List of practice Questions

If the angle between the lines joining the origin to the points of intersection of \( x + 2y + \lambda = 0 \) and \( 2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0 \) is \( \dfrac{\pi}{2} \), then a value of \( \lambda \) is:
The interval in which the curve represented by \( f(x) = 2x + \log\left(\frac{x}{2 + x}\right) \) is increasing is
The velocity acquired by an electron at rest when subjected to a uniform electric field of potential difference $180~\text{V}$ is:
(Mass of electron $= 9 \times 10^{-31}$ kg and charge of electron $= 1.6 \times 10^{-19}$ C)
A coil of 45 turns and radius 4 cm is placed in a uniform magnetic field such that its plane is perpendicular to the direction of the field. If the magnetic field increases from 0 to 0.70 T at a constant rate in a time interval of 220 s, then the induced emf in the coil is
If $ A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix} $ is symmetric and $ B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & 4 \\ -3 & r & s \end{bmatrix} $ is skew-symmetric, then find $ |A| + |B| - |AB| $
If $ 0 \le x \le 3,\ 0 \le y \le 3 $, then the number of solutions $(x, y)$ for the equation: $$ \left( \sqrt{\sin^2 x - \sin x + \frac{1}{2}} \right)^{\sec^2 y} = 1 $$
In triangle $ ABC $, $ 2A + C = 300^\circ $. If the circumradius is 8 times the inradius, then $ \sin\frac{C}{2} = ? $
If the equation of the circle having the common chord to the circles $$ x^2 + y^2 + x - 3y - 10 = 0 $$ and $$ x^2 + y^2 + 2x - y - 20 = 0 $$ as its diameter is $$ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, $$ then find $ \alpha + 2\beta + \gamma $.
Which of the following statements regarding adsorption theory of heterogeneous catalysis is not correct?
If the transformed equation of the equation \[ 2x^2 + 3xy - 2y^2 - 17x + 6y + 8 = 0 \] after translating the coordinate axes to a new origin \((\alpha, \beta)\) is \[ aX^2 + 2h XY + bY^2 + c = 0, \] then find \(3\alpha + c\).
Arrange the products I, II, III from the following reactions in decreasing order of their acid strength.
Arrange the products I, II, III
The number of ways of distributing 3 dozen fruits (no two fruits are identical) to 9 persons such that each gets the same number of fruits is
If a particle of mass 'm' covers half of the horizontal circle with constant speed 'v', then the change in its kinetic energy is
A small quantity of water of mass 'm' at temperature \( \theta^\circ \text{C} \) is mixed with a large mass 'M' of ice which is at its melting point. If 's' is the specific heat capacity of water and 'L' is the latent heat of fusion of ice, then the mass of ice melted is
In Young's double-slit experiment, if the distance between the slits is increased to 3 times its initial distance, then the ratio of initial and final fringe widths is.

$\left( 1 + \sqrt{5} + i \sqrt{10 - 2\sqrt{5}} \right)^5$

The final product (C) in the given reaction sequence is

If \( z = x+iy \) and \(x^2+y^2=1\), then \( \frac{1+x+iy}{1+x-iy} = \)
If \( {}^{27}P_{r+7} = 7722 \cdot {}^{25}P_{r+4} \), then r =
If \(2\sin x - \cos 2x = 1\), then \( (3 - 2\sin^2x) = \)
If the equation of the tangent of the hyperbola \( 5x^2 - 9y^2 - 20x - 18y - 34 = 0 \) which makes an angle \( 45^\circ \) with the positive X-axis in positive direction is \( x+by+c=0 \) then \( b^2+c^2 = \)
Consider the reaction sequence Dimethyl ketone \( \xrightarrow{\text{(i)CH}_3\text{MgCl (ii)H}_2\text{O}} \) X \( \xrightarrow{\text{(i)Na (ii)CH}_3\text{Br}} \) Y How many sp\(^3\) carbons are present in Y?
In triangle \(ABC\), if \(a = 6\), \(b = 8\), and \(c = 10\), then \(\dfrac{2r_1}{r_2} =\)
The triangle formed by the lines \( 2x^2 + xy - 6y^2 = 0 \) and \( x+y-1=0 \) is:
A molecule has a T-shape. The total number of electron pairs in the valence shell of its central atom is: