COMEDK 2024 Evening Shift

\(\text { The rate of change of the volume of a sphere with respect to its surface area } \mathrm{S} \text { is }\)

The turning point of the function \(y=\frac{a x-b}{(x-1)(x-4)}\) at the point \(P(2,-1)\) is

The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is

What is the nature of the function \(f(x)=x^3-3 x^2+4 x\) on real numbers?

The area bounded by the curve \(y=\cos x, x=0\) and \(x=\pi\) is

In the expansion \(\left(\frac{1}{x}+x \sin x\right)^{10}, \quad\) the co - efficient of \(6^{\text {th }}\) term is equal to \(7 \frac{7}{8}\), then the principal value of \(x\) is

The equation of the circle which touches the \(x\)-axis, passes through the point \((1,1)\) and whose centre lies on the line \(x+y=3\) in the first quadrant is

\(\text { If }(1-4 i)^3=a+i b \text { then the value of } \mathrm{a} \text { and } \mathrm{b} \text { is }\)

If \(a\) is a real number such that \(\int_\limits0^a x d x \leq a+4\) then

\(\text { If } I_n=\int_\limits0^{\frac{\pi}{4}} \tan ^n x d x \text {, for } n \geq 2 \text {, then } I_n+I_{n-2}=\)

\(\text { The value of the integral } \int_\limits{\frac{1}{3}}^1 \frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4} d x \text { is }\)

\(\text { If } \frac{d y}{d x}=y+3>0 \text { and } y(0)=2 \text { then } y(\log 2) \text { is equal to }\)

The general solution of the differential equation \(x \frac{d y}{d x}=y+x \tan \left(\frac{y}{x}\right)\) is

\(\text { The general solution of the differential equation }(1+\tan y)(d x-d y)+2 x d y=0 \text { is }\)

The sum of the order and degree of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^5+\frac{4\left(\frac{d^2 y}{d x^2}\right)^3}{\left(\frac{d^3 y}{d x^3}\right)}+\frac{d^3 y}{d x^3}=x^2-1\) is

\(\text { If } y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^2}}{13}\right) \text { then } \frac{d y}{d x} \text { equals }\)

\(\text { If } y=f(x), \quad p=\frac{d y}{d x} ; q=\frac{d^2 y}{d x^2} \text { then } \frac{d^2 x}{d y^2} \text { is equal to }\)

\(\text { If } y=\sqrt{\sin x+y} \text { then find } \frac{d y}{d x} \text { at } x=0, \quad y=1\)

\(\text { The area of the region enclosed by the curve }\left\{(x, y): 4 x^2+25 y^2=100\right\} \text { is }\)

The domain of the function \(y=\frac{1}{\log _{10}(3-x)}+\sqrt{x+7}\) is

\(\int e^x\left[\frac{x^2+1}{(x+1)^2}\right] d x \quad \text { is equal to }\)

\(\text { The value of } \int \frac{d x}{\sqrt{2 x-x^2}} \text { is }\)

\(\text { Evaluate: } \cot ^{-1}\left(-\frac{3}{\sqrt{3}}\right)-\sec ^{-1}\left(-\frac{2}{\sqrt{2}}\right)-\operatorname{cosec}^{-1}(-1)-\tan ^{-1}(1)\)

\(\text { Evaluate: } \cos ^{-1}\left(\cos \frac{35 \pi}{18}\right)-\sin ^{-1}\left(\sin \frac{35 \pi}{18}\right)\)

\(\text { The function } f(x)=\tan ^{-1}(\sin x+\cos x) \text { is an increasing function in }\)

Let \(\alpha\) and \(\beta\) be the distinct roots of \(a x^2+b x+c=0\), then \(\lim _\limits{x \rightarrow \alpha} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\alpha)^2}\) is equal to

\(\text { The value of } \lim _\limits{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}=\)

$$ \text { If } f(x)=\left\{\begin{array}{cc} x & , \quad 0 \leq x \leq 1 \\ 2 x-1 & , \quad x>1 \end{array}\right. \text { then } $$

The maximum value of \(P=500 x+400 y\) for the given constraints \(x+y \leq 200, \quad x \geq 20, \quad y \geq 4 x, \quad y \geq 0\) is

$$ \text { If } 3 A+4 B^t=\left(\begin{array}{ccc} 7 & -10 & 17 \\ 0 & 6 & 31 \end{array}\right) \text { and } 2 B-3 A^t=\left(\begin{array}{cc} -1 & 18 \\ 4 & -6 \\ -5 & -7 \end{array}\right) \text { then }(5 B)^t= $$

$$ \text { If } A=\left[\begin{array}{cc} 5 a & -b \\ 3 & 2 \end{array}\right] \text { and } A \operatorname{adj} A=A A^t \text {, then } 5 a+b \text { is equal to } $$

If $$A=\left[\begin{array}{ccc}0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x\end{array}\right]$$ is a singular matrix then \(x\) is equal to

If the matrix $A$ is such that $$A\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)=\left(\begin{array}{cc}-4 & 1 \\ 7 & 7\end{array}\right)$$ then \(A\) is equal to

In the parabola \(y^2=4 a x\) the length of the latus rectum is 6 units and there is a chord passing through its vertex and the negative end of the latus rectum. Then the equation of the chord is

For an examination a candidate has to select 7 questions from three different groups \(\mathrm{A}, \mathrm{B}\) and C. The three groups contain 4, 5 and 6 questions respectively. In how many different ways can a candidate make his selection...

The letters of the word "COCHIN" are permuted and all the permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word "COCHIN" is

Let \(\mathrm{A}\) and \({B}\) be two events such that \(P(A / B)=\frac{1}{2}\) and \(P(B / A)=\frac{1}{3}\) and \(P(A \cap B)=\frac{1}{6}\) then, which one of the following is not true?

Suppose we have three cards identical in form except that both sides of the first card are coloured red, both sides of the second are coloured black, and one side of the third card is coloured red and the other side is coloured black.
The t...

While shuffling a pack of cards, 3 cards were accidently dropped, then find the probability that the missing cards belong to different suits?

What is the probability of a randomly chosen 2 digit number being divisible by 3 ?

A coin is tossed until a head appears or until the coin has been tossed three times. Given that 'head' does not appear on the first toss, what is the probability that the coin is tossed thrice?

\(\text { The solution set for the inequality } 13 x-5<15 x+4<7 x+12 ; x \in W \text { is }\)

A number consists of three digits in geometric progression. The sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of left hand and middle digits is two third of the sum of the middle and right hand d...

Consider an infinite geometric series with first term '\(a\)' and common ratio '\(r\)'.
If the sum of infinite geometric series is 4 and the second term is \(\frac{3}{4}\) then

If two positive numbers are in the ratio \(3+2 \sqrt{2}: 3-2 \sqrt{2}\), then the ratio between their A.M (arithmetic mean) and G.M (geometric mean) is

Two finite sets have '\(m\)' and '\(n\)' number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of \(\mathrm{m}\) and \(\mathrm{n}\) are ...

Which of the following relations on the set of real numbers \(\mathrm{R}\) is an equivalence relation?

The shaded region in the Venn diagram represents

The mean of five observations is 4 and their variance is 5.2 . If three of these observations are 1, 2 and 6, then the other two observations are

The line joining two points \(A(2,0) B(3,1)\) is rotated about \(A\) in anticlockwise direction through an angle of \(15^{\circ}\). If \(B\) goes to \(C\) in the new position, then the coordinates of \(C\) is