An electron mass (m) with initial velocity
Question: An electron (mass $\mathrm{m}$ ) with initial velocity $\vec{v}=v_{0} \hat{i}+v_{0} \hat{j}$ is in an electric field $\vec{E}=-E_{0} \hat{k}$. If $\lambda_{0}$ is initial de-Broglie wavelength of electron, its de-Broglie wave length at time $t$ is given by:(1) $\frac{\lambda_{0} \sqrt{2}}{\sqrt{1+\frac{e^{2} E^{2} t^{2}}{m^{2} v_{0}^{2}}}}$(2) $\frac{\lambda_{0}}{\sqrt{1+\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$(3) $\frac{\lambda_{0}}{\sqrt{1+\frac{e^{2} E^{2} t^{2}}{2 m^{2} v_{...
Read More →In a shower, 5 cm of rain falls. What is the volume of water that falls on 2 hectares of ground?
Question: In a shower, 5 cm of rain falls. What is the volume of water that falls on 2 hectares of ground?(a) 500 m3(b) 750 m3(c) 800 m3(d) 1000 m3 Solution: (d) $1000 \mathrm{~m}^{3}$ Area of the land $=2 \mathrm{sq}$ hec $=2000 \mathrm{sq} \mathrm{m}$ Amount of rainfall $=5 \mathrm{~cm}=0.05 \mathrm{~m}$ $\therefore$ Volume of the water $=$ area of the land $x$ amount of rainfall $=2000 \times 0.05$ $=1000 \mathrm{~m}^{3}$...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: , 4 Solution:...
Read More →Three cubes of metal with edges 3 cm, 4 cm and 5 cm respectively are melted to form a single cube.
Question: Three cubes of metal with edges 3 cm, 4 cm and 5 cm respectively are melted to form a single cube. The lateral surface area of the new cube formed is(a) 72 cm2(b) 144 cm2(c) 128 cm2(d) 256 cm2 Solution: (b) $144 \mathrm{~cm}^{2}$ Volume of the new cube formed = total volume of the three cubesSuppose thatacm is the edge of the new cube, then $a^{3}=3^{3}+4^{3}+5^{3}$ $=27+64+125$ $=216$ $=6 \mathrm{~cm}$ $\therefore$ Lateral surface area of the new cube $=4 a^{2}=4 \times 6 \times 6=144...
Read More →If each edge of a cube is increased by 50%,
Question: If each edge of a cube is increased by 50%, then the percentage increase in its surface area is(a) 50%(b) 75%(c) 100%(d) 12% Solution: Letabe the edge of the cube. Then the surface area is $6 a^{2}=S$ (say) Now, increased edge $=\left(a+\frac{50}{100} a\right)=\frac{150}{100} a=\frac{3}{2} a$ Then, new surface area $=6\left(\frac{3}{2} a\right)^{2}=6 \times \frac{9}{4} a^{2}=\frac{9}{4} S$ In crease in surface area $=\frac{9}{4} S-S=\frac{5}{4} S$ $\therefore$ Percentage increase in su...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: , 3 Solution: Reaction involved:...
Read More →The area (in sq. units) bounded by the parabola
Question: The area (in sq. units) bounded by the parabola $y=x^{2}-1$, the tangent at the point $(2,3)$ to it and the $y$-axis is:(1) $\frac{8}{3}$(2) $\frac{32}{3}$(3) $\frac{56}{3}$(4) $\frac{14}{3}$Correct Option: 1 Solution: $\because \quad$ Curve is given as : $y=x^{2}-1$ $\Rightarrow \quad \frac{d y}{d x}=2 x$ $\Rightarrow \quad\left(\frac{d y}{d x}\right)_{(2,3)}=4$ equation of tangent at $(2,3)$ $(y-3)=4(x-2)$ $\Rightarrow y=4 x-5$ but $x=0$ $\Rightarrow y=-5$ Here the curve cuts $\mathr...
Read More →If the length of diagonal of a cube is
Question: If the length of diagonal of a cube is $8 \sqrt{3} \mathrm{~cm}$, then its surface area is (a) $192 \mathrm{~cm}^{2}$ (b) $384 \mathrm{~cm}^{2}$ (c) $512 \mathrm{~cm}^{2}$ (d) $768 \mathrm{~cm}^{2}$ Solution: (b) $384 \mathrm{~cm}^{2}$ We have: $\sqrt{3} a=8 \sqrt{3}$ $\Rightarrow a=8 \mathrm{~cm}$ $\therefore$ Surface area of the cube $=6 a^{2}=6 \times 8 \times 8=384 \mathrm{~cm}^{2}$...
Read More →The length of the longest rod that can fit in a cubical vessel of side 10 cm, is
Question: The length of the longest rod that can fit in a cubical vessel of side 10 cm, is(a) 10 cm(b) 20 cm (c) $10 \sqrt{2} \mathrm{~cm}$ (d) $10 \sqrt{3} \mathrm{~cm}$ Solution: (d) $10 \sqrt{3} \mathrm{~cm}$ Length of the longest rod = body diagonal of the vessel $=\sqrt{3} a$ $=\sqrt{3} \times 10$ $=10 \sqrt{3} \mathrm{~cm}$...
Read More →If the area (in sq. units) bounded by the parabola
Question: If the area (in sq. units) bounded by the parabola $y^{2}=4 \lambda x$ and the line $y=\lambda x, \lambda0$, is $\frac{1}{9}$, then $\lambda$ is equal to :(1) $2 \sqrt{6}$(2) 48(3) 24(4) $4 \sqrt{3}$Correct Option: , 3 Solution: Given parabola $y^{2}=4 \lambda x$ and the line $y=\lambda x$ Putting $y=\lambda$ in $v^{2}=4 \lambda x$, we get $x=0.4$ $\therefore$ required area $=\int_{0}^{\frac{4}{\lambda}}(2 \sqrt{\lambda x}-\lambda x) d x$ $=\frac{2 \sqrt{\lambda} \cdot x^{3 / 2}}{3 / 2...
Read More →The graph which depicts the results of Rutherford gold foil
Question: The graph which depicts the results of Rutherford gold foil experiment with $\alpha$-particles is: $\theta$ : Scattering angle $Y$ : Number of scattered $\alpha$-particles detected (Plots are schematic and not to scale)(1) (2) (3) (4) Correct Option: , 4 Solution: (4 )As per Rutherford $\alpha$-particle scattering experiment, number of $\alpha$-particles scattered $(N)$ related with deflection angle $\theta$ as $N \propto \frac{1}{\sin ^{4} \frac{\theta}{2}}$ i.e. $Y \propto \frac{1}{\...
Read More →The volume of a cube is 512 cm3.
Question: The volume of a cube is 512 cm3. Its total surface area is(a) 256 cm2(b) 384 cm2(c) 512 cm2(d) 64 cm2 Solution: (b) $384 \mathrm{~cm}^{2}$ Suppose thatacm is the edge of the cube.We have: $a^{3}=512$ $\Rightarrow a=\sqrt[3]{512}=8 \mathrm{~cm}$ $\therefore$ Total surface area of cube $=6 a^{2} \mathrm{~cm}^{2}$ $=6 \times 8 \times 8 \mathrm{~cm}^{2}=384 \mathrm{~cm}^{2}$...
Read More →If the area (in sq. units) of the region
Question: If the area (in sq. units) of the region $\left\{(x, y): y^{2} \leq 4 x\right.$, $x+y \leq 1, x \geq 0, y \geq 0\}$ is $a \sqrt{2}+b$, then $a-b$ is equal to:(1) $\frac{10}{3}$(2) 6(3) $\frac{8}{3}$(4) $-\frac{2}{3}$Correct Option: , 2 Solution: Consider $y^{2}=4 x$ and $x+y=1$ Substituting $x=1-y$ in the equation of parabola, $y^{2}=4(1-y) \Rightarrow y^{2}+4 y-4=0$ $\Rightarrow(y+2)^{2}=8 \Rightarrow y+2=\pm 2 \sqrt{2}$ Hence, required area $=\int_{0}^{3-2 \sqrt{2}} 2 \sqrt{x} d x+\f...
Read More →The total surface area of a cube is 96 cm2.
Question: The total surface area of a cube is 96 cm2. The volume of the cube is(a) 8 cm3(b) 27 cm3(c) 64 cm3(d) 512 cm3 Solution: (c) $64 \mathrm{~cm}^{3}$ Letacm be the edge of the cube.We have: $6 a^{2}=96$ $\Rightarrow a^{2}=16$ $\Rightarrow a=4 \mathrm{~cm}$ $\therefore$ Volume of the cube $=a^{3} \mathrm{~cm}^{3}=4^{3} \mathrm{~cm}^{3}=64 \mathrm{~cm}^{3}$...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: 1 Solution: For the given reaction condition, the major product is:...
Read More →The lateral surface area of a cube is 256 m2. The volume of the cube is
Question: The lateral surface area of a cube is 256 m2. The volume of the cube is(a) 64 m3(b) 216 m3(c) 256 m3(d) 512 m3 Solution: (d) $512 \mathrm{~m}^{3}$ Suppose thatam be the edge of the cube.We have: $4 a^{2}=256$ $\Rightarrow a^{2}=\frac{256}{4}=64$ $\Rightarrow a=8 \mathrm{~m}$ $\therefore$ Volume of the cube $=a^{3} \mathrm{~m}^{3}=8^{3} \mathrm{~m}^{3}=512 \mathrm{~m}^{3}$...
Read More →A river 1.5 m deep and 30 m wide is flowing at the rate of 3 km per hour.
Question: A river 1.5 m deep and 30 m wide is flowing at the rate of 3 km per hour. The volume of water that runs into the sea per minute is(a) 2000 m3(b) 2250 m3(c) 2500 m3(d) 2750 m3 Solution: (b) 2250 m3 Length of the river = 1.5 mBreadth of the river = 30 mDepth of the river = 3 km = 3000 m Now, volume of water that runs into the sea $=1.5 \times 30 \times 3000 \mathrm{~m}^{3}$ $=135000 \mathrm{~m}^{3}$ $\therefore$ Volume of water that runs into the sea per minute $=\frac{135000}{60}=2250 \...
Read More →The area (in sq. units) of the region bounded by the curves
Question: The area (in sq. units) of the region bounded by the curves $y=2^{x}$ and $y=|x+1|$, in the first quadrant is :(1) $\log _{e} 2+\frac{3}{2}$(2) $\frac{3}{2}$(3) $\frac{1}{2}$(4) $\frac{3}{2}-\frac{1}{\log _{e} 2}$Correct Option: 4, Solution: Area $=\int_{0}^{1}\left((x+1)-2^{x}\right) d x$ $\left(\because\right.$ Area $\left.=\int y d x\right)$ $=\left[\frac{x^{2}}{2}+x-\frac{2^{x}}{\ln 2}\right]_{0}^{1}=\left(\frac{1}{2}+1-\frac{2}{\ln 2}\right)-\left(\frac{-1}{\ln 2}\right)=\frac{3}{...
Read More →The increasing order of the reactivity of the following
Question: The increasing order of the reactivity of the following compounds towards electrophilic aromatic substitution reactions is : $\mathrm{II}\mathrm{I}\mathrm{III}$$\mathrm{III}\mathrm{II}\mathrm{I}$$\mathrm{III}\mathrm{I}\mathrm{II}$$\mathrm{I}\mathrm{III}\mathrm{II}$Correct Option: Solution: $\mathrm{CH}_{3}$ group when bonded to benzene increases the electron density of benzene ring due to $+I$ and hyper conjugation effects. $-\mathrm{Cl}$ group decreases the electron density of benzene...
Read More →How many persons can be accommodated in a dining hall of dimensions (20 m × 15 m × 4.5 m),
Question: How many persons can be accommodated in a dining hall of dimensions (20 m 15 m 4.5 m), assuming that each person requires 5 m3of air?(a) 250(b) 270(c) 320(d) 300 Solution: (b) 270 Number of persons $=\frac{\text { volume of the hall }}{\text { volume of air required by } 1 \text { person }}$ $=\frac{20 \times 15 \times 4.5}{5}$ $=20 \times 3 \times 4.5$ $=270$ 270 persons can be accommodated....
Read More →The region represented by
Question: The region represented by $|x-y| \leq 2$ and $|x+y| \leq 2$ is bounded by a :(1) square of side length $2 \sqrt{2}$ units(2) rhombus of side length 2 units(c) square of area $16 \mathrm{sq}$. units(4) rhombus of area $8 \sqrt{2}$ sq. unitsCorrect Option: 1 Solution: Let, $\mathrm{C}_{1}:|y-x| \leq 2$ $\mathrm{C}_{2}:|y+x| \leq 2$ By the diagram, region is square Now, length of side $=\sqrt{2^{2}+2^{2}}=2 \sqrt{2}$...
Read More →How many bricks will be required to construct a wall 8 m long, 6 m high and 22.5 cm
Question: How many bricks will be required to construct a wall 8 m long, 6 m high and 22.5 cm thick if each brick measures (25 cm 11.25 cm 6 cm)?(a) 4800(b) 5600(c) 6400(d) 5200 Solution: (c) 6400Length of the wall = 8 m = 800 cmBreadth of the wall = 6 m = 600 cmHeight of the wall = 22.5 cmLength of the brick = 25 cmBreadth of the brick = 11.25 cmHeight of the brick = 6 cm $\therefore$ Number of bricks required $=\frac{\text { volume of the wall }}{\text { volume of } 1 \text { brick }}$ $=\frac...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $\mathrm{A}=\left\{(x, \mathrm{y}): \frac{y^{2}}{2} \leq x \leq y+4\right\}$ is:(1) $\frac{53}{3}$(2) 30(3) 16(4) 18Correct Option: , 4 Solution: Given region, $A=\left\{(x, y): \frac{y^{2}}{2} \leq x \leq y+4\right\}$ Hence, area $=\int_{-2}^{4} x d y=\int_{-2}^{4}\left(y+4-\frac{y^{2}}{2}\right) d y$ $=\left[\frac{y^{2}}{2}+4 y-\frac{y^{3}}{6}\right]_{-2}^{4}=\left(8+16-\frac{64}{6}\right)-\left(2-8+\frac{8}{6}\right)$ $=\left(24-\frac{32}{3}\rig...
Read More →How many planks of dimensions (5 m × 25 cm × 10 cm) can be stored in a pit which is 20 m long,
Question: How many planks of dimensions (5 m 25 cm 10 cm) can be stored in a pit which is 20 m long, 6 m wide and 50 cm deep?(a) 480(b) 450(c) 320(d) 360 Solution: (a) 480Length of the pit = 20 mBreadth of the pit = 6 mHeight of the pit = 50 cm = 0.5 mLength of the plank = 5mBreadth of the plank = 25 cm = 0.25 mHeight of the plank = 10 cm = 0.1 m $\therefore$ Number of planks $=\frac{\text { volume of the pit }}{\text { volume of } 1 \mathrm{plank}}$ $=\frac{20 \times 6 \times 0.5}{5 \times 0.25...
Read More →p-Hydroxybenzophenone upon reaction
Question: p-Hydroxybenzophenone upon reaction with bromine in carbon tetrachloride gives:Correct Option: , 2 Solution:...
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