The numbers 42, 43, 44, 44, (2x + 3), 45, 45, 46, 47 have been arranged in an ascending order and their median is 45.
Question: The numbers 42, 43, 44, 44, (2x+ 3), 45, 45, 46, 47 have been arranged in an ascending order and their median is 45. Find the value ofx. Hence, find the mode of the above data. Solution: Arranging the given data in ascending order:42, 43, 44, 44, (2x+ 3), 45, 45, 46, 47Number of terms = 9 (odd) $\therefore$ Median $=\left(\frac{n+1}{2}\right)^{\text {th }}$ term $\Rightarrow 45=\left(\frac{9+1}{2}\right)^{\text {th }}$ term $\Rightarrow 45=(5)^{\text {th }}$ term $\Rightarrow 45=2 x+3$...
Read More →A circular hole of radius
Question: A circular hole of radius $\left(\frac{2}{2}\right)$ is cut out of a circular dise of radius 'a' shown in ligure. The centroid of the remaining circular portion with respect to point ' $\mathrm{O}^{\prime}$ will be : (1) $\frac{10}{11} a$(2) $\frac{2}{3} a$(3) $\frac{1}{6} a$(4) $\frac{5}{6} a$Correct Option: , 4 Solution: (4) Let $\sigma$ is the surface mass density of disc. $X_{\mathrm{com}}=\frac{\left(0 \times \pi a^{2} \times A\right)-\left(\frac{\pi^{2}}{\frac{1}{4}} \times \frac...
Read More →Amongst the following, the linear species is:
Question: Amongst the following, the linear species is:$\mathrm{NO}_{2}$$\mathrm{Cl}_{2} \mathrm{O}$$\mathrm{O}_{3}$$\mathrm{N}_{3}^{-}$Correct Option: , 4 Solution:...
Read More →Let z1 and z2 be any two non-zero complex numbers such that
Question: Let $z_{1}$ and $z_{2}$ be any two non-zero complex numbers such that $3\left|z_{1}\right|=4\left|z_{2}\right|$. If $z=\frac{3 z_{1}}{2 z_{2}}+\frac{2 z_{2}}{3 z_{1}}$ then:(1) $3 / 2 \leq|z| \leq 5 / 2$(2) $1 / 2 \leq|z| \leq 5 / 2$(3) $3 / 2 \leq|z| \leq 7 / 2$(4) $1 \leq|z| \leq 5 / 2$Correct Option: 1 Solution: Let $z_{1}=r_{1} e^{i \theta}$ and $z_{2}=r_{2} e^{i \phi}$ $3\left|z_{1}\right|=4\left|z_{2}\right| \Rightarrow 3 r_{1}=4 r_{2}$ $z=\frac{3 z_{1}}{2 z_{2}}+\frac{2 z_{2}}{3...
Read More →For what value of x is the mode of the data 24, 15, 40, 23, 27, 26, 22, 25, 20, x + 3 found 25?
Question: For what value ofxis the mode of the data 24, 15, 40, 23, 27, 26, 22, 25, 20,x+ 3 found 25? Using this value ofx, find the median. Solution: Given: Mode = 25 25 occurs maximum number of times.Arranging the given data in ascending order:15, 20, 22, 23, 24,x+ 3, 25, 26, 27, 40x+ 3 = 25⇒x= 25 3⇒x= 22Hence, the value ofxis 22.Arranging the given data in ascending order:15, 20, 22, 23, 24, 25, 25, 26, 27, 40Number of terms = 10 (even) $\therefore$ Median $=$ mean of $\left[\left(\frac{n}{2}...
Read More →A central atom in a molecule has two lone pairs of electrons and forms three single bonds.
Question: A central atom in a molecule has two lone pairs of electrons and forms three single bonds. The shape of this molecule is:see-sawplanar triangularT-shapedtrigonal pyramidalCorrect Option: , 3 Solution: $\mathrm{sp}^{3} \mathrm{~d}$ hybridised T-shaped...
Read More →Let Z0 be a root of the quadratic equation,
Question: Let $z_{0}$ be a root of the quadratic equation, $x^{2}+x+1=0$. If $z=3+6 i z_{0}^{81}-3 i z_{0}^{93}$, then arg $z$ is equal to:(1) $\frac{\pi}{4}$(2) $\frac{\pi}{6}$(3) $\frac{\pi}{3}$(4) 0Correct Option: 1 Solution: $\because z_{0}$ is a root of quadratic equation $x^{2}+x+1=0$ $\therefore \quad z_{0}=\omega$ or $\omega^{2} \Rightarrow z_{0}^{3}=1$ $\therefore \quad z=3+6 i z_{0}^{81}-3 i z_{0}^{93}$ $=3+6 i\left(\left(z_{0}\right)^{3}\right)^{27}-3 i\left(\left(z_{0}\right)^{3}\rig...
Read More →A ball with a speed
Question: A ball with a speed of $9 \mathrm{~m} / \mathrm{s}$ collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of $30^{\circ}$ with the original direction. The ratio of velocities of the balls after collision is $\mathrm{x}: \mathrm{y}$, where $\mathrm{x}$ is Solution: Momentum is conserved just before and just after the collision in both $x-y$ direction. In $y$-direction $\mathrm{p}_{\mathrm{i}}=0$ $P_{f}=m \times \frac{1}{2} v_{1}-m \...
Read More →The numbers 52, 53, 54, 54, (2x + 1), 55, 55, 56, 57 have been arranged in an ascending order and their median is 55.
Question: The numbers 52, 53, 54, 54, (2x+ 1), 55, 55, 56, 57 have been arranged in an ascending order and their median is 55. Find the value ofxand hence find the mode of the given data. Solution: Arranging the given data in ascending order:52, 53, 54, 54, (2x+ 1), 55, 55, 56, 57Number of terms = 9 (odd) $\therefore$ Median $=\left(\frac{n+1}{2}\right)^{\text {th }}$ term $\Rightarrow 55=\left(\frac{9+1}{2}\right)^{\text {th }}$ term $\Rightarrow 55=(5)^{\text {th }}$ term $\Rightarrow 55=2 x+1...
Read More →Given below are two statements: one is labelled as Assertion A and the other is labelled as
Question: Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason $\mathrm{R}$ : Assertion $\mathbf{A}$ : The $\mathrm{H}-\mathrm{O}-\mathrm{H}$ bond angle in water molecule is $104.5^{\circ}$ Reason $\mathbf{R}$ : The lone pair - lone pair repulsion of electrons is higher thanthe bond pair - bond pair repulsion.$A$ is false but $R$ is trueBoth $\mathrm{A}$ and $\mathrm{R}$ are true, but $\mathrm{R}$ is not the correct correct explanation of $\mathrm{A}...
Read More →Given below are two statements: one is labelled as Assertion A and the other is labelled as
Question: Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason $\mathrm{R}$ : Assertion $\mathbf{A}$ : The $\mathrm{H}-\mathrm{O}-\mathrm{H}$ bond angle in water molecule is $104.5^{\circ}$ Reason $\mathbf{R}$ : The lone pair - lone pair repulsion of electrons is higher thanthe bond pair - bond pair repulsion. $A$ is false but $R$ is trueBoth $\mathrm{A}$ and $\mathrm{R}$ are true, but $\mathrm{R}$ is not the correct correct explanation of $\mathrm{A...
Read More →Let alpha and beta be two roots of the equation
Question: Let $\alpha$ and $\beta$ be two roots of the equation $x^{2}+2 x+2=0$, then $\alpha^{15}+\beta^{15}$ is equal to:(1) $-256$(2) 512(3) $-512$(4) 256Correct Option: 1 Solution: Consider the equation $x^{2}+2 x+2=0$ $x=\frac{-2 \pm \sqrt{4-8}}{2}=-1 \pm i$ Let $\alpha=-1+i, \beta=-1-i$ $\alpha^{15}+\beta^{15}=(-1+i)^{15}+(-1-i)^{15}$ $=\left(\sqrt{2} e^{l \frac{3 \pi}{4}}\right)^{15}+\left(\sqrt{2} e^{-i \frac{3 \pi}{4}}\right)^{15}$ $=(\sqrt{2})^{15}\left[e^{\frac{i 45 \pi}{4}}+e^{\frac{...
Read More →If the mean of the data 3, 21, 25, 17, (x + 3), 19, (x – 4) is 18, find the value of x.
Question: If the mean of the data 3, 21, 25, 17, (x+ 3), 19, (x 4) is 18, find the value ofx. Using this value ofx, find the mode of the data. Solution: We know that, Mean $=\frac{\text { Sum of observations }}{\text { Number of observations }}$ The given data is 3, 21, 25, 17, (x+ 3), 19, (x 4). Mean of the given data $=\frac{3+21+25+17+(x+3)+19+(x-4)}{7}$ $\Rightarrow 18(7)=84+2 x$ $\Rightarrow 126-84=2 x$ $\Rightarrow 2 x=42$ $\Rightarrow x=21$ Hence, the value ofxis 21.Now, the given data is...
Read More →Then the sum of the elements in A is:
Question: Let $\mathrm{A}=\left\{\theta \in\left(-\frac{\pi}{2}, \pi\right): \frac{3+2 \mathrm{i} \sin \theta}{1-2 \mathrm{i} \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in $A$ is:(1) $\frac{5 \pi}{6}$(2) $\pi$(3) $\frac{3 \pi}{4}$(4) $\frac{2 \pi}{3}$Correct Option: , 4 Solution: Suppose $z=\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ Since, $z$ is purely imaginary, then $z+\bar{z}=0$ $\Rightarrow \frac{3+2 i \sin \theta}{1-2 i \sin \theta}+\frac{3-2 i \sin \thet...
Read More →The projectile motion of a particle of mass
Question: The projectile motion of a particle of mass $5 \mathrm{~g}$ is shown in the figure. The initial velocity of the particle is $5 \sqrt{2} \mathrm{~ms}^{-1}$ and the air resistance is assumed to be negligible. The magnitude of the change in momentum between the points $\mathrm{A}$ and $\mathrm{B}$ is $\mathrm{x} \times 10^{-2} \mathrm{kgms}^{-1}$. The value of $\mathrm{x}$, to the nearest integer, is_________ Solution: (5) $|\overrightarrow{\mathrm{u}}|=|\overrightarrow{\mathrm{v}}| \ldot...
Read More →A cricket player scored the following runs in 12 one-day matches:
Question: A cricket player scored the following runs in 12 one-day matches:50, 30, 9, 32, 60, 50, 28, 50, 19, 50, 27, 35.Find his modal score. Solution: On arranging the runs in ascending order, we get:9, 19, 27, 28, 30, 32, 35, 50, 50, 50, 50, 60Clearly, 50 occurs maximum number of times. Modal score = 50...
Read More →Calculate the mode of the following sizes of shoes sold by a shop on a particular day.
Question: Calculate the mode of the following sizes of shoes sold by a shop on a particular day.5, 9, 8, 6, 9, 4, 3, 9, 1, 6, 3, 9, 7, 1, 2, 5, 9 Solution: On arranging the shoe sizes in ascending order, we get:1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 9, 9, 9Clearly, 9 occurs maximum number of times. Mode = 9...
Read More →Determine the mode of the following values of a variable.
Question: Determine the mode of the following values of a variable.23, 15, 25, 40, 27, 25, 22, 25, 20 Solution: On arranging the values in ascending order, we get:15, 20, 22, 23, 25, 25, 25, 27, 40Clearly, 25 occurs maximum number of times. Mode = 25...
Read More →Let z
Question: Let $z \in \mathrm{C}$ with $\operatorname{Im}(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2 i-1$ for some natural number $n$. Then :(1) $n=20$ and $\operatorname{Re}(z)=-10$(2) $n=40$ and $\operatorname{Re}(z)=10$(3) $n=40$ and $\operatorname{Re}(z)=-10$(4) $n=20$ and $\operatorname{Re}(z)=10$Correct Option: 3, Solution: Let $\operatorname{Re}(z)=x$ i.e., $z=x+10 i$ $2 z-n=(2 i-1)(2 z+n)$ $(2 x-n)+20 i=(2 i-1)((2 x+n)+20 i)$ On comparing real and imaginary parts, $-(2 x+n)-40=2 x-n$ ...
Read More →An object of mass
Question: An object of mass $\mathrm{m}_{1}$ collides with another object of mass $\mathrm{m}_{2}$, which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses $\mathrm{m}_{2}: \mathrm{m}_{1}$ is :(1) $3: 1$(2) $2: 1$(3) $1: 2$(4) $1: 1$Correct Option: 1 Solution: $\mathrm{m}_{1} \mathrm{v}_{1}=-\mathrm{m}_{1} \mathrm{v}+\mathrm{m}_{2} \mathrm{v}$ $\mathrm{v}_{1}=-\mathrm{v}+\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}} \mathrm{v}$ $\frac{\lef...
Read More →Find the mode of the following items.
Question: Find the mode of the following items.0, 6, 5, 1, 6, 4, 3, 0, 2, 6, 5, 6 Solution: On arranging the items in ascending order, we get:0, 0, 1, 2, 3, 4, 5, 5, 6, 6, 6, 6Clearly, 6 occurs maximum number of times. Mode = 6...
Read More →Find the median of the data
Question: Find the median of the data46, 41, 77, 58, 35, 64, 87, 92, 33, 55, 90.In the above data, if 41 and 55 are replaced by 61 and 75 respectively, what will be the new median? Solution: Arranging the given data in ascending order:33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92Number of terms = 11 (odd) $\therefore$ Median $=\left(\frac{n+1}{2}\right)^{\text {th }}$ term $=\left(\frac{11+1}{2}\right)^{\text {th }}$ term $=6^{\text {th }}$ term $=58$ Hence, the median of the data is 58.Now, In the...
Read More →The major product obtained in the following reaction is:
Question: The major product obtained in the following reaction is: Correct Option: , 3 Solution:...
Read More →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$.............(1) and $\arg \left(\frac{z}{\omega}\right)=\frac{\pi}{2}$.............(2) $\therefore \frac{z}{\omega}+\frac{\bar{z}}{\omega}=0$ $\left[\because \operatorname{Re}\left(\frac{z}{\om...
Read More →The decreasing order of ease of alkaline hydrolysis for the following esters is
Question: The decreasing order of ease of alkaline hydrolysis for the following esters is III II IV I$\mathrm{III}\mathrm{II}\mathrm{I}\mathrm{IV}$$\mathrm{IV}\mathrm{II}\mathrm{III}\mathrm{I}$$\mathrm{II}\mathrm{III}\mathrm{I}\mathrm{IV}$Correct Option: , 2 Solution: Rate of reaction $\propto$ Electrophilicity of carbonyl carbon, so, E.W.G. increases rate while E.R.G. decreases the rate....
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