A cylindrical bucket, 28 cm in diameter and 72 cm and high, is full of water.

Question: A cylindrical bucket, 28 cm in diameter and 72 cm and high, is full of water. The water is emptied into a rectangular tank, 66 cm long and 28 cm wide. Find the height of the water level in the tank. Solution: Given: Diameter of the cylindrical bucket = 28 cmi.e., radius = 14 cmHeight of the cylindrical bucket,h1= 72 cmLength of the rectangular tank,l= 66 cmBreadth of the rectangular tank,b= 28 cmLet the height of the rectangular tank behcm.The water from the cylindrical bucket is empti...

An electron of mass $m$ and a photon have same

Question: An electron of mass $m$ and a photon have same energy E. The ratio of wavelength of electron to that of photon is: (c being the velocity of light)(1) $\frac{1}{c}\left(\frac{2 m}{E}\right)^{1 / 2}$(2) $\frac{1}{c}\left(\frac{E}{2 m}\right)^{1 / 2}$(3) $\left(\frac{E}{2 m}\right)^{1 / 2}$(4) $\mathrm{c}(2 \mathrm{mE})^{1 / 2}$Correct Option: , 2 Solution: (2) $\lambda_{1}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mE}}}$ $\lambda_{2}=\frac{\mathrm{hc}}{\mathrm{E}}$ $\frac{\lambda_{1}}{\lambda_{2...

The major product(s) obtained in the following reaction is

Question: The major product(s) obtained in the following reaction is/ are : Correct Option: 1 Solution:...

The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2.

Question: The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Find the volume of the cylinder if its total surface area is 616 cm2. Solution: Suppose that the curved surface area and the total surface area of the right circular cylinder arexcm2and 2xcm2.Then we have:2x= 616x= 308 sq cmHence, the curved surface area of the cylinder is 308 sq cm.Letrcm andhcm be the radius and height of the cylinder, respectively. Then $2 \pi r h+2 \pi r^{2}=...

Let and,

Question: Let $d \in \mathbf{R}$, and $A=\left[\begin{array}{ccc}-2 4+d (\sin \theta)-2 \\ 1 (\sin \theta)+2 d \\ 5 (2 \sin \theta)-d (-\sin \theta)+2+2 d\end{array}\right]$ $\theta \in[0,2 \pi] .$ If the minimum value of $\operatorname{det}(\mathrm{A})$ is 8, then a value of $d$ is: (1) $-5$(2) $-7$(3) $2(\sqrt{2}+1)$(4) $2(\sqrt{2}+2)$Correct Option: 1 Solution: $\operatorname{det}(A)=\left|\begin{array}{ccc}-2 4+d \sin \theta-2 \\ 1 \sin \theta+2 d \\ 5 2 \sin \theta-d -\sin \theta+2+2 d\end{...

If an electron is moving in the

Question: If an electron is moving in the $\mathrm{n}^{\text {th }}$ orbit of the hydrogen atom, then its velocity $\left(\mathrm{v}_{\mathrm{n}}\right)$ for the $n^{\text {th }}$ orbit is given as:(1) $\mathrm{v}_{\mathrm{n}} \propto \mathrm{n}$(2) $v_{n} \propto \frac{1}{n}$(3) $v_{n} \propto n^{2}$(4) $v_{n} \propto \frac{1}{n^{2}}$Correct Option: , 2 Solution: $(2)$ We know velocity of electron in $\mathrm{n}^{\text {th }}$ shell of hydrogen atom is given by $v=\frac{2 \pi k Z e^{2}}{n h}$ $...

The total surface area of a solid cylinder is 231 cm2 and its curved surface area is

Question: The total surface area of a solid cylinder is $231 \mathrm{~cm}^{2}$ and its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder. Solution: Curved surface area $=\frac{2}{3} \times$ total surface area $=\frac{2}{3} \times 231=2 \times 77=154 \mathrm{~cm}^{2}$ Now, total surface area - curved surface area $=2 \pi r h+2 \pi r^{2}-2 \pi r h$ Then $231-154=2 \pi r^{2}$ $\Rightarrow 2 \times \frac{22}{7} \times r^{2}=77$ $\Rightarrow r^{2}=\frac{7...

The major product ' Y ' in the following reaction is:

Question: The major product ' $Y$ ' in the following reaction is: Correct Option: 1 Solution:...

The de-Broglie wavelength associated with an electron and a proton were

Question: The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of $100 \mathrm{~V}$. What should nearly be the ratio of their wavelengths? $\left(\mathrm{m}_{\mathrm{P}}=1.00727 \mathrm{u}, \mathrm{m}_{\mathrm{e}}=0.00055 \mathrm{u}\right)$(1) $1860: 1$(2) $(1860)^{2}: 1$(3) $41.4: 1$(4) $43: 1$Correct Option: , 4 Solution: (4) $\lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mK}}}=\frac{\mathrm{h}...

Solve the following

Question: Compound $\mathrm{A}\left(\mathrm{C}_{9} \mathrm{H}_{10} \mathrm{O}\right)$ shows positive iodoform test. Oxidation of $\mathrm{A}$ with $\mathrm{KMnO}_{4} / \mathrm{KOH}$ gives acid $\mathrm{B}\left(\mathrm{C}_{8} \mathrm{H}_{6} \mathrm{O}_{4}\right)$. Anhydride of $\mathrm{B}$ is used for the preparation of phenolphthalein. Compound $\mathrm{A}$ is :Correct Option: , 4 Solution:...

The shortest distance between the point

Question: The shortest distance between the point $\left(\frac{3}{2}, 0\right)$ and the curve $y=\sqrt{x},(x0)$, is:(1) $\frac{\sqrt{5}}{2}$(2) $\frac{\sqrt{3}}{2}$(3) $\frac{3}{2}$(4) $\frac{5}{4}$Correct Option: 1 Solution: Here the curve is para bola with $a=\frac{1}{4}$, Let $\mathrm{P}\left(\mathrm{at}^{2}\right.$, 2at) i.e., $P\left(\frac{t^{2}}{4}, \frac{t}{2}\right)$ be a point on the curve. Now, $y^{2}=x$ $\Rightarrow 2 y \frac{d y}{d x}=1 \Rightarrow \frac{d y}{d x}=\frac{1}{2 \sqrt{x}...

The first three spectral lines of H-atom in the Balmer series are given

Question: The first three spectral lines of $\mathrm{H}$-atom in the Balmer series are given $\lambda_{1}, \lambda_{2}, \lambda_{3}$ considering the Bohr atomic model, the wave lengths of first and third spectral lines $\left(\frac{\lambda_{1}}{\lambda_{3}}\right)$ are related by afactor of approximately ' $x^{\prime} \times 10^{-1}$ .The value of $\mathrm{x}$, to the nearest integer, is Solution: $(15)$ For 1 st line $\frac{1}{\lambda_{1}}=\operatorname{Rz}^{2}\left(\frac{1}{2^{2}}-\frac{1}{3^{...

The total surface area of a cylinder is 462 cm2.

Question: The total surface area of a cylinder is 462 cm2. Its curved surface area is one-third of its total surface area. Find the volume of the cylinder. Solution: Total surface area = 462 cm2 Given: Curved surface area $=\frac{1}{3} \times$ total surface area $=\frac{1}{3} \times 462=154 \mathrm{~cm}^{2}$ Now, total surface area - curved surface area $=2 \pi \mathrm{rh}+2 \pi \mathrm{r}^{2}-2 \pi \mathrm{rh}$ $\Rightarrow \Rightarrow 462-154=2 \pi r^{2}$ $\Rightarrow 308=2 \times \frac{22}{7}...

Major products of the following reaction are :

Question: Major products of the following reaction are : $\mathrm{CH}_{3} \mathrm{OH}$ and $\mathrm{HCO}_{2} \mathrm{H}$Correct Option: , 3 Solution:...

The radius of the base and the height of a cylinder are in the ratio 2 : 3. If its volume is 1617 cm3,

Question: The radius of the base and the height of a cylinder are in the ratio 2 : 3. If its volume is 1617 cm3, find the total surface area of the cylinder. Solution: Suppose that the radius of the base and the height of the cylinder are 2xcm and 3xcm, respectively. Then $1617=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times(2 \mathrm{x})^{2} \times 3 \mathrm{x}$ $=\frac{22}{7} \times 12 \mathrm{x}^{3}$ $\Rightarrow \mathrm{x}^{3}=\frac{1617 \times 7}{22 \times 12}=42.875$ $\Rightarrow \mathr...

The stopping potential in the context of photoelectric effect depends on the

Question: The stopping potential in the context of photoelectric effect depends on the following property of incident electromagnetic radiation :(1) Phase(2) Intensity(3) Amplitude(4) FrequencyCorrect Option: , 4 Solution: (4) Stopping potential changes linearly with frequency of incident radiation....

If denotes the acute angle between the curves,

Question: If $\theta$ denotes the acute angle between the curves, $y=10-x^{2}$ and $y=2+x^{2}$ at a point of their intersection, then $|\tan \theta|$ is equal to:(1) $\frac{4}{9}$(2) $\frac{8}{15}$(3) $\frac{7}{17}$(4) $\frac{8}{17}$Correct Option: , 2 Solution: Since, the equation of curves are $y=10-x^{2}$........(1) $y=2+x^{2}$.........(2) Adding eqn (1) and (2), we get $2 y=12 \Rightarrow y=6$ Then, from eqn (1) $x=\pm 2$ Differentiate equation (1) with respect to $x$ $\frac{d y}{d x}=-2 x \...

The curved surface area of a cylinder is 4400 cm2 and the circumference of its base is 110 cm.

Question: The curved surface area of a cylinder is 4400 cm2and the circumference of its base is 110 cm. Find the height and the volume of the cylinder. Solution: Letrbe the radius andh be the height of the cylinder. Circumference of its base(circle) = 110 cm. $\Rightarrow 2 \pi r=110 \Rightarrow r=\frac{110}{2 \pi} \Rightarrow r=\frac{110}{2 \times \frac{22}{7}} \Rightarrow r=\frac{110 \times 7}{2 \times 22} \Rightarrow r=\frac{35}{2} \mathrm{~cm}$ Curved surface area of a cylinder $=4400 \mathr...

Question: The stopping potential in the context of photoelectric effect depends on the following property of incident electromagnetic radiation :(1) Phase(2) Intensity(3) Amplitude(4) FrequencyCorrect Option: , 4 Solution: (4) Stopping potential changes linearly with frequency of incident radiation....

In the following reaction

Question: In the following reaction Rate of the reaction is the highest for:Acetone as substrate and methanol in excess.Propanal as substrate and methanol in stoichiometric amount.Propanal as substrate and methanol in excess.Acetone as substrate and methanol in stoichiometric amount.Correct Option: , 3 Solution: Generally, aldehydes are more reactive than ketones in nucleophilic addition reactions. $\therefore$ Rate of reaction with alcohol to form acetal and ketal is...

The curved surface area of a cylinder is 1210 cm2 and its diameter is 20 cm.

Question: The curved surface area of a cylinder is 1210 cm2and its diameter is 20 cm. Find its height and volume. Solution: Curved surface area = 1210 cm2Suppose that the height of cylinder ishcm.We haver= 10 cm Now, $1210=2 \pi r h$ $\Rightarrow 1210=2 \times \frac{22}{7} \times 10 \times h$ $\Rightarrow h=\frac{1210 \times 7}{2 \times 22 \times 10}=\frac{11 \times 7}{2 \times 2}=19.25 \mathrm{~cm}$ $\therefore$ Volume of the cylinder $=\pi r^{2} h$ $=\frac{22}{7} \times 10^{2} \times 19.25 \ma...

Find the weight of a solid cylinder of radius 10.5 cm and height 60 cm if the material of the cylinder weighs 5 g per cm2.

Question: Find the weight of a solid cylinder of radius 10.5 cm and height 60 cm if the material of the cylinder weighs 5 g per cm2. Solution: Here,r= 10.5 cm;h= 60 cm Now, volume of the cylinder $=\pi r^{2} h$ $=\frac{22}{7} \times(10.5)^{2} \times 60 \mathrm{~cm}^{3}$ $=22 \times 10.5 \times 1.5 \times 60 \mathrm{~cm}^{3}=20790 \mathrm{~cm}^{3}$ $\therefore$ Weight of cylinder = volume of cylinder $\times$ weight of cylinder per gram $=20790 \times 5 \mathrm{~g}$ $=103950 \mathrm{~g}$ $=103.95...

The maximum volume (in cu.m) of the right circular cone having slant height

Question: The maximum volume (in cu.m) of the right circular cone having slant height $3 \mathrm{~m}$ is:(1) $6 \pi$(2) $3 \sqrt{3} \pi$(3) $\frac{4}{3} \pi$(4) $2 \sqrt{3} \pi$Correct Option: , 4 Solution: $h^{2}+r^{2}=\ell^{2}=9$ ........(1) Volume of cone $V=\frac{1}{3} \pi r^{2} h$ ..........(2) From (1) and (2), $\Rightarrow \quad V=\frac{1}{3} \pi\left(9-h^{2}\right) h$ $\Rightarrow \quad V=\frac{1}{3} \pi\left(9 h-h^{3}\right)$$\Rightarrow \frac{d v}{d h}=\frac{1}{3} \pi\left(9-3 h^{2}\ri...

The major product obtained in the following reaction is :

Question: The major product obtained in the following reaction is : Correct Option: , 2 Solution:...

In a water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm.

Question: In a water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system. Solution: Length of the cylindrical pipe,h= 28 m Radius of the cylindrical pipe, $r=\frac{5}{2}=2.5 \mathrm{~cm}=0.025 \mathrm{~m} \quad(1 \mathrm{~m}=100 \mathrm{~cm})$ Total radiating surface in the system= Curved surface area of the cylindrical pipe $=2 \pi r h$ $=2 \times \frac{22}{7} \times 0.025 \times 28$ $=4.4 \mathrm{~m}^{2}$ Thus, the total ...