COMEDK 2023 Evening Shift

\(f(x)=2 x-\tan ^{-1} x-\log (x+\sqrt{x^2+1}) \text { is monotonically increasing, when }\)

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is

The altitude of a cone is \(20 \mathrm{~cm}\) and its semi vertical angle is \(30^{\circ}\). If the semi vertical angle is increasing at the rate of \(2^0\) per second, then the radius of the base is increasing at the rate of

The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minimum at

The area bounded by the curve \(y^2=4 a^2(x-1)\) and the lines \(x=1, y=4 a\) is

The area of the upper half of the circle whose equation is \((x-1)^2+y^2=1\) is given by

The centre of the circle passing through \((0,0)\) and \((1,0)\) and touching the circle \(x^2+y^2=9\) is

If the conjugate of \((x+i y)(1-2 i)\) be \(1+i\), then

\(\int_0^2\left|x^2+2 x-3\right| d x \text { is equal to }\)

The sum of the degree and order of the following differential equation \(\left[1-\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}=k x \frac{d^2 y}{d x^2}\)

The general solution of the differential equation \(\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y\)

The solution of the differential equation \(\frac{d y}{d x}+y \cos x=\frac{1}{2} \sin 2 x\)

The particular solution of \(e^{\frac{d y}{d x}}=2 x+1\) given that \(y=1\) when \(x=0\) is

\(\text { If } f(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right) \text { then } f^{\prime}(0) \text { is equal to }\)

If \(f(x)=\frac{(x+1)^7 \sqrt{1+x^2}}{\left(x^2-x+1\right)^6}\) then the value of \(f^{\prime}(0)\) is equal to

If the length of the major axis of an ellipse is 3 times the length of the minor axis, then its eccentricity is

The range of \(x\) which satisfy the inequation : \(-5 \leq \frac{2-3 x}{4} \leq 9\) is

The distance between the foci of a hyperbola is 16 and its eccentricity is \(\sqrt{2}\). Then its equation is

\(\int x^x(1+\log x) d x\) is equal to

\(\int e^x\left(1+\tan x+\tan ^2 x\right) d x \text { is equal to }\)

\(\int \sqrt{\operatorname{cosec} x-1} d x=\)

\(\int \frac{\cos 4 x+1}{\cot x-\tan x} d x=\)

\(\text { The value of } \sin ^{-1}\left[\cos \left(39 \frac{\pi}{5}\right)\right] \text { is }\)

\(\text { Let } f(x)=\cos ^{-1}(3 x-1) \text {, then domain of } f(x) \text { is equal to }\)

$$
\text { The function defined by } f(x)=\left\{\begin{array}{cc}
\frac{\sin x}{x}+\cos x & x>0 \\
-5 k & x=0 \\
\frac{4(1-\sqrt{1-x})}{x} & x<0
\end{array} \quad \text { is continous at } x=0, \quad \text { then } k\right. \text { equals ...

\(\lim _\limits{x \rightarrow 0} \frac{a^x-b^x}{x} \text { is equal to }\)

The minimum value of \(Z=3 x+5 y\), given subject to the constraints \(x+y \geq 2, x+3 y \geq 3, x, y \geq 0\) is

If $$2 A+3 B=\left[\begin{array}{ccc}2 & -1 & 4 \\ 3 & 2 & 5\end{array}\right]$$ and $$A+2 B=\left[\begin{array}{lll}5 & 0 & 3 \\ 1 & 6 & 2\end{array}\right]$$ then \(B=\)

$$
\text { If } A=\left(\begin{array}{ll}
1 & 2 \\
0 & 1
\end{array}\right) \quad P=\left(\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right) \quad Q=P^T A P, \quad \text { then } P Q^{2014} P^T \tex...

\(A\) and \(B\) are invertible matrices of the same order such that \(\left|(A B)^{-1}\right|=8\) if \(|A|=2\) then \(|B|\) is

Solution of \(x-y+z=4 ; x-2 y+2 z=9\) and \(2 x+y+3 z=1\) is

If $$\left[\begin{array}{ccc}2+x & 3 & 4 \\ 1 & -1 & 2 \\ x & 1 & -5\end{array}\right]$$ is a singular matrix, then \(x\) is

How many factors of \(2^5 \times 3^6 \times 5^2\) are perfect squares?

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. The number of ways in which he can choo...

In a 12 storey house, 10 people enter a lift cabin. It is known that they will leave the lift in pre-decided groups of 2, 3 & 5 people at different storeys. The number of ways they can do so if the lift does not stop up to the second storey...

Bag A contains 3 white and 2 red balls. Bag B contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from bag A and put into bag B. However if tail appears then 2 balls are drawn at random from bag...

\(\text { If } P(B)=\frac{3}{5} \quad P(A / B)=\frac{1}{2} \text { and } P(A \cup B)=\frac{4}{5} \text { then } P(A \cup B)^{\prime}+P\left(A^{\prime} \cup B\right)=\)

The probability distribution of a discrete random variable X is given as

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overfl...

A die is thrown twice and the sum of numbers appearing is observed to be 8 . What is the conditional probability that the number 5 has appeared atleast once?

18 Points are indicated on the perimeter of a triangle \(\mathrm{ABC}\) as shown below. If three points are chosen then probability that it will from a triangle is

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length \(x\). The maximum area enclosed by the park is

If three numbers \(a, b, c\) constitute both an A.P and G.P, then

Le \(x\) be the arithmetic mean and \(y, z\) be the two geometric means between any two positive numbers, then \(\frac{y^3+z^3}{x y z}=\) -----------

In the set \(\mathrm{W}\) of whole numbers an equivalence relation \(\mathrm{R}\) is defined as follows \(\mathrm{aRb}\) iff both \(\mathrm{a}\) & \(\mathrm{~b}\) leave the same reminder when divided by 5. The equivalence class of 1 is give...

Which of the following is a singleton set?

Let \(X\) and \(Y\) be the set of all positive divisors of 400 and 1000 respectively (including 1 and the number). Then \(n(X \cap Y)\) is equal to

Consider the first 10 natural numbers. If we multiply each number by \(-\)1 and add 1 to each number, the variance of the numbers so obtained is

What can be said regarding a line if its slope is negative?

The ratio in which the line \(3 x+4 y+2=0\) divides the distance between the lines \(3 x+4 y+5=0\) and \(3 x+4 y-5=0\) is

In a \(\triangle A B C\), if coordinates of point \(A\) is \((1,2)\) and equation of the medians through \(B\) and \(C\) are \(x+y=5\) and \(x=4\) respectively, then the coordinates of \(B\) is