The exact volumes of

Question: The exact volumes of $1 \mathrm{MNaOH}$ solution required to neutralise $50 \mathrm{~mL}$ of $1 \mathrm{MH}_{3} \mathrm{PO}_{3}$ solution and $100 \mathrm{~mL}$ of $2 \mathrm{MH}_{3} \mathrm{PO}_{2}$ solution, respectively, are:$100 \mathrm{~mL}$ and $100 \mathrm{~mL}$$100 \mathrm{~mL}$ and $50 \mathrm{~mL}$$100 \mathrm{~mL}$ and $200 \mathrm{~mL}$$50 \mathrm{~mL}$ and $50 \mathrm{~mL}$Correct Option: , 3 Solution: $\mathrm{H}_{3} \mathrm{PO}_{3}+2 \mathrm{NaOH} \rightarrow \mathrm{Na}...

If for some

Question: If for some $\alpha \in \mathbf{R}$, the lines $L_{1}: \frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $L_{2}: \frac{x+2}{\alpha}=\frac{y+1}{5-\alpha}=\frac{z+1}{1}$ are coplanar, then the line $L_{2}$ passes through the point :$(10,2,2)$$(2,-10,-2)$$(10,-2,-2)$$(-2,10,2)$Correct Option: , 2 Solution: Since, lince are coplanar $\therefore\left|\begin{array}{ccc}1 3 2 \\ 2 -1 1 \\ \alpha 5-\alpha 1\end{array}\right|=0$ $\Rightarrow 1(-1-5+\alpha)-3(2-\alpha)+2(10-2 \alpha+\alpha)=0$ $\t...

Question: In order to determine the Young's Modulus of a wire of radius $0.2 \mathrm{~cm}$ (measured using a scale of least count $=0.001 \mathrm{~cm}$ ) and length $1 \mathrm{~m}$ (measured using a scale of least count $=1 \mathrm{~mm}$ ), a weight of mass $1 \mathrm{~kg}$ (measured using a scale of least count $=1 \mathrm{~g}$ ) was hanged to get the elongation of $0.5 \mathrm{~cm}$ (measured using a scale of least count $0.001 \mathrm{~cm}$ ). What will be the fractional error in the value of...

Solve the following

Question: $2 \mathrm{MnO}_{4}^{-}+\mathrm{bC}_{2} \mathrm{O}_{4}^{2-}+\mathrm{cH}^{+} \rightarrow \mathrm{xMn}^{2+}+\mathrm{yCO}_{2}+\mathrm{zH}_{2} \mathrm{O}$ If the above equation is balanced with integer coefficients, the value of $c$ is _________________ .(Round off to the Nearest Integer). Solution: (16) Writting the half reaction oxidation half reaction $\mathrm{MnO}_{4}^{-} \rightarrow \mathrm{Mn}^{2+}$ balancing oxygen $\mathrm{MnO}_{4}^{-} \rightarrow \mathrm{Mn}^{2+}+4 \mathrm{H}_{2} ...

If (a, b, c) is the image of the point

Question: If $(a, b, c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}$, then $a+b+c$ is equals to:2$-1$31Correct Option: 1 Solution: $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}=\lambda$ Any point on line $=Q(2 \lambda-1,-2 \lambda+3,-\lambda)$ $\therefore$ D.r. of $P Q=[2 \lambda-2,-2 \lambda+1,-\lambda+3]$ D.r. of given line $=[2,-2,-1]$ $\because P Q$ is perpendicular to line $L$ $\therefore 2(2 \lambda-2)-2(-2 \lambda+1)-1(-\lambda+3)=0$ $\Righta...

The distance of the point (1,-2,3) from

Question: The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is :$\frac{7}{5}$1$\frac{1}{7}$7Correct Option: , 2 Solution: Equation of line through point $P(1,-2,3)$ and parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-3}{-6}=\lambda$ So, any point on line $=Q(2 \lambda+1,3 \lambda-2,-6 \lambda+3)$ Since, this point lies on plane $x-y+2=5$ $\therefore 2 \lambda+1-3 \...

A spring mass system (mass m, spring constant k and natural length l ) rests in equilibrium on a horizontal disc.

Question: A spring mass system (mass $m$, spring constant $k$ and natural length $l$ ) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about it's axis with an angular velocity $\omega,\left(km \omega^{2}\right)$ the relative change in the length of the spring is best given by the option:(1) $\sqrt{\frac{2}{3}}\left(\frac{m \omega^{2}}{k}\right)$(2) $\frac{2 m \omega^{2}}{k}$(3) $\frac{...

If the equation of a plane P, passing

Question: If the equation of a plane $P$, passing through the intersection of the planes, $x+4 y-z+7=0$ and $3 x+y+5 z=8$ is $a x+b y$ Solution: Equation of plane $P$ is $(x+4 y-z+7)+\lambda(3 x+y+5 z-8)=0$ $\Rightarrow x(1+3 \lambda)+y(4+\lambda)+z(-1+5 \lambda)+(7-8 \lambda)=0$ $\Rightarrow \frac{1+3 \lambda}{a}=\frac{4+\lambda}{b}=\frac{5 \lambda-1}{6}=\frac{7-8 \lambda}{-15}$ From last two ratios, $\lambda=-1$ $\Rightarrow \frac{-2}{a}=\frac{3}{b}=-1$ $\therefore a=2, b=-3$ $\therefore$ Equa...

Let a plane P contain two lines

Question: Let a plane $P$ contain two lines $\vec{r}=\hat{i}+\lambda(\hat{i}+\hat{j}), \lambda \in \mathbf{R}$ and $\vec{r}=-\hat{j}+\mu(\hat{j}-\hat{k}), \mu \in \mathbf{R}$. If $Q(\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from the point $M(1,0,1)$ to $P$, then $3(\alpha+\beta+\gamma)$ equals__________. Solution: Normal of plane $=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 1 1 0 \\ 0 1 -1\end{array}\right|$ $\vec{n}=-\hat{i}+\hat{j}+\hat{k}$ Direction ratios of norm...

An object of mass m is suspended at the end of a massless wire of length L and area of cross-section,

Question: An object of mass $m$ is suspended at the end of a massless wire of length $L$ and area of cross-section, $A$. Young modulus of the material of the wire is $Y$. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:$f=\frac{1}{2 \pi} \sqrt{\frac{m L}{Y A}}$$f=\frac{1}{2 \pi} \sqrt{\frac{Y A}{m L}}$$f=\frac{1}{2 \pi} \sqrt{\frac{m A}{Y L}}$$f=\frac{1}{2 \pi} \sqrt{\frac{Y L}{m A}}$Correct Option: 1 Solution: (1) An elastic wire can be treated a...

The plane which bisects the line

Question: The plane which bisects the line joining the points $(4,-2,3)$ and $(2,4,-1)$ at right angles also passes through the point:$(4,0,1)$$(0,-1,1)$$(4,0,-1)$$(0,1,-1)$Correct Option: , 3 Solution: Direction ratios of normal to the plane are $1,-3,2$. Plane passes through $(3,1,1)$. Equation of plane is, $1(x-3)-3(y-1)+2(z-1)=0$ $\Rightarrow x-3 y+2 z-2=0$...

Kjeldahl's method cannot be used to estimate nitrogen for which of the following

Question: Kjeldahl's method cannot be used to estimate nitrogen for which of the following compounds?$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{2}$$\mathrm{CH}_{3} \mathrm{CH}_{2}-\mathrm{C} \equiv \mathrm{N}$$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NO}_{2}$Correct Option: , 3 Solution: Kjeldahl's method can not be used for nitrogen determination of compounds having nitro group or azo group or nitrogen present in rings as the nitrogen of these compounds can not be converted to $\left(\mathrm{NH}_...

A ring is hung on a nail.

Question: A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period $T_{1}$ and, (ii) back and forth in a direction perpendicular to its plane, with a period $T_{2} .$ The ratio $\frac{T_{1}}{T_{2}}$ will be :$\frac{2}{\sqrt{3}}$$\frac{2}{3}$$\frac{3}{\sqrt{2}}$$\frac{\sqrt{2}}{3}$Correct Option: 1 Solution: (1) Let $I_{1}$ and $I_{2}$ be the moment of inertia in first and second case respectively. $I_{1}=2 M R^{2}$ $I_{2}=M R^{2}+\frac{M R^{2}}{...

Prove the following

Question: The lines $r=(\hat{i}-\hat{j})+l(2 \hat{i}+k)$ and $\vec{r}=(2 \hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$do not intersect for any values of $l$ and $m$intersect for all values of $l$ and $m$intersect when $l=2$ and $m=\frac{1}{2}$intersect when $l=1$ and $m=2$Correct Option: 1 Solution: $L_{1} \equiv \vec{r}=(\hat{i}-\hat{j})+\ell(2 \hat{i}+k)$ $L_{2} \equiv \vec{r}=(2 \hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$ Equating coeff. of $\hat{i}, \hat{j}$ and $\hat{k}$ of $L_{1}$ and $L_{...

The strength of an aqueous NaOH solution is most accurately determined by titrating:

Question: The strength of an aqueous $\mathrm{NaOH}$ solution is most accurately determined by titrating: (Note: consider that an appropriate indicator is used)Aq. $\mathrm{NaOH}$ in a pipette and aqueous oxalic acid in a buretteAq. $\mathrm{NaOH}$ in a burette and aqueous oxalic acid in a conical flaskAq. $\mathrm{NaOH}$ in a burette and concentrated $\mathrm{H}_{2} \mathrm{SO}_{4}$ in a conical flaskAq. $\mathrm{NaOH}$ in a volumetric flask and concentrated $\mathrm{H}_{2} \mathrm{SO}_{4}$ in ...

While titrating dilute HCl solution

Question: While titrating dilute $\mathrm{HCl}$ solution with aqueous $\mathrm{NaOH}$, which of the following will not be required?Burette and porcelain tilePipette and distilled waterClamp and phenolphthaleinBunsen burner and measuring cylinderCorrect Option: , 4 Solution: In this acid base titration bunsen burner and measuring cylinder are of no use while other laboratory equipments will be required i.e., phenol phthalein, burette and pipette....

Nitrogen can be estimated by Kjeldah's method for which of the following

Question: Nitrogen can be estimated by Kjeldah's method for which of the following compound?Correct Option: , 2 Solution: Kjeldahl method is not applicable to compounds containing nitrogen in nitrogroup, Azo groups and nitrogen present in the ring (e.g Pyridine) as nitrogen of these compounds does not change to Ammonium sulphate under these conditions....

A block of mass m attached to a massless spring is performing oscillatory motion of amplitude ' A ' on a frictionless horizontal plane.

Question: A block of mass $m$ attached to a massless spring is performing oscillatory motion of amplitude ' $A$ ' on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining system become $f A$. The value of $f$ is :$\frac{1}{\sqrt{2}}$$\frac{2}{3}$$\frac{3}{\sqrt{2}}$$\frac{\sqrt{2}}{3}$Correct Option: 1, Solution: (1) Potential energy of spring $=\frac{1}{2} k x^{2}$ Here, $x=$ ...

The foot of the perpendicular

Question: The foot of the perpendicular drawn from the point $(4,2,3)$ to the line joining the points $(1,-2,3)$ and $(1,1,0)$ lies on the plane:(1) $2 x+y-z=1$(2) $x-y-2 z=1$(3) $x-2 y+z=1$(4) $x+2 y-z=1$Correct Option: 1 Solution: Equation of line through points $(1,-2,3)$ and $(1,1,0)$ is $\frac{x-1}{0}=\frac{y-1}{-3}=\frac{z-0}{3-0} \quad(=\lambda$ say $)$ $\therefore M(1,-\lambda+1, \lambda)$ Direction ratios of $\mathrm{PM}=[-3,-\lambda-1, \lambda-3]$ $\because P M \perp A B$ $\therefore(-...

Given below are two statements:

Question: Given below are two statements: Statement-I : Retardation factor $\left(\mathrm{R}_{\mathrm{f}}\right)$ can be measured in meter/centimeter. Statement-II : $\mathrm{R}_{\mathrm{f}}$ value of a compound remains constant in all solvents. Choose the most appropriate answer from the options given below:Statement-I is true but statement-II is falseBoth statement-I and statement-II are trueBoth statement-I and statement-II are falseStatement-I is false but statement-II is trueCorrect Option:...

The displacement time graph of a particle executing S.H.M. is given in figure : (sketch is schematic and not to scale)

Question: The displacement time graph of a particle executing S.H.M. is given in figure : (sketch is schematic and not to scale) Which of the following statements is/are true for this motion? (a) The force is zero at $t=\frac{3 T}{4}$ (b) The acceleration is maximum at $t=T$ (c) The speed is maximum at $t=\frac{T}{4}$ (d) The P.E. is equal to K.E. of the oscillation at $t=\frac{T}{2}$(1) (a), (b) and (d)(2) (b), (c) and (d)(3) (a), (b) and (c)(4) (a) and (d)Correct Option: , 3 Solution: (3) From...

In Duma's method of estimation of nitrogen,

Question: In Duma's method of estimation of nitrogen, $0.1840 \mathrm{~g}$ of an organic compound gave $30 \mathrm{~mL}$ of nitrogen collected at $287 \mathrm{~K}$ and $758 \mathrm{~mm}$ of $\mathrm{Hg}$ pressure. The percentage composition of nitrogen in the compound is______________. (Round off to the Nearest Integer). [Given : Aqueous tension at $287 \mathrm{~K}=14 \mathrm{~mm}$ of $\mathrm{Hg}$ ] Solution: In Duma's method of estimation of Nitrogen. $0.1840 \mathrm{gm}$ of organic compound g...

A plane passing through the point

Question: A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $1,-2,2$ and $2,3,-1$ respectively. If this plane also passes through the point $(\alpha,-3,5)$, then $\alpha$ is equal to :(1) 5(2) $-10$(3) 10(4) $-5$Correct Option: 1 Solution: $\because$ Plane contains two lines $\therefore \vec{n}=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 1 -2 2 \\ 2 3 -1\end{array}\right|$ $=i(2-6)-j(-1-4)+k(3+4)=-4 i+5 j+7 k$ So, equation of plane is $-4(x-3)+5(y-1...

Match List-I with List-II

Question: Match List-I with List-II The correct match is:$(a)-(i i i),(b)-(i),(c)-(i i),(d)-(i v)$(a) - (i), (b) - (iv), (c) - (iii), (d) - (ii)(a) $-($ iii $),(b)-(i),(c)-(i v),(d)-(i i)$(a) $-($ i $),($ b $)-($ ii $),($ c $)-($ iv $),($ d $)-($ iii $)$Correct Option: , 3 Solution: Match list:- Option-(a)-(iii) ; (b)-(i); (c)-(iv); (d)-(ii)...

The plane passing through the points

Question: The plane passing through the points $(1,2,1),(2,1,2)$ and parallel to the line, $2 x=3 y, z=1$ also through the point:(1) $(0,6,-2)$(2) $(-2,0,1)$(3) $(0,-6,2)$(4) $(2,0,-1)$Correct Option: , 2 Solution: Let plane passes through $(2,1,2)$ be $a(x-2)+b(y-1)+(z-2)=0$ It also passes through $(1,2,1)$ $\therefore-a+b-c=0 \Rightarrow a-b+c=0$ The given line is $\frac{x}{3}=\frac{y}{2}=\frac{z-1}{0}$ is parallel to plane $\therefore 3 a+2 b+c(0)=0$ $\Rightarrow \frac{a}{0-2}=\frac{b}{3-0}=\...