List of top Mathematics Questions

Find the direction cosine of Z-axis.
Obtain the projection of the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + 5\hat{k} \) on the vector \( \vec{b} = \hat{i} + 3\hat{j} + \hat{k} \).
Prove that \( \sin^{-1} x = \cos^{-1} \sqrt{1 - x^2} \).
If y = 5x\(^2\) + 4, then at the point with x-coordinate 2, the slope is
Solve by matrix method the following system of equations:
\(2x + y + z = 1\)
\(x - 2y - z = \frac{3}{2}\)
\(3y - 5z = 9\)
Prove that the semi-vertical angle of the cone of given slant height and maximum volume is tan\(^{-1}\) \(\sqrt{2}\).
Find the general solution of the differential equation \(y - x \frac{dy}{dx} = x + y \frac{dy}{dx}\).
Find the area of the region bounded by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
A die was thrown twice and the sum of the numbers which appeared was found to be 6. Find the conditional probability that the number 4 appears at least once.
Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).
If \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), prove that \( \sin\frac{\theta}{2} = \frac{1}{2} |\hat{a} - \hat{b}| \).
Maximize \(Z = x + 2y\) by graphical method under the constraints \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).
Sand is falling from a pipe at the rate of 12 cm\(^3\)/second. The falling sand forms such a cone on the ground that its height is always one-sixth of the radius of its base. At which rate is the height of the cone formed by sand increasing while its height is 4 cm?
Find the shortest distance between the lines whose vector equations are \(\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}-3\hat{j}+2\hat{k})\) and \(\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu(2\hat{i}+3\hat{j}+\hat{k})\).
Find the area of the parallelogram whose diagonals are \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} \).
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
Let a relation R be defined in the set \(\mathbb{N} \times \mathbb{N}\) as follows: \((a, b) R (c, d)\) if and only if \(a + d = b + c\). Prove that R is an equivalence relation.
In a leap year, selected at random, find the probability that there are 53 Tuesdays.
Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x>0\).
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
Find the direction-cosines of the y-axis.
The order of the differential equation \( \left( \frac{d^3y}{dx^3} \right)^2 + \log x \left( \frac{d^2y}{dx^2} \right)^3 + 5y = \cos x \) will be
Let [x] represents the greatest integer which is less than or equal to x. Then the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = [x]\) will be