The correct match between Item I and Item II is:
Question: The correct match between Item I and Item II is: $(\mathrm{A}) \rightarrow(\mathrm{Q}, \mathrm{R}) ;(\mathrm{B}) \rightarrow(\mathrm{S}) ;(\mathrm{C}) \rightarrow(\mathrm{P})$$(\mathrm{A}) \rightarrow(\mathrm{R}) ;(\mathrm{B}) \rightarrow(\mathrm{Q}) ;(\mathrm{C}) \rightarrow(\mathrm{P})$$(\mathrm{A}) \rightarrow(\mathrm{R}) ;(\mathrm{B}) \rightarrow(\mathrm{S}) ;(\mathrm{C}) \rightarrow(\mathrm{Q})$$(\mathrm{A}) \rightarrow(\mathrm{Q}) ;(\mathrm{B}) \rightarrow(\mathrm{S}) ;(\mathrm{C...
Read More →greatest common divisor of the least values of m and n is
Question: If $\left(\frac{1+i}{1-i}\right)^{m / 2}=\left(\frac{1+i}{i-1}\right)^{n / 3}=1,(m, n \in \mathbf{N})$, then the greatest common divisor of the least values of $m$ and $n$ is_________. Solution: Given that $\left(\frac{1+i}{1-i}\right)^{m / 2}=\left(\frac{1+i}{i-1}\right)^{n / 3}=1$ $\Rightarrow\left(\frac{(1+i)^{2}}{2}\right)^{m / 2}=\left(\frac{(1+i)^{2}}{-2}\right)^{n / 3}=1$ $\Rightarrow i^{m / 2}=(-i)^{n / 3}=1$ $m$ (least) $=8, n$ (least) $=12$ $\operatorname{GCD}(8,12)=4$...
Read More →Find the missing frequencies in the following frequency distribution, whose mean is 50.
Question: Find the missing frequencies in the following frequency distribution, whose mean is 50. Solution: We will prepare the following table: Thus, we have: Mean $=\frac{\sum f_{i} x_{i}}{\sum x_{i}}$ $\Rightarrow 50=\frac{3480+30 f_{1}+70 f_{2}}{120}$ $\Rightarrow 6000=3480+30 f_{1}+70 f_{2}$ $\Rightarrow 30 f_{1}+70 f_{2}=2520 \ldots \ldots$ (i) Also,Given: $17+f_{1}+32+f_{2}+19=120$ $\Rightarrow 68+f_{1}+f_{2}=120$ $\Rightarrow f_{1}+f_{2}=52$ or, $f_{2}=52-f_{1} \ldots$ (ii) By putting th...
Read More →The imaginary part of
Question: The imaginary part of $(3+2 \sqrt{-54})^{1 / 2}-(3-2 \sqrt{-54})^{1 / 2}$ can be :(1) $-\sqrt{6}$(2) $-2 \sqrt{6}$(3) 6(4) $\sqrt{6}$Correct Option: , 2 Solution: $3+2 \sqrt{-54}=3+6 \sqrt{6} i$ Let $\sqrt{3+6 \sqrt{6} i}=a+i b$ $\Rightarrow a^{2}-b^{2}=3$ and $a b=3 \sqrt{6}$ $\Rightarrow a^{2}+b^{2}=\sqrt{\left(a^{2}-b^{2}\right)^{2}+4 a^{2} b^{2}}=15$ So, $a=\pm 3$ and $b=\pm \sqrt{6}$ $\sqrt{3+6 \sqrt{6} i}=\pm(3+\sqrt{6} i)$ Similarly, $\sqrt{3-6 \sqrt{6} i}=\pm(3-\sqrt{6} i)$ $\o...
Read More →The homopolymer formed from 4-hydroxy-butanoic acids is :
Question: The homopolymer formed from 4-hydroxy-butanoic acids is :Correct Option: , 4 Solution:...
Read More →The value of
Question: The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^{3}$ is : (1) $\frac{1}{2}(1-i \sqrt{3})$(2) $\frac{1}{2}(\sqrt{3}-i)$(3) $-\frac{1}{2}(\sqrt{3}-i)$(4) $-\frac{1}{2}(1-i \sqrt{3})$Correct Option: , 3 Solution: $\left(\frac{1+\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}}{1+\cos \frac{5 \pi}{18}-i \sin \frac{5 \pi}{18}}\right)^{3}$ $=\left(\frac{2 \cos ^{2} \frac{5 \pi}{36}+i 2 \sin \frac{5 \pi}{36} \cdot \co...
Read More →Find the missing frequencies in the following frequency distribution whose mean is 34.
Question: Find the missing frequencies in the following frequency distribution whose mean is 34. Solution: We know that, Mean $=\frac{\sum x_{i} f_{i}}{\sum f_{i}}$ For the following data: Mean $=\frac{(10 \times 4)+\left(20 \times f_{1}\right)+(30 \times 8)+\left(40 \times f_{2}\right)+(50 \times 3)+(60 \times 4)}{35}$ $\Rightarrow 34=\frac{40+20 f_{1}+240+40 f_{2}+150+240}{35}$ $\Rightarrow 34(35)=670+20 f_{1}+40 f_{2}$ $\Rightarrow 1190-670=20 f_{1}+40 f_{2}$ $\Rightarrow 20 f_{1}+40 f_{2}=52...
Read More →The correct match between Item I and Item II is:
Question: The correct match between Item I and Item II is: $(\mathrm{A}) \rightarrow(\mathrm{R}) ;(\mathrm{B}) \rightarrow(\mathrm{P}) ;(\mathrm{C}) \rightarrow(\mathrm{Q}) ;(\mathrm{D}) \rightarrow(\mathrm{S})$$(\mathrm{A}) \rightarrow(\mathrm{P}) ;(\mathrm{B}) \rightarrow(\mathrm{R}) ;(\mathrm{C}) \rightarrow(\mathrm{Q}) ;(\mathrm{D}) \rightarrow(\mathrm{S})$$(\mathrm{A}) \rightarrow(\mathrm{R}) ;(\mathrm{B}) \rightarrow(\mathrm{P}) ;(\mathrm{C}) \rightarrow(\mathrm{S}) ;(\mathrm{D}) \rightarr...
Read More →Let z be those complex number which
Question: Let $z$ be those complex number which satisfy $|z+5| \leq 4$ and $z(1+i)+\bar{z}(1-i) \geq-10, i=\sqrt{-1}$ If the maximum value of $|z+1|^{2}$ is $\alpha+\beta \sqrt{2}$, then the value of $(\alpha+\beta)$ is Solution: Given, $|z+5| \leq 4$ $\Rightarrow(x+5)^{2}+y^{2} \leq 16 \ldots(1)$ Also, $z(1+i)+\bar{z}(1-i) \geq-10$ $\Rightarrow x-y \geq-5 \ldots(2)$ From (1) and (2) Locus of $z$ is the shaded region in the diagram. $|z+1|$ represents distance of $^{\prime} z^{\prime}$ from $Q(-...
Read More →Find the value of p for the following frequency distribution whose mean is 16.6
Question: Find the value ofpfor the following frequency distribution whose mean is 16.6 Solution: We will make the following table: Thus, we have: Mean $=\frac{\sum f_{i} x_{i}}{\sum x_{i}}$ $\Rightarrow 16.6=\frac{(1228+24 p)}{100}$ $\Rightarrow 1660=1228+24 p$ $\Rightarrow 24 p=432$ $\Rightarrow p=18$...
Read More →In the figure shown, after the switch
Question: In the figure shown, after the switch ' $\mathrm{S}$ ' is turned from position 'A' to position 'B', the energy dissipated in the 'circuit in terms of capacitance ' $\mathrm{C}$ ' and total charge ' $\mathrm{Q}$ ' is: (1) $\frac{1}{8} \frac{Q^{2}}{C}$(2) $\frac{3}{8} \frac{\mathrm{Q}^{2}}{\mathrm{C}}$(3) $\frac{5}{8} \frac{Q^{2}}{C}$(4) $\frac{3}{4} \frac{\mathrm{Q}^{2}}{\mathrm{C}}$Correct Option: , 2 Solution: (2) Energy stored in the system initially $\mathrm{U}_{\mathrm{i}}=\frac{1}...
Read More →The polymer obtained from the following reactions is
Question: The polymer obtained from the following reactions is Correct Option: , 3 Solution:...
Read More →The sum of
Question: The sum of $162^{\text {th }}$ power of the roots of the equation $x^{3}-2 x^{2}+2 x-1=0$ is Solution: Let roots of $x^{3}-2 x^{2}+2 x-1=0$ are $\alpha, \beta, \gamma$ $(x-1)\left(x^{2}-x+1\right)=0$ Now $\alpha^{162}+\beta^{162}+\gamma^{162}$ $=1+\omega^{162}+\left(\omega^{2}\right)^{162}$ $=1+\left(\omega^{3}\right)^{54}+\left(\omega^{3}\right)^{108}$ $=3$...
Read More →Among the following compounds, which one is found in RNA?
Question: Among the following compounds, which one is found in RNA?Correct Option: , 2 Solution: R.N.A contain uracil...
Read More →A parallel plate capacitor with plates of area
Question: A parallel plate capacitor with plates of area $1 \mathrm{~m}^{2}$ each, are at a separation of $0.1 \mathrm{~m}$. If the electric field between the plates is $100 \mathrm{~N} / \mathrm{C}$, the magnitude of charge on each plate is: (Take $\epsilon_{0}=8.85 \times 10^{-12} \frac{\mathrm{C}^{2}}{\mathrm{~N}-\mathrm{M}^{2}}$ )(1) $7.85 \times 10^{-10} \mathrm{C}$(2) $6.85 \times 10^{-10} \mathrm{C}$(3) $8.85 \times 10^{-10} \mathrm{C}$(4) $9.85 \times 10^{-10} \mathrm{C}$Correct Option: ...
Read More →Find the missing frequency p for the following frequency distribution whose mean is 28.25.
Question: Find the missing frequencypfor the following frequency distribution whose mean is 28.25. Solution: We will prepare the following table: Thus, we have: Mean $=\frac{\sum f_{i} x_{i}}{\sum x_{i}}$ $\Rightarrow 28.25=\frac{1445+25 p}{50+p}$ $\Rightarrow 28.25(50+p)=(1445+25 p)$ $\Rightarrow 1412.5+28.25 p=1445+25 p$ $\Rightarrow 3.25 p=32.5$ $\Rightarrow p=10$...
Read More →then its radius is :
Question: Let the lines $(2-\mathrm{i}) \mathrm{z}=(2+\mathrm{i}) \overline{\mathrm{z}}$ and $(2+\mathrm{i}) \mathrm{z}+(\mathrm{i}-2) \overline{\mathrm{z}}-4 \mathrm{i}=0$, (here $\mathrm{i}^{2}=-1$ ) be normal to a circle $\mathrm{C}$. If the line $\mathrm{iz}+\overline{\mathrm{z}}+1+\mathrm{i}=0$ is tangent to this circle $\mathrm{C}$, then its radius is :(1) $\frac{3}{\sqrt{2}}$(2) $3 \sqrt{2}$(3) $\frac{3}{2 \sqrt{2}}$(4) $\frac{1}{2 \sqrt{2}}$Correct Option: , 3 Solution: $(2-i) z=(2+i) \b...
Read More →If the mean of the following frequency distribution is 8, find the value of p.
Question: If the mean of the following frequency distribution is 8, find the value ofp. Solution: We will make the following table: We know: Mean $=\frac{\sum f_{i} x_{i}}{\sum x_{i}}$ Given:Mean = 8Thus, we have: $8=\frac{303+9 p}{41+p}$ $\Rightarrow 328+8 p=303+9 p$ $\Rightarrow p=25$...
Read More →Which of the following tests cannot be used for identifying amino acids?
Question: Which of the following tests cannot be used for identifying amino acids?Biuret testBarfoed testNinhydrin testXanthoproteic testCorrect Option: , 2 Solution: Barfoed test is used to test the reducing nature of sugar (carbohydrate)....
Read More →In the circuit shown,
Question: In the circuit shown, find $\mathrm{C}$ if the effective capacitance of the whole circuit is to be $0.5 \mu \mathrm{F}$. All values in the circuit are in $\mu \mathrm{F}$. (1) $\frac{7}{11} \mu \mathrm{F}$(2) $\frac{6}{5} \mu \mathrm{F}$(3) $4 \mu \mathrm{F}$(4) $\frac{7}{10} \mu \mathrm{F}$Correct Option: 1 Solution: For series combination $\frac{1}{\mathrm{C}_{\mathrm{eq}}}=\frac{1}{\mathrm{C}_{1}}+\frac{1}{\mathrm{C}_{2}}$ $\Rightarrow \frac{\frac{7 \mathrm{C}}{3}}{\frac{7}{3}+\math...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: 1 Solution: Reaction involved:...
Read More →If the mean of the following data is 20.2, find the value of p.
Question: If the mean of the following data is 20.2, find the value ofp. Solution: We know that, Mean $=\frac{\sum x_{i} f_{i}}{\sum f_{i}}$ For the following data: Mean $=\frac{(10 \times 6)+(15 \times 8)+(20 \times p)+(25 \times 10)+(30 \times 6)}{6+8+p+10+6}$ $\Rightarrow 20.2=\frac{60+120+20 p+250+180}{30+p}$ $\Rightarrow 20.2(30+p)=610+20 p$ $\Rightarrow 606+20.2 p=610+20 p$ $\Rightarrow 20.2 p-20 p=610-606$ $\Rightarrow 0.2 p=4$ $\Rightarrow p=\frac{4}{0.2}$ $\Rightarrow p=\frac{40}{2}$ $\...
Read More →be the greatest integral part of |k|
Question: Let $i=\sqrt{-1}$. If $\frac{(-1+i \sqrt{3})^{21}}{(1-i)^{24}}+\frac{(1+i \sqrt{3})^{21}}{(1+i)^{24}}=k$, and $\mathrm{n}=[|k|]$ be the greatest integral part of $|\mathrm{k}|$. Then $\sum_{j=0}^{n+5}(j+5)^{2}-\sum_{j=0}^{n+5}(j+5)$ is equal to Solution: $\frac{\left(2 e^{i \frac{2 \pi}{3}}\right)^{21}}{\left(\sqrt{2} e^{-i \frac{\pi}{4}}\right)^{21}}+\frac{\left(2 e^{i \frac{\pi}{3}}\right)^{21}}{\left(\sqrt{2} e^{i \frac{\pi}{4}}\right)^{24}}$ $\Rightarrow \frac{2^{21} \cdot e^{14 \p...
Read More →Solve this
Question: In the above circuit, $\mathrm{C}=\frac{\sqrt{3}}{2} \mu \mathrm{F}, \mathrm{R}_{2}=20 \Omega, \mathrm{L}=\frac{\sqrt{3}}{10} \mathrm{H}$ and $\mathrm{R}_{1}=10 \Omega$. Current in $\mathrm{L}-\mathrm{R}_{1}$ path is $\mathrm{I}_{1}$ and in $\mathrm{C}-\mathrm{R}_{2}$ path it is $\mathrm{I}_{2}$. The voltage of A.C source is given by, $\mathrm{V}=200 \sqrt{2} \sin (100 \mathrm{t})$ volts. The phase difference between $\mathrm{I}_{1}$ and $\mathrm{I}_{2}$ is :(1) $60^{\circ}$(2) $30^{\c...
Read More →If the least and the largest real values of
Question: If the least and the largest real values of $\alpha$, for which the equation $z+\alpha|z-1|+2 i=0$ $(z \in C$ and $i=\sqrt{-1})$ has a solution, are $p$ and $\mathrm{q}$ respectively; then $4\left(p^{2}+q^{2}\right)$ is equal to Solution: $x+i y+\alpha \sqrt{(x-1)^{2}+y^{2}}+2 i=0$ $\therefore y+2=0$ and $x+\alpha \sqrt{(x-1)^{2}+y^{2}}=0$ $y=-2 \ x^{2}=\alpha^{2}\left(x^{2}-2 x+1+4\right)$ $\alpha^{2}=\frac{x^{2}}{x^{2}-2 x+5} \Rightarrow x^{2}\left(\alpha^{2}-1\right)-2 x \alpha^{2}+...
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