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A takes 10 days than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days,

Question: A takes 10 days than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work. Solution: Let B takes $x$ days to complete the work. Therefore, A will take $(x-10)$ days. $\therefore \frac{1}{x}+\frac{1}{(x-10)}=\frac{1}{12}$ $\Rightarrow \frac{(x-10)+x}{x(x-10)}=\frac{1}{12}$ $\Rightarrow \frac{2 x-10}{x^{2}-10 x}=\frac{1}{12}$ $\Rightarrow x^{2}-10 x=12(2 x-10)$ $\Rightarrow x^{2}-10 x=24 x-120$...

If the lines x+y=a and x-y=b touch the curve

Question: If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^{2}-3 x+2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to Solution: The given curve $y=(x-1)(x-2)$, intersects the $x$-axis at $A(1,0)$ and $B(2,0)$. $\therefore \frac{d y}{d x}=2 x-3 ;\left(\frac{d y}{d x}\right)_{(x=1)}=-1$ and $\left(\frac{d y}{d x}\right)_{(x=2)}=1$ Equation of tangent at $A(1,0)$, $y=-1(x-1) \Rightarrow x+y=1$ Equation of tangent at $B(2,0)$, $y=1(x-2) \Rightarrow x-y=2$ S...

If the common tangent to the parabolas,

Question: If the common tangent to the parabolas, $y^{2}=4 x$ and $x^{2}=4 y$ also touches the circle, $x^{2}+y^{2}=c^{2}$, then $c$ is equal to:(1) $\frac{1}{2 \sqrt{2}}$(2) $\frac{1}{\sqrt{2}}$(3) $\frac{}{4}$(4) $\frac{1}{2}$Correct Option: , 2 Solution: Equation tangent to parabola $y^{2}=4 x$ with slope $m$ be: $y=m x+\frac{1}{m}$ ...(i) $\because$ Equation of tangent to $x^{2}=4 y$ with slope $m$ be : $y=m x-a m^{2}$ .......(ii) From eq. (i) and (ii), $\frac{1}{m}=-m^{2} \Rightarrow m=-1$ ...

The change in the magnitude of the volume of an ideal gas when a small additional pressure

Question: The change in the magnitude of the volume of an ideal gas when a small additional pressure $\Delta P$ is applied at a constant temperature, is the same as the change when the temperature is reduced by a small quantity $\Delta T$ at constant pressure. The initial temperature and pressure of the gas were $300 \mathrm{~K}$ and $2 \mathrm{~atm}$. respectively. If $|\Delta T|=C|\Delta P|$, then value of $C$ in (K/atm.) is_______ Solution: (150) In first case, From ideal gas equation $P V=n ...

The number of chiral centres present in threonine is

Question: The number of chiral centres present in threonine is __________________. Solution: (2)...

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes.

Question: A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream. Solution: Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Downstream speed $=(9+x) \mathrm{km} / \mathrm{hr}$ Upstream speed $=(9-x) \mathrm{km} / \mathrm{hr}$ Distance covered downstream $=$ Distance covered upstream $=15 \mathrm{~km}$ Total time taken $=3$ hours 45 minutes $=\left(3+\frac{45}{60}\right...

Among the following compounds, which one has the shortest

Question: Among the following compounds, which one has the shortest $\mathrm{C}-\mathrm{Cl}$ bond?Correct Option: , 4 Solution: Due to conjugation of lonepair of $\mathrm{Cl}$ with $\pi$ bond, partial double bond character decreases bond length that's why compound (d) has shortest $\mathrm{C}-\mathrm{Cl}$ bond length....

Let P be a point on the parabola,

Question: Let $P$ be a point on the parabola, $y^{2}=12 x$ and $N$ be the foot of the perpendicular drawn from $P$ on the axis of the parabola. A line is now drawn through the mid-point $M$ of $P N$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line $N Q$ is $\frac{4}{3}$, then :(1) $P N=4$(2) $M Q=\frac{1}{3}$(3) $M Q=\frac{1}{4}$(4) $P N=3$Correct Option: , 3 Solution: $\because y^{2}=12 x$ $\therefore a=3$ Let $P\left(a t^{2}, 2 a t\right)$ $\Rightarrow N\...

The IUPAC name of the following compound is :

Question: The IUPAC name of the following compound is : 5-Bromo-3-methylcyclopentanoic acid4-Bromo-2-methylcyclopentane carboxylic acid3 -Bromo-5-methylcyclopentanoic acid3-Bromo-5-methylcyclopentane carboxylic acidCorrect Option: , 2 Solution: 4-Bromo-2-methylcyclopentane carboxylic acid...

Match the Cp/Cv ratio for ideal gases with different type of molecules:

Question: Match the $\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}$ ratio for ideal gases with different type of molecules: (1) (A)-(IV), (B)-(II), (C)-(I), (D)-(III)(2) (A)-(III), (B)-(IV), (C)-(II), (D)-(I)(3) (A)-(IV), (B)-(I), (C)-(II), (D)-(III)(4) (A)-(II), (B)-(III), (C)-(I), (D)-(IV)Correct Option: , 3 Solution: (3) As we know, $\gamma=\frac{C_{p}}{C_{v}}=1+\frac{2}{f}$, where $f=$ degree of freedom (A) Monatomic, $f=3$ $\therefore \gamma=1+\frac{2}{3}=\frac{5}{3}$ (B) Diatomic rigid...

The speed of a boat in still water is 15 km/hr. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes.

Question: The speed of a boat in still water is 15 km/hr. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. Solution: Let, speed of stream bexkm/hSpeed of boat = 15 km/hDistance from each side = 30 km We know that time taken $=\frac{\text { distance covered }}{\text { Speed }}$ Total speed of the boat while going upstream $=15-x \mathrm{~km} / \mathrm{h}$ Time taken to go upstream $=\frac{30}{15-x}$ hrs Total speed of boat while going ...

The area (in sq. units) of an equilateral triangle inscribed

Question: The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^{2}=8 x$, with one of its vertices on the vertex of this parabola, is :(1) $64 \sqrt{3}$(2) $256 \sqrt{3}$(3) $192 \sqrt{3}$(4) $128 \sqrt{3}$Correct Option: , 3 Solution: Let $A=\left(2 t^{2}, 4 t\right)$ and $B=\left(2 t^{2},-4 t\right)$ For equilateral triangle $\left(\angle A O M=30^{\circ}\right)$ $\tan 30^{\circ}=\frac{4 t}{2 t^{2}} \Rightarrow \frac{1}{\sqrt{3}}=\frac{4 t}{2 t^{2}} \Rightarrow t=2 \s...

The total number of monohalogenated organic products in the following

Question: The total number of monohalogenated organic products in the following (including stereoisomers) reaction is _______________. Solution: 8...

Which one of the following compounds possesses the most acidic hydrogen?

Question: Which one of the following compounds possesses the most acidic hydrogen?Correct Option: , 4 Solution: Acidic strength $\propto-\mathrm{I},-\mathrm{M}$ effect. Due to strong $-\mathrm{I}$, and $-\mathrm{M}$ effect of $3-\mathrm{COOCH}_{3}$ group, it has most acidic Hydrogen....

A line is a common tangent to the circle

Question: A line is a common tangent to the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4 x$. If the two points of contact $(a, b)$ and $(c, d)$ are distinct and lie in the first quadrant, then $2(a+c)$ is equal to Solution: Circle: $(x-3)^{2}+y^{2}=9$ Parabola: $y^{2}=4 x$ Let tangent $y=m x+\frac{a}{m}$ $y=m x+\frac{1}{m}$ $m^{2} x-m y+1=0$ the above line is also tangent to circle $(x-3)^{2}+y^{2}=9$ $\therefore \perp$ from $(3,0)=3$ $\left|\frac{3 m^{2}-0+1}{\sqrt{m^{2}+m^{4}}}\right|=...

A motor boat whose speed in still water is 18 km/hr, takes 1 hour more to go 24 km upstream than o return to the same spot.

Question: A motor boat whose speed in still water is 18 km/hr, takes 1 hour more to go 24 km upstream than o return to the same spot. Find the speed of the stream. Solution: Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. Given : Speed of the boat $=18 \mathrm{~km} / \mathrm{hr}$ $\therefore$ Speed downstream $=(18+x) \mathrm{km} / \mathrm{hr}$ Speed upstream $=(18-x) \mathrm{km} / \mathrm{hr}$ $\therefore \frac{24}{(18-x)}-\frac{24}{(18+x)}=1$ $\Rightarrow \frac{1}{(18-x)}-\frac{...

If minimum possible work is done by a refrigerator in converting 100 grams

Question: If minimum possible work is done by a refrigerator in converting 100 grams of water at $0^{\circ} \mathrm{C}$ to ice, how much heat (in calories) is released to the surroundings at temperature $27^{\circ} \mathrm{C}$ (Latent heat of ice $=80 \mathrm{Cal} /$ gram ) to the nearest integer? Solution: (8791) Given, Heat absorbed, $Q_{2}=m L=80 \times 100=8000 \mathrm{Cal}$ Temperature of ice, $T_{2}=273 \mathrm{~K}$ Temperature of surrounding, $T_{1}=273+27=300 \mathrm{~K}$ Efficiency $=\f...

Arrange the following labelled hydrogens in decreasing order of acidity:

Question: Arrange the following labelled hydrogens in decreasing order of acidity: (ii) $$ (i) $$ (iii) $$ (iv)(iii) $$ (ii) $$ (iv) $$ (i)(ii) $$ (iii) $$ (iv) $$ (i)(iii) $$ (ii) $$ (i) $$ (iv)Correct Option: , 3 Solution: Acidic strength $\propto$ Stability of conjugate base General order of acidic strength is $\mathrm{R}-\mathrm{COOH}\mathrm{Ph}-\mathrm{OH}\mathrm{R}-\mathrm{C} \equiv \mathrm{CH}$ In between (iii) and (iv), (iii) is more acidic due to $-\mathrm{M}$ effect of $-\mathrm{NO}_{2...

To raise the temperature of a certain mass of gas

Question: To raise the temperature of a certain mass of gas by $50^{\circ} \mathrm{C}$ at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by $100^{\circ} \mathrm{C}$ at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)?(1) 5(2) 6(3) 3(4) 7Correct Option: , 2 Solution: (2) Let $C_{p}$ and $C_{v}$ be the specific heat capacity of the gas at constant pressure and volu...

The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train.

Question: The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speed of the two trains differ by 20 km/hr. Solution: Let the speed of the Deccan Queen be $x \mathrm{~km} / \mathrm{hr}$. According to the question: Speed of another train $=(\mathrm{x}-20) \mathrm{km} / \mathrm{h}$ $\therefore \frac{192}{x-20}-\frac{192}{x}=\frac{48}{60}$ $\Rightarrow \frac{4}{x-20}-\frac{4}{x}=\fr...

The number of chiral carbons present in the molecule given below is

Question: The number of chiral carbons present in the molecule given below is ________. Solution:...

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed.

Question: A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find its usual speed. Solution: Let the usual speed be $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\frac{300}{x}-\frac{300}{(x+5)}=2$ $\Rightarrow \frac{300(x+5)-300 x}{x(x+5)}=2$ $\Rightarrow \frac{300 x+1500-300 x}{x^{2}+5 x}=2$ $\Rightarrow 1500=2\left(x^{2}+5 x\right)$ $\Rightarrow 1500=2 x^{2}+10 x$ $\Rightarrow x^{2}+5 x-750=0$ $\Rightarrow x^{2}...

A tangent is drawn to the parabola

Question: A tangent is drawn to the parabola $y^{2}=6 x$ which is perpendicular to the line $2 x+y=1$. Which of the following points does NOT lie on it?(1) $(0,3)$(2) $(-6,0)$(3) $(4,5)$(4) $(5,4)$Correct Option: , 4 Solution: Equation of tangent $: y=m x+\frac{3}{2 m} \mathrm{~m}_{\mathrm{T}}=\frac{1}{2}(\because$ perpendicular to line $2 x+y=1$ ) : $\quad$ tangent is : $y=\frac{x}{2}+3 \quad \Rightarrow x-2 y+6=0$...

The IUPAC name for the following compound is :

Question: The IUPAC name for the following compound is : 2,5 -dimethyl-5-carboxy-hex-3-enal2,5 -dimethyl-6-carboxy-hex-3-enal2,5 -dimethyl-6-oxo-hex-3-enoic acid6-formyl-2-methyl-hex-3-enoic acidCorrect Option: , 3 Solution:...

If P is a point on the parabola

Question: If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y=4 x-1$, then the co-ordinates of $\mathrm{P}$ are : (1) $(-2,8)$(2) $(1,5)$(3) $(3,13)$(4) $(2,8)$Correct Option: , 4 Solution: $\left.\frac{d y}{d x}\right|_{p}=4$ $\therefore 2 x_{1}=4$ $\Rightarrow x_{1}=2$ $\therefore$ Point will be $(2,8)$...

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