List of practice Questions

The position vector of the point of intersection of the line joining the points \( \mathbf{i} - \mathbf{j} + \mathbf{k} \) and the line joining the points \( 2\mathbf{i} + \mathbf{j} - 6\mathbf{k} \), \( 3\mathbf{i} - \mathbf{j} - 7\mathbf{k} \) is:
P and Q are the points of trisection of the line segment AB. If the position vectors of A and B are \( 2\hat{i} - 5\hat{j} + 3\hat{k} \) and \( 4\hat{i} + \hat{j} - 6\hat{k} \) respectively, then the position vector of the point that divides PQ in the ratio 2:3 is:
The equation of the pair of transverse common tangents drawn to the circles \( x^2 + y^2 + 2x + 2y + 1 = 0 \) and \( x^2 + y^2 - 2x - 2y + 1 = 0 \) is:
Evaluate the integral: \[ \int (\log x)^3 \, dx \]
What is the structural formula of simplest ketone?

Two point charges M and N having charges +q and -q respectively are placed at a distance apart. Force acting between them is F. If 30% of charge of N is transferred to M, then the force between the charges becomes:

The coordinates of the point P which is equidistant from the three vertices of the △AOB as shown in the figure is
The coordinates of the point P which is equidistant from the three vertices of the △AOB
If the mode and mean of a data are 24 and 60 respectively, then the median of the data is
The electrical energy (in kWh) consumed in operating a bulb of 40 W for 5 hours a day in a month of 30 days is
A tree is broken by wind, its upper part touches the ground at a point 10 metre from the foot of the tree and makes an angle of 45° with the ground. Then what is the entire height of the tree?
In hypermetropia defect, the image is formed
If two towers of heights h1, and h2 subtend angles of 30° and 60° respectively at the midpoint of the line joining their feet, then the ratio of h1: h2 is
Reactivity increasing order of the following metals will be
An auto driver started an auto rickshaw with the help of pulling a rope. The device used by him to convert mechanical energy into electrical energy is
Choose the correct circuit which can achieve the bridge balance.
The general solution of the differential equation \[ (9x - 3y + 5) dy = (3x - y + 1) dx. \]
The HCF of 306 and 657 is
60 cm\(^3\) of SO\(_2\) gas diffused through a porous membrane in \( x \) minutes. Under similar conditions, 360 cm\(^3\) of another gas (molar mass 4 g mol\(^{-1}\)) diffused in \( y \) minutes. The ratio of \( x \) and \( y \) is:
As shown in the circuit, the initial voltage across the capacitor is \(10 \, {V}\), with the switch being open. The switch is then closed at \(t = 0\). The total energy dissipated in the ideal Zener diode \((V_Z = 5 \, {V})\) after the switch is closed (in mJ, rounded off to three decimal places) is \(\_\_\_\_\). \begin{center} \includegraphics[width=6cm]{35.png} \end{center}
Given below are two statements:
Statement I: Bromination of phenol in a solvent with low polarity such as CHCl\(_3\) or CS\(_2\) requires a Lewis acid catalyst.
Statement II: The Lewis acid catalyst polarizes the bromine to generate \( \text{Br}^+ \).
In light of the above statements, choose the correct answer from the options given below:
If the centroid of a triangle formed by the points \((a,b), (b, c)\) and \((c,a)\) is at the origin, then \(a^3+b^3+c^3 =\)
The remainder when the square of any prime number greater than 3 is divided by 6 is
A plane \( \pi_1 \) passing through the point \( 3\hat{i} - 7\hat{j} + 5\hat{k} \) is perpendicular to the vector \( \hat{i} + 2\hat{j} - 2\hat{k} \) and another plane \( \pi_2 \) passing through the point \( 2\hat{i} + 7\hat{j} - 8\hat{k} \) is perpendicular to the vector \( 3\hat{i} + 2\hat{j} + 6\hat{k} \). If \( p_1 \) and \( p_2 \) are the perpendicular distances from the origin to the planes \( \pi_1 \) and \( \pi_2 \) respectively, then \( p_1 - p_2 \) is:
In the given figure, PA and PB are the tangents to the circle with centre at O. If ∠APB = 36°, then ∠AOB =
In the given figure, PA and PB are the tangents to the circle with centre at O
The value of \(log_2 32\) is