If z and omega are two complex numbers such that
Question: If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and $\arg (z)-\arg (\omega)=\frac{\pi}{2}$, then:(1) $\bar{z} \omega=i$(2) $z \bar{\omega}=\frac{-1+i}{\sqrt{2}}$(3) $\bar{z} \omega=-i$(4) $z \bar{\omega}=\frac{1-i}{\sqrt{2}}$Correct Option: , 3 Solution: Given $|z \omega|=1$ and $\arg \left(\frac{z}{\omega}\right)=\frac{\pi}{2}$...
Read More →The numbers 50, 42, 35, (2x + 10), (2x – 8), 12, 11, 8 have been written in a descending order.
Question: The numbers 50, 42, 35, (2x+ 10), (2x 8), 12, 11, 8 have been written in a descending order. If their median is 25, find the value ofx. Solution: Arranging the given data in ascending order:8, 11, 12, (2x 8), (2x+ 10), 35, 42, 50Number of terms = 8 (even) $\therefore$ Median $=$ mean of $\left[\left(\frac{n}{2}\right)^{\text {th }}\right.$ term and $\left(\frac{n}{2}+1\right)^{\text {th }}$ term $]$ $\Rightarrow 25=$ mean of $\left[\left(\frac{8}{2}\right)^{\text {th }}\right.$ term an...
Read More →A ball of mass 10 kg moving with a velocity
Question: A ball of mass $10 \mathrm{~kg}$ moving with a velocity $10 \sqrt{3} \mathrm{~m} / \mathrm{s}$ along the $\mathrm{x}$-axis, hits another ball of mass $20 \mathrm{~kg}$ which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along $y$-axis with a speed of $10 \mathrm{~m} / \mathrm{s}$. The second piece starts moving at an angle of $30^{\circ}$ with respect to the $x$-axis. The velocity of the ball...
Read More →The equation
Question: The equation $|z-i|=|z-1|, i=\sqrt{-1}$, represents:(1) a circle of radius $\frac{1}{2}$.(2) the line through the origin with slope 1 .(3) a circle of radius 1 .(4) the line through the origin with slope $-1$.Correct Option: , 2 Solution: Given equation is, $|z-1|=|z-i|$ $\Rightarrow(x-1)^{2}+y^{2}=x^{2}+(y-1)^{2}$$[$ Here, $z=x+i y]$ $\Rightarrow 1-2 x=1-2 y \Rightarrow x-y=0$ Hence, locus is straight line with slope $1 .$...
Read More →Which dicarboxylic acid in presence of a dehydrating agent is least reactive to give an anhydride?
Question: Which dicarboxylic acid in presence of a dehydrating agent is least reactive to give an anhydride?Correct Option: 1 Solution: Adipic acid does not form anhydride....
Read More →The following observations are arranged in ascending order:
Question: The following observations are arranged in ascending order:26, 29, 42, 53,x, x+ 2, 70, 75, 82, 93.If the median is 65, find the value ofx. Solution: Arranging the given data in ascending order:26, 29, 42, 53,x, x+ 2, 70, 75, 82, 93Number of terms = 10 (even) $\therefore$ Median $=$ mean of $\left[\left(\frac{n}{2}\right)^{\text {th }}\right.$ term and $\left(\frac{n}{2}+1\right)^{\text {th }}$ term $]$ $\Rightarrow 65=$ mean of $\left[\left(\frac{10}{2}\right)^{\text {th }}\right.$ ter...
Read More →The disc of mass M with uniform surface mass density
Question: The disc of mass $M$ with uniform surface mass density $\sigma$ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position $\frac{x}{3} \frac{a}{\pi}, \frac{x}{3} \frac{a}{\pi}$ where $x$ is_____________ (Round off to the Nearest Integer) [a is an area as shown in the figure] Solution: (4) C.O.M of quarter disc is at $\frac{4 \mathrm{a}}{3 \pi}, \frac{4 \mathrm{a}}{3 \pi}$ $=4$...
Read More →The correct decreasing order for acid strength is:
Question: The correct decreasing order for acid strength is:$\mathrm{NO}_{2} \mathrm{CH}_{2} \mathrm{COOH}\mathrm{FCH}_{2} \mathrm{COOH}$$\mathrm{CNCH}_{2} \mathrm{COOH}\mathrm{ClCH}_{2} \mathrm{COOH}$$\mathrm{FCH}_{2} \mathrm{COOH}\mathrm{NCCH}_{2} \mathrm{COOH}$$\mathrm{NO}_{2} \mathrm{CH}_{2} \mathrm{COOH}\mathrm{ClCH}_{2} \mathrm{COOH}$$\mathrm{CNCH}_{2} \mathrm{COOH}\mathrm{O}_{2} \mathrm{NCH}_{2} \mathrm{COOH}$$\mathrm{FCH}_{2} \mathrm{COOH}\mathrm{ClCH}_{2} \mathrm{COOH}$$\mathrm{NO}_{2} ...
Read More →If 10, 13, 15, 18, x + 1, x + 3, 30, 32, 35, 41 are ten observation in an ascending order with median 24
Question: If 10, 13, 15, 18,x+ 1,x+ 3, 30, 32, 35, 41 are ten observation in an ascending order with median 24, find the value ofx. Solution: 10, 13, 15, 18,x+1,x+3, 30, 32, 35 and 41 are arranged in ascending order.Median = 24We have to find the value ofx.Here,nis 10, which is an even number.Thus, we have: Median $=$ Mean of $\left(\frac{n}{2}\right)$ th \ $\left(\frac{n}{2}+1\right)$ th observations $\therefore$ Median $=$ Mean of $\left(\frac{10}{2}\right)$ th $\\left(\frac{10}{2}+1\right)$ t...
Read More →If
Question: If $\mathrm{a}0$ and $\mathrm{z}=\frac{(1+i)^{2}}{\mathrm{a}-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\overline{\mathrm{z}}$ is equal to :(1) $-\frac{1}{5}-\frac{3}{5} i$(2) $-\frac{3}{5}-\frac{1}{5} i$(3) $\frac{1}{5}-\frac{3}{5} i$(4) $-\frac{1}{5}+\frac{3}{5} i$Correct Option: 1 Solution: $z=\frac{(1+i)^{2}}{a-i} \times \frac{a+i}{a+i}$ $z=\frac{(1-1+2 i)(a+i)}{a^{2}+1}=\frac{2 a i-2}{a^{2}+1}$ $|z|=\sqrt{\left(\frac{-2}{a^{2}+1}\right)^{2}+\left(\frac{2 a}{a^{2}+1}\right)^{2}...
Read More →The ages (in years) of 10 teachers in a school are:
Question: The ages (in years) of 10 teachers in a school are:32, 44, 53, 47, 37, 54, 34, 36, 40, 50.Find the median age Solution: Arranging the ages (in years) in ascending order, we have:32, 34, 36, 37, 40, 44, 47, 50, 53, 54Here,nis 10, which is an even number.Thus, we have: Median $=$ Mean of $\left(\frac{n}{2}\right)$ th $\left(\frac{n}{2}+1\right)$ th observations Median age $=$ Mean of $\left(\frac{10}{2}\right)$ th $\left(\frac{10}{2}+1\right)$ th observations $=$ Mean of 5 th 6 th observ...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: Correct Option: 1 Solution:...
Read More →The weights (in kg) of 8 children are:
Question: The weights (in kg) of 8 children are:13.4, 10.6, 12.7, 17.2, 14.3, 15, 16.5, 9.8.Find the median weight. Solution: Arranging the weights (in kg) in ascending order, we have:9.8, 10.6, 12.7, 13.4, 14.3, 15, 16.5, 17.2Here,nis 8, which is an even number.Thus, we have Median $=$ Mean of $\left(\frac{n}{2}\right)$ th $\left(\frac{n}{2}+1\right)$ th observations Median weight $=$ Mean of $\left(\frac{8}{2}\right)$ th $\left(\frac{8}{2}+1\right)$ th observations $=$ Mean of 4 th 5 th observ...
Read More →Two identical blocks A and B each of mass m
Question: Two identical blocks $A$ and $B$ each of mass $m$ resting on the smooth horizontal floor are connected by a light spring of natural length $L$ and spring constant $\mathrm{K}$. A third block $\mathrm{C}$ of mass $\mathrm{m}$ moving with a speed $\mathrm{v}$ along the line joining $\mathrm{A}$ and $\mathrm{B}$ collides with $\mathrm{A}$. The maximum compression in the spring is (1) $\sqrt{\frac{\mathrm{M}}{2 \mathrm{~K}}}$(2) $\sqrt{\frac{\mathrm{mv}}{2 \mathrm{~K}}}$(3) $\sqrt{\frac{\m...
Read More →Let z
Question: Let $z \in \mathrm{C}$ be such that $|z|1$. If $\omega=\frac{5+3 z}{5(1-z)}$, then :(1) $5 \operatorname{Re}(\omega)4$(2) $4 \operatorname{Im}(\omega)5$(3) $5 \operatorname{Re}(\omega)1$(4) $5 \operatorname{Im}(\omega)1$Correct Option: , 3 Solution: $\omega=\frac{5+3 z}{5-5 z} \Rightarrow 5 \omega-5 \omega z=5+3 z$ $\Rightarrow 5 \omega-5=z(3+5 \omega) \Rightarrow z=\frac{5(\omega-1)}{3+5 \omega}$ $\because|z|1, \therefore 5|\omega-1||3+5 \omega|$ $\Rightarrow 25(\omega \bar{\omega}-\o...
Read More →The major product of the following reaction is:
Question: The major product of the following reaction is: $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{CH}_{3}$$\mathrm{CH}_{3} \mathrm{CH}=\mathrm{CHCH}_{2} \mathrm{OH}$$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{OH}$$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CHO}$Correct Option: Solution:...
Read More →The heights (in cm) of 9 students of a class are 148, 144, 152, 155, 160, 147, 150, 149, 145.
Question: The heights (in cm) of 9 students of a class are 148, 144, 152, 155, 160, 147, 150, 149, 145.Find the median height. Solution: Arranging the given data in ascending order:144, 145, 147, 148, 149, 150, 152, 155, 160Number of terms = 9 (odd) $\therefore$ Median $=\left(\frac{n+1}{2}\right)^{\text {th }}$ term $=\left(\frac{9+1}{2}\right)^{\text {th }}$ term $=5^{\text {th }}$ term $=149$ Hence, the median height is 149....
Read More →All the points in the set
Question: All the points in the set $S=\left\{\frac{\alpha+i}{\alpha-i}: \alpha \in R\right\}(i=\sqrt{-1})$ lie on a:(1) straight line whose slope is 1 .(2) circle whose radius is 1 . (3) circle whose radius is $\sqrt{2}$.(4) straight line whose slope is $-1$.Correct Option: , 2 Solution: Let $\mathrm{z} \in \mathrm{S}$ then $z=\frac{\alpha+i}{\alpha-i}$ Since, $z$ is a complex number and let $z=x+i y$ Then, $x+i y=\frac{(\alpha+i)^{2}}{\alpha^{2}+1}$ (by rationalisation) $\Rightarrow x+i y=\fra...
Read More →The marks of 15 students in an examination are
Question: The marks of 15 students in an examination are:25, 19, 17, 24, 23, 29, 31, 40, 19, 20, 22, 26, 17, 35, 21.Find the median score. Solution: Arranging the marks of 15 students in ascending order, we get:17, 17, 19, 19, 20, 21, 22, 23, 24, 25, 26, 29, 31, 35, 40Here,nis 15, which is an odd number.We know Median $=$ Value of $\left(\frac{n+1}{2}\right)$ th observation Thus, we have: Median score $=$ Value of $\left(\frac{15+1}{2}\right)$ th observation $=$ Value of the 8 th observation $=2...
Read More →The most suitable reagent for the given conversion is:
Question: The most suitable reagent for the given conversion is: $\mathrm{B}_{2} \mathrm{H}_{6}$$\mathrm{NaBH}_{4}$$\mathrm{LiAlH}_{4}$$\mathrm{H}_{2} / \mathrm{Pd}$Correct Option: 1 Solution: $\mathrm{B}_{2} \mathrm{H}_{6}$ is a very selective reducing agent and usually used to reduce acid to alcohol....
Read More →Find the median of
Question: Find the median of(i) 17, 19, 32, 10, 22, 21, 9, 35(ii) 72, 63, 29, 51, 35, 60, 55, 91, 85, 82(iii) 10, 75, 3, 15, 9, 47, 12, 48, 4, 81, 17, 27 Solution: (i) Arranging the numbers in ascending order, we get:9, 10, 17, 19, 21, 22, 32, 35Here,nis 8, which is an even number.Ifnis an even number, we have: Median $=$ Mean of $\left(\frac{n}{2}\right)$ th $\\left(\frac{n}{2}+1\right)$ th observations Now, Median $=$ Mean of $\left(\frac{8}{2}\right)$ th \ $\left(\frac{8}{2}+1\right)$ th obse...
Read More →Consider the following molecules and statements related to them :
Question: Consider the following molecules and statements related to them : (a) (B) is more likely to be crystalline than (A) (b) (B) has higher boiling point than (A) (c) (B) dissolves more readily than (A) in water Identify the correct option from below :(a) and (b) are true(a) and (c) are trueonly (a) is true(a), (b) and (c) are trueCorrect Option: , 4 Solution: Molecule (A) shows intramolecular H-bonding while molecule (B) shows intermolecular H-bonding. Due to presence of intermolecular H-b...
Read More →Find the median of
Question: Find the median of(i) 2, 10, 9, 9, 5, 2, 3, 7, 11(ii) 15, 6, 16, 8, 22, 21, 9, 18, 25(iii) 20, 13, 18, 25, 6, 15, 21, 9, 16, 8, 22(iv) 7, 4, 2, 5, 1, 4, 0, 10, 3, 8, 5, 9, 2 Solution: (i) Arranging the numbers in ascending order, we get:2, 2, 3, 5, 7, 9, 9, 10, 11Here,nis 9, which is an odd number.Ifnis an odd number, we have: Median $=$ Value of $\left(\frac{n+1}{2}\right)$ th observation Now, Median $=$ Value of $\left(\frac{9+1}{2}\right)$ th observation $=$ Value of the 5 th observ...
Read More →If
Question: If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$, then $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ is equal to:(1) 0(2) 1(3) $(-1+2 i)^{9}$(4) $-1$Correct Option: , 4 Solution: $\frac{\sqrt{3}}{2}+\frac{i}{2}=-i\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)=-i \omega$ where $\omega$ is imaginary cube root of unity. Now, $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ $=\left(1+\omega-i \omega^{2}+i \omega^{2}\right)^{9}=(1+\omega)^{9}$ $=\left(-\omega^{2}\right)^{9}=-\omega^{18}=-1 \quad\left(\bec...
Read More →An organic compound [A], molecular formula
Question: An organic compound [A], molecular formula $\mathrm{C}_{10} \mathrm{H}_{20} \mathrm{O}_{2}$ was hydrolyzed with dilute sulphuric acid to give a carboxylic acid [B] and an alcohol [C]. Oxidation of [C] with $\mathrm{CrO}_{3}-\mathrm{H}_{2} \mathrm{SO}_{4}$ produced [B]. Which of the following structures are not possible for $[\mathrm{A}]$ ?Correct Option: 1 Solution:...
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