What is the common difference of an AP in which
Question: What is the common difference of an AP in which $a_{27}-a_{7}=84 ?$ Solution: $a_{27}=a+26 d$ $a_{7}=a+6 d$ And $a_{27}-a_{7}=84$ $\Rightarrow a+26 d-a-6 d=84$ $\Rightarrow 20 d=84$ $\Rightarrow d=\frac{21}{5}=4.2$...
Read More →Prove the following
Question: Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$. if $\lambda=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ and $\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}$, then the ordered pair, $(\lambda, \vec{d})$ is equal to:$\left(\frac{3}{2}, 3 \vec{a} \times \vec{c}\right)$$\left(-\frac{3}{2}, 3 \vec{c} \times \vec{b}\right)$$\left(\frac{3}{2}, 3 \vec{b} \times \vec{c}\right)...
Read More →The mass percentage of nitrogen in histamine is
Question: The mass percentage of nitrogen in histamine is____________ . Solution: (37.84) Molecular formula of histamine is $\mathrm{C}_{5} \mathrm{H}_{9} \mathrm{~N}_{3}$ Molecular mass of histamine $=5 \times 12+9 \times 1+3 \times 14=111$ Mass percentage of nitrogen in histamine $=\frac{42}{111} \times 100=37.84 \%$...
Read More →The molarity of
Question: The molarity of $\mathrm{HNO}_{3}$ in a sample which has density $1.4 \mathrm{~g} / \mathrm{mL}$ and mass percentage of $63 \%$ is_________. (Molecular Weight of $\mathrm{HNO}_{3}=63$ ) Solution: (14.00) Mass percent of $\mathrm{HNO}_{3}=63$ Thus, $100 \mathrm{~g}$ of nitric acid solution contains $63 \mathrm{~g}$ of nitric acid by mass. No. of moles $=\frac{63 \mathrm{~g}}{63 \mathrm{~g} \mathrm{~mol}^{-1}}=1$ Volume of $100 \mathrm{~g}$ of nitric acid solution $=\frac{\text { Mass }}...
Read More →Prove the following
Question: A vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4 \hat{k}$. If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c}$, then:$\vec{a} \cdot \hat{i}+3=0$$\vec{a} \cdot \hat{i}+1=0$$\vec{a} \cdot \hat{k}+2=0$$\vec{a} \cdot \hat{k}+4=0$Correct Option: , 3 Solution: Angle bisector between $\vec{b}$ and $\vec{c}$ can be $\vec{a}=\lambda(\hat{b}+\hat{c}) \quad$ or $\qua...
Read More →Ferrous sulphate heptahydrate is used to fortify foods with iron.
Question: Ferrous sulphate heptahydrate is used to fortify foods with iron. The amount (in grams) of the salt required to achieve $10 \mathrm{ppm}$ of iron in $100 \mathrm{~kg}$ of wheat is___________ .Atomic weight: $\mathrm{Fe}=55.85 ; \mathrm{S}=32.00 ; \mathrm{O}=16.00$ Solution: (4.96) $10=\frac{\text { Mass of Fe }(\text { in } \mathrm{g})}{100 \times 1000} \times 10^{6}$ $\Rightarrow$ Mass of $\mathrm{Fe}=1 \mathrm{~g}$ Molar mass of $\mathrm{FeSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}=27...
Read More →The predominant intermolecular forces present in ethyl acetate,
Question: The predominant intermolecular forces present in ethyl acetate, a liquid, are:London dispersion and dipole-dipolehydrogen bonding and London dispersionDipole-dipole and hydrogen bondingLondon dispersion, dipole-dipole and hydrogen bondingCorrect Option: 1 Solution: Ethyl acetate is polar molecule so dipole-dipole interaction and London dispersion will be present in its liquid state....
Read More →Very-Short and Short-Answer Questions
Question: Very-Short and Short-Answer QuestionsFind an AP whose 4th term is 9 and the sum of its 6th and 13th terms is 40. Solution: Letabe the first term anddbe the common difference of the AP. Then, $a_{4}=9$ $\Rightarrow a+(4-1) d=9 \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow a+3 d=9 \quad \ldots .(1)$ Now, $a_{6}+a_{13}=40$ (Given) $\Rightarrow(a+5 d)+(a+12 d)=40$ $\Rightarrow 2 a+17 d=40 \quad \ldots(2)$ From (1) and (2), we get $2(9-3 d)+17 d=40$ $\Rightarrow 18-6 d+17 d=40$ $\Rightarr...
Read More →The flocculation value of HCl for arsenic sulphide sol. is 30 m
Question: The flocculation value of $\mathrm{HCl}$ for arsenic sulphide sol. is $30 \mathrm{~m}$ mol $\mathrm{L}^{-1}$. If $\mathrm{H}_{2} \mathrm{SO}_{4}$ is used for the flocculation of arsenic sulphide, the amount, in grams, of $\mathrm{H}_{2} \mathrm{SO}_{4}$ in 250 $\mathrm{mL}$ required for the above purpose is______________ . (molecular mass of $\mathrm{H}_{2} \mathrm{SO}_{4}=98 \mathrm{~g} / \mathrm{mol}$ ) Solution: (0.37) For $1 \mathrm{~L}$ sol $30 \mathrm{~m} \mathrm{~mol}$ of $\math...
Read More →Very-Short and Short-Answer Questions
Question: Very-Short and Short-Answer QuestionsIf the sum of firstnterms is (3n2+ 5n), find its common difference. Solution: LetSndenotes the sum of firstnterms of the AP. $\therefore S_{n}=3 n^{2}+5 n$ $\Rightarrow S_{n-1}=3(n-1)^{2}+5(n-1)$ $=3\left(n^{2}-2 n+1\right)+5(n-1)$ $=3 n^{2}-n-2$ Now, $n^{\text {th }}$ term of the $\mathrm{AP}, a_{n}=S_{n}-S_{n-1}$ $=\left(3 n^{2}+5 n\right)-\left(3 n^{2}-n-2\right)$ $=6 n+2$ Letdbe the common difference of the AP. $\therefore d=a_{n}-a_{n-1}$ $=(6 ...
Read More →The relative strength of
Question: The relative strength of interionic/intermolecular forces in decreasing order is:dipole-dipole $$ ion-dipole $$ ion-ionion-dipole $$ ion-ion $$ dipole-dipoleion-dipole $$ dipole-dipole $$ ion-ionion-ion ion-dipole dipole-dipoleCorrect Option: , 4 Solution: Among given intermolecular forces, ionic interactions are stronger as compared to van der Waal interaction Thus, correct order is ion-ion $$ ion-dipole $$ dipoledipole...
Read More →Very-Short and Short-Answer Questions
Question: Very-Short and Short-Answer QuestionsIf the sum of firstpterms of an AP is (ap2+bp), find its common difference. Solution: LetSpdenotes the sum of firstpterms of the AP. $\therefore S_{p}=a p^{2}+b p$ $\Rightarrow S_{p-1}=a(p-1)^{2}+b(p-1)$ $=a\left(p^{2}-2 p+1\right)+b(p-1)$ $=a p^{2}-(2 a-b) p+(a-b)$ Now, $p^{\text {th }}$ term of the $\mathrm{AP}, a_{p}=S_{p}-S_{p-1}$ $=\left(a p^{2}+b p\right)-\left[a p^{2}-(2 a-b) p+(a-b)\right]$ $=a p^{2}+b p-a p^{2}+(2 a-b) p-(a-b)$ $=2 a p-(a-b...
Read More →The average molar mass of chlorine is
Question: The average molar mass of chlorine is $35.5 \mathrm{~g} \mathrm{~mol}^{-1}$. The ratio of ${ }^{35} \mathrm{Cl}$ to ${ }^{37} \mathrm{Cl}$ in naturally occrring chlorine is close to :$4: 1$$3: 1$$2: 1$$1: 1$Correct Option: , 2 Solution:...
Read More →A solution of two components containing
Question: A solution of two components containing $n_{1}$ moles of the $1^{\text {st }}$ component and $n_{2}$ moles of the $2^{\text {nd }}$ component is prepared. $M_{1}$ and $M_{2}$ are the molecular weights of component 1 and 2 respectively. If $d$ is the density of the solution in $g \mathrm{~mL}^{-1}, \mathrm{C}_{2}$ is the molarity and $x_{2}$ is the mole fraction of the $2^{\text {nd }}$ component, then $C_{2}$ can be expressed as:$\mathrm{C}_{2}=\frac{1000 x_{2}}{\mathrm{M}_{1}+x_{2}\le...
Read More →Solve this
Question: If $(2 p-1), 7$ and $3 p$ are in $\mathrm{AP}$, find the value of $p$. Solution: Let $(2 p-1), 7$ and $3 p$ be three consecutive terms of an AP. Then $7-(2 p-1)=3 p-7$ $\Rightarrow 5 p=15$ $\Rightarrow p=3$ $\therefore$ When $p=3,(2 p-1), 7$ and $3 p$ form three consecutive terms of an AP....
Read More →Solve this
Question: If $(2 p+1), 13,(5 p-3)$ are in AP, find the value of $p$. Solution: Let $(2 p+1), 13,(5 p-3)$ be three consecutive terms of an AP. Then $13-(2 p+1)=(5 p-3)-13$ $\Rightarrow 7 p=28$ $\Rightarrow p=4$ $\therefore$ When $p=4,(2 p+1), 13$ and $(5 p-3)$ form three consecutive terms of an AP....
Read More →Prove that
Question: If $\frac{4}{5}, a, 2$ are three consecutive terms of an $\mathrm{AP}$, find the value of $a$. Solution: If $\frac{4}{5}, a$ and 2 are three consecutive terms of an $\mathrm{AP}$, then we have: $a-\frac{4}{5}=2-a$ $\Rightarrow 2 a=2+\frac{4}{5}$ $\Rightarrow 2 a=\frac{14}{5}$ $\Rightarrow a=\frac{7}{5}$...
Read More →The minimum number of moles of
Question: The minimum number of moles of $\mathrm{O}_{2}$ required for complete combustion of 1 mole of propane and 2 moles of butane is ______ . Solution: (18) Complete combustion of hydrocarbons can be represented by the following reaction. $\mathrm{C}_{x} \mathrm{H}_{y}+\left(x+\frac{y}{4}\right) \mathrm{O}_{2} \longrightarrow x \mathrm{CO}_{2}+\frac{y}{2} \mathrm{H}_{2} \mathrm{O}$ For propane combustion reaction is $\mathrm{C}_{3} \mathrm{H}_{8}+\left(3+\frac{8}{4}\right) \mathrm{O}_{2} \lo...
Read More →The first term of an AP is p and its common difference is q.
Question: The first term of an AP ispand its common difference isq. Find its 10th term. Solution: Here,a=pandd=qNow, Tn= a + (n - 1)d⇒Tn=p+ (n- 1)q T10=p+ 9q...
Read More →Find the sum of first n even natural numbers.
Question: Find the sum of firstneven natural numbers. Solution: The firstneven natural numbers are 2 ,4, 6, 8, 10, ...,n. Here,a= 2 andd = (4 - 2) = 2 Sum of $n$ terms of an AP is given by $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $=\left(\frac{n}{2}\right) \times[2 \times 2+(n-1) \times 2]$ $=\left(\frac{n}{2}\right) \times[4+2 n-2]=\left(\frac{n}{2}\right) \times(2 n+2)=n(n+1)$ Hence, the required sum isn(n+1)....
Read More →Find the sum of first n natural numbers.
Question: Find the sum of firstnnatural numbers. Solution: The firstnnatural numbers are 1, 2, 3, 4, 5, ...,n. Here,a= 1 andd= (2 - 1) = 1 Sum of $n$ terms of an AP is given by $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $=\left(\frac{n}{2}\right) \times[2 \times 1+(n-1) \times 1]$ $=\left(\frac{n}{2}\right) \times[2+n-1]=\left(\frac{n}{2}\right) \times(n+1)=\frac{n(n+1)}{2}$...
Read More →A 100 mL solution was made by adding 1.43 g of
Question: A $100 \mathrm{~mL}$ solution was made by adding $1.43 \mathrm{~g}$ of $\mathrm{Na}_{2} \mathrm{CO}_{3} \cdot x \mathrm{H}_{2} \mathrm{O}$. The normality of the solution is $0.1 \mathrm{~N}$. The value of $x$ is ___________ . (The atomic mass of $\mathrm{Na}$ is $23 \mathrm{~g} / \mathrm{mol}$ ) Solution: (10) Normality $=\frac{\text { No. of equivalents of solute }}{\text { Volume of solution (in L) }}$ $0.1=\frac{1.43}{\frac{(106+18 x)}{2} \times 0.1} \Rightarrow \frac{106+18 x}{2}=1...
Read More →The mass of ammonia in grams produced when
Question: The mass of ammonia in grams produced when $2.8 \mathrm{~kg}$ of dinitrogen quantitatively reacts with $1 \mathrm{~kg}$ of dihydrogen is _____________. Solution: (3400)...
Read More →Which term of the AP 21, 18, 15,... is zero?
Question: Which term of the AP 21, 18, 15,... is zero? Solution: In the given AP, first term,a= 21 and common difference,d= (18 - 21) = -3Let's itsnthterm be 0.ThenTn= 0⇒a+ (n - 1)d = 0 $\Rightarrow 21+(n-1) \times(-3)=0$ $\Rightarrow 24-3 n=0$ $\Rightarrow 3 n=24$ $\Rightarrow n=8$ Hence, the8th term of the given AP is 0....
Read More →Write the next term of the AP
Question: Write the next term of the AP $\sqrt{2}, \sqrt{8}, \sqrt{18}, \ldots \ldots$ Solution: The given AP is $\sqrt{2}, \sqrt{8}, \sqrt{18}, \ldots$ On simplifying the terms, we get: $\sqrt{2}, 2 \sqrt{2}, 3 \sqrt{2}, \ldots$ Here, $a=\sqrt{2}$ and $d=(2 \sqrt{2}-\sqrt{2})=\sqrt{2}$ $\therefore$ Next term, $\mathrm{T}_{4}=a+3 d=\sqrt{2}+3 \sqrt{2}=4 \sqrt{2}=\sqrt{32}$...
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