List of top Mathematics Questions asked in COMEDK UGET

A bag contains \( (n + 1) \) coins. It is known that one of these coins has a head on both sides, whereas the other coins are fair. One of these coins is selected at random and tossed. If the probability that the toss results in heads is \( \frac{7}{12} \), then the value of \( n \) is :

The function \( f(x) = \left\{ \begin{array}{ll} \frac{|x|}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{array} \right. \) is discontinuous at

If \[ \binom{n+2}{8} : \, \binom{n-2}{4} = 57 : 16, \text{ then } n \text{ is } \]

Integrating factor of the differential equation \[ \frac{dy}{dx} + y = \frac{x^3 + y}{x} \]

If \[ y = (\sin^{-1} x)^2 + (\cos^{-1} x)^2, \] then \[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = \]

The equations \( x = a(\theta + \sin \theta) \) and \( y = a(1 - \cos \theta) \) represent the equation of a curve. If \( \theta \) changes at a constant rate \( k \), then the rate of change of the slope of the tangent to the curve at \( \theta = \frac{\pi}{3} \) is

The length of the latus rectum of a conic \( 49y^2 - 16x^2 = 784 \) is

If \( A = \frac{1}{\pi} \begin{bmatrix} \sin^{-1} \frac{1}{2} & \tan^{-1} \frac{x}{\pi} \sin^{-1} \frac{x}{\pi} & \cot^{-1} \sqrt{3} \end{bmatrix} \), then \( A - B \) is:

The curve \( 4y = 3x^4 - 2x^2 \) attains

If \( \cos A = \frac{3}{4} \), then \( 32 \sin \frac{A}{2} \sin \frac{5A}{2} = \text{?} \)

The area of the region enclosed by the lines \( 2x + y = 10 \), \( y = 1 \), \( y = 5 \) and the y-axis is

If \( 2y = \left[ \cot^{-1} \left( \frac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x} \right) \right]^2 \ \quad \forall x \in \left( 0, \frac{\pi}{2} \right), \text{ then } \frac{dy}{dx} \text{ is equal to:}\)}

For real numbers \( x \) and \( y \), \( xRy \iff x - y + \sqrt{2} \) is an irrational number. Then the relation \( R \) is:

The terms of an infinitely decreasing geometric progression in which all the terms are positive, the first term is 4, and the difference between third and fifth term is \( \frac{32}{81} \), then which of the following is not true?

Five persons entered the lift cabin on the ground floor of an eight-floor apartment. Suppose that each of them independently and with equal probability, can leave the cabin at any floor beginning with the first floor, then the probability of all five persons leaving at different floors is:

Let \( A = \{x : x = 4n + 1, n \in \mathbb{Z}, 0 \leq n < 4 \} \)
Let \( B = \{x : x = 15n + 4, n \in \mathbb{N}, n \leq 3 \} \)
Let \( C = \{x : x \text{ is a prime number}, x \in A \cup B \} \)
Then the cardinal number of set \( C \) is

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

If \( \frac{x-1}{3+i} + \frac{y-1}{3-i} = i \) then \( (y,x) = \)