List of practice Questions

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

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: \(ydx - (x + 2y^2)dy = 0\).

Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).

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 \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).

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.

Find the value of \(\frac{dy}{dx}\) if \(y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)\), where \(-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\).

Find the value of \(\int \frac{1}{x^2 - a^2} dx\).

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.

If two vectors \(\vec{a}\) and \(\vec{b}\) are such that \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) and \(\vec{a} \cdot \vec{b} = 4\), find \(|\vec{a} - \vec{b}|\).

The Cartesian equation of a line is \(\frac{x+3}{2} = \frac{y-5}{4} = \frac{z+6}{2}\). Find its vector equation.

A die is thrown once. If the event 'the number obtained on the die is a multiple of 3' is represented by E and 'the number obtained on the die is even' is represented by F, tell whether the events E and F are independent.

Find the principal value of \(\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\).

Find the value of \(\int x^3 e^{x^4} dx\).

Let \(\vec{a} = \hat{i} + 2\hat{j}\) and \(\vec{b} = 2\hat{i} + \hat{j}\). Is \(|\vec{a}| = |\vec{b}|\)? Are the vectors \(\vec{a}\) and \(\vec{b}\) equal?

If the orders of the matrices A and B are \(m \times n\) and \(n \times p\) respectively, then the order of AB will be

The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are

\[ \left( \sqrt{2} + 1 + i \sqrt{2} - 1 \right)^8 = ? \]

In which of the following places was the first atomic power station located?

Given below are statement and reason. Give answer by selecting correct option.
Statement: Alluvial soil is useful for paddy, wheat, and pulses.
Reason: Alluvial soil is famous for moisture.

Which of the following ports is not on the western coast of India?