Two water taps together fill a tank in
Question: Two water taps together fill a tank in $1 \frac{7}{8}$ hours. The tap with longer diameter takes 2 hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately. Solution: Let the smaller tap takesxhours to fill the tank.Then, the larger one takes (x 2) hoursto fill the tank Tank filled in 1 hour by smaller tap $=\frac{1}{x}$ Tank filled in 1 hour by larger tap $=\frac{1}{x-2}$ Tank filled in 1 hour by both the taps $...
Read More →Consider a frame that is made up of two thin massless rods AB and AC as shown in the figure.
Question: Consider a frame that is made up of two thin massless rods $\mathrm{AB}$ and $\mathrm{AC}$ as shown in the figure. A vertical force $\overrightarrow{\mathrm{P}}$ of magnitude $100 \mathrm{~N}$ is applied at point $\mathrm{A}$ of the frame. Suppose the force is $\overrightarrow{\mathrm{P}}$ resolved parallel to the arms $\mathrm{AB}$ and $\mathrm{AC}$ of the frame. The magnitude of the resolved component along the arm $\mathrm{AC}$ is $\mathrm{xN}$. The value of $\mathrm{x}$, to the nea...
Read More →The process that involves the removal of sulphur
Question: The process that involves the removal of sulphur from the ores is:SmeltingRoastingLeachingRefiningCorrect Option: Solution: In roasting process, metal sulphide (MS) ore are converted into metal oxide and sulphur is remove in the form of $\mathrm{SO}_{2}$ gas. $2 \mathrm{MS}+3 \mathrm{O}_{2} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{MO}+2 \mathrm{SO}_{2} \uparrow$...
Read More →If the letters of the word 'MOTHER' be permuted
Question: If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is__________. Solution: $\begin{array}{cccccc}\mathrm{M} \mathrm{O} \mathrm{T} \mathrm{H} \mathrm{E} \mathrm{R} \\ 3 4 6 2 1 5\end{array}$ $\Rightarrow 2 \times 5 !+2 \times 4 !+3 \times 3 !+2 !+1$ $=240+48+18+2+1=309$...
Read More →If the letters of the word 'MOTHER' be permuted
Question: If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is__________. Solution: $\begin{array}{cccccc}\mathrm{M} \mathrm{O} \mathrm{T} \mathrm{H} \mathrm{E} \mathrm{R} \\ 3 4 6 2 1 5\end{array}$ $\Rightarrow 2 \times 5 !+2 \times 4 !+3 \times 3 !+2 !+1$ $=240+48+18+2+1=309$...
Read More →The total number of 4 -digit numbers whose
Question: The total number of 4 -digit numbers whose greatest common divisor with 18 is 3 , is Solution: Since, required number has G.C.D with 18 as 3 . It must be odd multiple of ' $3^{\prime}$ but not a multiple of '9'. (i) Now, 4 -digit number which are odd multiple of ' $3^{\prime}$ are, $1005,1011,1017, \ldots \ldots \ldots \ldots .9999 \rightarrow 1499$ (ii) 4-digit number which are odd multiple of 9 are, $1017,1035, \ldots \ldots \ldots \ldots . .9999 \rightarrow 499$ $\because$ Required ...
Read More →A natural number has prime factorization given
Question: A natural number has prime factorization given by $\mathbf{n}=2^{x} 3^{y} 5^{z}$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6}, yz$. Then the number of odd divisors of $n$, including 1 , is:(1) 11(2) $6 x$(3) 12(4) 6Correct Option: , 3 Solution: $y+z=5 \ldots(1)$ $\frac{1}{y}+\frac{1}{z}=\frac{5}{6}$ $\Rightarrow \frac{y+z}{y z}=\frac{5}{6}$ $\Rightarrow \frac{5}{y z}=\frac{5}{6}$ $\Rightarrow y z=6$ Also $(y-z)^{2}=(y+z)^{2}-4 y z$ $\Rightarrow(y-z)^{2}=(y+z)...
Read More →Two pipes together can fill a tank in 12 hours.
Question: Two pipes together can fill a tank in 12 hours. If the first pipe can fill the tank 10 hours faster than the second then how many hours will the second pipe take to fill the tank? Solution: Let the time required to fill the tank by second pipebexhours.Then,the time required to fill the tankby first pipe is (x 10)hours.Given:Two pipes together can fill a tank in 12 hours.According to the question, $\frac{1}{x}+\frac{1}{x-10}=\frac{1}{12}$ $\Rightarrow \frac{(x-10)+(x)}{(x)(x-10)}=\frac{...
Read More →The number of seven digit integers
Question: The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only is(1) 77(2) 42(3) 35(4) 82Correct Option: 1 Solution: CASE-I: $1,1,1,1,1,2,3$ $\mathrm{WAYS}=\frac{7 !}{5 !}=42$ CASE-II: $1,1,1,1,2,2,2$ WAYS $=\frac{7 !}{4 ! \cdot 3 !}=35$ TOTAL WAYS $=42+35=77$...
Read More →The correct order for acid strength of compounds
Question: The correct order for acid strength of compounds $\mathrm{CH} \equiv \mathrm{CH}$, $\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{CH}$ and $\mathrm{CH}_{2}=\mathrm{CH}_{2}$ is as follows:$\mathrm{CH} \equiv \mathrm{CH}\mathrm{CH}_{2}=\mathrm{CH}_{2}\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{CH}$$\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{CH}\mathrm{CH} \equiv \mathrm{CH}\mathrm{CH}_{2}=\mathrm{CH}_{2}$$\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{CH}\mathrm{CH}_{2}=\mathrm{CH}_{2}\mathrm{HC} \e...
Read More →A block 200g mass moves with a uniform speed
Question: A block of $200 \mathrm{~g}$ mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius $20 \mathrm{~cm}$. If the block takes $40 \mathrm{~s}$ to complete one round, the normal force by the side walls of the groove is:(1) $0.0314 \mathrm{~N}$(2) $9.859 \times 10^{-2} \mathrm{~N}$(3) $6.28 \times 10^{-3} \mathrm{~N}$(4) $9.859 \times 10^{-4} \mathrm{~N}$Correct Option: , 4 Solution: (4) $\mathrm{N}=\mathrm{m} \omega^{2} \mathrm{R}$ $\mathrm{N}=\m...
Read More →A motorboat whose speed is 18 kmph in still water takes 1 hr 30 min more to go 36 km upstream than to return downstream to the same spot.
Question: A motorboat whose speed is 18 kmph in still water takes 1 hr 30 min more to go 36 km upstream than to return downstream to the same spot. Find the speed of the stream. Solution: Let speed of stream bexkm/h.Given:Speed of boat =18 km/hDistance covered upstream = 36 kmDistance covered downstream = 36 km It takes $\frac{3}{2}$ hours more to go $36 \mathrm{~km}$ upstream than to return downstream to the same spot Now, Speed of boat upstream = 18 xkm/hSpeed of boat downstream = 18 +x km/h $...
Read More →The total number of two digit numbers
Question: The total number of two digit numbers 'n', such that $3^{n}+7^{n}$ is a multiple of 10 , is Solution: $=10 p+(-1)^{t}$ $=10 p \pm 1$ $\therefore$ if $\mathrm{n}=$ even then $7^{n}+3^{n}$ will not be multiply of 10 So if $n$ is odd then only $7^{n}+3^{n}$ will be multiply of 10 $\therefore \mathrm{n}=11,13,15, \ldots \ldots, \ldots, 99$ $\therefore$ Ans 45...
Read More →Which compound (s) out of the following is/are not aromatic?
Question: Which compound (s) out of the following is/are not aromatic? (B), (C) and (D)(C) and (D)(B)(A) and (C)Correct Option: 1 Solution:...
Read More →A block of mass m slides along a floor while a force of magnitude F
Question: A block of mass $m$ slides along a floor while a force of magnitude $F$ is applied to it at an angle $\theta$ as shown in figure. The coefficient of kinetic friction is $\mu_{\mathrm{K}}$. Then, the block's acceleration ' $\mathrm{a}$ ' is given by : ( $\mathrm{g}$ is acceleration due to gravity) (1) $-\frac{\mathrm{F}}{\mathrm{m}} \cos \theta-\mu_{\mathrm{K}}\left(\mathrm{g}-\frac{\mathrm{F}}{\mathrm{m}} \sin \theta\right)$(2) $\frac{\mathrm{F}}{\mathrm{m}} \cos \theta-\mu_{\mathrm{K}...
Read More →The total number of numbers,
Question: The total number of numbers, lying between 100 and 1000 that can be formed with the digits $1,2,3,4,5$, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is Solution: $=12$ $4 \times 3$ $12 \rightarrow 3,4,5 \rightarrow 3 !=6$ $15 \rightarrow 2,3,4 \rightarrow 3 !=6$ $24 \rightarrow 1,3,5 \rightarrow 3 !=6$ $42 \rightarrow 1,2,3 \rightarrow 3 !=6$ Required No. $=24+12-4=32$...
Read More →A faster train takes one hour less than a slower train for a journey of 200 km.
Question: A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 kmph less than that of the faster train, find the speeds of two trains. Solution: Let the speed of faster train bexkm/h.Then,the speed of slower train is (x 10) km/h. Given:A faster train takes one hour less than a slower train for a journey of 200 km. $\frac{\text { Distance }}{\text { Speed }}=$ Time Time taken by faster train to cover $200 \mathrm{~km}=\frac{200}{x}...
Read More →What is the IUPAC name of the following compound?
Question: What is the IUPAC name of the following compound? 3-Bromo-1, 2-dimethylbut-1-ene3-Bromo-3-methyl-1, 2-dimethylprop-1-ene2-Bromo-3-methylpent-3-ene4-Bromo-3-methylpent-2-eneCorrect Option: , 4 Solution:...
Read More →The total number of positive
Question: The total number of positive integral solutions $(x, y, z)$ such that $x y z=24$ is(1) 36(2) 45(3) 24(4) 30Correct Option: , 4 Solution: $x \cdot y \cdot z=24$ $x \cdot y \cdot z=2^{3} \cdot 3^{1}$ Now using beggars method. 3 things to be distributed among 3 persons\ Each may receive none, one or more $\therefore{ }^{5} \mathrm{C}_{2}$ ways Similarly for $1^{\prime} \therefore{ }^{3} \mathrm{C}_{2}$ ways Total ways $={ }^{5} \mathrm{C}_{2} \cdot{ }^{3} \mathrm{C}_{2}=30$ ways...
Read More →Solve the following
Question: The students $S_{1}, S_{2}, \ldots, S_{10}$ are to be divided into 3 groups $A, B$ and $C$ such that each group has at least one student and the group $C$ has at most 3 students. Then the total number of possibilities of forming such groups is Solution: $={ }^{10} C_{1}\left[2^{9}-2\right]+{ }^{10} C_{2}\left[2^{8}-2\right]+{ }^{10} C_{3}\left[2^{7}-2\right]$ $=2^{7}\left[{ }^{10} C_{1} \times 4+{ }^{10} C_{2} \times 2+{ }^{10} C_{3}\right]-20-90-240$ $=128[40+90+120]-350$ $=(128 \time...
Read More →The increasing order of the
Question: The increasing order of the $\mathrm{p} K_{\mathrm{a}}$ values of the following compounds is: $\mathrm{C}\mathrm{B}\mathrm{A}\mathrm{D}$$\mathrm{B}\mathrm{C}\mathrm{D}\mathrm{A}$$\mathrm{D}\mathrm{A}\mathrm{C}\mathrm{B}$$\mathrm{B}\mathrm{C}\mathrm{A}\mathrm{D}$Correct Option: , 4 Solution: Electron withdrawing substituents increase the acidic strength while electron releasing groups decrease the acidic strength....
Read More →One mole of an ideal gas passes through a process where
Question: One mole of an ideal gas passes through a process where pressure and volume obey the relation $P=P_{0}\left[1-\frac{1}{2}\left(\frac{V_{0}}{V}\right)^{2}\right]$. Here $P_{o}$ and $V_{o}$ are constants. Calculate the charge in the temperature of the gas if its volume changes from $\mathrm{V}_{\mathrm{o}}$ to $2 \mathrm{~V}_{\mathrm{o}}$. (1) $\frac{1}{2} \frac{\mathrm{Po} \mathrm{V}_{o}}{\mathrm{R}}$(2) $\frac{5}{4} \frac{\mathrm{P}_{o} \mathrm{~V}_{o}}{\mathrm{R}}$(3) $\frac{3}{4} \fr...
Read More →The hypotenuse of a right-angled triangle is 1 metre less than twice the shortest side.
Question: The hypotenuse of a right-angled triangle is 1 metre less than twice the shortest side. If the third is 1 metre more than the shortest side, find the sides of the triangle. Solution: Let the shortest side be $x \mathrm{~m}$. Therefore, according to the question: Hypotenuse $=(2 x-1) \mathrm{m}$ Third side $=(x+1) \mathrm{m}$ On applying Pythagoras theorem, we get: $(2 x-1)^{2}=(x+1)^{2}+x^{2}$ $\Rightarrow 4 x^{2}-4 x+1=x^{2}+2 x+1+x^{2}$ $\Rightarrow 2 x^{2}-6 x=0$ $\Rightarrow 2 x(x-...
Read More →Which of the following compounds is not aromatic?
Question: Which of the following compounds is not aromatic?Correct Option: 1 Solution: Compounds $(2),(3)$ and $(4)$ are containing $6 \pi e^{-}$in complete conjugation and are aromatic....
Read More →A scientific committee is to formed from
Question: A scientific committee is to formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed is:(1) 560(2) 1050(3) 1625(4) 575Correct Option: , 3 Solution: $(2 \mathrm{I}, 4 \mathrm{~F})+(3 \mathrm{I}, 6 \mathrm{~F})+(4 \mathrm{I}, 8 \mathrm{~F})$ $={ }^{6} \mathrm{C}_{2}{ }^{8} \mathrm{C}_{4}+{ }^{6} \mathrm{C}_{3}{ }^{8} \mathrm{C}_{6}+{ }^{6} \mathrm{C}_{4}{ }^{8} \mathrm{C...
Read More →