List of practice Questions

The unit vector \( (ai + bj) \) is perpendicular to \( (i + j) \). The value of 'b' is

Figure shows three forces \( F_1, F_2, F_3 \) acting along the sides of an equilateral triangle. If the total torque acting at point 'O' (centre of the triangle) is zero then the magnitude of \( F_3 \) is

A pipe open at one end has length 0.8 m. At the open end of the tube a string 0.5 m long is vibrating in its 1st overtone and resonates with the fundamental frequency of the pipe. If tension in the string is 50 N, the mass of the string is (speed of sound = 320 m/s)

If the dimensions of a physical quantity are given by \( [L^{-1} M^0 T^1] \), then the physical quantity is

The equation of the circle whose end points of a diameter are the centres of the circles \[ x^2 + y^2 + 2x - 4y + 1 = 0 \quad \text{and} \quad x^2 + y^2 - 8x + 6y + 17 = 0 \] is

The area of the region bounded by the curve \( y = 4x - x^2 \) and the x-axis is

Evaluate the integral \[ \int \frac{(1 + \log x)}{\cos^2(\log x)} \, dx \]

If \[ \mathbf{a} = 2\hat{i} + 3\hat{j} - \hat{k}, \quad \mathbf{b} = -\hat{i} + 2\hat{j} - 4\hat{k}, \quad \mathbf{c} = \hat{i} + \hat{j} + \hat{k} \] then \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c}) = \)

The maximum value of the function \( \frac{\log x}{x}, x \neq 0 \) is

The approximate value of \( \cot^{-1} (1 \cdot 001) \) is

With usual notations, if in \( \triangle ABC \), \( s \) is the semi-perimeter and \( (s - a)(s - b) = (s - c) \), then \( \triangle ABC \) is

The direction cosines of a line which makes equal acute angles with the co-ordinate axes are

If \[ y = \left( \frac{x^2 + 1}{x} \right)^x \text{ and } \frac{dy}{dx} = y \left[ g(x) + \log \left( \frac{x^2}{x+1} \right) \right], \text{ then } g(x) = \]

For \( \theta \in \left( 0, \frac{\pi}{2} \right) \), if \( \tan 3\theta \cdot \tan 2\theta \cdot \tan \theta + \tan 2\theta + \tan \theta = 1 \), then \( \theta = \)

For the following shaded region, the linear constraints are:

A metal has half-life period of 10 days. A sample originally has a mass of 1000 mg, then the mass remaining after 50 days is

If \[ A = \{ x / x \text{ is a prime number}, 0 \leq x \leq 9 \}, \text{ then the number of elements of the power set of } A \text{ is} \]

The solution of \( rdx + (x - r^2) dr = 0 \) is

The general solution of \( \tan \theta + \tan 2\theta = \tan 3\theta \)

The symbolic form of the following circuit is (where \( p, q, r \) represent switches \( s_1, s_2, s_3 \) which are closed respectively)

The equations of the lines which make intercepts on the axes whose sum is 8 and product is 15 are

For the following reaction, the relation between \( \Delta H \) and \( \Delta U \) is
\[ \text{2SO}_3(\text{g}) + \text{O}_2(\text{g}) \rightarrow \text{2SO}_3(\text{g}) \]

Which among the following group -15 elements forms hydrogen bonding in its hydride compounds?

An element crystallizes in bcc type of unit cell having atomic radius \( 1.33 \times 10^{-8} \) cm, the edge length of unit cell will be

Which among the following compounds is treated with benzonitrile to obtain benzophenone in dry ether and then acid hydrolysis?