List of practice Questions

The integrating factor of the differential equation \[ x \frac{dy}{dx} + 2y = x e^x \] is:

The minimum value of the function \( f(x) = x^4 - 4x - 5 \), where \( x \in \mathbb{R} \), is:

\[ \int_0^{\frac{\pi}{4}} (\tan^3 x + \tan^5 x) \, dx \]

\[ \int \left( \frac{\log_e t}{1+t} + \frac{\log_e t}{t(1+t)} \right) dt \]

Evaluate the integral:

\[ \int \frac{4x \cos \left( \sqrt{4x^2 + 7} \right)}{\sqrt{4x^2 + 7}} \, dx \]

Let \( f(x) = \frac{x^2 + 40}{7x} \), \( x \neq 0 \), \( x \in [4,5] \). The value of \( c \) in \( [4,5] \) at which \( f'(c) = -\frac{1}{7} \) is equal to:

If \( f'(x) = 4x\cos^2(x) \sin\left(\frac{x}{4}\right) \), then \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:

Two statements, Assertion and Reason are given below. Choose the correct option. 
Assertion: $ \text{n-propyl tert-butyl ether can be readily prepared in the laboratory by Williamson's synthesis.} $ 
Reason: The reaction occurs by $ S_N1 $ attack of Primary alkoxide on Tert-alkyl halide to give a good yield of the product, $ \text{n-propyl tert-butyl ether.} $

Identify the compound which is non-aromatic in nature.

Identify the correct statement from the following.

One of the reactions A, B, C, D given below yields a product which will not answer Hinsberg's test when reacted with Benzene sulphonyl chloride. Identify the reaction.

If Planck's constant is \( 6.63 \times 10^{-34} \) Js, then the slope of a graph drawn between cut-off voltage and frequency of incident light in a photoelectric experiment is:

Study the graph between partial pressure and mole fraction of some gases and arrange the gases P, Q, R, and S dissolved in H2O, in the decreasing order of their KH values.

A compound having molecular formula $ C_4H_{11}N $, reacts with $ \text{CHCl}_3 $ in alcoholic KOH, on heating, to form a compound with a foul smell. Identify the optically active isomer of the compound which also shows the above reaction.

If $ A = \begin{bmatrix} 0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x \end{bmatrix} $ is a singular matrix, then $ x $ is equal to

If the straight lines $ \frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{-t} \quad \text{and} \quad \frac{x - 1}{t} = \frac{y - 4}{2} = \frac{z - 5}{1} \quad \text{are intersecting, then $ t $ can have:} $
The sum of the order and degree of the differential equation $ \left( \frac{d^5y}{dx^5} \right) + 4 \left( \frac{d^4y}{dx^4} \right) + \left( \frac{d^3y}{dx^3} \right) = x^2 - 1 $
In the expansion of $ \left( \frac{1}{x} + x \sin x \right)^{10} $, the coefficient of the 6th term is equal to $ 7^7g $, then the principal value of $ x $ is.
The letters of the word "COCHIN" are permuted and all the permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word "COCHIN" is
For an examination a candidate has to select 7 questions from three different groups A, B and C. The three groups contain 4, 5 and 6 questions respectively. In how many different ways can a candidate make his selection if he has to select at least 2 questions from each group?
The function $ f(x) = \tan^{-1}(\sin x + \cos x) $ is an increasing function in.
Suppose we have three cards identical in form except that both sides of the first card are coloured red, both sides of the second are coloured black, and one side of the third card is coloured red and the other side is coloured black. The three cards are mixed and a card is picked randomly. If the upper side of the chosen card is coloured red, what is the probability that the other side is coloured black?
The points on the $x\text{-axis whose perpendicular distance from the line } \frac{x}{3} + \frac{y}{4} = 1 \text{ is 4 units are}$
In the parabola $y^2 = 4ax$ the length of the latus rectum is 6 units and there is a chord passing through its vertex and the negative end of the latus rectum. Then the equation of the chord is
The turning point of the function $ y = \frac{ax-b}{(x-1)(x-4)} $ at the point $ P(2, -1) $ is