List of practice Questions

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) = -f(x)$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $h'(x) = f(x)^2 + (f'(x))^2$ for all $x \in \mathbb{R}$. If $h(0) = 1$ and $h\left(\frac{1}{3}\right) = \frac{5}{3}$, then the value of $h(100)$ is

Suppose $\lim_{x \to -2} \frac{bx^2 + 15x + 15 + b}{x^2 + x - 2} = L$, where $b$ and $L$ are real numbers. Then

$\text{PQ}_2$ is a sparingly soluble salt with solubility product $\text{K}_{sp} = 4 \times 10^{-12}$ in aqueous medium at some given temperature. It is observed that upon addition of a highly soluble salt RQ at the same temperature, the solubility of $\text{PQ}_2$ drops by a factor of 100. The concentration (in millimoles per litre) of added RQ in the solution is closest to:

Consider the following chemical reactions performed at identical temperature and volume:
Experiment I: $\text{X(g)} \rightleftharpoons 2\text{Y(g)}$
Experiment II: $\text{X(g)} \rightleftharpoons \text{Z(g)}$
The partial pressure of the reactant and product are denoted, respectively, by $p_{\text{reactant}}$ and $p_{\text{product}}$ during the course of the reactions. Assuming ideal gas behaviour, the correct plot of $p_{\text{reactant}}$ versus $p_{\text{product}}$ for Experiments I and II is:

If velocity of an electron in the first Bohr orbit is denoted by $v_0$, then the velocity ($v$) of the electron in other orbits (as a function of principle quantum number `n') is represented as:

Consider two reactions.
I. A zero order reaction: $\text{X} \rightarrow P_1$.
II. A first order reaction: $\text{Y} \rightarrow P_2$.
Both reactions begin simultaneously with the same initial concentration `C', that is $\text{X}_0 = \text{Y}_0 = \text{C}$. It is observed that the concentrations of the reactants fall to one-third of their initial values at the same time, provided the rate constants satisfy a certain ratio. If $k_0$ and $k_1$ denote, respectively, the zero and first order rate constants then the ratio $k_0/k_1$ necessary for this condition to be satisfied is:

A possible solution of the system of equations \[ x^2 - 8xy + 16y^2 = 0 \] \[ (\log_{10} x)^2 + 2(\log_{10} x)(\log_{10} y) + (\log_{10} y)^2 = 4 \] is

Consider a reversible thermodynamic process involving one mole of ideal gas represented by a line on the $P-V$ diagram connecting two states X (4 bar, 1 L, $\text{T}_1$) and Y (1 bar, 2 L, $\text{T}_2$) as shown in the figure:

During the process the change in temperature (in K) as a function of volume (in L) is best represented as:

Consider the following species:
(i) $\text{ClO}^-$
(ii) $\text{ClO}_2^-$
(iii) $\text{ClO}_3^-$
(iv) $\text{ClO}_4^-$
Among them, the species that undergoes(undergo) disproportionation reactions is(are):

X, Y, and Z are three volatile liquids, which when mixed make an ideal solution. At a given temperature $\text{T}_0$, the constituents X, Y, and Z have pure vapour pressures in the ratio 3:2:1. A mixture prepared with mole-fractions of X, Y, and Z in the ratio 1:2:3 starts to boil at temperature $\text{T}_0$ at 1.5 atm pressure. At temperature $\text{T}_0$, the partial vapour pressure (in atm) of Y is:

1,2-dimethylbenzene on treatment with chromic oxide in acetic anhydride at 273-283 K followed by hydrolysis produces P. P on heating with concentrated NaOH followed by hydrolysis provides Q. The functional groups present in the product Q are:

The increasing order of acidic strength for the aqua complexes $[\text{V}(\text{H}_2\text{O})_6]^{2+}$, $[\text{Fe}(\text{H}_2\text{O})_6]^{2+}$, $[\text{Fe}(\text{H}_2\text{O})_6]^{3+}$, and $[\text{Co}(\text{H}_2\text{O})_6]^{3+}$ is:

Consider the following table of three lanthanoid ions X, Y, and Z and their properties.

Atomic numbers of Ce, Eu, and Lu are 58, 63, and 71, respectively. Given these atomic numbers, the lanthanoid ions X, Y, and Z, respectively, are:

Consider the following complex ions:
$[\text{Ni(CN)}_4]^{2-}$, $[\text{Fe(CN)}_6]^{3-}$, $[\text{Cu(H}_2\text{O)}_6]^{2+}$, and $[\text{Co(CN)}_6]^{3-}$.
The complex ions that are expected to show diamagnetic behavior at room temperature are:

Consider the following compound.

The compound undergoes reaction with two equivalents of diisobutyl aluminium hydride followed by acid hydrolysis to provide P as the major product. The structure of P is:

tert-Butylbenzene reacts with 1-chloro-2-methylpropane in the presence of anhydrous $\text{AlCl}_3$. The major product in this reaction is:

Consider the following compound.

The compound reacts with $\text{SOCl}_2$ to produce P. P reacts with dimethyl cadmium to produce Q. Q upon reaction with ethyl magnesium bromide (3 equiv) followed by water workup produces R. The structure of R is:

Acetylene is passed through red hot iron tube at 873 K to produce P. P reacts with chlorine gas in the presence of anhydrous $\text{AlCl}_3$ to produce Q. Q reacts with benzyl chloride and sodium to produce R as the major product. The structure of R is:

The reaction that produces butanoic acid as a product is:

The compound that produces aromatic species upon deprotonation of the indicated hydrogen is:

Two students (P and Q) independently made three measurements (denoted by I, II, and III) of the same property. The expected correct value is 9.0. The observed values by the students are provided in the given table.

Based on the given data, the correct statement is:

Species that are both isoelectronic and isostructural are:

Two galvanic cells, Cell I and Cell II, operate at the same temperature and involve two different redox systems. It is observed that at a particular value of the reaction quotient $\text{Q}_{\text{I}} = \text{Q}_{\text{II}} = \text{Q}_0$, the ratio of EMF of the two cells ($\text{E}_{\text{I}}/\text{E}_{\text{II}}$) as well as the ratio of change of Gibbs free energy for the two cells ($\Delta\text{G}_{\text{I}}/\Delta\text{G}_{\text{II}}$) are both 1/3. Assuming standard cell reactions written in the spontaneous direction and reversible operation, the pair of cells can satisfy these conditions is:

Desert ants are known to use landmarks (reference points), pedometers (memorising step counts), and optic flow (rate of passing objects across its eye to estimate navigation speed) to navigate across deserts. A desert ant is trained to walk 100 m from its nest to a feeder in an artificial arena as depicted in the schematic. The arena had a single tall landmark placed at 40 m from the nest.

At the feeder, certain manipulations were performed before allowing the ant to return to its nest with the food.
Manipulation 1: Stilts were attached to its legs, which doubled the step length.
Manipulation 2: The optic flow of the arena was doubled.
While the ant uses the landmark exclusively for long-distance ($\gt $55 m) navigation, it switches to optic flow and pedometer simultaneously and exclusively for short-distance navigation. Due to the manipulations, the distance from the feeder where the ant will search its nest is:

The table shows the presence (+) / absence (-) of features 1 - 5 for animals P - U.

Based on the table, the correct statement is: