Solve the following
Question: If $\mathrm{C}_{\mathrm{r}} \equiv{ }^{25} \mathrm{C}_{\mathrm{r}}$ and $\mathrm{C}_{0}+5 \cdot \mathrm{C}_{1}+9 \cdot \mathrm{C}_{2}+\ldots+(101) \cdot \mathrm{C}_{25}=2^{25} \cdot k$ then $k$ is equal to________. Solution: $\sum_{r=0}^{25}(4 r+1){ }^{25} C_{r}=4 \sum_{r=0}^{25} r \cdot{ }^{25} C_{r}+\sum_{r=0}^{25}{ }^{25} C_{r}$ $=4 \sum_{r=1}^{25} r \times \frac{25}{r}{ }^{24} C_{r-1}+2^{25}=100 \sum_{r=1}^{25}{ }^{24} C_{r-1}+2^{25}$ $=100.2^{24}+2^{25}=2^{25}(50+1)=51.2^{25}$ Hen...
Read More →If the sum of the roots of the equation
Question: If the sum of the roots of the equation $k x^{2}+2 x+3 x=0$ is equal to their product, then the value of $k$ is (a) $\frac{1}{3}$ (b) $\frac{-1}{3}$ (C) $\frac{2}{3}$ (d) $\frac{-2}{3}$ Solution: (d) $\frac{-2}{3}$ Given: $k x^{2}+2 x+3 k=0$ Sum of the roots $=$ Product of the roots $\Rightarrow \frac{-2}{k}=\frac{3 k}{k}$ $\Rightarrow 3 k=-2$ $\Rightarrow k=\frac{-2}{3}$...
Read More →Match List-I with List-II
Question: Match List-I with List-II Choose the correct answer from the options given below:$(\mathrm{a})-(\mathrm{i}),(\mathrm{b})-(\mathrm{ii}),(\mathrm{c})-(\mathrm{v}),(\mathrm{d})-(\mathrm{iii})$$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{v}),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{iv})$$(\mathrm{a})-(\mathrm{i}),(\mathrm{b})-(\mathrm{ii}),(\mathrm{c})-(\mathrm{iii}),(\mathrm{d})-(\mathrm{i} v)$$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{v}),(\...
Read More →If the number of five digit numbers with
Question: If the number of five digit numbers with distinct digits and 2 at the $10^{\text {th }}$ place is $336 \mathrm{k}$, then $\mathrm{k}$ is equal to:(1) 4(2) 6(3) 7(4) 8Correct Option: , 4 Solution: Number of five digit numbers with 2 at $10^{\text {th }}$ place $=8 \times 8 \times 7 \times 6=2688$ $=8 \times 8 \times 7 \times 6=2688$ $\because \quad$ It is given that, number of five digit number with 2 at $10^{\text {th }}$ place $=336 k$ $\therefore \quad 336 k=2688 \Rightarrow k=8$...
Read More →A bullet of mass 0.1 kg is fired on a wooden block to pierce through it,
Question: A bullet of mass $0.1 \mathrm{~kg}$ is fired on a wooden block to pierce through it, but it stops after moving a distance of $50 \mathrm{~cm}$ into it. If the velocity of bullet before hitting the wood is $10 \mathrm{~m} / \mathrm{s}$ and it slows down with uniform deceleration, then the magnitude of effective retarding force on the bullet is 'x' N. The value of ' $x$ ' to the nearest integer is______ Solution: (10) $\mathrm{v}^{2}=\mathrm{u}^{2}+2 \mathrm{as}$ $0=(10)^{2}+2(-\mathrm{a...
Read More →Match List-I with List-II.
Question: Match List-I with List-II. Choose the most appropriate answer from the option given below :$(a)-(i i),(b)-(i v),(c)-(i),(d)-(i i i)$$(a)-(i i),(b)-(i),(c)-(i v),(d)-(i i i)$$(a)-(i),(b)-(i i i),(c)-(i v),(d)-(i i)$$(a)-(i i i),(b)-(i),(c)-(i i),(d)-(i v)$Correct Option: , 2 Solution: Fact...
Read More →The number of 4 letter words
Question: The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is_______. Solution: EXAMINATION $2 \mathrm{~N}, 2 \mathrm{~A}, 2 \mathrm{I}, \mathrm{E}, \mathrm{X}, \mathrm{M}, \mathrm{T}, \mathrm{O}$ Case I : If all are different, then ${ }^{8} p_{4}=\frac{8 !}{4 !}=8.7 .6 .5=1680$ Case II : If two are same and two are different, then ${ }^{3} C_{1} \cdot{ }^{7} C_{2} \cdot \frac{4 !}{2 !}=3.21 .12=756$ Case III : If two are...
Read More →If one root of
Question: If one root of $5 x^{2}+13 x+k=0$ be the reciprocal of the other root, then the value of $k$ is (a) 0(b) 1(c) 2(d) 5 Solution: (d) 5 Let the roots of the equation $\left(5 x^{2}+13 x+k=0\right)$ be $\alpha$ and $\frac{1}{\alpha}$. $\therefore$ Product of the roots $=\frac{c}{a}$ $\Rightarrow \alpha \times \frac{1}{\alpha}=\frac{k}{5}$ $\Rightarrow 1=\frac{k}{5}$ $\Rightarrow k=5$...
Read More →A boy of mass 4 kg is standing on a piece of wood having mass
Question: A boy of mass $4 \mathrm{~kg}$ is standing on a piece of wood having mass $5 \mathrm{~kg}$. If the coefficient of friction between the wood and the floor is $0.5$, the maximum force that the boy can exert on the rope so that the piece of wood does not move from its place is N.(Round off to the Nearest Integer) $\left[\right.$ Take $\left.g=10 \mathrm{~ms}^{-2}\right]$ Solution: $\mathrm{N}+\mathrm{T}=90$ $\mathrm{~T}=\mu \mathrm{N}=0.5(90-\mathrm{T})$ $1.5 \mathrm{~T}=45$ $\mathrm{~T}=...
Read More →The method used for the purification of Indium is :
Question: The method used for the purification of Indium is :van Arkel methodvapour phase refiningzone refiningLiquationCorrect Option: , 3 Solution: Fact Ga, In, Si, Ge are refined by zone refining or vaccume refining....
Read More →An urn contains 5 red marbles,
Question: An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is_________. Solution: 0 Red, 1 Red, 2 Red, 3 Red Number of ways of selecting atmost three red balls $={ }^{7} C_{4}+{ }^{5} C_{1} \cdot{ }^{7} C_{3}+{ }^{5} C_{2} \cdot{ }^{7} C_{2}+{ }^{5} C_{3} \cdot{ }^{7} C_{1}$ $=35+175+210+70=490$...
Read More →The major components of German Silver are :
Question: The major components of German Silver are :$\mathrm{Cu}, \mathrm{Zn}$ and $\mathrm{Ag}$Ge, $\mathrm{Cu}$ and $\mathrm{Ag}$$\mathrm{Zn}, \mathrm{Ni}$ and $\mathrm{Ag}$$\mathrm{Cu}, \mathrm{Zn}$ and $\mathrm{Ni}$Correct Option: , 4 Solution: Fact German silver is alloy which does not have silver. $\mathrm{Cu}-50 \% ; \mathrm{Ni}-30 \% ; \mathrm{Zn}-20 \%$...
Read More →A body of mass 1 kg rests on a horizontal floor
Question: A body of mass $1 \mathrm{~kg}$ rests on a horizontal floor with which it has a coefficient of static friction $\frac{1}{\sqrt{3}}$. It is desired to make the body move by applying the minimum possible force $\mathrm{FN}$. The value of $\mathrm{F}$ will be (Round off to the Nearest Integer) $\left[\right.$ Take $\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right]$ Solution: $F \cos \theta=\mu N$ $F \sin \theta+N=m g$ $\Rightarrow F=\frac{\mu m g}{\cos \theta+\mu \sin \theta}$ $F_{\min }=\frac...
Read More →If $a, b$ and $c$ are the greatest values
Question: If $a, b$ and $c$ are the greatest values of ${ }^{19} C_{p},{ }^{20} C_{q}$ and ${ }^{21} C_{r}$ respectively, then:(1) $\frac{a}{11}=\frac{b}{22}=\frac{c}{21}$(2) $\frac{a}{10}=\frac{b}{11}=\frac{c}{21}$(3) $\frac{a}{11}=\frac{b}{22}=\frac{c}{42}$(4) $\frac{a}{10}=\frac{b}{11}=\frac{c}{42}$Correct Option: , 3 Solution: We know ${ }^{n} C_{r}$ is greatest at middle term. So, $a=\left({ }^{19} C_{p}\right)_{\max }={ }^{19} C_{10}={ }^{19} C_{9}$ $b=\left({ }^{20} C_{q}\right)_{\max }={...
Read More →Ellingham diagram is a graphical representation of:
Question: Ellingham diagram is a graphical representation of:$\Delta \mathrm{G}$ vs $\mathrm{T}$$(\Delta G-T \Delta S) v s T$$\Delta \mathrm{H}$ vs $\mathrm{T}$$\Delta G$ vs PCorrect Option: 1 Solution: Ellingham diagram tells us about the spontanity of a reaction with temperature....
Read More →If one root of the equation
Question: If one root of the equation $3 x^{2}-10 x+3=0$ is $\frac{1}{3}$, then the other root is (a) $\frac{-1}{3}$ (b) $\frac{1}{3}$ (c) $-3$ (d) 3 Solution: (d) 3 Given: $3 x^{2}-10 x+3=0$ One root of the equation is $\frac{1}{3}$. Let the other root be $\alpha$. We know that: Product of the roots $=\frac{c}{a}$ $\Rightarrow \frac{1}{3} \times \alpha=\frac{3}{3}$ $\Rightarrow \alpha=3$...
Read More →Match List-I with List-II
Question: Match List-I with List-II Choose the correct answer from the options given below:$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{iii}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{i})$(a) - (i), (b) - (ii), (c) - (iii), (d) - (iv)$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{i} v),(\mathrm{d})-(\mathrm{ii})$$(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{i} v),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{iii})$Correct Option: , 3 Solution:...
Read More →Match List-I with List-II
Question: Match List-I with List-II Choose the correct answer from the options given below:$(\mathrm{a})-(\mathrm{iv}),(\mathrm{b})-(\mathrm{iii}),(\mathrm{c})-(\mathrm{ii}),(\mathrm{d})-(\mathrm{i})$(a) - (i), (b) - (ii), (c) - (iii), (d) - (iv)$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{i} v),(\mathrm{d})-(\mathrm{ii})$$(\mathrm{a})-(\mathrm{ii}),(\mathrm{b})-(\mathrm{i} v),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{iii})$Correct Option: , 3 Solution:...
Read More →Two blocks m=0.5 kg and M=4.5 kg
Question: Two blocks $(\mathrm{m}=0.5 \mathrm{~kg}$ and $\mathrm{M}=4.5 \mathrm{~kg})$ are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is $\frac{3}{7}$. Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is__________$\mathrm{N}$. (Round off to the NearestInteger) [Take g as $9.8 \mathrm{~ms}^{-2}$ ] Solution: (21) $a_{\max }=\mu g=\frac{3}{7} \times 9.8$ $F=(M+m) ...
Read More →The ratio of the sum and product of the roots of the equation
Question: The ratio of the sum and product of the roots of the equation $7 x^{2}-12 x+18=0$ (a) 7 : 12(b) 7 : 18(c) 2 : 3(d) 3 : 2 Solution: (c) 2 : 3 Given: $7 x^{2}-12 x+18=0$ $\therefore \alpha+\beta=\frac{12}{7}$ and $\alpha \beta=\frac{18}{7}$, where $\alpha$ and $\beta$ are the roots of the equation $\therefore$ Ratio of the sum and product of the roots $=\frac{12}{7}: \frac{18}{7}$ $=12: 18$ $=2: 3$...
Read More →The number of ordered pairs
Question: The number of ordered pairs $(r, k)$ for which 6. ${ }^{35} C_{r}=\left(k^{2}-3\right) \cdot{ }^{36} C_{r+1}$, where $k$ is an integer, is:(1) 3(2) 2(3) 6(4) 4Correct Option: , 4 Solution: $\frac{36}{r+1} \times{ }^{35} \mathrm{C}_{r}\left(k^{2}-3\right)={ }^{35} \mathrm{C}_{r} \cdot 6$ $\Rightarrow \quad k^{2}-3=\frac{r+1}{6}$ $\Rightarrow \quad k^{2}=3+\frac{r+1}{6}$ $r$ can be 5,35 for $k \in I$ $r=5, k=\pm 2$ $r=35, k=\pm 3$ Hence, number of ordered pairs $=4$....
Read More →Solve the following
Question: $\mathrm{Al}_{2} \mathrm{O}_{3}$ was leached with alkali to get $\mathrm{X}$. The solution of $\mathrm{X}$ on passing of gas $\mathrm{Y}$ , forms Z. X, Y and $Z$ respectively are :$\mathrm{X}=\mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_{4}\right], \mathrm{Y}=\mathrm{CO}_{2}, \mathrm{Z}=\mathrm{Al}_{2} \mathrm{O}_{3} \cdot \mathrm{xH}_{2} \mathrm{O}$$\mathrm{X}=\mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_{4}\right], \mathrm{Y}=\mathrm{SO}_{2}, \mathrm{Z}=\mathrm{Al}_{2} \mathrm{O}_{3}$$\mat...
Read More →Total number of 6-digit numbers
Question: Total number of 6-digit numbers in which only and all the five digits $1,3,5,7$ and 9 appear, is:(1) $\frac{1}{2}(6 !)$(2) $6 !$(3) $5^{6}$(4) $\frac{5}{2}(6 !)$Correct Option: , 4 Solution: Five digits numbers de $1,3,5,7,9$ For selection of one digit, we have ${ }^{5} C_{1}$ choice. And six digits can be arrange in $\frac{6 !}{2 !}$ ways. Hence, total such numbers $=\frac{5.6 !}{2 !}=\frac{5}{2} \cdot 6 !$...
Read More →If the product of the roots of the equation
Question: If the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$, then the value of $k$ is (a) 2(b) 8(c) 8(d) 12 Solution: (c) 8 It is given that the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$. The equation can be rewritten as : $x^{2}-3 x+(k-10)=0$ Product of the roots of a quadratic equation $=\frac{c}{a}$ $\Rightarrow \frac{c}{a}=-2$ $\Rightarrow \frac{(k-10)}{1}=-2$ $\Rightarrow k=8$...
Read More →The major components in "Gun Metal" are:
Question: The major components in "Gun Metal" are:$\mathrm{Al}, \mathrm{Cu}, \mathrm{Mg}$ and $\mathrm{Mn}$$\mathrm{Cu}, \mathrm{Sn}$ and $\mathrm{Zn}$$\mathrm{Cu}, \mathrm{Zn}$ and $\mathrm{Ni}$$\mathrm{Cu}, \mathrm{Ni}$ and $\mathrm{Fe}$Correct Option: , 2 Solution: Gun metal' is alloy of copper with tin and zinc....
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