Let
Question: Let $A=\{2,3,4,5, \ldots, 17,18\}$. Let ' $\simeq$ ' be the equivalence relation on $A \times A$, cartesian product of $A$ with itself, defined by ( $a, b$ ) $\simeq$ $(c, d)$ if $a d=b c$. Then, the number of ordered pairs of the equivalence class of $(3,2)$ is (a) 4(b) 5(c) 6(d) 7 Solution: (c) 6 The ordered pairs of the equivalence class of (3, 2) are {(3, 2), (6, 4), (9, 6), (12, 8), (15, 10), (18, 12)}.We observe that these are 6 pairs....
Read More →If x+iy=
Question: If $x+i y=\frac{a+i b}{a-i b}$, prove that $x^{2}+y^{2}=1$ Solution: $x+i y=\frac{a+i b}{a-i b}$ Taking mod on both the sides: $|x+i y|=\left|\frac{a+i b}{a-i b}\right|$ $\Rightarrow \sqrt{x^{2}+y^{2}}=\frac{\sqrt{a^{2}+b^{2}}}{\sqrt{a^{2}+b^{2}}}$ $\Rightarrow \sqrt{x^{2}+y^{2}}=1$ $\Rightarrow x^{2}+y^{2}=1$ Hence proved....
Read More →If A = {a, b, c}, then the relation R = {(b, c)} on A is
Question: IfA= {a,b,c}, then the relationR= {(b,c)} onAis (a) reflexive only(b) symmetric only(c) transitive only(d) reflexive and transitive only Solution: (c) transitive only The relationR= {(b,c)} is neither reflexive nor symmetric because every element ofAis not related to itself. Also, the ordered pair ofRobtained by interchanging its elements is not contained inR.We observe thatRis transitive onAbecause there is only one pair....
Read More →Find the modulus of
Question: Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$ Solution: $\frac{1+i}{1-i}-\frac{1-i}{1+i}$ $=\frac{(1+i)(1+i)-(1-i)(1-i)}{(1-i)(1+i)}$ $=\frac{1+i^{2}+2 i-1-i^{2}+2 i}{1^{2}-i^{2}}$ $=\frac{4 i}{2} \quad\left(\because i^{2}=-1\right)$ $=2 i$ $\therefore|2 i|=\sqrt{0^{2}+2^{2}}$ $=2 \quad\left(\because|a+b i|=\sqrt{a^{2}+b^{2}}\right)$ $\Rightarrow\left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=2$...
Read More →Write
Question: Write (i) the coefficient of $x^{3}$ in $x+3 x^{2}-5 x^{3}+x^{4}$. (ii) the coefficient of $x$ in $\sqrt{3}-2 \sqrt{2} x+6 x^{2}$. (iii) the coefficient of $x^{2}$ in $2 x-3+x^{3}$. (iv) the coefficient of $x$ in $\frac{3}{8} x^{2}-\frac{2}{7} x+\frac{1}{6}$. (v) the constant term in $\frac{\pi}{2} x^{2}+7 x-\frac{2}{5} \pi$. Solution: (i) The coefficient of $x^{3}$ in $x+3 x^{2}-5 x^{3}+x^{4}$ is $-5$. (ii) The coefficient of $x$ in $\sqrt{3}-2 \sqrt{2} x+6 x^{2}$ is $-2 \sqrt{2}$. (i...
Read More →Let R be the relation over the set of all straight lines in a plane such that
Question: Let $R$ be the relation over the set of all straight lines in a plane such that $I_{1} R I_{2} \Leftrightarrow I_{1} \perp I_{2}$. Then, $R$ is (a) symmetric(b) reflexive(c) transitive(d) an equivalence relation Solution: (a) symmetricA= Set of all straight lines in the plane $R=\left\{\left(l_{1}, l_{2}\right): l_{1}, l_{2} \in A: l_{1} \perp l_{2}\right\}$ Reflexivity: $l_{1}$ is not $\perp l_{1}$ $\Rightarrow\left(l_{1}, l_{1}\right) \notin R$ So, $R$ is not reflexive on $A$. Symmet...
Read More →The following table gives the frequency distribution of married women by age at marriage:
Question: The following table gives the frequency distribution of married women by age at marriage: Calculate the median and interpret the results. Solution: Here, the frequency table is given in inclusive form. So, we first transform it into exclusive form by subtracting and addingh/2 to the lower and upper limits respectively of each class, wherehdenotes the difference of lower limit of a class and upper limit of the previous class. We have,N= 357So,N/2 = 178.5 Thus, the cumulative frequency j...
Read More →Solve the following
Question: If $z_{1}=2-i, z_{2}=-2+i$, find (i) $\operatorname{Re}\left(\frac{z_{1} z_{2}}{z_{1}}\right)$ (ii) $\operatorname{lm}\left(\frac{1}{z_{1} \bar{z}_{1}}\right)$ Solution: (i) $z_{1}=2-i, z_{2}=-2+i, \overline{z_{1}}=2+i$ $\therefore\left(\frac{z_{1} z_{2}}{z_{1}}\right)=\left(\frac{[2-i][-2+i]}{2+i}\right)$ $=\left(\frac{-4+2 i+2 i-i^{2}}{2+i}\right)$ $=\left(\frac{-3+4 i}{2+i}\right)$ $=\left[\frac{-3+4 i}{2+i} \times\left(\frac{2-i}{2-i}\right)\right]$ $=\left(\frac{-6+3 i+8 i-4 i^{2}...
Read More →Solve the following
Question: If $z_{1}=2-i, z_{2}=1+i$, find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|$ Solution: Given: $z_{1}=2-i, z_{2}=1+i$ $\therefore\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|=\left|\frac{2-i+1+i+1}{2-i-1-i+i}\right|$ $=\left|\frac{4}{1-i}\right|$ $=\frac{4}{|1-i|}$ Also, $|1-i|=\sqrt{1^{2}+i^{2}} \quad\left(\because|a+b i|=\sqrt{a^{2}+b^{2}}\right)$ $=\sqrt{2}$ $\therefore\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|=\frac{4}{\sqrt{2}}$...
Read More →An incomplete distribution is given below :
Question: An incomplete distribution is given below : You are given that the median value is 46 and the total number of items is 230.(i) Using the median formula fill up missing frequencies.(ii) Calculate the AM of the completed distribution. Solution: (i) Let the missing frequencies bexandy.First we prepare the following cummulative table. Here, $N=230$ So, $\frac{N}{2}=115$ It is given that the median is 46. Therefore, $40-50$ is the median class. $I=40, f=65, F=42+x$ and $h=10$ We know that M...
Read More →Identify constant, linear, quadratic, cubic and quartic polynomials from the following.
Question: Identify constant, linear, quadratic, cubic and quartic polynomials from the following. (i) $-7+x$ (ii) $6 y$ (iii) $-z^{3}$ (iv) $1-y-y^{3}$ (v) $x-x^{3}+x^{4}$ (vi) $1+x+x^{2}$ (vii) $-6 x^{2}$ (viii) $-13$ (ix) $-p$ Solution: (i) $-7+x$ is a polynomial with degree $1 .$ So, it is a linear polynomial. (ii) $6 y$ is a polynomial with degree $1 .$ So, it is a linear polynomial. (iii) $-z^{3}$ is a polynomial with degree $3 .$ So, it is a cubic polynomial. (iv) $1-y-y^{3}$ is a polynomi...
Read More →Find the multiplicative inverse of the following complex numbers:
Question: Find the multiplicative inverse of the following complex numbers: (i) $1-i$ (ii) $(1+i \sqrt{3})^{2}$ (iii) $4-3 i$ (iv) $\sqrt{5}+3 i$ Solution: (i) Let $z=1-i$. Then, $\frac{1}{z}=\frac{1}{1-i}$ $=\frac{1}{1-i} \times \frac{1+i}{1+i}$ $=\frac{1+i}{1-i^{2}}$ $=\frac{1}{2}(1+i)$ $=\frac{1}{2}+\frac{1}{2} i$ (ii) $z=(1+\sqrt{3} i)^{2}$ $=1+3 i^{2}+2 \sqrt{3} i$ $=-2+2 \sqrt{3} i$ Then, $\frac{1}{z}=\frac{1}{-2+2 \sqrt{3} i} \times \frac{-2-2 \sqrt{3} i}{-2-2 \sqrt{3} i}$ $=\frac{-2-2 \s...
Read More →The relation R defined on the
Question: The relationRdefined on the setA= {1, 2, 3, 4, 5} by $R=\left\{(a, b):\left|a^{2}-b^{2}\right|16\right\}$ is given by (a) $\{(1,1),(2,1),(3,1),(4,1),(2,3)\}$ (b) $\{(2,2),(3,2),(4,2),(2,4)\}$ (c) $\{(3,3),(4,3),(5,4),(3,4)\}$ (d) none of these Solution: (d) none of these R is given by {(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4), (1, 3), (3, 1), (1, 4), (4, 1) ,(2, 4), (4, 2)}, which is not mentioned in (a), (b) or (c)....
Read More →Find the multiplicative inverse of the following complex numbers:
Question: Find the multiplicative inverse of the following complex numbers: (i) $1-i$ (ii) $(1+i \sqrt{3})^{2}$ (iii) $4-3 i$ (iv) $\sqrt{5}+3 i$ Solution: (i) Let $z=1-i$. Then, $\frac{1}{z}=\frac{1}{1-i}$ $=\frac{1}{1-i} \times \frac{1+i}{1+i}$ $=\frac{1+i}{1-i^{2}}$ $=\frac{1}{2}(1+i)$ $=\frac{1}{2}+\frac{1}{2} i$ (ii) $z=(1+\sqrt{3} i)^{2}$ $=1+3 i^{2}+2 \sqrt{3} i$ $=-2+2 \sqrt{3} i$ Then, $\frac{1}{z}=\frac{1}{-2+2 \sqrt{3} i} \times \frac{-2-2 \sqrt{3} i}{-2-2 \sqrt{3} i}$ $=\frac{-2-2 \s...
Read More →R is a relation on the set Z of integers and it is given by
Question: Ris a relation on the setZof integers and it is given by $(x, y) \in R \Leftrightarrow|x-y| \leq 1$. Then, $R$ is (a) reflexive and transitive(b) reflexive and symmetric(c) symmetric and transitive(d) an equivalence relation Solution: (b) reflexive and symmetric Reflexivity: Let $x \in R$. Then, $x-x=01$ $\Rightarrow|x-x| \leq 1$ $\Rightarrow(x, x) \in R$ for all $x \in Z$ So, $R$ is reflexive on $Z$. Symmetry: Let $(x, y) \in R$. Then, $|x-y| \leq 0$ $\Rightarrow|-(y-x)| \leq 1$ $\Rig...
Read More →Which of the following expressions are polynomials? In case of a polynomial, write its degree.
Question: Which of the following expressions are polynomials? In case of a polynomial, write its degree. (i) $x^{5}-2 x^{3}+x+\sqrt{3}$ (ii) $y^{3}+\sqrt{3} y$ (iii) $t^{2}-\frac{2}{5} t+\sqrt{5}$ (iv) $x^{100}-1$ (v) $\frac{1}{\sqrt{2}} x^{2}-\sqrt{2} x+2$ (vi) $x^{-2}+2 x^{-1}+3$ (vii) 1 (viii) $\frac{-3}{5}$ (ix) $\frac{x^{2}}{2}-\frac{2}{x^{2}}$ (x) $\sqrt[3]{2} x^{2}-8$ (xi) $\frac{1}{2 x^{2}}$ (xii) $\frac{1}{\sqrt{5}} x^{\frac{1}{2}}+1$ (xiii) $\frac{3}{5} x^{2}-\frac{7}{3} x+9$ (xiv) $x^...
Read More →Find the conjugates of the following complex numbers:
Question: Find the conjugates of the following complex numbers: (i) $4-5 i$ (ii) $\frac{1}{3+5 i}$ (iii) $\frac{1}{1+i}$ (iv) $\frac{(3-i)^{2}}{2+i}$ (v) $\frac{(1+i)(2+i)}{3+i}$ (vi) $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$ Solution: (i) Let $z=4-5 i$ $\therefore \bar{z}=4+5 i \quad(z=a+i b$, so $\bar{z}=a-i b)$ (ii) Let $z=\frac{1}{3+5 i}$ $=\frac{1}{3+5 i} \times \frac{3-5 i}{3-5 i}$ $=\frac{3-5 i}{9-25 i^{2}}$ $=\frac{3-5 i}{9+25}$ $=\frac{3-5 i}{34}$ $\therefore \bar{z}=\frac{3+5 i}{34}$ (iii)...
Read More →Find the conjugates of the following complex numbers:
Question: Find the conjugates of the following complex numbers: (i) $4-5 i$ (ii) $\frac{1}{3+5 i}$ (iii) $\frac{1}{1+i}$ (iv) $\frac{(3-i)^{2}}{2+i}$ (v) $\frac{(1+i)(2+i)}{3+i}$ (vi) $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$ Solution: (i) Let $z=4-5 i$ $\therefore \bar{z}=4+5 i \quad(z=a+i b$, so $\bar{z}=a-i b)$ (ii) Let $z=\frac{1}{3+5 i}$ $=\frac{1}{3+5 i} \times \frac{3-5 i}{3-5 i}$ $=\frac{3-5 i}{9-25 i^{2}}$ $=\frac{3-5 i}{9+25}$ $=\frac{3-5 i}{34}$ $\therefore \bar{z}=\frac{3+5 i}{34}$ (iii)...
Read More →Find the missing frequencies and the median for the following distribution if the mean is 1.46.
Question: Find the missing frequencies and the median for the following distribution if the mean is 1.46. Solution: (1) Let the missing frequencies bexandy. Given: $N=200$ $86+x+y=200$ $x=114-y \quad \ldots . .(1)$ We know that mean, $\bar{X}=\frac{\sum f_{i} x_{i}}{\sum f_{i}}$ $1.46=\frac{140+x+2 y}{200}$ $x+2 y+140=292$ $x+2 y=152$......(2) Solving (1) and (2), we get $114-y+2 y=152$ $y=38$ Therefore, $x=114-38$ $=76$ Hence, the missing frequencies are 38 and 76. (2) Calculation of median. No...
Read More →If a relation R is defined on the set Z of integers as follows:
Question: If a relationRis defined on the set Z of integers as follows: $(a, b) \in R \Leftrightarrow a^{2}+b^{2}=25$. Then, domain (R) is (a) $\{3,4,5\}$ (b) $\{0,3,4,5\}$ (c) $\{0, \pm 3, \pm 4, \pm 5\}$ (d) none of these Solution: (c) $\{0, \pm 3, \pm 4, \pm 5\}$ $R=\left\{(a, b): a^{2}+b^{2}=25, a, b \in Z\right\}$ $\Rightarrow a \in\{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $b \in\{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ So, domain $(R)=\{0, \pm 3, \pm 4, \pm 5\}$...
Read More →Calculate the missing frequency from the following distribution,
Question: Calculate the missing frequency from the following distribution, it being given that the median of the distribution is 24. Solution: Let the frequency of the class $20-30$ be $f_{1}$. It is given that median is 35 which lies in the class $20-30$. So $20-30$ is the median class. Now, lower limit of median class $(l)=20$ Height of the class $(h)=10$ Frequency of median class $(f)=f_{1}$ Cumulative frequency of preceding median class $(F)=5+25$ Total frequency $(N)=55+f_{1}$ Formula to be...
Read More →Let R be a relation on the set N given by
Question: LetRbe a relation on the setNgiven by R= {(a,b) :a=b 2,b 6}. Then, (a) $(2,4) \in R$ (b) $(3,8) \in R$ (c) $(6,8) \in R$ (d) $(8,7) \in R$ Solution: (c) $(6,8) \in R$ $(6,8) \in R$ Then, $a=b-2$ $\Rightarrow 6=8-2$ and $b=86$ Henc $e,(6,8) \in R$...
Read More →Find the real values of x and y, if
Question: Find the real values ofxandy, if (i) $(x+i y)(2-3 i)=4+i$ (ii) $(3 x-2 i y)(2+i)^{2}=10(1+i)$ (iii) $\frac{(1+i) x-2 i}{3+i}+\frac{(2-3 i) y+i}{3-i}$ (iv) $(1+i)(x+i y)=2-5 i$ Solution: (i) $(x+i y)(2-3 i)=4+i$ $2 x-3 i x+2 i y-3 i^{2} y=4+i$ $2 x+3 y+i(-3 x+2 y)=4+i$ Comparing both the sides: $2 x+3 y=4 \quad \ldots .(1)$ $-3 x+2 y=1 \quad \ldots .(2)$ Multiplying equation (1) by 3 and equation (2) by 2 : $6 x+9 y=12 \quad \ldots(3)$ $-6 x+4 y=2 \quad \ldots(4)$ Adding equations$(3)$ ...
Read More →Find the real values of x and y, if
Question: Find the real values ofxandy, if (i) $(x+i y)(2-3 i)=4+i$ (ii) $(3 x-2 i y)(2+i)^{2}=10(1+i)$ (iii) $\frac{(1+i) x-2 i}{3+i}+\frac{(2-3 i) y+i}{3-i}$ (iv) $(1+i)(x+i y)=2-5 i$ Solution: (i) $(x+i y)(2-3 i)=4+i$ $2 x-3 i x+2 i y-3 i^{2} y=4+i$ $2 x+3 y+i(-3 x+2 y)=4+i$ Comparing both the sides: $2 x+3 y=4 \quad \ldots .(1)$ $-3 x+2 y=1 \quad \ldots .(2)$ Multiplying equation (1) by 3 and equation (2) by 2 : $6 x+9 y=12 \quad \ldots(3)$ $-6 x+4 y=2 \quad \ldots(4)$ Adding equations$(3)$ ...
Read More →An incomplete distribution is given as follows:
Question: An incomplete distribution is given as follows: You are given that the median value is 35 and the sum of all the frequencies is 170. Using the median formula, fill up the missing frequencies. Solution: Let the frequency of the class 20-30 be $f_{1}$ and that of class 40-50 be $f_{2}$. The total frequency is 170. So, $10+20+f_{1}+40+f_{2}+25+15=170$ So, $f_{1}+f_{2}=60$.....(1) It is given that median is 35 which lies in the class 30-40. So 30-40 is the median class. Now, lower limit of...
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