If n > 2 is a positive integer, then the sum of the series
Question: If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{\mathrm{n}+1} \mathrm{C}_{2}+2\left({ }^{2} \mathrm{C}_{2}+{ }^{3} \mathrm{C}_{2}+{ }^{4} \mathrm{C}_{2}+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{2}\right)$ is :(1) $\frac{\mathrm{n}(\mathrm{n}+1)^{2}(\mathrm{n}+2)}{12}$(2) $\frac{\mathrm{n}(\mathrm{n}-1)(2 \mathrm{n}+1)}{6}$(3) $\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}$(4) $\frac{\mathrm{n}(2 \mathrm{n}+1)(3 \mathrm{n}+1)}{6}$Correct Option: , 3 Solution: ${ }^...
Read More →An excited He+ion emits two photons in succession,
Question: An excited $\mathrm{He}^{+}$ion emits two photons in succession, with wavelengths $108.5 \mathrm{~nm}$ and $30.4 \mathrm{~nm}$, in making a transition to ground state. The quantum number $n$, corresponding to its initial excited state is (for photon of wavelength $\lambda$, energy $\mathrm{E}=\frac{1240 \mathrm{eV}}{\lambda(\text { innm })}$(1) $n=4$(2) $n=5$(3) $n=7$(4) $n=6$Correct Option: , 2 Solution: (2) $E=E_{1}+E_{2}$ $13.6 \frac{z^{2}}{n^{2}}=\frac{1240}{\lambda_{1}}+\frac{1240...
Read More →The curved surface area of one cone is twice that of the other while the slant height of the latter is twice that of the former.
Question: The curved surface area of one cone is twice that of the other while the slant height of the latter is twice that of the former. The ratio of their radii is(a) 2 : 1(b) 4 : 1(c) 8 : 1(d) 1 : 1 Solution: (b) 4 : 1If the slant height of the first cone isl, then the slant height of the second cone will be 2l.Let the radii of the first and second cones berandR,respectively.Then we have: $\pi r l=2 \times(\pi R \times 2 l)$ $\Rightarrow r=4 R$ $\Rightarrow \frac{r}{R}=\frac{4}{1}$...
Read More →When light of wavelength 248 nm falls on
Question: When light of wavelength $248 \mathrm{~nm}$ falls on a metal of threshold energy $3.0 \mathrm{eV}$, the de-Broglie wavelength of emitted electrons is ____________\AA (Round off to the Nearest Integer). $\left[\right.$ Use: $\sqrt{3}=1.73, \mathrm{~h}=6.63 \times 10^{-34} \mathrm{~J}_{\mathrm{S}}$$\mathrm{m}_{\mathrm{e}}=9.1 \times 10^{-31} \mathrm{~kg} ; \mathrm{c}=3.0 \times 10^{8} \mathrm{~ms}^{-1}$ $\left.\mathrm{leV}=1.6 \times 10^{-19} \mathrm{~J}\right]$ Solution: (9) Energy $=\f...
Read More →If the height of a cone is doubled, then its volume is increased by
Question: If the height of a cone is doubled, then its volume is increased by(a) 100%(b) 200%(c) 300%(d) 400% Solution: (a) 100 %Suppose that height of the cone becomes 2hand let its radius ber. Then new volume of the cone $=\frac{1}{3} \pi r^{2}(2 h)=\frac{2}{3} \pi r^{2} h=2 \times$ volume of the cone Increase in volume $=\frac{2}{3} \pi r^{2} h-\frac{1}{3} \pi r^{2} h=\frac{1}{3} \pi r^{2} h$ $\therefore$ Percentage increase $=\frac{\text { increase in volume }}{\text { initial volume }} \tim...
Read More →The value of
Question: The value of $-{ }^{15} \mathrm{C}_{1}+2 \cdot{ }^{15} \mathrm{C}_{2}-3{ }^{15} \mathrm{C}_{3}+\ldots . .-15 \cdot{ }^{15} \mathrm{C}_{15}+{ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+{ }^{14} \mathrm{C}_{5}+\ldots .+{ }^{14} \mathrm{C}_{11}$ is:(1) $2^{14}$(2) $2^{13}-13$(3) $2^{16}-1$(4) $2^{13}-14$Correct Option: , 4 Solution: $\mathrm{S}_{1}=-{ }^{15} \mathrm{C}_{1}+2 \cdot{ }^{15} \mathrm{C}_{2}-\ldots \ldots \ldots . . .15^{15} \mathrm{C}_{15}$ $=\sum_{r=1}^{15}(-1)^{r} \cdot ...
Read More →If the volumes of two cones be in the ratio 1 : 4 and the radii of their bases be in the ratio 4 : 5
Question: If the volumes of two cones be in the ratio 1 : 4 and the radii of their bases be in the ratio 4 : 5, then the ratio of their heights is(a) 1 : 5(b) 5 : 4(c) 25 : 16(d) 25 : 64 Solution: (d) 25 : 64Suppose that the radii of the cones are 4rand 5rand their heights arehandH, respectively .It is given that the ratio of the volumes of the two cones is 1:4Then we have: $\frac{\frac{1}{3} \pi(4 r)^{2} h}{\frac{1}{3} \pi(5 r)^{2} H}=\frac{1}{4}$ $\Rightarrow \frac{16 r^{2} h}{25 r^{2} H}=\fra...
Read More →The stopping potential
Question: The stopping potential $V_{0}$ (in volt) as a function of frequency $(v)$ for a sodium emitter, is shown in the figure. The work function of sodium, from the data plotted in the figure, will be : (Given : Planck's constant $(h)=6.63 \times 10^{-34} \mathrm{Js}$, electron charge $e=1.6 \times 10^{-19} \mathrm{C}$ ) (1) $1.82 \mathrm{eV}$(2) $1.66 \mathrm{eV}$(3) $1.95 \mathrm{eV}$(4) $2.12 \mathrm{eV}$Correct Option: , 2 Solution: (2) $f_{0}=4 \times 10^{14} \mathrm{~Hz}$ $W_{0}=h f_{0}...
Read More →Let P be a plane containing the line
Question: Let $P$ be a plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7} .$ If the point $(1,-1, \alpha)$ lies on the plane $\mathrm{P}$, then the value of $|5 \alpha|$ is equal to______. Solution: Equation of plane is $\left|\begin{array}{ccc}x-1 y+6 z+5 \\ 3 4 2 \\ 4 -3 7\end{array}\right|=0$ $, \alpha)$ lies on it so Now $(1,-1,$, $\left|\begin{array}{ccc}0 5 \alpha+5 \\ 3 4 2 \\ 4 -3 7\end{array}\right|=...
Read More →The volume of a right circular cone of height 24 cm is 1232 cm3.
Question: The volume of a right circular cone of height 24 cm is 1232 cm3. Its curved surface area is(a) 1254 cm2(b) 704 cm2(c) 550 cm2(d) 462 cm2 Solution: (c) 550 cm2Letrcm be the radius of the cone.Volume of the right circular cone = 1232 cm3Then we have: $\frac{1}{3} \times \frac{22}{7} \times r^{2} \times 24=1232$ $\Rightarrow r^{2}=\frac{1232 \times 21}{22 \times 24}$ $\Rightarrow r^{2}=49$ $\Rightarrow r=7 \mathrm{~cm}$ $\therefore$ Curved surface area of the cone $=\pi r l$ $=\frac{22}{7...
Read More →Let ${ }^{n} C_{r}$ denote the binomial coefficient of
Question: Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$. If $\sum_{k=0}^{10}\left(2^{2}+3 k\right)^{n} C_{k}=\alpha \cdot 3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R$, then $\alpha+\beta$ is equal to Solution: Instead of ${ }^{n} C_{k}$ it must be ${ }^{10} C_{k}$ i.e. $\sum_{k=0}^{10}\left(2^{2}+3 k\right)^{10} C_{k}=\alpha .3^{10}+\beta .2^{10}$ $\mathrm{LHS}=4 \sum_{\mathrm{k}-0}^{10}{ }^{10} \mathrm{C}_{\mathrm{k}}+3 \sum_{\mathrm{k}=0}^{10...
Read More →The height of a cone is 21 cm and its slant height is 28 cm.
Question: The height of a cone is 21 cm and its slant height is 28 cm. The volume of the cone is(a) 7356 cm3(b) 7546 cm3(c) 7506 cm3(d) 7564 cm3 Solution: (b) $7546 \mathrm{~cm}^{3}$ Radius of the cone, $r=\sqrt{l^{2}-h^{2}}$ $=\sqrt{28^{2}-21^{2}}$ $=\sqrt{784-441}$ $=\sqrt{343} \mathrm{~cm}$ $\therefore$ Volume of the cone $=\frac{1}{3} \pi r^{2} h$ $=\frac{1}{3} \times \frac{22}{7} \times 343 \times 21$ $=22 \times 343$ $=7546 \mathrm{~cm}^{3}$...
Read More →The term independent of
Question: The term independent of $\mathrm{x}$ in the expansion of $\left\lceil\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1$, is equal to___________. Solution: $\left(\left(x^{1 / 3}+1\right)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)^{10}$ $\left(x^{1 / 3}-x^{-1 / 2}\right)^{10}$ $T_{r+1}=10 C_{r}\left(x^{1 / 3}\right)^{10-r}\left(-x^{-1 / 2}\right)^{r}$ $\frac{10-r}{3}-\frac{r}{2}=0 \Rightarrow 20-2 r-3 r=0$ $\Rightarrow r=4$ $T_{5}={ }^{10} C_{4}=\frac{...
Read More →The volume of a cone is 1570 cm3 and its height is 15 cm.
Question: The volume of a cone is 1570 cm3and its height is 15 cm. What is the radius of the cone?(a) 10 cm(b) 9 cm(c) 12 cm(d) 8.5 cm Solution: (a) 10 cmLetrcm be the radius of the cone.Volume = 1570 cm3 Then $\frac{1}{3} \times 3.14 \times r^{2} \times 15=1570$$\Rightarrow r^{2}=\frac{1570}{3.14 \times 5}=100$ $\Rightarrow r=10 \mathrm{~cm}$...
Read More →In Li++, electron in first {Bohr} orbit is excited to a level by
Question: In $\mathrm{Li}^{++}$, electron in first $\mathrm{Bohr}$ orbit is excited to a level by a radiation of wavelength $\lambda$. When the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of $\lambda$ ? (Given : $h=6.63 \times 10^{-34} \mathrm{Js} ; c=3 \times 10^{8} \mathrm{~ms}^{-1}$ )(1) $11.4 \mathrm{~nm}$(2) $9.4 \mathrm{~nm}$(3) $12.3 \mathrm{~nm}$(4) $10.8 \mathrm{~nm}$Correct...
Read More →Let
Question: Let $\left(1+x+2 x^{2}\right)^{20}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{40} x^{40}$ then $\mathrm{a}_{1}+\mathrm{a}_{3}+\mathrm{a}_{5}+\ldots+\mathrm{a}_{37}$ is equal to (1) $2^{20}\left(2^{20}-21\right)$(2) $2^{19}\left(2^{20}-21\right)$(3) $2^{19}\left(2^{20}+21\right)$(4) $2^{20}\left(2^{20}+21\right)$Correct Option: , 2 Solution: $\left(1+x+2 x^{2}\right)^{20}=a_{0}+a_{1} x+\ldots+a_{40} x^{40}$ put $\mathrm{x}=1,-1$ $\Rightarrow a_{0}+a_{1}+a_{2}+\ldots+a_{40}=2^{20}$ $\Rightarrow...
Read More →Consider the following reactions:
Question: Consider the following reactions: Correct Option: , 4 Solution: Vinyl halides and aryl halides are unreactive towards Friedel Craft's reaction. Therefore reactions $(\mathrm{A})$ and $(\mathrm{C})$ are not possible....
Read More →How much cloth 2.5 m wide will be required to make a conical tent having base radius 7 m and height 24 m?
Question: How much cloth 2.5 m wide will be required to make a conical tent having base radius 7 m and height 24 m?(a) 120 m(b) 180 m(c) 220 m(d) 550 m Solution: (c) 220 mLet the length of the required cloth beLm and its breadth beBm.Here,B= 2.5 mRadius of the conical tent =7 mHeight of the tent = 24 mArea of cloth required = curved surface area of the conical tent $\Rightarrow L \times B=\pi \mathrm{rl}$ $\Rightarrow \mathrm{L} \times 2.5=\frac{22}{7} \times 7 \times \sqrt{7^{2}+24^{2}}$ $\Righ...
Read More →The volume of a right circular cone of height 12 cm and base radius 6 cm, is
Question: The volume of a right circular cone of height 12 cm and base radius 6 cm, is (a) $(12 \pi) \mathrm{cm}^{3}$ (b) $(36 \pi) \mathrm{cm}^{3}$ (c) $(72 \pi) \mathrm{cm}^{3}$ (d) $(144 \pi) \mathrm{cm}^{3}$ Solution: (d) $(144 \pi) \mathrm{cm}^{3}$ Volume of the cone $=\frac{1}{3} \pi r^{2} h$ $=\frac{1}{3} \pi \times 6^{2} \times 12$ $=\pi \times 36 \times 4$ $=144 \pi \mathrm{cm}^{3}$...
Read More →Let the coefficients of third, fourth and fifth terms in the expansion
Question: Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0$, be in the ratio 12: $8: 3$. Then the term independent of $x$ in the expansion, is equal to Solution: $\mathrm{T}_{\mathrm{r}+1}={ }^{n} \mathrm{C}_{\mathrm{r}}(\mathrm{x})^{\mathrm{n}-\mathrm{r}}\left(\frac{\mathrm{a}}{\mathrm{x}^{2}}\right)^{\mathrm{r}}$ $={ }^{n} \mathrm{C}_{\mathrm{r}} \mathrm{a}^{\mathrm{r}} \mathrm{x}^{\mathrm{n}-3 \mathrm{r}}$ ${ }^{\mathrm{n}...
Read More →The height of a cone is 24 cm and the diameter of its base is 14 cm.
Question: The height of a cone is 24 cm and the diameter of its base is 14 cm. The curved surface area of the cone is(a) 528 cm2(b) 550 cm2(c) 616 cm2(d) 704 cm2 Solution: (b) $550 \mathrm{~cm}^{2}$ $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{7^{2}+24^{2}}$ $=\sqrt{49+576}$ $=\sqrt{625}$ $=25 \mathrm{~cm}$ $\therefore$ Curved surface area of the cone $=\pi r l$ $=\frac{22}{7} \times 7 \times 25$ $=550 \mathrm{~cm}^{2}$...
Read More →The value of
Question: The value of $\sum_{r=0}^{6}\left({ }^{6} \mathrm{C}_{r}{ }^{-6} \mathrm{C}_{6-\mathrm{r}}\right)$ is equal to :(1) 1124(2) 1324(3) 1024(4) 924Correct Option: , 4 Solution: $\sum_{r=0}^{6}{ }^{6} \mathrm{C}_{\mathrm{r}} \cdot{ }^{6} \mathrm{C}_{6-\mathrm{r}}$ Now, $(1+x)^{6}(1+x)^{6}$ $=\left({ }^{6} C_{0}+{ }^{6} C_{1} x+{ }^{6} C_{2} x^{2}+\ldots \ldots+{ }^{6} C_{6} x^{6}\right)$ $\left({ }^{6} C_{0}+{ }^{6} C_{1} x+{ }^{6} C_{2} x^{2}+\ldots \ldots+{ }^{6} C_{6} x^{6}\right)$ Compa...
Read More →A 2 m W laser operates at a wavelength of 500 nm.
Question: A $2 \mathrm{~mW}$ laser operates at a wavelength of $500 \mathrm{~nm}$. The number of photons that will be emitted per second is : [Given Planck's constant $\mathrm{h}=6.6 \times 10^{-34} \mathrm{Js}$, speed of light $c=3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}$ ](1) $5 \times 10^{15}$(2) $1.5 \times 10^{16}$(3) $2 \times 10^{16}$(4) $1 \times 10^{16}$Correct Option: 1, Solution: (1) Energy of photon (E) is given by $E=\frac{h c}{\lambda}$ Number of photons of wavelength $\lambda$ em...
Read More →The lateral surface area of a cylinder is
Question: The lateral surface area of a cylinder is (a) $\pi r^{2} h$ (b) $\pi r h$ (c) $2 \pi \mathrm{rh}$ (d) $2 \pi r^{2}$ Solution: (c) $2 \pi r h$ The lateral surface area of a cylinder is equal to its curved surface area. $\therefore$ Lateral surface area of a cylinder $=2 \pi r h$...
Read More →if is divided by 17 , then the remainder is_____
Question: If $(2021)^{3762}$ is divided by 17 , then the remainder is_______. Solution: $(2023-2)^{3762}=2023 \mathrm{k}_{1}+2^{3762}$ $=17 \mathrm{k}_{2}+2^{3762}($ as $2023=17 \times 17 \times 9)$ $=17 \mathrm{k}_{2}+4 \times 16^{940}$ $=17 \mathrm{k}_{2}+4 \times(17-1)^{940}$ $=17 \mathrm{k}_{2}+4\left(17 \mathrm{k}_{3}+1\right)$ $=17 \mathrm{k}+4 \Rightarrow$ remainder $=4$...
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