List of practice Questions

Let \( f(x) = \begin{cases} \frac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac{3}{2}, \frac{1}{2} \end{cases} \) be continuous at \( x = -\frac{3}{2} \). If \( f(x) = \frac{7}{5} \), then \( x \) is equal to :

Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right) \). Then \( \alpha + 2\beta \) is equal to :

Let the direction cosines of two lines satisfy the equations : \( 4l + m - n = 0 \) and \( 2mn + 5nl + 3lm = 0 \). Then the cosine of the acute angle between these lines is :

Number of solutions of \( \sqrt{3} \cos 2\theta + 8 \cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi] \) is:

Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:

What are the constituents of Natalite?

Consider the following redox reaction taking place in acidic medium: \[ \mathrm{BH_4^- (aq) + ClO_3^- (aq) \rightarrow H_2BO_3^- (aq) + Cl^- (aq)} \] If the Nernst equation for the above balanced reaction is \[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q, \] then the value of \(n\) is ________ (Nearest integer).

\(0.53\ \text{g}\) of an organic compound \(X\) when heated with excess concentrated nitric acid and then with silver nitrate gave \(0.75\ \text{g}\) of silver bromide precipitate. \(1.0\ \text{g}\) of \(X\) gave \(1.32\ \text{g}\) of \(\mathrm{CO_2}\) on combustion. Find the percentage of hydrogen in compound \(X\). (Nearest integer) [Given: Atomic masses (g mol\(^{-1}\)): H = 1, C = 12, Br = 80, Ag = 108, O = 16]

Consider the dissociation equilibrium of the following weak acid: \[ \mathrm{HA \rightleftharpoons H^+(aq) + A^-(aq)} \] If the \(pK_a\) of the acid is \(4\), then the pH of a \(10\ \text{mM}\) HA solution is ________ (Nearest integer). (Given: The degree of dissociation can be neglected with respect to unity)

500 mL of \(1.2\) M KI solution is mixed with 500 mL of \(0.2\) M \( \mathrm{KMnO_4} \) solution in basic medium. The liberated iodine is titrated with standard \(0.1\) M \( \mathrm{Na_2S_2O_3} \) solution in the presence of starch indicator till the blue colour disappears. The volume (in L) of \( \mathrm{Na_2S_2O_3} \) consumed is ________ (nearest integer).

In the given pentapeptide, find out an essential amino acid (\(Y\)) and the sequence present in the pentapeptide.
Choose the correct answer from the options given below:

The correct statement among the following is:

Given below are the four isomeric compounds \(P, Q, R, S\):
\(P\): Aromatic compound containing an \(-\mathrm{OH}\) group
\(Q\): Aromatic compound containing an \(-\mathrm{CHO}\) group (aldehyde)
\(R\): Aromatic compound containing a ketone group
\(S\): Aromatic compound containing a ketone group Identify the correct statements from below:
[A.] \(Q, R\) and \(S\) will give precipitate with \(2,4\)-DNP.
[B.] \(P\) and \(Q\) will give positive Baeyer’s test.
[C.] \(Q\) and \(R\) will give sooty flame.
[D.] \(R\) and \(S\) will give yellow precipitate with \(I_2/\mathrm{NaOH}\).
[E.] \(Q\) alone will deposit silver with Tollens’ reagent. Choose the correct option.

The figures below show: Which of the following points in Figure 2 most accurately represents the nodal surface shown in Figure 1?

Consider the following reaction sequence:

Given: Compound (x) has percentage composition \(76.6%\ \text{C}\), \(6.38%\ \text{H}\) and vapour density \(=47\). Compound (y) develops a characteristic colour with neutral \(\mathrm{FeCl_3}\) solution. Identify the {INCORRECT statement.}

Consider the following reactions giving major product. Identify the correct reaction.

An organic compound undergoes first order decomposition. The time taken for decomposition to \(\dfrac{1}{8}\) and \(\dfrac{1}{10}\) of its initial concentration are \(t_{1/8}\) and \(t_{1/10}\) respectively. Find the value of \[ \frac{t_{1/8}}{t_{1/10}}\times10 \] (Given: \(\log 2 = 0.3\))

The wave numbers of three spectral lines of hydrogen atom are considered. Identify the set of spectral lines belonging to the {Balmer series. (\(R\) = Rydberg constant)}

Consider the reaction: \[ \text{Ph–CH=CH}_2 \xrightarrow[\text{peroxide}]{\text{HBr}} \text{Product} \] Which of the following statements are correct?
[A.] The reaction proceeds through a more stable radical intermediate.
[B.] The role of peroxide is to generate \(\mathrm{H^\bullet}\) radical.
[C.] During this reaction, benzene is formed as a byproduct.
[D.] \(1\)-Bromo-\(2\)-phenylethane is formed as a minor product.
[E.] The same reaction in absence of peroxide proceeds via a carbocation intermediate. Choose the correct answer.

At temperature \(T\) K, \(2\) moles of liquid \(A\) and \(3\) moles of liquid \(B\) are mixed. The vapour pressure of the ideal solution formed is \(320\) mm Hg. At this stage, one mole of \(A\) and one mole of \(B\) are added to the solution. The vapour pressure is now measured as \(328.6\) mm Hg. The vapour pressures (in mm Hg) of pure \(A\) and pure \(B\) respectively are:

In period 4 of the periodic table, the elements with highest and lowest atomic radii respectively are:

Regarding the hydrides of group 15 elements \(EH_3\) \((E = N, P, As, Sb)\), select the correct statement(s):
[A.] The stability of hydrides decreases down the group.
[B.] The basicity of hydrides decreases down the group.
[C.] The reducing character increases down the group.
[D.] The boiling point increases down the group. Choose the correct answer:

Consider a weak base \(B\) of \(pK_b = 5.699\). \(x\) mL of \(0.02\) M HCl and \(y\) mL of \(0.02\) M weak base \(B\) are mixed to make \(100\) mL of a buffer of pH \(=9\) at \(25^\circ\text{C}\). The values of \(x\) and \(y\) respectively are

\(20.0\,\text{dm}^3\) of an ideal gas \(X\) at \(600\) K and \(0.5\) MPa undergoes isothermal reversible expansion until the pressure of the gas becomes \(0.2\) MPa. Which of the following option is correct? (Given: \(\log 2 = 0.3010\), \(\log 5 = 0.6989\))

Given below are two statements: Statement I: The number of species among \(\mathrm{BF_4^-}\), \(\mathrm{SiF_4}\), \(\mathrm{XeF_4}\) and \(\mathrm{SF_4}\), that have unequal E–F bond lengths is three. Here, E is the central atom. Statement II: Among \(\mathrm{O_2^-}\), \(\mathrm{O_2^{2-}}\), \(\mathrm{F_2}\) and \(\mathrm{O_2^+}\), \(\mathrm{O_2^+}\) has the highest bond order. In the light of the above statements, choose the correct answer.