List of top Mathematics Questions asked in COMEDK UGET

If \( f(x) = \left(\frac{3+x}{1+x}\right)^{2+3x} \), then \( f'(0) = \)
The value of \( \lambda \) for which the angle between lines \( \vec{r} = \hat{i} + \hat{j} + \hat{k} + p(2\hat{i} + \hat{j} + 2\hat{k}) \) and \( \vec{r} = (1+q)\hat{i} + (1+q\lambda)\hat{j} + (1+q)\hat{k} \) is \( \frac{\pi}{2} \)
If \( A(t) = \begin{pmatrix} \cos t & \sin t \\ -\sin t & \cos t \end{pmatrix} \), then the product of \( A(t) \) and \( A(-t) \) is
The range of \( x \) for which the equation \( \sin^{-1}\left(\frac{2x}{1+x^2}\right) = 2\tan^{-1}(x) \) holds true
The value of \( \displaystyle \lim_{x \to 0} \frac{(1-x)^n - 1}{x} \) is
If \( f(x) = \begin{cases} \frac{1-x^m}{1-x}, & \text{for } x \neq 1 \\ 2m-1, & \text{for } x = 1 \end{cases} \) and the function is discontinuous at \( x = 1 \), then
If \( \displaystyle \int \frac{dx}{(x+2)(x^2+1)} = p\log|x+2| + q\log|x^2+1| + r\tan^{-1}x + c \), then \( p+q+r = \)
If the third and fourth terms in the expansion \( (2x + \frac{1}{8})^{10} \) are equal, then the value of \( x \) is
In an entrance test, there are multiple choice questions. There are four possible answers to each question of which only one is correct. The probability that a student knows the answer to a question is 90%. If he gets the correct answer to a question, then the probability that he was guessing is
If \( \frac{x-1}{3+i} + \frac{y-1}{3-i} = i \) then \( (y,x) = \)
If the distance between the foci is equal to the length of the latus rectum, then the eccentricity of the ellipse is
A solid \(S\) is made from a cylinder surmounted by a hemisphere on top with both its circular faces sharing a common centre. The radius of cylinder and radius of hemisphere are \(x\) cm. The height of the cylinder is \((20-4x)\) cm and the volume of \(S\) is \(V=\frac{1}{3}\pi y\). Find the maximum value of \(y\).
A line \(L_1\) passing through the point \(A\) with position vector \( \vec{a} = 4\hat{i} + 2\hat{j} + 2\hat{k} \) is parallel to the vector \( \vec{b} = 2\hat{i} + 3\hat{j} + 6\hat{k} \). The length of the perpendicular drawn from a point \(P\) with position vector \( \vec{p} = \hat{i} + 2\hat{j} + 3\hat{k} \) to \(L_1\) is
The magnitude of the projection of the vector \( -\hat{i} + 2\hat{j} - \hat{k} \) on the \(z\)-axis is
Identify the correct statement
Oil from a conical funnel is dripping at the rate of \(5\ \text{cm}^3/\text{s}\). If the radius and height of the funnel are 10 cm and 20 cm respectively, then the rate at which the oil level drops when it is 5 cm from the top is
Let \(R\) be a relation on natural numbers defined by \(x+2y=8,\ x,y\in N\). The domain of \(R\) is
Vasant and Jothi play a game with a coin. Vasant stakes INR 1 and throws the coin four times. If he throws four heads, he gets his stake and INR3 from Jothi. If he throws only three heads and they are consecutive, he gets his stake and INR 2 from Jothi. If he throws only two heads and they are consecutive, he gets his stake and INR 1 from Jothi. In all other cases Jothi takes the stake money. Find the expectation of Vasant's gain.
The solution of the differential equation \( \frac{dy}{dx}+y\log y\cot x=0 \) is
If \( \tan x^\circ \tan 2^\circ \tan 3^\circ \cdots \tan 88^\circ \tan y^\circ = 1 \), then \( \cot(x+y)= \)
A shopkeeper sells three varieties of fruit juice. He has a large number of bottles of same size of each variety. The number of different ways of displaying all the three varieties on the shelf with 5 places in a row and each display must have at least one bottle of each variety is
For a given Linear Programming problem, the objective function is \( z = 3x + 2y \). Subject to constraints are: \( 4x + 3y \leq 60,\ x \geq 3,\ y \leq 2x,\ y \geq 0 \). \(P\) is one of the corner points of the feasible region. Then the coordinate of \(P\) is
\( \displaystyle \int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}\,dx \) is equal to
The curve \( ax^3+bx^2+cx+d \) has a point of minima at \( x=1 \), then
A spherical snowball is melting such that its volume is decreasing at the rate of \(1\ \text{cm}^3/\text{min}\). The rate at which the diameter is decreasing when the diameter is 10 cm is