List of practice Questions

Find the differential equation by eliminating arbitrary constants from the relation \( y = Ae^{3x} + Be^{-3x} \).
Using the truth table, verify \( p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \).
If \( x = \frac{4t}{1+t^2} \), \( y = 3\left(\frac{1-t^2}{1+t^2}\right) \), then show that \( \frac{dy}{dx} = -\frac{9x}{4y} \).
Divide the number 84 into two parts such that the product of one part and square of the other is maximum.
Solve the following differential equation \( (x^2 - yx^2)dy + (y^2 + xy^2)dx = 0 \).
An agent who gives guarantee to his principal that the party will pay the sale price of goods is called --
In an ordinary annuity, payments or receipts occur at
\( \int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx = ............. + c \)
If \( f'(x) = x^2 + 5 \) and \( f(0) = -1 \) then \( f(x) = ..............\)
Write the converse, inverse and contrapositive of the statement "If a triangle is equilateral then it is equiangular".
Solve the following equations by the method of inversion: \( 2x-y+z=1 \), \( x+2y+3z=8 \), \( 3x+y-4z=1 \).
Find MPC, MPS, APC and APS, if the expenditure \(E_c\) of a person with income I is given as \( E_c = (0.0003)I^2 + (0.075)I \); when \( I=1000 \).
Evaluate: \( \int_1^2 \frac{dx}{x^2+6x+5} \)
Find \( \frac{dy}{dx} \) if \( y = x^x + a^x \).
The area of the region bounded by the curve \(y=x^2\) and the line \(y=4\) is:
The order and degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = a^x \) are ............. respectively.
The integrating factor of the differential equation \( \frac{dy}{dx} + \frac{y}{x} = x^3 - 3 \) is:
If A is a matrix and K is a constant, then \( (KA)^T = K A^T \).
The value of \( \int \log x \, dx = x \log x + x + c \).
The differential equation obtained by eliminating arbitrary constants from \( bx + ay = ab \) is \( \frac{d^2y}{dx^2} = 0 \).
The average revenue \( R_A \) is 50 and elasticity of demand \( \eta \) is 5, the marginal revenue \( R_M \) is ..............
If \( p \): He is intelligent, \( q \): He is strong. Then, symbolic form of statement "It is wrong that he is intelligent or strong" is:
The value of \( \int \left(x + \frac{1}{x}\right)^3 dx \) is equal to:
The value of the definite integral \( \int_2^7 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{9-x}} dx \) is:
Interpret the given diagrams A and B. Enlist the changes occurring during inspiration and expiration.