List of top Mathematics Questions

A box is formed by a 3 m x 8 m rectangular steel-sheet on cutting the squares of length x m from its each corner to form the box without cover. Then find the maximum volume of the box so formed.
If the ordered pairs (2x - 3, 5) and (x, y - 1) are equal, then find the numbers x and y.
Obtain the differential equation of the family of curves \(y = \frac{2ce^{2x}}{1+ce^{2x}}\).
Prove that \( 0 \leq P(E) \leq 1 \), where P(E) is the probability of the event E.
Find the area of a parallelogram whose adjacent sides are the vectors \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k} \).
The modulus of two vectors \(\vec{a}\) and \(\vec{b}\) are \(\sqrt{3}\) and 4 respectively, and \(\vec{a} \cdot \vec{b} = 6\). Then find the angle between the vectors \(\vec{a}\) and \(\vec{b}\).
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 the function f: N \(\rightarrow\) N, is defined by f(x) = x - 1 for all x > 2 and f(1) = f(2) = 1, then f is
If y = 5x\(^2\) + 4, then at the point with x-coordinate 2, the slope is
Find the direction cosine of Z-axis.
Find the area of the region bounded by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Prove that the semi-vertical angle of the cone of given slant height and maximum volume is tan\(^{-1}\) \(\sqrt{2}\).
Solve by matrix method the following system of equations:
\(2x + y + z = 1\)
\(x - 2y - z = \frac{3}{2}\)
\(3y - 5z = 9\)
Find the general solution of the differential equation \(y - x \frac{dy}{dx} = x + y \frac{dy}{dx}\).
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 area of the parallelogram whose diagonals are \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} \).
If \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), prove that \( \sin\frac{\theta}{2} = \frac{1}{2} |\hat{a} - \hat{b}| \).
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 \(\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu(2\hat{i}+3\hat{j}+\hat{k})\).
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.
Maximize \(Z = x + 2y\) by graphical method under the constraints \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).
Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
Find the projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
In a leap year, selected at random, find the probability that there are 53 Tuesdays.