{3}\) represents which of the following interval when \(x \in R\)

\(\text { If } 6^{\text {th }} \text { term of a geometric progression is }-\frac{1}{32} \text { and } 9^{\text {th }} \text { term is } \frac{1}{256} \text { then } r \text { is }\)

The sum of four numbers in a geometric progression is 60 , and the arithmetic mean of the first and the last number is 18 . Then the numbers are

The sum of first three terms of a geometric progression is 16 and the sum of next three terms is 128 . The sum to \(\mathrm{n}\) terms of the geometric progression is

\(\text { Let } \mathrm{A} \text { and } \mathrm{B} \text { be two sets then } A-(A \cap B) \text { is equal to }\)

\(\text { If } a \mathcal{N}=\{a x: x \in \mathcal{N}\} \text {, then } 3 \mathcal{N} \cap 7 \mathcal{N} \text { is }\)

A relation \(R\) is defined from \(\{2,3,4\}\) to \(\{3,6,7,10\}\). If \(x R y \Leftrightarrow x\) and \(y\) are co prime numbers. Then range of \(R\) is

Let 'P' be the mean deviation of the first five odd natural numbers about their mean and 'Q' be the mean deviation of the first five prime numbers about their mean. The \(Q-P=\)