COMEDK 2024 Morning Shift

\(\text { The point on the curve } x^2=x y \text { which is closest to }(0,5) \text { is }\)

For a given curve \(y=2 x-x^2\), when \(x\) increases at the rate of 3 units/sec, then how does the slope of the curve change?

The most economical proportion of the height of a covered box of fixed volume whose base is a rectangle with one side three times as long as the other, is

If \(f(x)=2 x^3+9 x^2+\lambda x+20\) is a decreasing function of \(x\) in the largest possible interval \((-2,-1)\), then \(\lambda\) is equal to

The side of an equilateral triangle expands at the rate of \(\sqrt{3} \mathrm{~cm} / \mathrm{sec}\). When the side is \(12 \mathrm{~cm}\), the rate of increase of its area is

The coefficient of the third term in the expansion of \(\left(x^2-\frac{1}{4}\right)^n\), when expanded in the descending power of \(x\) is 31, then \(n\) is

The area (in sq units) of the minor segment bounded by the circle \(x^2+y^2=a^2\) and the line \(x=\frac{a}{\sqrt{2}}\) is

The equation of a circle passing through the origin is \(x^2+y^2-6 x+2 y=0\). The equation of one of its diameter is

\(\text { The modulus of the following complex number } \frac{1+i}{1-i}-\frac{1-i}{1+i} \text { is }\)

\(\int_\limits{-1}^1 \frac{d}{d x}\left(\tan ^{-1} \frac{1}{x}\right) d x \text { is }\)

The particular solution of \(\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0\), when \(x=0, y=\frac{1}{2}\) is

The particular solution of the differential equation \(\cos x \frac{d y}{d x}+y=\sin x\) at \(y(0)=1\)

\(\text { If } y=\log _e\left(\frac{x^2}{e^2}\right) \text {, then } \frac{d^2 y}{d x^2} \text { is equal to }\)

\(\text { If } y=\tan ^{-1}\left(\frac{3-2 x}{1+6 x}\right) \text { then } \frac{d y}{d x} \text { is }\)

If \(f(x)=f^{\prime}(x)\) and \(f(1)=2\), then \(f(3)\) is

\(\text { If } \sin y=x(\cos (a+y)) \text {, then find } \frac{d y}{d x} \text { when } x=0\)

The equation of an ellipse, whose focus is \((1,0)\), directrix is \(x=4\) and whose eccentricity is a root of the quadratic equation \(2 x^2-3 x+1=0\), is

Let \(f: R \rightarrow R\) be a function defined by \(f=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}\) then

Which of the following is not a homogenous function of \(x\) and \(y\)

\(\text { The value of } \int \frac{1}{x+\sqrt{x-1}} d x \text { is }\)

If \(\int \frac{1}{\sqrt{\sin ^3 x \cos x}} d x=\frac{k}{\sqrt{\tan x}}+c\) then the value of \(k\) is

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

\(\int \frac{x}{x^4-16} d x=\)

Evaluate : \(\cos ^{-1}\left[\cos \left(-680^{\circ}\right)\right]+\sin ^{-1}\left[\sin \left(-600^{\circ}\right)\right]-\cos ^{-1}\left(\sin 270^{\circ}\right)\)

\(\text { The value of } \sin ^{-1}\left[\cot \left(\frac{1}{2} \tan ^{-1} \frac{1}{\sqrt{3}}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sin ^{-1} \frac{1}{\sqrt{2}}\right)\right]\) is

\(\lim _\limits{x \rightarrow 0} \frac{\sqrt{a+x}-\sqrt{a}}{x \sqrt{a(a+x)}}\) equals to

$$ \text { If } f(x)=\left\{\begin{array}{cc} \frac{1-\sin x}{(\pi-2 x)^2} & , \quad \text { if } x \neq \frac{\pi}{2} \\ \lambda, & \text { if } x=\frac{\pi}{2} \end{array}\right. $$
Then \(f(x)\) will be continues function at $$x=\frac{\p...

\(\lim _\limits{x \rightarrow 0}\left(\frac{\sin a x}{\sin b x}\right)^k \text { equals }\)

The minimum value of \(Z=150 x+200 y\) for the given constraints
$$\begin{aligned} & 3 x+5 y \geq 30 \\ & x+y \geq 8 ; x \geq 0, y \geq 0 \text { is } \end{aligned}$$

$$ \text { If } x, y, z \text { are non zero real numbers, then inverse of matrix } A=\left[\begin{array}{lll} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{array}\right] \text { is } $$

$$ \text { If the matrix } A=\left(\begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array}\right) \text { then } A^{n+1}= $$

If $$A=\left[\begin{array}{ccc}-1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$$ then the inverse of \((A I)^t\) (where \(\mathrm{I}\) is an identity matrix) is

If $$ \left[\begin{array}{lll} 1 & x & 1 \end{array}\right]\left[\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{array}\right]\left[\begin{array}{l} 1 \\ 2 \\ x \end{array}\right]=[0] $$ then x is equal to

$$ \text { If } P=\left[\begin{array}{lll} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{array}\right] \text { is the adjoint of a } 3 \times 3 \text { matrix } A \text { and }|A|=4 \text { then } \alpha \text { is equal to } $$

\(\text { The area (in sq units) enclosed by the parabola } y^2=8 x \text {, its latus-rectum and the } x \text {-axis is }\)

If \(\left[{ }^{n+1} C_{r+1}\right]:\left[{ }^n C_r\right]:\left[{ }^{n-1} C_{r-1}\right]=11: 6: 3\) then \(n r=\)

In how many ways can the word "CHRISTMAS" be arranged so that the letters '\(\mathrm{C}\)' and '\(\mathrm{M}\)' are never adjacent?

If the events A and B are mutually exclusive events such that \(P(A)=\frac{1}{3}(3 x+1)\) and \(P(B)=\frac{1}{4}(1-x)\) then the possible values of $x$ lies in the interval

A random variable X with probability distribution is given below

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An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white it is not replaced into the urn. Otherwise it is replaced along with another ball of the same colour. The process is repeated. The probability that the thi...

A and B each have a calculator which can generate a single digit random number from the set \(\{1,2,3,4,5,6,7,8\}\). They can generate a random number on their calculator. Given that the sum of the two numbers is 12 , then the probability t...

The probability that a randomly chosen number from one to twelve is a divisor of twelve is

\(\text { If } \frac{1}{2}\left(\frac{3 x}{5}+4\right) \geq \frac{1}{3}(x-6), x \in R \text { then }\)

Given \(a, b, c\) are three unequal numbers such that \(\mathrm{b}\) is arithmetic mean of \(a\) and \(c\) and \((b-a),(c-b), a\) are in geometric progression. Then \(a: b: c\) is

\((32) \times(32)^{\frac{1}{6}} \times(32)^{\frac{1}{36}} \times-----\infty \text { is equal to }\)

A geometric progression consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio of the G.P is

Express the set \(A=\{1,7,17,31,49\}\) in set builder form

\(\text { If } A=\{1,2,3,4,5\} \text { and } B=\{2,3,6,7\} \text { then number of elements in the set }(A \times B) \cap(B \times A) \text { is equal to }\)

$$\begin{aligned}
&\begin{aligned}
& \text { A, B, C are subsets of the Universal set U } \\
& \text { If } \mathrm{A}=\{x: x \text { is even number, } x \leq 20\} \\
& \mathrm{B}=\{x: x \text { is multiple of } 3, x \leq 15\} \\
& \mathrm{...

The mean of the numbers \(a, b, 8,5,10\) is 6 and the variance is 6.80 , then which of the following gives possible values of \(a\) & \(b\)