List of top Mathematics Questions

Find the direction-cosines of the sum of the vectors \( \vec{a} = 3\hat{i} + 4\hat{j} - 3\hat{k} \) and \( \vec{b} = -2\hat{i} - 3\hat{j} + \hat{k} \).
If the functions \( f:\mathbb{R} \to \mathbb{R} \) and \( g:\mathbb{R} \to \mathbb{R} \) are defined as \( f(x)=\cos x \) and \( g(x)=3x^2 \) respectively then find gof.
Prove that \( \sin^{-1} x = \tan^{-1} [x / \sqrt{1-x^2}] \).
If two dice are thrown together then find the probability of getting the sum eight.
Find the differential equation representing the family of curves \(y=2mx\).
If the position vectors of the points A and B are \(\hat{i}+\hat{j}+\hat{k}\) and \(2\hat{i}+5\hat{j}\) respectively, then find the unit vector along the straight line AB.
The number of arbitrary constants in a general solution of a differential equation of fifth order is
If the function \( f : \mathbb{R} \to \mathbb{R} \) is defined as \( f(x) = 3x \) then f is
If \( X = \{ a, b, c \} \) and \( Y = \{ 1, 2, 3 \} \) and the mapping f is given by \( f(a)=2, f(b)=3, f(c)=1 \), then
If \( \int \log x \, dx = x \log x + k(x) + c \) then
Find the area enclosed by the curve \( y = x^2 \) and the line \( y = 16 \).
Solve the following system of equations by matrix method:
\(x + y + 2z = 1\)
\(3x + 2y + z = 7\)
\(2x + y + 3z = 2\)
Solve : \( \frac{dy}{dx} = \frac{x + 2y - 3}{2x + y - 3} \).
If the function \(f: [0, \frac{\pi}{2}] \rightarrow \mathbb{R}\) is given by \(f(x) = \sin x\) and function \(g: [0, \frac{\pi}{2}] \rightarrow \mathbb{R}\) is given by \(g(x) = \cos x\), then prove that f and g are one-one but \(f + g\) is not one-one.
Find the shortest distance between the lines \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \) and \( \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \).
Find the differential equation of the family of curves denoted by \(y = a \sin(x+b)\), where a and b are arbitrary constants.
Differentiate the function \(x^{\cos x}\) with respect to x.
Minimize Z = 3x + 2y by graphical method under the following constraints: \(x + 2y \leq 10\), \(3x + y \leq 15\), \(x \geq 0\), \(y \geq 0\).
A car is started from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by \( x = t^2(2 - \frac{t}{3}) \). Find the time required by the car to reach at point Q and also find the distance between P and Q.
Find the area of a triangle \( \triangle ABC \) whose vertices are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1).
A person has a contract of construction. The probability of being a strike is 0.65. The probabilities of completing the construction work on time in both conditions are 0.80 and 0.32 whether the strike is not happened and it is happened respectively. Then find the probability of completing the construction work in due time.
If the ordered pairs (2x - 3, 5) and (x, y - 1) are equal, then find the numbers x and y.
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 \(\vec{b}\).
Prove that \( 0 \leq P(E) \leq 1 \), where P(E) is the probability of the event E.