List of top Mathematics Questions

If \(\sin \theta = \frac{1}{9}\), then \(\tan \theta\) is equal to
If the points \(A(6, 1)\), \(B(p, 2)\), \(C(9, 4)\), and \(D(7, q)\) are the vertices of a parallelogram \(ABCD\), then find the values of \(p\) and \(q\). Hence, check whether \(ABCD\) is a rectangle or not.

A manufacturer makes two types of toys A and B. Three machines are needed for production with the following time constraints (in minutes): \[ \begin{array}{|c|c|c|} \hline \text{Machine} & \text{Toy A} & \text{Toy B} \\ \hline M1 & 12 & 6 \\ M2 & 18 & 0 \\ M3 & 6 & 9 \\ \hline \end{array} \] Each machine is available for 6 hours = 360 minutes. Profit on A = Rupee 20, on B = Rupee 30.
Formulate and solve the LPP graphically.

Evaluate the definite integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin x}{1 + \cos^2 x} \, dx \]
A die is thrown three times. Find the probability of getting a number greater than 4 at least once.
Find the shortest distance between the lines: \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}, \quad \frac{x + 2}{1} = \frac{y - 3}{-2} = \frac{z + 1}{2} \]
The cost and revenue functions are given as: \[ C(x) = 3x^2 + 5x + 200, \quad R(x) = 50x \] Find the number of items that should be sold to maximize the profit.
Two dice are thrown. Defined are the following two events A and B: \[ A = \{(x, y) : x + y = 9\}, \quad B = \{(x, y) : x \neq 3\}, \] where \( (x, y) \) denote a point in the sample space. Check if events \( A \) and \( B \) are independent or mutually exclusive.
Find the shortest distance between the lines: \[ \mathbf{r}_1 = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda (\hat{i} - 2\hat{j} + 3\hat{k}) \] \[ \mathbf{r}_2 = (\hat{i} + 4\hat{k}) + \mu (3\hat{i} - 6\hat{j} + 9\hat{k}) \]
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \lambda \hat{k}$, then the value of $\lambda$ is:
Find the angle at which the given lines are inclined to each other: \[ l_1: \frac{x - 5}{2} = \frac{y + 3}{1} = \frac{z - 1}{-3} \] \[ l_2: \frac{x}{3} = \frac{y - 1}{2} = \frac{z + 5}{-1} \]
Evaluate: \[ \int_0^\pi \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \]
\[ \int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha} \, dx \] is equal to :
Find the distance of the point $P(2, 4, -1)$ from the line \[ \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}. \]
(a) Find the point Q on the line \( \frac{2x + 4}{6} = \frac{y + 1}{2} = \frac{-2z + 6}{-4} \) at a distance of \( \frac{\sqrt{5}}{2} \) from the point \( P(1, 2, 3) \).
Solve the differential equation \( 2(y + 3) - xy \frac{dy}{dx} = 0 \); given \( y(1) = -2 \).
The integrating factor of the differential equation \( \frac{dx}{dy} = \frac{x \log x}{2 \log x - y} \) is:

Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is

Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]
The corner points of the feasible region in graphical representation of a L.P.P. are \( (2, 72), (15, 20) \) and \( (40, 15) \). If \( Z = 1x + 9y \) be the objective function, then:
If a line makes angles of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) with the positive directions of \( x \), \( y \), and \( z \)-axes respectively, then \( \theta \) is:
If \( x = e^{\frac{x}{y}} \), then prove that \( \frac{dy}{dx} = \frac{x - y}{x \log x} \).
A man needs to hang two lanterns on a straight wire whose end points have coordinates A (4, 1, -2) and B (6, 2, -3). Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.
Let the volume of a metallic hollow sphere be constant. If the inner radius increases at the rate of 2 cm/s, find the rate of increase of the outer radius when the radii are 2 cm and 4 cm respectively.
Sum of two skew-symmetric matrices of the same order is always a/an: