The area of the region, enclosed by the circle
Question: The area of the region, enclosed by the circle $x^{2}+y^{2}=2$ which is not common to the region bounded by the parabola $y^{2}=x$ and the straight line $y=x$, is:(1) $\frac{1}{6}(24 \pi-1)$(2) $\frac{1}{3}(6 \pi-1)$(3) $\frac{1}{3}(12 \pi-1)$(4) $\frac{1}{6}(12 \pi-1)$Correct Option: , 4 Solution: Total area - enclosed area between line and parabola $=2 \pi-\int_{0}^{1} \sqrt{x}-x d x$ $=2 \pi-\left(\frac{2 x^{3 / 2}}{3}-\frac{x^{2}}{2}\right)_{0}^{1}$ $=2 \pi-\left(\frac{2}{3}-\frac{...
Read More →Which of these will produce the highest yield in Friedel Crafts reaction?
Question: Which of these will produce the highest yield in Friedel Crafts reaction?Correct Option: , 3 Solution: Aniline and phenol form complex with lewis acid. Chlorobenzene produces highest yield in Friedel craft reaction among the given options....
Read More →The area (in sq. units) of the region enclosed by the curves
Question: The area (in sq. units) of the region enclosed by the curves $y=x^{2}-1$ and $y=1-x^{2}$ is equal to:(1) $\frac{4}{3}$(2) $\frac{8}{3}$(3) $\frac{7}{2}$(4) $\frac{16}{3}$Correct Option: , 2 Solution: Required area Area $=2 \int_{0}^{1}\left(\left(1-x^{2}\right)-\left(x^{2}-1\right)\right) d x=4 \int_{0}^{1}\left(1-x^{2}\right) d x$ $=\left.4\left(x-\frac{x^{3}}{3}\right)\right|_{0} ^{1}=4\left(1-\frac{1}{3}\right)=4 \cdot \frac{2}{3}=\frac{8}{3}$ sq. units...
Read More →Assuming the nitrogen molecule is moving with r.m.s.velocity
Question: Assuming the nitrogen molecule is moving with r.m.s.velocity at $400 \mathrm{~K}$, the de-Broglie wavelength of nitrongen molecule is close to : (Given : nitrogen molecule weight : $4.64 \times 10^{-26} \mathrm{~kg}$, Boltzman constant : $1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$, Planck constant : $\left.6.63 \times 10^{-34} \mathrm{~J} . \mathrm{s}\right)$\text { (1) } 0.24 A(2) $0.20 A(3) $0.34 A(4) $0.44 ACorrect Option: 1, Solution: (1) Rms speed of gas molecule, $V_{r m s}=\...
Read More →In the following sequence of reactions the maximum number
Question: In the following sequence of reactions the maximum number of atoms present in molecule ' $C$ ' in one plane is __________ (A is a lowest molecular weight alkyne) Solution:...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $\mathrm{A}=\left\{(x, y):|x|+|y| \leq 1,2 y^{2} \geq|x|\right\}$ is:(1) $\frac{1}{3}$(2) $\frac{7}{6}$(3) $\frac{1}{6}$(4) $\frac{5}{6}$Correct Option: , 4 Solution: Required area $=4\left[\int_{0}^{\frac{1}{2}} 2 y^{2} d y+\frac{1}{2} \operatorname{area}(\Delta P A B)\right]$ $=4\left[\frac{2}{3}\left[y^{3}\right]_{0}^{\frac{1}{2}}+\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\right]=4\left[\frac{2}{3} \times \frac{1}{8}+\frac{1}{8}\right]$ $...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $A=\{(x, y):(x-1)[x] \leq y \leq 2 \sqrt{x}, 0 \leq x \leq 2\}$, where $[t]$ denotes the greatest integer function, is :(1) $\frac{8}{3} \sqrt{2}-\frac{1}{2}$(2) $\frac{4}{3} \sqrt{2}+1$(3) $\frac{8}{3} \sqrt{2}-1$(4) $\frac{4}{3} \sqrt{2}-\frac{1}{2}$Correct Option: Solution: $[x]=0$ when $x \in[0,1)$ and $[x]=1$ when $x \in[1,2)$ $y=\left\{\begin{array}{cc}0 0 \leq x1 \\ x-1 1 \leq x2\end{array}\right.$ $\therefore A=\int_{0}^{2} 2 \sqrt{x} d x-\...
Read More →An electron, a doubly ionized helium ion
Question: An electron, a doubly ionized helium ion $\left(\mathrm{He}^{++}\right)$and a proton are having the same kinetic energy. The relation between their respective de-Broglie wavelengths $\lambda_{e}, \lambda_{\mathrm{He}+t}$ and $\lambda_{\mathrm{p}}$ is :(1) $\lambda_{\mathrm{e}}\lambda_{\mathrm{He}++}\lambda_{\mathrm{p}}$(2) $\lambda_{\mathrm{e}}\lambda_{\mathrm{He}++}=\lambda_{\mathrm{p}}$(3) $\lambda_{e}\lambda_{p}\lambda_{\mathrm{He}++}$(4) $\lambda_{\mathrm{e}}\lambda_{\mathrm{p}}\la...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $\left\{(x, y): 0 \leq y \leq x^{2}+1,0 \leq y \leq x+1, \frac{1}{2} \leq x \leq 2\right\}$ is : (1) $\frac{23}{16}$(2) $\frac{79}{24}$(3) $\frac{79}{16}$(4) $\frac{23}{6}$Correct Option: , 2 Solution: Required area $=\int_{\frac{1}{2}}^{1}\left(x^{2}+1\right) d x+\int_{1}^{2}(x+1) d x$ $=\left[\frac{x^{3}}{3}+x\right]_{\frac{1}{2}}^{1}+\left[\frac{x^{2}}{2}+x\right]_{1}^{2}$ $=\left[\frac{4}{3}-\frac{13}{24}\right]+\frac{5}{2}=\frac{79}{24}$...
Read More →The surface of a metal is illuminated alternately with photons of energies
Question: The surface of a metal is illuminated alternately with photons of energies $E_{1}=4 \mathrm{eV}$ and $E_{2}=2.5 \mathrm{eV}$ respectively. The ratio of maximum speeds of the photoelectrons emitted in the two cases is 2 . The work function of the metal in $(\mathrm{eV})$ is Solution: From the Einstein's photoelectric equation Energy of photon $=$ Kinetic energy of photoelectrons + Work function $\Rightarrow$ Kinetic energy $=$ Energy of Photon $-$ Work Function Let $\phi_{0}$ be the wor...
Read More →In a photoelectric effect experiment,
Question: In a photoelectric effect experiment, the graph of stopping potential $V$ versus reciprocal of wavelength obtained is shown in the figure. As the intensity of incident radiation is increased : (1) Straight line shifts to right(2) Slope of the straight line get more steep(3) Straight line shifts to left(4) Graph does not changeCorrect Option: , 4 Solution: (4) According to Einstein's photoelectric equation $K_{\max }=h v-\phi_{0}$ $\Rightarrow e V_{s}=\frac{h c}{\lambda}-\phi_{0}$ $\Rig...
Read More →Solve this
Question: Note Take $\pi=\frac{22}{7}$, unless stated otherwise. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl. Solution: Inner radius of the bowl,r= 5 cmLet the outer radius of the bowl beRcm.Thickness of the bowl = 0.25 cm (Given)Rr= 0.25 cm⇒R= 0.25 +r= 0.25 + 5 = 5.25 cm $\therefore$ Outer curved surface area of the bowl $=2 \pi r^{2}=2 \times \frac{22}{7} \times(5.25)^{2}=173.25 \mathrm{~cm}^{2}$ Thu...
Read More →Consider the following reactions:
Question: Consider the following reactions: Which of these reactions are possible?(A) and (B)(A) and (D) (B), (C) and (D)(B) and (D)Correct Option: Solution: The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are given below: The correct plot for $3 \mathrm{~s}$ orbital is: (1) $\mathrm{D}$ (2) B (3) $\mathrm{A}$ (4) $\mathrm{C}$...
Read More →Consider a region
Question: Consider a region $R=\left\{(x, y) \in \mathbf{R}^{2}: x^{2} \leq y \leq 2 x\right\}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true?(1) $\alpha^{3}-6 \alpha^{2}+16=0$(2) $3 \alpha^{2}-8 \alpha^{3 / 2}+8=0$(3) $3 \alpha^{2}-8 \alpha+8=0$(4) $\alpha^{3}-6 \alpha^{3 / 2}-16=0$Correct Option: , 2 Solution: Let $y=x^{2}$ and $y=2 x$ According to question $\therefore \int_{0}^{\alpha}\left(\sqrt{y}-\frac{y}{2}\right) d y=\int_{...
Read More →Consider the following reactions:
Question: Consider the following reactions: Which of these reactions are possible?(A) and (B)(A) and (D) (B), (C) and (D)(B) and (D)Correct Option: Solution: The plots of radial distribution functions for various orbitals of hydrogen atom against 'r' are given below: The correct plot for $3 \mathrm{~s}$ orbital is: (1) $\mathrm{D}$ (2) B (3) $\mathrm{A}$ (4) $\mathrm{C}$...
Read More →A hemispherical bowl is made of steel 0.5 cm thick.
Question: A hemispherical bowl is made of steel 0.5 cm thick. The inside radius of the bowl is 4 cm. Find the volume of steel used in making the bowl. Solution: Internal radius of the hemispherical bowl = 4 cmThickness of a the bowl = 0.5 cmNow, external radius of the bowl = (4 + 0.5 ) cm = 4.5 cmNow, volume of steel used in making the bowl = volume of the shell $=\frac{2}{3} \pi\left(4.5^{3}-4^{3}\right)$ $=\frac{2}{3} \times \frac{22}{7} \times(91.125-64)$ $=\frac{2}{3} \times \frac{22}{7} \ti...
Read More →A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into cylindrical shaped small bottles of diameter 3 cm and height 4 cm.
Question: A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into cylindrical shaped small bottles of diameter 3 cm and height 4 cm. How many bottles are required to empty the bowl? Solution: Internal radius of the hemispherical bowl = 9 cmRadius of a cylindrical shaped bottle = 1.5 cmHeight of a bottle = 4 cm Number of bottles required to empty the bowl $=\frac{\text { volume of the hemispherical bowl }}{\text { volume of a cylindrical shaped bottle }}$ ...
Read More →Area (in sq. units) of the region outside
Question: Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is : (1) $6(\pi-2)$(2) $3(\pi-2)$(3) $3(4-\pi)$(4) $6(4-\pi)$Correct Option: 1 Solution: $-4$ (Area of triangle OPQ) $=6 \pi-4\left(\frac{1}{2} \times 2 \times 3\right)$ $=6 \pi-12=6(\pi-2)$ sq. units...
Read More →A hemisphere of lead of radius 9 cm is cast into a right circular cone of height 72 cm.
Question: A hemisphere of lead of radius 9 cm is cast into a right circular cone of height 72 cm. Find the radius of the base of the cone. Solution: Radius of the hemisphere = 9 cmHeight of the right circular cone = 72 cmSuppose that the radius of the base of the cone isrcm.Volume of the hemisphere = volume of the cone $\Rightarrow \frac{2}{3} \pi \times 9^{3}=\frac{1}{3} \pi \times r^{2} \times 72$ $\Rightarrow r^{2}=\frac{2 \times 9 \times 9 \times 9}{72}=\frac{81}{4}$ $\Rightarrow r=\frac{9}{...
Read More →A hollow spherical shell is made of a metal of density 4.5 g per cm3.
Question: A hollow spherical shell is made of a metal of density 4.5 g per cm3. If its internal and external radii are 8 cm and 9 cm respectively, find the weight of the shell. Solution: Internal radius of the hollow spherical shell,r= 8 cmExternal radius of the hollow spherical shell,R= 9 cm Volume of the shell $=\frac{4}{3} \pi\left(R^{3}-r^{3}\right)$ $=\frac{4}{3} \pi\left(9^{3}-8^{3}\right)$ $=\frac{4}{3} \times \frac{22}{7} \times(729-512)$ $=\frac{4 \times 22 \times 217}{21}$ $=\frac{88 \...
Read More →Let A_{1} be the area of the region bounded by the curves
Question: Let $A_{1}$ be the area of the region bounded by the curves $y=\sin x, y=\cos x$ and $y$-axis in the first quadrant. Also, let $A_{2}$ be the area of the region bounded by the curves $y=\sin x, y=\cos x, x$-axis and $x=\frac{\pi}{2}$ in the first quadrant. Then, (1) $\mathrm{A}_{1}=\mathrm{A}_{2}$ and $\mathrm{A}_{1}+\mathrm{A}_{2}=\sqrt{2}$(2) $\mathrm{A}_{1}: \mathrm{A}_{2}=1: 2$ and $\mathrm{A}_{1}+\mathrm{A}_{2}=1$(3) $2 \mathrm{~A}_{1}=\mathrm{A}_{2}$ and $\mathrm{A}_{1}+\mathrm{A...
Read More →The outer diameter of a spherical shell is 12 cm and its inner diameter is 8 cm.
Question: The outer diameter of a spherical shell is 12 cm and its inner diameter is 8 cm. Find the volume of metal contained in the shell. Also, find its outer surface area. Solution: Outer radius of the spherical shell = 6 cmInner radius of the spherical shell = 4 cm Volume of metal contained in the shell $=\frac{4}{3} \times \frac{22}{7}\left(6^{3}-4^{3}\right)$ $=\frac{88}{21} \times(216-64)$ $=\frac{88}{21} \times 152$ $=636.95 \mathrm{~cm}^{3}$ $\therefore$ Outer surface area $=4 \times \f...
Read More →In the line spectra of hydrogen atom,
Question: In the line spectra of hydrogen atom, difference between the largest and the shortest wavelengths of the Lyman series is $304 A. The corresponding difference for the Paschan series in A is :______ Solution: (10553.14) From Bohr's formula for hydrogen atom, $\frac{1}{\lambda}=R\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)$ $R=1.097 \times 10^{7} \mathrm{~m}^{-1}$ For Lyman series : $\frac{1}{\lambda_{\min .}}=R(1)=R \quad \because n_{2}=\infty$ and $n_{1}=1$ $\frac{1}{\lambda_{\m...
Read More →A cylindrical bucket with base radius 15 cm is filled with water up to a height of 20 cm.
Question: A cylindrical bucket with base radius 15 cm is filled with water up to a height of 20 cm. A heavy iron spherical ball of radius 9 cm is dropped into the bucket to submerge completely in the water. Find the increase in the level of water. Solution: Lethcm be the increase in the level of water.Radius of the cylindrical bucket = 15 cmHeight up to which water is being filled = 20 cmRadius of the spherical ball = 9 cmNow, volume of the sphere = increased in volume of the cylinder $\Rightarr...
Read More →The area bounded by the lines
Question: The area bounded by the lines $y=|| x-1|-2|$ is Note: NTA has dropped this question in the final official answer key. Solution:...
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