List of practice Questions

If \(y = t^2 + t^3\) and \(x = t - t^4\), then \(\frac{d^2y}{dx^2}\) at \(t = 1\) is:

If \( n \) is an integer and \( Z = \cos \theta + i \sin \theta, \theta \neq (2n + 1)\frac{\pi}{2}, \) then: \[ \frac{1 + Z^{2n}}{1 - Z^{2n}} = ? \]

If $a^2 x^4 + b^2 y^4 = c^6$, then the maximum value of $xy$ is:
Evaluate the integral \[ \int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx. \]
Evaluate the integral \[ I = \int_{-1}^{1} \left( \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right) \, dx \]

Match the following (f is the number of degrees of freedom):

Monoatomic Diatomic (rigid) Diatomic Polyatomic (non-rigid)
The current passing through a coil of 120 turns and inductance \( 40 \) mH is \( 30 \) mA. The magnetic flux linked with the coil is:
Consider the system of linear equations: \[ x + 2y + z = -3, \] \[ 3x + 3y - 2z = -1, \] \[ 2x + 7y + 7z = -4. \] Determine the nature of its solutions.
If the vectors \( a\hat{i} + \hat{j} + 3\hat{k} \), \( 4\hat{i} + 5\hat{j} + \hat{k} \), and \( 4\hat{i} + 2\hat{j} + 6\hat{k} \) are coplanar, then \( a \) is:
Angle between the circles \( x^2 + y^2 - 4x - 6y - 3 = 0 \) and \( x^2 + y^2 + 8x - 4y + 11 = 0 \) is:
The line \( 21x + 5y = k \) touches the hyperbola \( 7x^2 - 5y^2 = 232 \), then \( k \) is:
Which of the following solutions is used in the styptic action which prevents bleeding of blood?
The number of 5-digit odd numbers greater than 40,000 that can be formed by using 3, 4, 5, 6, 7, 0 such that at least one of its digits must be repeated is:
The equation of the pair of asymptotes of the hyperbola \( 4x^2 - 9y^2 - 24x - 36y - 36 = 0 \) is:
Evaluate the integral: \[ \int e^x \left( \frac{x + 2}{(x+4)} \right)^2 dx. \]
The solution of the differential equation $$ (x + 2y^3) \frac{dy}{dx} = y $$ is: 
X and Y are two circuits having coefficient of mutual inductance 3 mH and resistances 10 \( \Omega \) and 4 \( \Omega \) respectively. To have induced current 60 \( \times 10^{-4} \, {A} \) in circuit Y, the amount of current to be changed in circuit X in 0.02 sec is:

The number of \( d \) electrons in Fe is equal to which of the following?
(i) Total number of \( s \)-electrons of Mg
(ii) Total number of \( p \)-electrons of Cl 
(iii) Total number of \( p \)-electrons of Ne

If \( x - 4 = 0 \) is the radical axis of two orthogonal circles out of which one is \( x^2 + y^2 = 36 \), then the centre of the other circle is:
If \( P = (0,1,2) \), \( Q = (4,-2,-1) \) and \( O = (0,0,0) \), then \( \angle POQ \) is:
What is the enthalpy change (in J) for converting 9 g of H\(_2\)O (l) at \( +10^\circ C \) to H\(_2\)O (l) at \( +20^\circ C \)? Given \( C_p \) of water = 75 J/mol K and density of water = 1 g/mL.
If \( 5 \sinh x - \cosh x = 5 \), then one of the values of \( \tanh x \) is:
When the load applied to a wire is increased from 5 kg wt to 8 kg wt, the elongation of the wire increases from 1 mm to 1.8 mm. The work done during the elongation of the wire is (Acceleration due to gravity = 10 m/s\(^2\)):

The density of nitric acid solution is 1.5 g mL\(^{-1}\). Its weight percentage is 68. What is the approximate concentration (in mol L\(^{-1}\)) of nitric acid? (N = 14 u; O = 16 u; H = 1 u)

A car is moving on a horizontal curved road of radius 50 m. If the friction coefficient between tyres and road is 0.34, the approximate maximum speed of the car will be nearly: