The major product [ R ] in the following sequence of reactions is :
Question: The major product [ $R$ ] in the following sequence of reactions is : Correct Option: , 3 Solution:...
Read More →The period of oscillation of a simple pendulum
Question: The period of oscillation of a simple pendulum is $T=2 \pi \sqrt{\frac{\bar{L}}{g}}$. Measured value of ' $L^{\prime}$ is $1.0 \mathrm{~m}$ from meter scale having a minimum division of $1 \mathrm{~mm}$ and time of one complete oscillation is $1.95 \mathrm{~s}$ measured from stopwatch of $0.01 \mathrm{~s}$ resolution. The percentage error in the determination of ' $g$ ' will be :(1) $1.33 \%$(2) $1.30 \%$(3) $1.13 \%$(4) $1.03 \%$Correct Option: , 3 Solution: $(3)$ $\mathrm{T}=2 \pi \s...
Read More →Let S be the set of all real roots of the equation
Question: Let $S$ be the set of all real roots of the equation, $3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right|$. Then $S$ :(1) contains exactly two elements.(2) is a singleton.(3) is an empty set.(4) contains at least four elements.Correct Option: , 2 Solution: Let $3^{x}=y$ $\therefore \quad y(y-1)+2=|y-1|+|y-2|$ Case 1: when $y2$ $y^{2}-y+2=y-1+y-2$ $y^{2}-3 y+5=0$ $\because \quad D0[\therefore$ Equation not satisfy. $]$ Case 2: when $1 \leq y \leq 2$ $y^{2}-y^{2}+2=y-1...
Read More →Find the common difference of an AP whose first term is 5 and the sum of its first
Question: Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms Solution: Let the common difference of the AP bed.First term,a= 5Now, $a_{1}+a_{2}+a_{3}+a_{4}=\frac{1}{2}\left(a_{5}+a_{6}+a_{7}+a_{8}\right)$ (Given) $\Rightarrow a+(a+d)+(a+2 d)+(a+3 d)=\frac{1}{2}[(a+4 d)+(a+5 d)+(a+6 d)+(a+7 d)] \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow 4 a+6 d=\frac{1}{2}(4 a+22 d)$ $\Rightarrow 8 a+12 d=4 a+22 d$ $\Rightarrow...
Read More →The radius of a sphere is measured to be
Question: The radius of a sphere is measured to be $(7.50 \pm 0.85) \mathrm{cm}$. Suppose the percentage error in its volume is $x$. The value of $x$, to the nearest $x$, is_______ Solution: (34) $\because \mathrm{v}=\frac{4}{3} \pi \mathrm{r}^{3}$ taking log \ then differentiate $\frac{\mathrm{dV}}{\mathrm{V}}=3 \frac{\mathrm{dr}}{\mathrm{r}}$ $=\frac{3 \times 0.85}{7.5} \times 100 \%=34 \%$...
Read More →The radius of a sphere is measured to be
Question: The radius of a sphere is measured to be $(7.50 \pm 0.85) \mathrm{cm}$. Suppose the percentage error in its volume is $x$. The value of $x$, to the nearest $x$, is_______ Solution: (34) $\because \mathrm{v}=\frac{4}{3} \pi \mathrm{r}^{3}$ taking log \ then differentiate $\frac{\mathrm{dV}}{\mathrm{V}}=3 \frac{\mathrm{dr}}{\mathrm{r}}$ $=\frac{3 \times 0.85}{7.5} \times 100 \%=34 \%$...
Read More →The major product in the following reaction is :
Question: The major product in the following reaction is : Correct Option: , 3 Solution:...
Read More →Prove the following
Question: Let $\alpha=\frac{-1+i \sqrt{3}}{2}$. If $a=(1+\alpha) \sum_{k=0}^{100} \alpha^{2 k}$ and $b=\sum_{k=0}^{100} \alpha^{3 k}$, then $a$ and $b$ are the roots of the quadratic equation:(1) $x^{2}+101 x+100=0$(2) $x^{2}-102 x+101=0$(3) $x^{2}-101 x+100=0$(4) $x^{2}+102 x+101=0$Correct Option: , 2 Solution: Let $\alpha=\omega, b=1+\omega^{3}+\omega^{6}+\ldots . .=101$ $a=(1+\omega)\left(1+\omega^{2}+\omega^{4}+\ldots \ldots \omega^{198}+\omega^{200}\right)$ $=(1+\omega) \frac{\left(1-\left(...
Read More →The time period of a simple pendulum is given by
Question: The time period of a simple pendulum is given by $\mathrm{T}=2 \pi \sqrt{\frac{\ell}{\mathrm{g}}} .$ The measured value of the length of pendulum is $10 \mathrm{~cm}$ known to a $1 \mathrm{~mm}$ accuracy. The time for 200 oscillations of the pendulum is found to be 100 second using a clock of 1 s resolution. The percentage accuracy in the determination of ' $g$ ' using this pendulum is ' $x$ '. The value of ' $x$ ' to the nearest integer is:-(1) $2 \%$(2) $3 \%$(3) $5 \%$(4) $4 \%$Corr...
Read More →A student writes general characteristics based on the given mechanism as :
Question: The mechanism of $S_{N} 1$ reaction is given as : A student writes general characteristics based on the given mechanism as : (a) The reaction is favoured by weak nucleophiles. (b) $R^{\oplus}$ would be easily formed if the substituents are bulky. (c) The reaction is accompanied by racemization. (d) The reaction is favoured by non-polar solvents. Which observations are correct?(a) and (b)(a) and (c)(a), (b) and (c)(b) and (d)Correct Option: , 3 Solution: Above reaction is $S_{N} 1$ reac...
Read More →The resistance
Question: The resistance $\mathrm{R}=\frac{\mathrm{V}}{\mathrm{I}}$, where $\mathrm{V}=(50 \pm 2) \mathrm{V}$ and $\mathrm{I}=(20 \pm 0.2) \mathrm{A} .$ The percentage error in $\mathrm{R}$ is ' $\mathrm{x}$ ' \\%. The value of ' $\mathrm{x}$ ' to the nearest integer is______ Solution: (5) $\frac{\Delta \mathrm{R}}{\mathrm{R}} \times 100=\frac{\Delta \mathrm{V}}{\mathrm{V}} \times 100+\frac{\Delta \mathrm{I}}{\mathrm{I}} \times 100$ $\%$ error in $\mathrm{R}=\frac{2}{50} \times 100+\frac{0.2}{20...
Read More →Which of the following compounds produces an optically inactive compound
Question: Which of the following compounds produces an optically inactive compound on hydrogenation?Correct Option: , 4 Solution:...
Read More →Solve the following
Question: Let $\alpha=\frac{-1+i \sqrt{3}}{2} .$ If $a=(1+\alpha) \sum_{k=0}^{100} \alpha^{2 k}$ and $b=\sum_{k=0}^{100} \alpha^{3 k}$, then $a$ and $b$ are the roots of the quadratic equation:(1) $x^{2}+101 x+100=0$(2) $x^{2}-102 x+101=0$(3) $x^{2}-101 x+100=0$(4) $x^{2}+102 x+101=0$Correct Option: , 2 Solution: Let $\alpha=\omega, b=1+\omega^{3}+\omega^{6}+\ldots . .=101$ $a=(1+\omega)\left(1+\omega^{2}+\omega^{4}+\ldots . . \omega^{198}+\omega^{200}\right)$ $=(1+\omega) \frac{\left(1-\left(\o...
Read More →Consider the reaction sequence given below :
Question: Consider the reaction sequence given below : Which of the following statements is true?(1) Changing the base from $\mathrm{OH}^{\ominus}$ to $^{\ominus} \mathrm{OR}$ will have no effect on reaction (2).Changing the concentration of base will have no effect on reaction (1).Doubling the concentration of base will double the rate of both the reactions.Changing the concentration of base will have no effect on reaction (2).Correct Option: , 2 Solution: First reaction is $\mathrm{S}_{\mathrm...
Read More →The least positive value of
Question: The least positive value of ' $a$ ' for which the equation, $2 x^{2}+(a-10) x+\frac{33}{2}=2 a$ has real roots is Solution: Since, $2 x^{2}+(a-10) x+\frac{33}{2}=2 a$ has real roots, $\therefore \quad D \geq 0$ $\Rightarrow \quad(a-10)^{2}-4(2)\left(\frac{33}{2}-2 a\right) \geq 0$ $\Rightarrow \quad(a-10)^{2}-4(33-4 a) \geq 0$ $\Rightarrow a^{2}-4 a-32 \geq 0$ $\Rightarrow \quad(a-8)(a+4) \geq 0$ $\Rightarrow \quad a \leq-4 \cup a \geq 8$ $\Rightarrow \quad a \in(-\infty,-4] \cup[8, \i...
Read More →The major product obtained from E2-elimination of
Question: The major product obtained from E2-elimination of 3 bromo-2-fluoropentane is : Correct Option: , 2 Solution:...
Read More →A moving coil galvanometer, having a resistance G,
Question: A moving coil galvanometer, having a resistance $\mathrm{G}$, produces full scale deflection when a current $I_{g}$ flows through it. This galvanometer can be converted into (i) an ammeter of range 0 to $\mathrm{I}_{0}\left(\mathrm{I}_{0}\mathrm{I}_{g}\right)$ by connecting a shunt resistance $R_{A}$ to it and (ii) into a voltmeter of range 0 to $V$ $\left(\mathrm{V}=\mathrm{GI}_{0}\right)$ by connecting a series resistance $\mathrm{R}_{\mathrm{v}}$ to it. Then,(1) $R_{A} R_{V}=G^{2}\l...
Read More →A galvanometer of resistance
Question: A galvanometer of resistance $100 \Omega$ has 50 divisions on its scale and has sensitivity of $20 \mu \mathrm{A} /$ division. It is to be converted to a voltmeter with three ranges, of $0-2 \mathrm{~V}$, $0-10 \mathrm{~V}$ and $0-20 \mathrm{~V}$. The appropriate circuit to do so is :Correct Option: , 3 Solution: (3) $i_{g}=20 \times 50=1000 \mu A=1 \mathrm{~mA}$ Using, $V=i_{g}(G+R)$, we have $2=10^{-3}\left(100+R_{1}\right)$ $R_{1}=1900 \Omega$ when, $V=10$ volt $10=10^{-3}\left(100+...
Read More →A magnetic compass needle oscillates 30 times per minute at a place where
Question: A magnetic compass needle oscillates 30 times per minute at a place where the dip is $45^{\circ}$, and 40 times per minute where the dip is $30^{\circ}$. If $\mathrm{B}_{1}$ and $\mathrm{B}_{2}$ are respectively the total magnetic field due to the earth and the two places, then the ratio $\mathrm{B}_{1} / \mathrm{B}_{2}$ is best given by :(1) $1.8$(2) $0.7$(3) $3.6$(4) $0.46$Correct Option: , 4 Solution: (4) We have, $T=2 \pi \sqrt{\frac{I}{M B_{x}}}$ $\therefore \frac{T_{1}^{2}}{T_{2}...
Read More →A magnetic compass needle oscillates 30 times per minute at a place where
Question: A magnetic compass needle oscillates 30 times per minute at a place where the dip is $45^{\circ}$, and 40 times per minute where the dip is $30^{\circ}$. If $\mathrm{B}_{1}$ and $\mathrm{B}_{2}$ are respectively the total magnetic field due to the earth and the two places, then the ratio $\mathrm{B}_{1} / \mathrm{B}_{2}$ is best given by :(1) $1.8$(2) $0.7$(3) $3.6$(4) $0.46$Correct Option: , 4 Solution: (4) We have, $T=2 \pi \sqrt{\frac{I}{M B_{x}}}$ $\therefore \frac{T_{1}^{2}}{T_{2}...
Read More →If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.
Question: If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero. Solution: Letabe the first term anddbe the common difference of the AP. Then, $10 \times a_{10}=15 \times a_{15} \quad$ (Given) $\Rightarrow 10(a+9 d)=15(a+14 d) \quad\left[a_{n}=a+(n-1) d\right]$ $\Rightarrow 2(a+9 d)=3(a+14 d)$ $\Rightarrow 2 a+18 d=3 a+42 d$ $\Rightarrow a=-24 d$ $\Rightarrow a+24 d=0$ $\Rightarrow a+(25-1) d=0$ $\Rightarrow a_{25}=0$ Hence, the 25th term of the A...
Read More →If the equation,
Question: If the equation, $x^{2}+b x+45=0(b \in R)$ has conjugate complex roots and they satisfy $|z+1|=2 \sqrt{10}$, then:(1) $b^{2}-b=30$(2) $b^{2}+b=72$(3) $b^{2}-b=42$(4) $b^{2}+b=12$Correct Option: 1 Solution: Let $z=\alpha \pm$ i $\beta$ be the complex roots of the equation So, sum of roots $=2 \alpha=-b$ and Product of roots $=\alpha^{2}+\beta^{2}=45$ $(\alpha+1)^{2}+\beta^{2}=40$ Given, $|z+1|=2 \sqrt{10}$ $\Rightarrow \quad(\alpha+1)^{2}-\alpha^{2}=-5$ $\left[\because \beta^{2}=45-\alp...
Read More →A moving coil galvanometer allows a full scale current
Question: A moving coil galvanometer allows a full scale current of $10^{-4} \mathrm{~A}$. A series resistance of $2 \mathrm{M} \Omega$ is required to convert the above galvanometer into a voltmeter of range $0-5 \mathrm{~V}$. Therefore the value of shunt resistance required to convert the above galvanometer into an ammeter of range $0-10 \mathrm{~mA}$ is:(1) $500 \Omega$(2) $100 \Omega$(3) $200 \Omega$(4) $-195 \times 10^{4} \Omega$Correct Option: , 4 Solution: (4) $\quad \mathrm{v}=\mathrm{i}_...
Read More →The major product in the following reaction is :
Question: The major product in the following reaction is : Correct Option: , 2 Solution:...
Read More →Solve the following
Question: Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-x-1=0$. If $p_{k}=(\alpha)^{k}+(\beta)^{k}, k \geq 1$, then which one of the following statements is not true ?(1) $p_{3}=p_{5}-p_{4}$(2) $P_{5}=11$(3) $\left(p_{1}+p_{2}+p_{3}+p_{4}+p_{5}\right)=26$(4) $p_{5}=p_{2} \cdot p_{3}$Correct Option: , 4 Solution: $\alpha^{5}=5 \alpha+3$ $\beta^{5}=5 \beta+3$ $p_{5}=5(\alpha+\beta)+6=5(1)+6$ $\left[\because\right.$ from $\left.x^{2}-x-1=0, \alpha+\beta=\frac{-b}{a}=1\right]$ $p_{5}=...
Read More →