Chapter 16

The total number of diprotic acids among the following is ${\mathrm{H}}_3{\mathrm{PO}}_4$ ${\mathrm{H}}_2{\mathrm{SO}}_4{\mathrm{H}}_3{\mathrm{PO}}_3$ ${\mathrm{H}}_2{\mathrm{CO}}_3$ ${\mathrm{H}}_2{\mathrm{S}}_2{\mathrm{O}}_7$ $\begin{array}{llll}{\mathrm{H}}_3{\mathrm{BO}}_3& {\mathrm{H}}_3{\mathrm{PO}}_2& {\mathrm{H}}_2{\mathrm{CrO}}_4& {\mathrm{H}}_2{\mathrm{SO}}_3\end{array}$c

The hydrogen-like species ${\mathrm{Li}}^{2+}$ is in a spherically symmetric state ${\mathrm{S}}_1$ with one radial node. Upon absorbing light the ion undergoes transition to a state $S_2$. The state $S_2$ has one radial node and its energy is equal to the ground state energy of the hydrogen atom. Energy of the state $S_1$ in units of the hydrogen atom ground state energy is A) $0.75$ B) $1.50$ C) $2.25$ D) $4.50$

For $\mathrm{r}=0,1,\dots .10$, let ${\mathrm{A}}_{\mathrm{r}},{\mathrm{B}}_{\mathrm{r}}$ and ${\mathrm{C}}_{\mathrm{r}}$ denote, respectively, the coefficient of ${\mathrm{x}}^{\mathrm{r}}$ in the expansions of $(1+x{)}^{10},(1+x{)}^{20}$ and $(1+x{)}^{30}$. Then $$\sum _{\mathrm{r}=1}^{10}{\mathrm{A}}_{\mathrm{r}}({\mathrm{B}}_{10}{\mathrm{B}}_{\mathrm{r}}-{\mathrm{C}}_{10}{\mathrm{A}}_{\mathrm{r}})$$ is equal to A) $B_{10}-C_{10}$ B) ${\mathrm{A}}_{10}({\mathrm{B}}_{10}^2-{\mathrm{C}}_{10}{\mathrm{A}}_{10})$ C) 0 D) $C_{10}-B_{10}$

Let $$\begin{gathered} S= \\ 1,2,3,4 \end{gathered}$$. The total number of unordered pairs of disjoint subsets of $S$ is equal to A) 25 B) 34 C) 42 D) 41

Two adjacent sides of a parallelogram $\mathrm{ABCD}$ are given by $\overrightarrow{\mathrm{AB}}=2\hat{\mathrm{i}}+10\hat{\mathrm{j}}+11\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AD}}=-\hat{\mathrm{i}}+2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ The side $\mathrm{AD}$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $\mathrm{AD}$ becomes ${\mathrm{AD}}^{\prime }$. If ${\mathrm{AD}}^{\prime }$ makes a right angle with the side $\mathrm{AB}$, then the cosine of the angle $\alpha$ is given by A) $\frac{8}{9}$ B) $\frac{\sqrt{17}}{9}$ C) $\frac{1}{9}$ D) $\frac{4\sqrt{5}}{9}$

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station $A$ and then transmitted to station $B.$ The probability of each station receiving the signal correctly is $\frac{3}{4}.$ If the signal received at station $B$ is green, then the probability that the original signal was green is A) $\frac{3}{5}$ B) $\frac{6}{7}$ (C) $\frac{20}{23}$ D) $\frac{9}{20}$

Tangents are drawn from the point $\mathrm{P}(3,4)$ to the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$ touching the ellipse at points $\mathrm{A}$ and $\mathrm{B}$.The coordinates of $\mathrm{A}$ and $\mathrm{B}$ are A) $(3,0)$ and $(0,2)$ B) $(-\frac{8}{5},\frac{2\sqrt{161}}{15})$ and $(-\frac{9}{5},\frac{8}{5})$ C) $(-\frac{8}{5},\frac{2\sqrt{161}}{15})$ and $(0,2)$ D) $(3,0)$ and $(-\frac{9}{5},\frac{8}{5})$

A Vernier calipers has $1\mathrm{mm}$ marks on the main scale. It has 20 equal divisions on the Vernier scale which match with 16 main scale divisions. For this Vernier calipers, the least count is A) $0.02\mathrm{mm}$ B) $0.05\mathrm{mm}$ C) $0.1\mathrm{mm}$ D) $0.2\mathrm{mm}$

A hollow pipe of length $0.8\mathrm{m}$ is closed at one end. At its open end a $0.5\mathrm{m}$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is $50\mathrm{N}$ and the speed of sound is $320{\mathrm{ms}}^{-1}$, the mass of the string is A) 5 grams B) 10 grams C) 20 grams D) 40 grams

A biconvex lens of focal length $15\mathrm{cm}$ is in front of a plane mirror. The distance between the lens and the mirror is $10\mathrm{cm}$. A small object is kept at a distance of $30\mathrm{cm}$ from the lens. The final image is A) virtual and at a distance of $16\mathrm{cm}$ from the mirror B) real and at a distance of $16\mathrm{cm}$ from the mirror C) virtual and at a distance of $20\mathrm{cm}$ from the mirror D) real and at a distance of $20\mathrm{cm}$ from the mirror