List of practice Questions

Sum of the last 40 coefficients in the expansion of $ (1 + x)^{79} $, when expanded in ascending power of $ x $ is:
$ (666 \ldots \text{up to } n \text{ digits})^2 + (888 \ldots \text{up to } n \text{ digits}) $ is:
Out of 6 boys and 4 girls, a committee of 5 members is to be formed. In how many ways can this be done, if at least 2 girls are included?
For what value of $ n $, the curve \[ \left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2 \text{ touches the straight line } \frac{x}{a} + \frac{y}{b} = 2 \text{ at the point } (a, b)? \]
If $ f(x) $ and $ g(x) $ are two functions such that $ f(x) + g(x) = e^x $ and $ f(x) - g(x) = e^{-x} $, then:
The range of the function $ f(x) = 7 - x\cdot P_{x-3} $ is:

Let the function $ f(x) $ be defined as follows: $$ f(x) = \begin{cases} (1 + | \sin x |)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}<x<0 \\b, & x = 0 \\ \frac{\tan 2x}{\tan 3x}, & 0<x<\frac{\pi}{6} \end{cases} $$ Then the values of $ a $ and $ b $ are:

The value(s) of $ c \in (1, 2) $, where the conclusion of Lagrange’s M.V.T. is satisfied for the function $ f(x) = x^2 + 3x + 2 $ in $[1,2]$, is/are:
Let $ \phi_1(x) = e^{\sin x, \phi_2(x)} = e^{\phi_1(x)}, \ldots, \phi_{n+1}(x) = e^{\phi_n(x)}, \forall n \geq 1$. Then for any fixed $ n $, the expression $ \frac{d}{dx} \left\{ \phi_n(x) \right\} $ is:
The equation of the circle of radius 3 units which touches the circles $ x^2 + y^2 - 6|x| = 0 $ is:
An aqueous HCl solution has the pH = 5. If 1 ml of this solution is being diluted to 1 L, then predict the pH of the resulting solution.
1 molal NaCl solution has the density 1.21 gm/cc. Find the molar concentration of the same solution.
At 30°C, a substance being dissolved in CCl$_4$ (solvent) exhibits its dissociation half-life as 2.5 hrs. If 256 g of the substance is taken initially, then tell how much of it will remain as such after 20 hrs.
At 12°C and 756 mm atmospheric pressure, a balloon contains 450 ml air. If the balloon is shifted to a place of temperature 5°C and atmospheric pressure 765 mm, then indicate the nature (shape) and degree (volume) of change of the balloon.
First ionisation potential energy of Li(m) is 520 kJ/mol$^{-1}$, hence calculate the energy required to convert Li-atoms present in 70 mg Li(m) into Li$^+$ ion.
In HF molecule, internuclear distance is 0.92 Å and dipole moment approximates 2 D. Estimate the percentage ionic character of HF molecule.
In solid state, nitrogen exists in two different allotropic forms ($\alpha$ and $\beta$). Give an idea regarding their shapes and stability.
How would you differentiate between $ \text{O}_3 $ and $ \text{H}_2\text{O}_2 $ (chemically)?

Identify the product(s) of the following two reactions (i) and (ii):

Write the end product of the following reaction:

Ascertain the products $B_1$ and $C_1$ of the following reaction:

Analyzing the following reaction, ascertain the final product:

In aqueous medium, cations like $ \text{H}^+ $ and $ \text{Li}^+ $ undergo hydration and exist in the hydrated form. Predict the numbers of water molecules surrounded with $ \text{H}^+ $ and $ \text{Li}^+ $ ions to form stable complex structure respectively.
Upon addition of phenolphthalein to the aqueous solution of borax, pink colour generates i.e. solution becomes pink. At this stage, if glycerol is being added to the pink coloured solution, it turns colourless. Highlight the phenomenon.
1 mole gas at standard pressure and at 27°C temperature is heated such that its volume and pressure both become doubled. The final temperature will be