List of top Questions asked in COMEDK UGET

The integral \[ \int e^x \left( 1 + \tan x + \tan^2 x \right) dx \] is equal to

In a \( \triangle ABC \), if coordinates of point A are \( (1, 2) \) and the equations of the medians through B and C are \[ x + y = 5 \quad \text{and} \quad x = 4 \quad \text{respectively, then the coordinates of B are} \]

The distance of the point \( (2, 3, 4) \) from the line \[ 1 - x = \frac{y}{2} = \frac{1}{3}(1 + z) \] is

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

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

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

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

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

The solution of the differential equation \[ \frac{dy}{dx} + y \cos x = \frac{1}{2} \sin 2x \] is

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

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

P is a point on the line segment joining the points \( (3, 2, -1) \) and \( (6, 2, -1) \).

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

In the set \( W \) of whole numbers, an equivalence relation \( R \) is defined as follows: \( a R b \) iff both \( a \) and \( b \) leave the same remainder when divided by 5. The equivalence class of 1 is given by

The function defined by \[ f(x) = \begin{cases} \frac{\sin x}{x} + \cos x & \text{if } x>0 \\ -5k & \text{if } x = 0 \\ \frac{4(1 - \sqrt{1 - x})}{x} & \text{if } x<0 \end{cases} \] is continuous at

If \[ f(x) = 2x - \tan^{-1} x - \log(x + \sqrt{x^2 + 1}) \] is monotonically increasing, then

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

cos\(^6\) A - sin\(^6\) A is equal to

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

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

If \( 2A + 3B = \begin{bmatrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{bmatrix} \) and \( A + 2B = \begin{bmatrix} 5 & 0 & 3 \\ 1 & 6 & 2 \end{bmatrix} \), then \( B = \)

If \( \csc(90^\circ + A) + x \cos A \cot(90^\circ + A) = \sin(90^\circ + A) \), then the value of

The scalar components of a unit vector which is perpendicular to each of the vectors \( \hat{i} + 2\hat{j} - \hat{k} \) and \( 3\hat{i} - \hat{j} + 2\hat{k} \) are:

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 choose the 7 questions is:

Evaluate the integral \[ \int \sqrt{\csc x - 1} \, dx = \]