List of top Mathematics Questions

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 \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).
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.
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.
Find the projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
In a leap year, selected at random, find the probability that there are 53 Tuesdays.
Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).
Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x>0\).
Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).
Find the direction-cosines of the y-axis.
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).
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
If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) 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
Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).
Minimize Z = 200x + 500y by graphical method subject to the following constraints:
x + 2y \(\ge\) 10, 3x + 4y \(\le\) 24, x \(\ge\) 0, y \(\ge\) 0.
For two vectors \(\vec{a}\) and \(\vec{b}\) prove that \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\).
Solve: \(ydx - (x + 2y^2)dy = 0\).
For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the curve passing through the point \((1, -1)\).
A person has taken the contract of a construction work. The probability of strike is 0.65. The probabilities of the construction work being completed on time in the circumstances of no strike and strike are respectively 0.80 and 0.32. Find the probability of the construction work being completed on time.
If \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).
Show that the function \(f(x) = \log \sin x\) is increasing in the interval \((0, \frac{\pi}{2})\) and decreasing in \((\frac{\pi}{2}, \pi)\).
If \(R_1\) and \(R_2\) are equivalence relations in the set A, prove that \(R_1 \cap R_2\) is also an equivalence relation in A.