List of practice Questions

For \( a \in \mathbb{R} \setminus \{0\} \), if \( a \cos x + a \sin x + a = 2K + 1 \) has a solution, then \( K \) lies in the interval:
The equation \( x^4 - x^3 - 6x^2 + 4x + 8 = 0 \) has two equal roots. If \( \alpha, \beta \) are the other two roots of this equation, then \( \alpha^2 + \beta^2 = \):
If \( m, n \) are respectively the least positive and greatest negative integer values of such that \( (\frac{1-i}{1+i})^k = -i\), then \( m - n = \):
A block of mass 100 g is attached to the end of a string of unknown material of 50 cm long and cross-sectional area of \(2 \, {cm}^2\). When the block is whirled horizontally at a constant angular speed of 40 m/s, it moves along a circular path of radius 52 cm. Young's modulus of the material is:

A metal rod of 50 cm length is clamped at its midpoint and is set to vibrations. The density of that metal is \( 2 \times 10^3 \, {kg/m}^3 \).
Young's modulus of that metal is \( 8 \times 10^8 \, {Nm}^2 \). The fundamental frequency of the vibration is 
 

The element which is important in determining solute concentration and oxygen evolution:
The domain of the function $ y = \frac{1}{\log_{10}(3 - x)} + \sqrt{x + 7} $ is:
The radii of two conducting spheres A and B of each charge +90 µC are 8 cm and 10 cm respectively. When the two spheres are connected by a conducting wire, then the charge flowing from sphere A to sphere B is:
Evaluate the integral \[ \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. \]
If the plane \[ x - y + z + 4 = 0 \] divides the line joining the points \[ P(2,3,-1) \quad {and} \quad Q(1,4,-2) \] in the ratio \( l:m \), then \( l + m \) is:
Let \( \vec{a} \times \vec{b} = 7\hat{i} - 5\hat{j} - 4\hat{k} \) and \( \vec{a} = \hat{i} + 3\hat{j} - 2\hat{k} \), if the length of projection of \( \vec{b} \) on \( \vec{a} \) is \( \frac{8}{\sqrt{14}} \), then \( |\vec{b}| \) is:
The tangent drawn to the magnetic field line at a point gives
Two masses \( m_1 \) and \( m_2 \) are connected by a light string passing over a smooth pulley. When set free, \( m_1 \) moves downwards by 3 m in 3 s. The ratio of \( \frac{m_1}{m_2} \) is \((g = 10 \text{ m/s}^2)\).
What will come in place of the question mark (?) in the following question ? \[ \frac{120 - 22 \times 6 \div 2}{\frac{1}{4} \text{ of } 256 \div 16 + 2} = ? \]

A hydrogen atom falls from nth higher energy orbit to first energy orbit (n = 1). 

The energy released is equal to 12.75 eV. The nth orbit is

The longest wavelength of light that can initiate photo electric effect in the metal of work function 9 eV is
An electromagnetic wave travels in a medium with a speed of \( 2 \times 10^8 \, \text{ms}^{-1} \). The relative permeability of the medium is 1. Then the relative permittivity is.
Magnetic field at a distance \(r\) from z axis is \( B = B_0 r \, \text{kt} \) present in the region. \( B_0 \) is constant and \(t\) is time. The magnitude of induced electric field at a distance \(r\) from z-axis is.

Select a figure from the given four alternatives which when placed in the blank space of the given figure would complete the pattern.

Select a figure from the given four alternatives
The slope of a line L passing through the point \((-2, -3)\) is not defined. If the angle between the lines L and \( ax - 2y + 3 = 0 \) (where \( a>0 \)) is 45°, then the angle made by the line \( x + ay - 4 = 0 \) with positive X-axis in the anticlockwise direction is:
Which of the following hormones is responsible for preparing the uterus for implantation of a fertilized egg?
If \( (a, b) \) is the stationary point of the curve \( y = 2x - x^2 \), then the area bounded by the curves \( y = 2x - x^2 \), \( y = x^2 - 2x \), and \( x = a \) is:
A boy gave a bundle of RS.100 notes to a guy. The serial number of notes are from CJJ103440 to CJJ103490. What is the total amount given to the guy
Evaluate \( \int_{1}^{2} \frac{x^4 - 1}{x^6 - 1} dx \):
Identify the major product \((P)\) in the following reaction sequence: \[ (\text{CH}_3)_3\text{CBr} \xrightarrow[\Delta]{\text{Alcoholic KOH}} X \xrightarrow{\text{HBr}} P \]