Calculate the time interval between 33 % decay and 67 % decay if half-life of a substance is 20 minutes.

Question: Calculate the time interval between $33 \%$ decay and $67 \%$ decay if half-life of a substance is 20 minutes.(1) 60 minutes(2) 20 minutes(3) 40 minutes(4) 13 minutesCorrect Option: , 2 Solution: (2) $\mathrm{N}_{1}=\mathrm{N}_{0} \mathrm{e}^{-\lambda t_{1}}$ $\frac{\mathrm{N}_{1}}{\mathrm{~N}_{0}}=\mathrm{e}^{-\lambda t_{1}}$ $0.67=\mathrm{e}^{-\lambda t_{1}}$ $\ln (0.67)=-\lambda t_{1}$ $\mathrm{N}_{2}=\mathrm{N}_{0} \mathrm{e}^{-\lambda t_{2}}$ $\frac{\mathrm{N}_{2}}{\mathrm{~N}_{0}...

Find A, B and C in the following reaction:

Question: Find $A, B$ and $C$ in the following reaction: $\mathrm{NH}_{3}+\mathrm{A}+\mathrm{CO}_{2} \rightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}$ $\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O}+\mathrm{B} \rightarrow \mathrm{NH}_{4} \mathrm{HCO}_{3}$ $\mathrm{NH}_{4} \mathrm{HCO}_{3}+\mathrm{NaCl} \rightarrow \mathrm{NH}_{4} \mathrm{Cl}+\mathrm{C}$$\mathrm{A}-\mathrm{H}_{2} \mathrm{O} ; \mathrm{B}-\mathrm{CO}_{2} ; \mathrm{C}-\mathrm{NaHCO}_{3}$$\mathr...

Find A, B and C in the following reaction:

Question: Find $A, B$ and $C$ in the following reaction: $\mathrm{NH}_{3}+\mathrm{A}+\mathrm{CO}_{2} \rightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}$ $\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O}+\mathrm{B} \rightarrow \mathrm{NH}_{4} \mathrm{HCO}_{3}$ $\mathrm{NH}_{4} \mathrm{HCO}_{3}+\mathrm{NaCl} \rightarrow \mathrm{NH}_{4} \mathrm{Cl}+\mathrm{C}$$\mathrm{A}-\mathrm{H}_{2} \mathrm{O} ; \mathrm{B}-\mathrm{CO}_{2} ; \mathrm{C}-\mathrm{NaHCO}_{3}$ $\math...

All possible values of

Question: All possible values of $\theta \in[0,2 \pi]$ for which $\sin 2 \theta+\tan 2 \theta0$ lie in:(1) $\left(0, \frac{\pi}{2}\right) \cup\left(\pi, \frac{3 \pi}{2}\right)$(2) $\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \cup\left(\pi, \frac{5 \pi}{4}\right) \cup\left(\frac{3 \pi}{2}, \frac{7 \pi}{4}\right)$(3) $\left(0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$(4) $\left(0, \frac{\pi}{4}\righ...

Which of the following compound is added to the sodium

Question: Which of the following compound is added to the sodium extract before addition of silver nitrate for testing of halogens ?Nitric acidSodium hydroxideHydrochloric acidAmmoniaCorrect Option: 1 Solution: $\mathrm{NaCN}+\mathrm{HNO}_{3} \rightarrow \mathrm{NaNO}_{3}+\mathrm{HCN} \uparrow$ $\mathrm{Na}_{2} \mathrm{~S}+\mathrm{HNO}_{3} \rightarrow \mathrm{NaNO}_{3}+\mathrm{H}_{2} \mathrm{~S} \uparrow$ Nilnic acid decomposed $\mathrm{NaCN} \ \mathrm{Na}_{2} \mathrm{~S}$, else they precipitate...

The half-life

Question: The half-life of $\mathrm{Au}^{198}$ is $2.7$ days. The activity of $1.50 \mathrm{mg}$ of $\mathrm{Au}^{198}$ if its atomic weight is $198 \mathrm{~g} \mathrm{~mol}^{-1}$ is, $\left(\mathrm{N}_{\mathrm{A}}=6 \times 10^{23} / \mathrm{mol}\right)$(1) $240 \mathrm{Ci}$(2) $357 \mathrm{Ci}$(3) $535 \mathrm{Ci}$(4) $252 \mathrm{Ci}$Correct Option: , 2 Solution: (2) $\mathrm{A}=\lambda \mathrm{N}$ $\mathrm{N}=\mathrm{nN}_{\mathrm{A}}$ $\mathrm{N}=\left(\frac{1.5 \times 10^{-3}}{198}\right) \...

Prove the following

Question: If $\mathrm{e}^{\left(\cos ^{2} x+\cos ^{4} x+\cos ^{6} x+\ldots \infty\right) \log _{e} 2}$ satisfies the equation $t^{2}-9 t+8=0$, then the value of $\frac{2 \sin x}{\sin x+\sqrt{3} \cos x}\left(0x\frac{\pi}{2}\right)$ is(1) $\frac{3}{2}$(2) $2 \sqrt{3}$(3) $\frac{1}{2}$(4) $\sqrt{3}$Correct Option: , 3 Solution: $\mathbf{e}^{\left(\cos ^{2} x+\cos ^{4} x+\ldots \ldots \infty\right) \ln 2}=2^{\cos ^{2} x+\cos ^{4} x+\ldots \ldots \infty}$ $=2^{\cot ^{2} x}$ $t^{2}-9 t+8=0 \Rightarrow...

Given below are two statements :

Question: Given below are two statements : Statement I : $\alpha$ and $\beta$ forms of sulphur can change reversibly between themselves with slow heating ot slow cooling. Statement II : At room temperature the stable crystalline form of sulphur is monoclinic sulphur. In the light of the above statements, choose the correct answer from the options given below.Both statement I and statement II are falseStatement I is true but statement II is falseBoth statement I and statement II are trueStatement...

The number of solutions

Question: The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2 \pi]$ is Solution: If $\cot x0 \Rightarrow \frac{1}{\sin x}=0$ (Not possible) If $\cot x0 \Rightarrow 2 \cot x+\frac{1}{\sin x}=0$ $\Rightarrow 2 \cos x=-1$ $\Rightarrow \mathrm{x}=\frac{2 \pi}{3}$ or $\frac{4 \pi}{3}$ (reject)...

The number of solutions of the equation

Question: The number of solutions of the equation $\mathrm{x}+2 \tan \mathrm{x}=\frac{\pi}{2}$ in the interval $[0,2 \pi]$ is :(1) 3(2) 4(3) 2(4) 5Correct Option: 1 Solution: $x+2 \tan x=\frac{\pi}{2}$ $\Rightarrow 2 \tan x=\frac{\pi}{2}-x$ $\Rightarrow \tan x=-\frac{1}{2} x+\frac{\pi}{4}$ Number of soluitons of the given eauation is ' 3 '....

Among the following, the number of

Question: Among the following, the number of halide(s) which is/are inert to hydrolysis is$\mathrm{BF}_{3}$$\mathrm{SiCl}_{4}$$\mathrm{PCl}_{5}$$\mathrm{SF}_{6}$Correct Option: 1 Solution: Due to crowding $\mathrm{SF}_{6}$ is not hydrolysed....

The correct statement about

Question: The correct statement about $\mathrm{B}_{2} \mathrm{H}_{6}$ is:All B-H-B angles are of $120^{\circ}$.Its fragment, $\mathrm{BH}_{3}$, behaves as a Lewis base.Terminal B-H bonds have less p-character when compared to bridging bonds.The two $\mathrm{B}-\mathrm{H}-\mathrm{B}$ bonds are not of same length.Correct Option: , 3 Solution: Terminal bond angle is greater than that of bridge bond angle Bond angle Bond order $\propto$ S-character...

Prove the following

Question: If for $x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1$ and $\log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), n0$ then the value of $\mathrm{n}$ is equal to :(1) 20(2) 12(3) 9(4) 16Correct Option: , 2 Solution: $\mathrm{x} \in\left(0, \frac{\pi}{2}\right)$ $\log _{10} \sin x+\log _{10} \cos x=-1$ $\Rightarrow \quad \log _{10} \sin x \cdot \cos x=-1$ $\Rightarrow \quad \sin x \cdot \cos x=\frac{1}{10}$ $\log _{10}(\sin x+\cos x)=\frac{1}{2}\l...

Among the following allotropic forms of sulphur,

Question: Among the following allotropic forms of sulphur, the number of allotropic forms, which will show paramagnetism is$\alpha$-sulphur$\beta$-sulphur$\mathrm{S}_{2}$-form.Correct Option: 1 Solution: $\mathrm{S}_{2}$ is like $\mathrm{O}_{2} \mathrm{i} ;$ e paramagnetic as per molecular orbital theory....

Question: Among the following allotropic forms of sulphur, the number of allotropic forms, which will show paramagnetism is$\alpha$-sulphur$\beta$-sulphur$\mathrm{S}_{2}$-form.Correct Option: 1 Solution: $\mathrm{S}_{2}$ is like $\mathrm{O}_{2} \mathrm{i} ;$ e paramagnetic as per molecular orbital theory....

Given below are two statements :

Question: Given below are two statements : Statement I : Colourless cupric metaborate is reduced to cuprous metaborate in a luminous flame. Statement II : Cuprous metaborate is obtained by heating boric anhydride and copper sulphate in a non-luminous flame. In the light of the above statements, choose the most appropriate answer from the options given below.Statement I is false but statement II is true.Statement I is true but Statement II is false.Both Statement I and Statement II are true.Both ...

A plane is inclined at an angle

Question: A plane is inclined at an angle $\alpha=30^{\circ}$ with respect to the horizontal. A particle is projected with a speed $\mathrm{u}=2 \mathrm{~ms}^{-1}$, from the base of the plane, as shown in figure. The distance from the base, at which the particle hits the plane is close to : (Take $\mathrm{g}=10 \mathrm{~ms}^{-2}$ ) (1) $20 \mathrm{~cm}$(2) $18 \mathrm{~cm}$(3) $26 \mathrm{~cm}$(4) $14 \mathrm{~cm}$Correct Option: 1 Solution: (1) On an inclined plane, time of flight (T) is given ...

The stream of a river is flowing with a speed of 2 km/ h.

Question: The stream of a river is flowing with a speed of $2 \mathrm{~km} / \mathrm{h}$. A swimmer can swim at a speed of $4 \mathrm{~km} / \mathrm{h}$. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?(1) $90^{\circ}$(2) $150^{\circ}$(3) $120^{\circ}$(4) $60^{\circ}$Correct Option: , 3 Solution: (3) To cross the river straight $\mathrm{V}_{\mathrm{s}} \sin \theta=\mathrm{V}_{\mathrm{r}} \therefore \sin \theta=\frac{v_{r}}{v_{s}}=\fra...

The stream of a river is flowing with a speed of 2 km/ h.

Question: The stream of a river is flowing with a speed of $2 \mathrm{~km} / \mathrm{h}$. A swimmer can swim at a speed of $4 \mathrm{~km} / \mathrm{h}$. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?(1) $90^{\circ}$(2) $150^{\circ}$(3) $120^{\circ}$(4) $60^{\circ}$Correct Option: , 3 Solution: (3) To cross the river straight $\mathrm{V}_{\mathrm{s}} \sin \theta=\mathrm{V}_{\mathrm{r}} \therefore \sin \theta=\frac{v_{r}}{v_{s}}=\fra...

A xenon compound 'A' upon partial hydrolysis gives

Question: A xenon compound 'A' upon partial hydrolysis gives $\mathrm{XeO}_{2} \mathrm{~F}_{2}$. The number of lone pair of electrons present in compound $\mathrm{A}$ is_______________ . (Round off to the Nearest integer) Solution: (19) Total l.p. on $(\mathrm{A})=19$...

A ball is dropped from the top of a 100 m high tower on a planet.

Question: A ball is dropped from the top of a $100 \mathrm{~m}$ high tower on a planet. In the last $\frac{1}{2} \mathrm{~s}$ before hitting the ground, it covers a distance of $19 \mathrm{~m}$. Acceleration due to gravity $\left(\right.$ in $\left.\mathrm{ms}^{-2}\right)$ near the surface on that planet is Solution: $(08.00)$ Let the ball takes time t to reach the ground Using, $S=u t+\frac{1}{2} g t^{2}$ $\Rightarrow S=0 \times t+\frac{1}{2} g t^{2}$ $\Rightarrow \quad 200=g t^{2}$ $[\because ...

A particle of mass m is dropped from a height h above the ground.

Question: A particle of mass $m$ is dropped from a height $h$ above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of $\sqrt{2 g h}$. If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of $\sqrt{\frac{h}{g}}$ is:(1) $\sqrt{\frac{1}{2}}$(2) $\sqrt{\frac{3}{4}}$(3) $\frac{1}{2}$(4) $\sqrt{\frac{3}{2}}$Correct Option: , 4 Solution: (4) Let 1 ve ue ume taken uy ...

Slope of a line passing through

Question: Slope of a line passing through $\mathrm{P}(2,3)$ and intersecting the line $x+y=7$ at a distance of 4 units from $P$, is:(1) $\frac{1-\sqrt{5}}{1+\sqrt{5}}$(2) $\frac{1-\sqrt{7}}{1+\sqrt{7}}$(3) $\frac{\sqrt{7}-1}{\sqrt{7}+1}$(4) $\frac{\sqrt{5}-1}{\sqrt{5}+1}$Correct Option: , 2 Solution: Since point at 4 units from $\mathrm{P}(2,3)$ will be $\mathrm{A}(4 \cos \theta+2,4 \sin (\theta+3)$ and this point will satisfy the equation of line $x+y=7$ $\Rightarrow \cos \theta+\sin \theta=\fr...

Suppose that the points

Question: Suppose that the points $(h, k),(1,2)$ and $(-3,4)$ lie on the line $\mathrm{L}_{1}$. If a line $\mathrm{L}_{2}$ passing through the points $(h, k)$ and $(4,3)$ is perpendicular on $\mathrm{L}_{1}$, then $\frac{k}{h}$ equals :(1) $\frac{1}{3}$(2) 0(3) 3(4) $-\frac{1}{7}$Correct Option: 1 Solution: $\because(h, k),(1,2)$ and $(-3,4)$ are collinear $\therefore\left|\begin{array}{lll}h k 1 \\ 1 2 1 \\ -3 4 1\end{array}\right|=0 \Rightarrow-2 h-4 k+10=0$ $\Rightarrow h+2 k=5$ .....(i) Now,...

Prove the following

Question: Let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two fixed points. Then the locus of a point $\mathrm{P}$ such that the perimeter of $\triangle \mathrm{AOP}$ is 4 , is :(1) $8 x^{2}-9 y^{2}+9 y=18$(2) $9 x^{2}-8 y^{2}+8 y=16$(3) $9 x^{2}+8 y^{2}-8 y=16$(4) $8 x^{2}+9 y^{2}-9 y=18$Correct Option: , 3 Solution: Let point $P(h, k)$ $O A=1$ So, $O P+A P=3$ $\Rightarrow \sqrt{h^{2}+k^{2}}+\sqrt{h^{2}+(k-1)^{2}}=3$ $\Rightarrow h^{2}+(k-1)^{2}=9+h^{2}+k^{2}-6 \sqrt{h^{2}+k^{2}}$ $\Rightarrow 6...