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If y is a function of x and $\log(x + y) = 2xy$, what is the value of $y'(0)$?

Evaluate the integral: $\int \cos^3(x) dx$

Solve the following differential equation: $\frac{dx}{dt} = \frac{x}{\log(x) \cdot t}$

A random variable X has the probability mass function $P(X = x) = {4 \choose x} (\frac{5}{9})^x (\frac{4}{9})^{4-x}$, for $x = 0, 1, 2, 3, 4$. What is the expected value $E(X)$?

What is the joint equation of the coordinate axes?

Find $dy/dx$ if $x = at^2$ and $y = 2at$.

Find the area of the region enclosed between the curves $y^2 = 4ax$ and $x^2 = 4ay$.

Find the particular solution of the differential equation $\frac{dy}{dx} + y = e^{-x}$, given that $y = 0$ when $x = 0$.

Determine the values of $x$ for which the matrix $A = \begin{bmatrix} 1 & x & 1 \ x & 1 & 1 \ 1 & 1 & x \end{bmatrix}$ is singular.

Find the equation of the plane that passes through the point $(1, 2, 3)$ and is perpendicular to the line with direction ratios 2, -1, and 1.

Evaluate $\int_{0}^{\pi/2} \sin^2(x) \cos^3(x) , dx$.

Given the differential equation $\frac{dy}{dx} = x + y$ with the initial condition $y(0) = 1$, use Euler's method with a step size of $h = 0.1$ to approximate $y(0.2)$.

A fair coin is tossed 3 times. What is the probability of getting at least two heads?

Find the general solution of the differential equation $\frac{dy}{dx} + 2y = e^{-x}$.

Determine the angle between the vectors $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - 2\hat{k}$.

A manufacturer produces two products, A and B. Each product requires processing time on two machines: Machine 1 and Machine 2. Product A requires 1 hour on Machine 1 and 2 hours on Machine 2. Product B requires 3 hours on Machine 1 and 1 hour on Machine 2. Machine 1 is available for a maximum of 9 hours, and Machine 2 is available for a maximum of 8 hours. If the profit per unit of A is $4 and per unit of B is $5, formulate this as a linear programming problem to maximize profit.

What is the mean of numbers randomly selected from 1 to 15?

Find the area of the region bounded by the curve $y = x^2$ and the line $y = 4$.

Find the general solution of $\sin \theta + \sin 3\theta + \sin 5\theta = 0$.

If $-1 \leq x \leq 1$, then prove that $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$.

Find the direction ratios of a vector perpendicular to two lines whose direction ratios are -2, 1, -1 and -3, -4, 1.

Find the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$.

If $y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + ... + \infty}}}$, then show that $\frac{dy}{dx} = \frac{\sec^2 x}{2y - 1}$. Find $\frac{dy}{dx}$ at $x = 0$.

Evaluate $\int x \tan^{-1} x dx$

Given the vector $\vec{u} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, find the value of c that satisfies the equation $c \vec{u} = 3\hat{k}$.

Determine the integral of cotangent x with respect to x: $\int cot(x) dx$?

What is the degree of the following differential equation? $e^{\frac{dy}{dx}} + \frac{dy}{dx} = x$

Given the statement "If x < y then $x^2 < y^2$", write its inverse and contrapositive.

Given the matrix $A = \begin{bmatrix} x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z \end{bmatrix}$, where A is non-singular, determine $A^{-1}$ using elementary row transformations. Hence, give the inverse of the specific matrix $\begin{bmatrix} 2 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \end{bmatrix}$.

Find the Cartesian coordinates of the point whose polar coordinates are $\left(2, \frac{\pi}{4}\right)$.

If $ax^2 + 2hxy + by^2 = 0$ represents a pair of lines and $h^2 = ab \neq 0$, find the ratio of their slopes.

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are the position vectors of points A, B, C respectively, and $5\vec{a} + 3\vec{b} - 8\vec{c} = 0$, find the ratio in which point C divides the line segment AB.

Solve the differential equation $\frac{dy}{dx} = e^{2y} \cos x$ with the initial condition $x = \frac{\pi}{6}$, $y = 0$.

Given the probability density function $f(x) = \frac{x+2}{18}$ for $-2 < x < 4$ and $0$ otherwise, find $P(|X| < 1)$.

A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of at least 5 successes?

Simplify the logic circuit with switches $S_1$ and $S_2$ connected in the configuration $((S_1 \land S_2') \lor (S_1' \land S_2))$.

If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, verify that $A(\text{adj }A) = (\text{adj }A)A = |A|I$, where $I$ is the identity matrix.

Prove that volume of a tetrahedron with coterminus edges $\vec{a}$, $\vec{b}$ and $\vec{c}$ is $\frac{1}{6} |[\vec{a} \ \vec{b} \ \vec{c}]|$.

Find the length of the perpendicular drawn from the point $P(3, 2, 1)$ to the line given by $\vec{r} = (7\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda(-2\hat{i} + 2\hat{j} + 3\hat{k})$.

If $y = \cos(m \cos^{-1} x)$, then show that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + m^2y = 0$.