List of top Mathematics Questions

If \( y = e^{2x} \), then find \( \dfrac{dy}{dx} \).
Evaluate: \[ \int (3x^2 + 4x - 5) \, dx \]
Find the maximum value of the linear objective function \( Z = 3x + 4y \) subject to the system constraints: \( x + y \le 4 \), \( x \ge 0 \), and \( y \ge 0 \).
If the straight line equation \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) runs completely parallel to the plane surface given by \( A x + 2y + 3z = 5 \), find the value of the coefficient \( A \).

Determine the degree of the following differential equation: \[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}} = \frac{d^2y}{dx^2} \] 

Under what condition will a linear programming system containing objective function variables have no common feasible region?
A straight vector line makes equal acute angles \( \alpha = \beta = \gamma \) with all three primary coordinate axes. Find the absolute value of \( \cos\alpha \).
Determine the sum of the order and the degree of the differential equation given by: \[ y = x\frac{dy}{dx} + \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
Find the total area of the region bounded by the parabola \( y^2 = 4x \) and the straight line \( y = x \).
A pair of standard unbiased six-sided dice are rolled simultaneously. Given that the sum of the numbers showing is exactly 6, what is the conditional probability that one of the dice shows the number 2?
If the direction vectors of two straight lines are given by \( \vec{m} = \hat{i} + \hat{j} + 0\hat{k} \) and \( \vec{n} = \hat{i} + 0\hat{j} + \hat{k} \), calculate the value of \( \cos\phi \) representing the acute angle between them.
Find the total area of the bounded region enclosed between the parabola curve \( y^2 = 8x \) and its vertical latus rectum boundary line.
Let a relation \( R \) be defined on the set of all natural numbers \( \mathbb{N} \) by the rule: \( (a, b) \in R \) if and only if \( a + b \) is an even number. This relation \( R \) is classified as an:
Find the angle \( \theta \) between the line \( \frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-3}{2} \) and the plane \( 2x - y + 2z = 7 \).
Find the general solution of the following homogeneous differential equation: \[ \frac{dy}{dx} = \frac{x^2 + y^2}{xy} \]
Two independent events \( A \) and \( B \) have individual probabilities of occurring given by \( P(A) = 0.4 \) and \( P(B) = 0.5 \). Find the probability that at least one of these two events occurs.
If \( A \) is a skew-symmetric matrix of order \( 5 \times 5 \), what is the value of its determinant \( |A| \)?
Evaluate the definite integral using standard definite integral properties: \[ \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} \, dx \]

Calculate the mean (expected value) of a random variable \( X \) given its probability distribution setup below: 

\( X \)123
\( P(X) \)\( 0.2 \)\( 0.5 \)\( 0.3 \)
Simplify the following expression using inverse trigonometric properties: \( \tan\left(\sin^{-1}x + \cos^{-1}x\right) \) where \( |x| \le 1 \).
Find the interval where the function \( f(x) = x^2 - 4x + 6 \) is strictly decreasing.
If a square matrix \( A \) satisfies the condition \( A^T = -A \), then the matrix \( A \) is classified as a:
If two vectors \( \vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k} \) and \( \vec{b} = 4\hat{i} - 2\hat{j} - 2\hat{k} \) are perpendicular to each other, determine the value of the scalar constant \( \lambda \).

A random variable \( X \) has the following probability distribution table: 

\( X \)012
\( P(X) \)\( k \)\( 2k \)\( 3k \)

Find the exact value of the unknown parameter \( k \).

Find the slope of the normal to the curve \( y = 2x^2 + 3\sin x \) at the coordinate point where \( x = 0 \).