If z be a complex number satisfying
Question: If $z$ be a complex number satisfying $|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=4$, then $|z|$ cannot be:(1) $\sqrt{\frac{17}{2}}$(2) $\sqrt{10}$(3) $\sqrt{7}$(4) $\sqrt{8}$Correct Option: , 3 Solution: $z=x+i y$ $|x|+|y|=4$ $|z|=\sqrt{x^{2}+y^{2}}$ Minimum value of $|z|=4$ $|z| \in[\sqrt{8}, \sqrt{16}]$ So, $|z|$ can't be $\sqrt{7}$...
Read More →The number of compound/s given below which contain
Question: The number of compound/s given below which contain/s - COOH group isSulphanilic acidPicric acidAspirinAscorbic acidCorrect Option: 1 Solution:...
Read More →Let z be a complex number such that
Question: Let $z$ be a complex number such that $\left|\frac{z-i}{z+2 i}\right|=1$ and $|z|=\frac{5}{2}$. Then the value of $|z+3 i|$ is :(1) $\sqrt{10}$(2) $\frac{7}{2}$(3) $\frac{15}{4}$(4) $2 \sqrt{3}$Correct Option: , 2 Solution: Let $z=x+i y$ Then, $\left|\frac{z-i}{z+2 i}\right|=1 \Rightarrow x^{2}+(y-1)^{2}$ $=x^{2}+(y+2)^{2} \Rightarrow-2 y+1=4 y+4$ $\Rightarrow \quad 6 y=-3 \Rightarrow y=-\frac{1}{2}$ $\because \quad|z|=\frac{5}{2} \Rightarrow x^{2}+y^{2}=\frac{25}{4}$ $\Rightarrow x^{2...
Read More →if
Question: If $\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi]$, is $a$ real number, then an argument of $\sin \theta+i \cos \theta$ is:(1) $\pi-\tan ^{-1}\left(\frac{4}{3}\right)$(2) $\pi-\tan ^{-1}\left(\frac{3}{4}\right)$(3) $-\tan ^{-1}\left(\frac{3}{4}\right)$(4) $\tan ^{-1}\left(\frac{4}{3}\right)$Correct Option: , 2 Solution: Let $z=\frac{3+i \sin \theta}{4-i \cos \theta}$, after rationallsing $z=\frac{(3+i \sin \theta)}{(4-i \cos \theta)} \times \frac{(4+i \cos \theta)}{(4+i ...
Read More →Which one of the following reactions will not form acetaldehyde?
Question: Which one of the following reactions will not form acetaldehyde?Correct Option: 1 Solution:...
Read More →if Re
Question: If $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)=1$, where $z=x+i y$, then the point $(x, y)$ lies on $a$ : (1) circle whose centre is at $\left(-\frac{1}{2},-\frac{3}{2}\right)$.(2) straight line whose slope is $-\frac{2}{3}$.(3) straight line whose slope is $\frac{3}{2}$.(4) circle whose diameter is $\frac{\sqrt{5}}{2}$.Correct Option: , 4 Solution: $\because \quad z=x+i y$ $\left(\frac{z-1}{2 z+i}\right)=\frac{(x-1)+i y}{2(x+i y)+i}$ $=\frac{(x-1)+i y}{2 x+(2 y+1) i} \times \frac...
Read More →Match List-I and List-II.I.
Question: Match List-I and List-II.I. Choose the correct answer from the options given below :$(a)-(i i),(b)-(i),(c)-(i v),(d)-(i i i)$$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i} v),(\mathrm{c})-(\mathrm{i}),(\mathrm{d})-(\mathrm{ii})$$(\mathrm{a})-(\mathrm{iii}),(\mathrm{b})-(\mathrm{i}),(\mathrm{c})-(\mathrm{i} v),(\mathrm{d})-(\mathrm{ii})$$(a)-(i i),(b)-(i v),(c)-(i),(d)-(i i i)$Correct Option: , 4 Solution:...
Read More →Let z=x+i y be a non-zero comp lex number such that
Question: Let $z=x+i y$ be a non-zero comp lex number such that $z^{2}=i|z|^{2}$, where $i=\sqrt{-1}$, then $z$ lies on the:(1) line, $y=-x$(2) imaginaryaxis(3) line, $y=x$(4) real axisCorrect Option: , 3 Solution: Let $z=x+i y$ $\because z^{2}=i|z|^{2}$ $\therefore x^{2}-y^{2}+2 i x y=i\left(x^{2}+y^{2}\right)$ $\Rightarrow x^{2}-y^{2}=0$ and $2 x y=x^{2}+y^{2}$ $\Rightarrow(x-y)(x+y)=0$ and $(x-y)^{2}=0$ $\Rightarrow x=y$...
Read More →The region represented by
Question: The region represented by $\{z=x+i y \in \mathrm{C}:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality:(1) $y^{2} \geq 2(x+1)$(2) $y^{2} \leq 2\left(x+\frac{1}{2}\right)$(3) $y^{2} \leq x+\frac{1}{2}$(4) $y^{2} \geq x+1$Correct Option: 2, Solution: $\because|z|-\operatorname{Re}(z) \leq 1 \quad(\because z=x+i y)$ $\Rightarrow \sqrt{x^{2}+y^{2}}-x \leq 1 \Rightarrow \sqrt{x^{2}+y^{2}} \leq 1+x$ $\Rightarrow x^{2}+y^{2} \leq 1+x^{2}+2 x$ $\Rightarrow y^{2} \leq 1+2 x \Rig...
Read More →In the above reaction, the reagent "A" is:
Question: In the above reaction, the reagent "A" is: $\mathrm{NaBH}_{4}, \mathrm{H}_{3} \mathrm{O}^{+}$$\mathrm{LiAlH}_{4}$Alkaline $\mathrm{KMnO}_{4}, \mathrm{H}^{+}$$\mathrm{HCl}, \mathrm{Zn}-\mathrm{Hg}$Correct Option: , 3 Solution:...
Read More →The value of
Question: The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is(1) $-2^{15}$(2) $2^{15} i$(3) $-2^{15} i$(4) $6^{5}$Correct Option: , 3 Solution: $\because-1+\sqrt{3} i=2 \cdot e^{\frac{2 \pi}{3} i}$ and $1-i=\sqrt{2} \cdot e^{-\frac{i \pi}{4}}$ $\therefore\left(\frac{-1+\sqrt{3} i}{1-i}\right)^{30}=\left(\sqrt{2} e^{\left(\frac{2 \pi}{3}+\frac{\pi}{4}\right)}\right)^{30}$ $=2^{15} \cdot e^{-\frac{\pi}{2} i}=-2^{15} \cdot i$...
Read More →If the four complex numbers z,
Question: If the four complex numbers $z, \bar{z}, \bar{z}-2 \operatorname{Re}(\bar{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to :(1) $4 \sqrt{2}$(2) 4(3) $2 \sqrt{2}$(4) 2Correct Option: , 3 Solution: Let $z=x+i y$ $\because$ Length of side of square $=4$ units Then, $|z-\bar{z}|=4 \Rightarrow|2 i y|=4 \Rightarrow|y|=2$ Also, $|z-(z-2 \operatorname{Re}(z))|=4$ $\Rightarrow|2 \operatorname{Re}(z)|=4 \Rightarrow...
Read More →The product "P" in the above reaction is :
Question: The product "P" in the above reaction is : Correct Option: , 2 Solution: DIBAL can not reduce double bond It can reduce cyclic ester....
Read More →If a and b are real numbers such that
Question: If $a$ and $b$ are real numbers such that $(2+\alpha)^{4}=a+b \alpha$, where $\alpha=\frac{-1+i \sqrt{3}}{2}$, then $a+b$ is equal to :(1) 9(2) 24(3) 33(4) 57Correct Option: 1 Solution: Given that, $\alpha=\frac{-1+\sqrt{3} i}{2}=\omega$ $\therefore(2+\omega)^{4}=a+b \omega \Rightarrow\left(4+\omega^{2}+4 \omega\right)^{2}=a+b \omega$ $\Rightarrow\left(\omega^{2}+4(1+\omega)\right)^{2}=a+b \omega$ $\Rightarrow\left(\omega^{2}-4 \omega^{2}\right)^{2}=a+b \omega$ $\Rightarrow\left(-3 \om...
Read More →A large block of wood of mass
Question: A large block of wood of mass $\mathrm{M}=5.99 \mathrm{~kg}$ is hanging from two long massless cords. A bullet of mass $\mathrm{m}=10 \mathrm{~g}$ is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance $\mathrm{h}=9.8 \mathrm{~cm}$ before the (block $+$ bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is : (Take $\mathrm{g}=9.8 \mathrm{~ms}^{...
Read More →Let
Question: Let $u=\frac{2 z+i}{z-k i}, z=x+i y$ and $k0 .$ If the curve represented by $\operatorname{Re}(\mathrm{u})+\operatorname{Im}(\mathrm{u})=1$ intersects the $y$-axis at the points $P$ and $Q$ where $P Q=5$, then the value of $k$ is :(1) $3 / 2$(2) $1 / 2$(3) 4(4) 2Correct Option: 2, 4, Solution: $u=\frac{2(x+i y)+i}{(x+i y)-k i}=\frac{2 x+i(2 y+1)}{x+i(y-k)}$ Real part of $u=\operatorname{Re}(u)=\frac{2 x^{2}+(y-K)(2 y+1)}{x^{2}+(y-K)^{2}}$ Imaginary part of $u$ $=\operatorname{Im}(u)=\f...
Read More →A ball of mass 10 kg moving with a velocity
Question: A ball of mass $10 \mathrm{~kg}$ moving with a velocity $10 \sqrt{3} \mathrm{~ms}^{-1}$ along X-axis, hits another ball of mass $20 \mathrm{~kg}$ which is at rest. After collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along Y-axis at a speed of $10 \mathrm{~m} / \mathrm{s}$. The second piece starts moving at a speed of $20 \mathrm{~m} / \mathrm{s}$ at an angle $\theta$ (degree) with respect to the $X$-axis....
Read More →If z1, z2 are complex numbers such that
Question: If $z_{1}, z_{2}$ are complex numbers such that $\operatorname{Re}\left(z_{1}\right)=\left|z_{1}-1\right|$, $\operatorname{Re}\left(z_{2}\right)=\left|z_{2}-1\right|$ and $\arg \left(z_{1}-z_{2}\right)=\frac{\pi}{6}$, then $\operatorname{Im}\left(z_{1}+z_{2}\right)$ is equal to :(1) $\frac{2}{\sqrt{3}}$(2) $2 \sqrt{3}$(3) $\frac{\sqrt{3}}{2}$(4) $\frac{1}{\sqrt{3}}$Correct Option: , 2 Solution: Let $z_{1}=x_{1}+i y_{1}$ and $z_{2}=x_{2}+i y_{2}$' $\because\left|z_{1}-1\right|=\operator...
Read More →Solve the following
Question: Poly- $\beta$-hydroxybutyrate-co- $\beta$-hydroxy valerate $(\mathrm{PHBV})$ is a copolymer of ______________3-Hydroxybutanoic acid and 4-Hydroxypentanoic acid2-Hydroxybutanoic acid and 3-Hydroxypentanoic acid3-Hydroxybutanoic acid and 2-Hydroxypentanoic acid3-Hydroxybutanoic acid and 3-Hydroxypentanoic acidCorrect Option: , 4 Solution:...
Read More →Among the following compounds most basic amino acid is:
Question: Among the following compounds most basic amino acid is:AsparagineLysineSerineHistidineCorrect Option: , 2 Solution: Lysine...
Read More →The mean of the following distribution is 50.
Question: The mean of the following distribution is 50. Find the value ofaand hence the frequencies of 30 and 70. Solution: We know that, Mean $=\frac{\sum x_{i} f_{i}}{\sum f_{i}}$ For the following data: Mean $=\frac{(10 \times 17)+(30 \times(5 a+3))+(50 \times 32)+(70 \times(7 a-11))+(90 \times 19)}{17+5 a+3+32+7 a-11+19}$ $\Rightarrow 50=\frac{170+150 a+90+1600+490 a-770+1710}{60+12 a}$ $\Rightarrow 50(60+12 a)=2800+640 a$ $\Rightarrow 3000+600 a=2800+640 a$ $\Rightarrow 640 a-600 a=3000-280...
Read More →The two monomers for the synthesis of nylon 6,6 are :
Question: The two monomers for the synthesis of nylon 6,6 are :$\mathrm{HOOC}\left(\mathrm{CH}_{2}\right)_{4} \mathrm{COOH}, \mathrm{H}_{2} \mathrm{~N}\left(\mathrm{CH}_{2}\right)_{6} \mathrm{NH}_{2}$$\mathrm{HOOC}\left(\mathrm{CH}_{2}\right)_{6} \mathrm{COOH}, \mathrm{H}_{2} \mathrm{~N}\left(\mathrm{CH}_{2}\right)_{6} \mathrm{NH}_{2}$$\mathrm{HOOC}\left(\mathrm{CH}_{2}\right)_{4} \mathrm{COOH}, \mathrm{H}_{2} \mathrm{~N}\left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2}$$\mathrm{HOOC}\left(\mathr...
Read More →The correct structure of histidine in a strongly acidic solution
Question: The correct structure of histidine in a strongly acidic solution $(\mathrm{pH}=2)$ is :Correct Option: , 3 Solution:...
Read More →Find the value of p, when the mean of the following distribution is 20.
Question: Find the value ofp, when the mean of the following distribution is 20. Solution: We know that, Mean $=\frac{\sum x_{i} f_{i}}{\sum f_{i}}$ For the following data: Mean $=\frac{(15 \times 2)+(17 \times 3)+(19 \times 4)+((20+p) \times 5 p)+(23 \times 6)}{2+3+4+5 p+6}$ $\Rightarrow 20=\frac{30+51+76+100 p+5 p^{2}+138}{15+5 p}$ $\Rightarrow 20(15+5 p)=5 p^{2}+100 p+295$ $\Rightarrow 300+100 p=5 p^{2}+100 p+295$ $\Rightarrow 5 p^{2}=300-295$ $\Rightarrow 5 p^{2}=5$ $\Rightarrow p^{2}=\frac{...
Read More →Find the value of p, when the mean of the following distribution is 20.
Question: Find the value ofp, when the mean of the following distribution is 20. Solution: We know that, Mean $=\frac{\sum x_{i} f_{i}}{\sum f_{i}}$ For the following data:/span/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/10/image98089.png" alt="" Mean $=\frac{(15 \times 2)+(17 \times 3)+(19 \times 4)+((20+p) \times 5 p)+(23 \times 6)}{2+3+4+5 p+6}$ $\Rightarrow 20=\frac{30+51+76+100 p+5 p^{2}+138}{15+5 p}$ $\Rightarrow 20(15+5 p)=5 p^{2}+100 p+295$ $\Rightarrow 300+100 p=5 p^{2}+100 p...
Read More →