The four arms of a Wheatstone bridge have resistances as shown in the figure.
Question: The four arms of a Wheatstone bridge have resistances as shown in the figure. A galvanometer of $15 \Omega$ resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of $10 \mathrm{~V}$ is maintained across $\mathrm{AC}$. (1) $2.44 \mu \mathrm{A}$(2) $2.44 \mathrm{~mA}$(3) $4.87 \mathrm{~mA}$(4) $4.87 \mu \mathrm{A}$Correct Option: , 3 Solution: $\frac{x-10}{100}+\frac{x-y}{15}+\frac{x-0}{10}=0$ $53 x-20 y=30 \ldots . .(1)$ $\frac{y-1...
Read More →The rate of growth of bacteria in a culture is
Question: The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time $t=0$. The number of bacteria is increased by $20 \%$ in 2 hours. If the population of bacteria is 2000 after $\frac{\mathrm{k}}{\log _{\mathrm{e}}\left(\frac{6}{5}\right)}$ hours, then $\left(\frac{\mathrm{k}}{\log _{\mathrm{e}} 2}\right)^{2}$ is equal to(1) 4(2) 2(3) 16(4) 8Correct Option: 1 Solution: $\frac{\mathrm{dx}}{\mathrm{dt}} \propto \ma...
Read More →Two cells of emf 2 E
Question: Two cells of emf $2 \mathrm{E}$ and $\mathrm{E}$ with internal resistance $\mathrm{r}_{1}$ and $\mathrm{r}_{2}$ respectively are connected in series to an external resistor $R$ (see figure). The value of $\mathrm{R}$, at which the potential difference across the terminals of the first cell becomes zero is (1) $\mathrm{r}_{1}+\mathrm{r}_{2}$(2) $\frac{r_{1}}{2}-r_{2}$(3) $\frac{\mathrm{r}_{1}}{2}+\mathrm{r}_{2}$(4) $\mathrm{r}_{1}-\mathrm{r}_{2}$Correct Option: 2, Solution: (2) $\mathrm...
Read More →The equivalent resistance of series combination of two resistors
Question: The equivalent resistance of series combination of two resistors is 's'. When they are ponnected in parallel, the equivalent resistance is ' $\mathrm{p}$ '. If $s=n p$, then the minimum value for $n$ is (Round off to the Nearest Integer) Solution: (4) $\mathrm{R}_{1}+\mathrm{R}_{2}=\mathrm{s} \quad \ldots(1)$ $\frac{\mathrm{R}_{1} \mathrm{R}_{2}}{\mathrm{R}_{1}+\mathrm{R}_{2}}=\mathrm{p} \ldots$ $\mathrm{R}_{1} \mathrm{R}_{2}=\mathrm{sp}$ $\mathrm{R}_{1} \mathrm{R}_{2}=\mathrm{np}^{2}$...
Read More →State fundamental theorem of arithmatic.
Question: State fundamental theorem of arithmatic. Solution: The fundamental theorem of arithmetic, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and this product is unique....
Read More →State Euclid's division lemma.
Question: State Euclid's division lemma. Solution: Euclid's division lemma, states that for any two positive integersaandb, there exist unique whole numbersqandr,such thata=bq+rwhere 0rb...
Read More →Prove that
Question: Prove that $\frac{3}{\sqrt{5}}$ is irrational, given that $\sqrt{5}$ is irrational. Solution: Let us assume that $\frac{3}{\sqrt{5}}$ is a rational number. Thus, $\frac{3}{\sqrt{5}}$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers. $\frac{3}{\sqrt{5}}=\frac{p}{q}$ $\Rightarrow p \sqrt{5}=3 q$ $\Rightarrow \sqrt{5}=\frac{3 q}{p}$ Since, $\frac{3 q}{p}$ is rational $\Rightarrow \sqrt{5}$ is rational But, it is gi...
Read More →A current
Question: A current of $10 \mathrm{~A}$ exists in a wire of crosssectional area of $5 \mathrm{~mm}^{2}$ with a drift velocity of $2 \times 10^{-3} \mathrm{~ms}^{-1}$. The number of free electrons in each cubic meter of the wire is(1) $2 \times 10^{6}$(2) $625 \times 10^{25}$(3) $2 \times 10^{25}$(4) $1 \times 10^{23}$Correct Option: , 2 Solution: (2) $\mathrm{i}=10 \mathrm{~A}, \mathrm{~A}=5 \mathrm{~mm}^{2}=5 \times 10^{-6} \mathrm{~m}^{2}$ and $v_{\mathrm{d}}=2 \times 10^{-3} \mathrm{~m} / \ma...
Read More →Prove that
Question: Prove that $\frac{(3-4 \sqrt{2})}{7}$ is an irrational number, given that $\sqrt{2}$ is an irrational number. Solution: Let us assume that $\frac{(3-4 \sqrt{2})}{7}$ is a rational number. Thus, $\frac{(3-4 \sqrt{2})}{7}$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers. $\frac{3-4 \sqrt{2}}{7}=\frac{p}{q}$ $\Rightarrow 3-4 \sqrt{2}=\frac{7 p}{q}$ $\Rightarrow 4 \sqrt{2}=3-\frac{7 p}{q}$ $\Rightarrow 4 \sqrt{2}=\...
Read More →The energy dissipated by a resistor
Question: The energy dissipated by a resistor is $10 \mathrm{~mJ}$ in $1 \mathrm{~s}$ when an electric current of $2 \mathrm{~mA}$ flows through it. The resistance is________$\Omega$ (Round off to the Nearest Integer) Solution: $(2500)$ $\mathrm{Q}=\mathrm{i}^{2} \mathrm{RT}$ $\mathrm{R}=\frac{\mathrm{Q}}{\mathrm{i}^{2} \mathrm{t}}=\frac{10 \times 10^{-3}}{4 \times 10^{-6} \times 1}=2500 \Omega$...
Read More →Prove that
Question: Prove that $(2+3 \sqrt{5})$ is an irrational number, given that $\sqrt{5}$ is an irrational number. Solution: Let us assume that $(2+3 \sqrt{5})$ is a rational number. Thus, $(2+3 \sqrt{5})$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers. $2+3 \sqrt{5}=\frac{p}{q}$ $\Rightarrow 3 \sqrt{5}=\frac{p}{q}-2$ $\Rightarrow 3 \sqrt{5}=\frac{p-2 q}{q}$ $\Rightarrow \sqrt{5}=\frac{p-2 q}{3 q}$ Since, $\frac{p-2 q}{3 q}$...
Read More →Prove that
Question: Prove that $(3+5 \sqrt{2})$ is an irrational number, given that $\sqrt{2}$ is an irrational number. Solution: Let us assume that $(3+5 \sqrt{2})$ is a rational number. Thus, $(3+5 \sqrt{2})$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers. $3+5 \sqrt{2}=\frac{p}{q}$ $\Rightarrow 5 \sqrt{2}=\frac{p}{q}-3$ $\Rightarrow 5 \sqrt{2}=\frac{p-3 q}{q}$ $\Rightarrow \sqrt{2}=\frac{p-3 q}{5 q}$ Since, $\frac{p-3 q}{5 q}$...
Read More →A resistor develops
Question: A resistor develops $500 \mathrm{~J}$ of thermal energy in 20 s when a current of $1.5 \mathrm{~A}$ is passed through it. If the current is increased from $1.5 \mathrm{~A}$ to $3 \mathrm{~A}$, what will be the energy developed in $20 \mathrm{~s}$.(1) $1500 \mathrm{~J}$(2) $1000 \mathrm{~J}$(3) $500 \mathrm{~J}$(4) $2000 \mathrm{~J}$Correct Option: , 4 Solution: (4) $500=(1.5)^{2} \times \mathrm{R} \times 20$ $\mathrm{E}=(3)^{2} \times \mathrm{R} \times 20$ $\mathrm{E}=2000 \mathrm{~J}$...
Read More →Prove that
Question: Prove that $(4-\sqrt{3})$ is an irrational number, given that $\sqrt{3}$ is an irrational number. Solution: Let us assume that $(4-\sqrt{3})$ is a rational number. Thus, $(4-\sqrt{3})$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers. $4-\sqrt{3}=\frac{p}{q}$ $\Rightarrow-\sqrt{3}=\frac{p}{q}-4$ $\Rightarrow-\sqrt{3}=\frac{p-4 q}{q}$ $\Rightarrow \sqrt{3}=\frac{4 q-p}{q}$ Since, $\frac{4 q-p}{q}$ is rational $\R...
Read More →If the curve y=y(x) represented by the solution of
Question: If the curve $y=y(x)$ represented by the solution of the differential equation $\left(2 x y^{2}-y\right) d x+x d x=0$, passes through the intersection of the lines, $2 x-3 y=1$ and $3 x+2 y=8$, then $|y(1)|$ is equal to Solution: Given, $\left(2 x y^{2}-y\right) d x+x d x=0$ $\Rightarrow \frac{d y}{d x}+2 y^{2}-\frac{y}{x}=0$ $\Rightarrow-\frac{1}{y^{2}} \frac{d y}{d x}+\frac{1}{y}\left(\frac{1}{x}\right)=2$ $\frac{1}{y}=z$ $-\frac{1}{y^{2}} \frac{d y}{d x}=\frac{d z}{d x}$ $\Rightarro...
Read More →The value of $x$ to the nearest integer is
Question: The value of $x$ to the nearest integer is In the figure given, the electric current flowing through the $5 \mathrm{k} \Omega$ resistor is 'x' mA. The value of $x$ to the nearest integer is Solution: (3) $I=\frac{21}{5+1+1}=3 \mathrm{~mA}$...
Read More →Prove that
Question: Prove that $(2+\sqrt{3})$ is an irrational number, given that $\sqrt{3}$ is an irrational number. Solution: Let us assume that $(2+\sqrt{3})$ is a rational number. Thus, $(2+\sqrt{3})$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers. $2+\sqrt{3}=\frac{p}{q}$ $\Rightarrow \sqrt{3}=\frac{p}{q}-2$ $\Rightarrow \sqrt{3}=\frac{p-2 q}{q}$ Since, $\frac{p-2 q}{q}$ is rational $\Rightarrow \sqrt{3}$ is rational But, it...
Read More →If a curve passes through the origin and the slope
Question: If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$ is $\frac{x^{2}-4 x+y+8}{x-2}$, then this curve also passes through the point:(1) $(4,5)$(2) $(5,4)$(3) $(4,4)$(4) $(5,5)$Correct Option: , 4 Solution: $\frac{d y}{d x}=\frac{(x-2)^{2}+y+4}{(x-2)}=(x-2)+\frac{y+4}{(x-2)}$ Let $x-2=t \Rightarrow d x=d t$ and $y+4=u \Rightarrow d y=d u$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{du}}{\mathrm{dt}}$ $\frac{d u}{d t}=t+\frac{u}{t} \Rightarrow \...
Read More →The given plots represents the variation of the concentration
Question: The given plots represents the variation of the concentration of a reactant $R$ with time for two different reactions (i) and (ii). The respective orders of the reactions are: 1,01,10,10,2Correct Option: 1 Solution: In graph (i), In [Reactant vs time is linear with positive intercept and negative slope. Hence it is $1^{\text {st }}$ order In graph (ii), [Reactant] vs time is linear with positive intercept and negative slope. Hence, it is zero order....
Read More →If a curve passes through the origin and the slope
Question: If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$ is $\frac{x^{2}-4 x+y+8}{x-2}$, then this curve also passes through the point:(1) $(4,5)$(2) $(5,4)$(3) $(4,4)$(4) $(5,5)$Correct Option: , 4 Solution: $\frac{d y}{d x}=\frac{(x-2)^{2}+y+4}{(x-2)}=(x-2)+\frac{y+4}{(x-2)}$...
Read More →Find a rational number between
Question: Find a rational number between $\sqrt{2}$ and $\sqrt{3}$. Solution: $\sqrt{2}=1.41421 .$ $\sqrt{3}=1.73205 \ldots$ One of the rational number between them is $1.5=\frac{3}{2}$ Hence, a rational number between $\sqrt{2}$ and $\sqrt{3}$ is $\frac{3}{2}$....
Read More →For a reaction scheme
Question: For a reaction scheme $\mathrm{A} \stackrel{k_{1}}{\longrightarrow} \mathrm{B} \stackrel{k_{2}}{\longrightarrow} \mathrm{C}$, if the rate of formation of $B$ is set to be zero then the concentration of $B$ is given by:$\left(k_{1}-k_{2}\right)[\mathrm{A}]$$k_{1} k_{2}[\mathrm{~A}]$$\left(k_{1}+k_{2}\right)[\mathrm{A}]$$\left(\frac{k_{1}}{k_{2}}\right)[\mathrm{A}]$Correct Option: , 4 Solution: $\mathrm{A} \stackrel{\mathrm{k}_{1}}{\longrightarrow} \mathrm{B} \stackrel{\mathrm{k}_{2}}{\l...
Read More →A conducting wire of length
Question: A conducting wire of length $^{\prime} l$, area of crosssection $\mathrm{A}$ and electric resistivity $\rho$ is connected between the terminals of a battery. A potential difference $\mathrm{V}$ is developed between its ends, causing an electric current. If the length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current would be :(1) $\frac{1}{4} \frac{\mathrm{VA}}{\rho l}$(2) $\frac{3}{4} \frac{\mathrm{VA}}{\rho l}$(3) $\frac{1}{4} ...
Read More →If a curve y=f(x) passes through the point
Question: If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x \frac{\mathrm{d} y}{\mathrm{~d} x}+y=\mathrm{b} x^{4}$, then for what value of b, $\int_{1}^{2} f(x) \mathrm{d} x=\frac{62}{5} ?$(1) 5(2) $\frac{62}{5}$(3) $\frac{31}{5}$(4) 10Correct Option: 4, Solution: $\frac{d y}{d x}+\frac{y}{x}=b x^{3} \cdot I . F .=e^{\int \frac{d x}{x}}=x$ $\therefore y x=\int b x^{4} d x=\frac{b x^{5}}{5}+c$ Passes through $(1,2)$, we get $2=\frac{b}{5}+C \ldots \ldots(i)$ Also, $\int_{1}^{2...
Read More →Classify the following numbers as rational or irrational:
Question: Classify the following numbers as rational or irrational: (i) $\frac{22}{7}$ (ii) $3.1416$ (iii) $\pi$ (iv) $3 . \overline{142857}$ (v) $5.636363 \ldots$ (vi) $2.040040004 \ldots$ (vii) $1.535335333 \ldots$ (viii) $3.121221222 \ldots$ (ix) $\sqrt{21}$ (x) $\sqrt[3]{3}$ Solution: (i) $\frac{22}{7}$ is a rational number because it is of the form of $\frac{p}{q}, q \neq 0$. (ii) 3.1416 is a rational number because it is a terminating decimal. (iii) $\pi$ is an irrational number because it...
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