For all z C, prove that
Question: For all z C, prove that (i) $\frac{1}{2}(z+\bar{z})=\operatorname{Re}(z)$ (ii) $\frac{1}{2}(z+\bar{z})=\operatorname{Re}(z)$ (iii) $Z \bar{Z}=|z|^{2}$ (iv) $(z+\bar{z})$ is real (v) $(z-\bar{z})$ is 0 or imaginary. Solution: Let z = a + ib $\Rightarrow \bar{z}=a-i b$ Now,$\frac{z+\bar{z}}{2}=\frac{(a+i b)+(a-i b)}{2}=\frac{2 a}{2}=a=\operatorname{Re}(z)$ Hence Proved. (ii) Let $z=a+i b$ $\Rightarrow \bar{z}=a-i b$ $w, \frac{z+\bar{z}}{2}$ $=\frac{(a+i b)+(a-i b)}{2}$ $=\frac{2 a}{2}=\f...
Read More →The perimeter of a rhombus is 180 cm and one of its diagonals is 72 cm.
Question: The perimeter of a rhombus is 180 cm and one of its diagonals is 72 cm. Solution: Let $A B C D$ be a rhombus whose diagonals $A C$ and $B D$ intersect at a point $O$. Let the length of the diagonal AC be $72 \mathrm{~cm}$ and the side of the rhombus be $x \mathrm{~cm}$. $P$ erimeter of the rhombus $=4 x \mathrm{~cm}$ But it is given that the perimeter of the rhombus is $180 \mathrm{~cm}$. $\therefore 4 x=180$ $\Rightarrow x=\frac{180}{4}$ $\Rightarrow x=45$ Hence, the length of the sid...
Read More →What is the central angle of the sector
Question: What is the central angle of the sector (in the above pie chart) representing hormones enzymes and other proteins. (a) 120o (b) 144o (c) 156o (d) 176o Solution: (b) 144o Distribution of protein in the skin = 1/10 Distribution of protein in the bones = 1/6 Distribution of protein in the Muscles = 1/3 Central angle of the sector representing skin, muscles and bones, = 1/10 + 1/6 + 1/3 = (3 + 5 + 10)/30 = 18/30 360o = 216o Then, Central angle of the sector representing hormones enzymes an...
Read More →What is the central angle of the sector
Question: What is the central angle of the sector (in the above pie chart) representing skin and bones together? (a) 36o (b) 60o (c) 90o (d) 96o Solution: (d) 96o Distribution of protein in the skin = 1/10 Distribution of protein in the bones = 1/6 Central angle of the sector representing skin and bones, = 1/10 + 1/6 = (3 + 5)/30 = 8/30 360o = 96o...
Read More →Find the area of an equilateral triangle of side 6 cm.
Question: Find the area of an equilateral triangle of side 6 cm. Solution: Area of an equilateral traingle $=\left(\frac{\sqrt{3}}{4} \times(\text { side })^{2}\right)$ square units $=\left(\frac{\sqrt{3}}{4} \times 6 \times 6\right) \mathrm{cm}^{2}$ $=\left(\frac{\sqrt{3}}{4} \times 36\right) \mathrm{cm}^{2}$ $=9 \sqrt{3} \mathrm{~cm}^{2}$ Hence, the area of an equilateral triangle is $9 \sqrt{3} \mathrm{~cm}^{2}$....
Read More →Data collected in a survey shows that 40%
Question: Data collected in a survey shows that 40% of the buyers are interested in buying a particular brand of toothpaste. The central angle of the sector of the pie chart representing this information is (a) 120o (b) 150o (c) 144o (d) 40 Solution: (c) 144o = (40/100) 360o = 0.4 360o = 144o...
Read More →The base of a triangular field is three times its height and its area is 1350 m2.
Question: The base of a triangular field is three times its height and its area is 1350 m2. Find the base and height of the field. Solution: Let the base of the triangular field be $3 x \mathrm{~cm}$ and its height be $x \mathrm{~cm}$. Then, area of the triangle $=\left(\frac{1}{2} \times 3 x \times x\right) m^{2}$ $=\frac{3 x^{2}}{2} \mathrm{~m}^{2}$ But it is given that the area of the triangular field is $1350 \mathrm{~m}^{2}$. $\therefore \frac{3 x^{2}}{2}=1350$ $\Rightarrow x^{2}=\left(1350...
Read More →A coin is tossed 200 times
Question: A coin is tossed 200 times and head appeared 120 times. The probability of getting a head in this experiment is (a) 2/5 (b) 3/5 (c) 1/5 (d) 4/5 Solution: (b) 3/5 Probability = Number of times head appeared / Total number of times coin is tossed = 120/200 = 12/20 [divide both numerator and denominator by 4] = 3/5...
Read More →In a frequency distribution with classes
Question: In a frequency distribution with classes 0 10, 10 20 etc., the size of the class intervals is 10. The lower limit of fourth class is (a) 40 (b) 50 (c) 20 (d) 30 Solution: (d) 30 The lower value of the class interval is called its Lower Class Limit. First class = 0 -10 Second class = 10 20 Third class = 20 30 Fourth class = 30 40 Fifth class = 40 50 and Tenth class = 90 100...
Read More →If z1 is a complex number other than –1 such that
Question: If $z_{1}$ is a complex number other than $-1$ such that $\left|z_{1}\right|=1$ and $z_{2}={ }^{\frac{z_{1}-1}{z_{1}+1}}$ then show that $z 2$ is purely imaginary. Solution: Let $z_{1}=a+i b$ such that $\left.\left|z_{1}\right|=\sqrt{(} a^{2}+b^{2}\right)=1$ Now, $z_{2}=\frac{z_{1}-1}{z_{1}+1}=\frac{a+i b-1}{a+i b+1}=\frac{(a-1)+i b}{(a+1)+i b}$ $\Rightarrow \frac{(a-1)+i b}{(a+1)+i b} \times \frac{(a+1)-i b}{(a+1)-i b}$ $=\frac{a^{2}+a-i a b-a-1+i b+i a b+i b-i^{2} b^{2}}{(a+1)^{2}+b^...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer: In the given figure,AB||DCandDAAB. IfDC= 7 cm,BC= 10 cm,AB= 13 cm andCLAB, the area of trap.ABCDis (a) 84 cm2 (b) 72 cm2 (c) 80 cm2 (d) 91 cm2 Solution: (c) $80 \mathrm{~cm}^{2}$ From the given trapezium, we find: $D C=A L=7 \mathrm{~cm} \quad[$ since $D A \perp A B$ and $C L \perp A B]$ From the right $\Delta$ CBL, we have : $C L^{2}=C B^{2}-L B^{2}$ $\Rightarrow C L^{2}=(10)^{2}-(6)^{2}$ $\Rightarrow C L^{2}=100-36$ $\Rightarrow C L^{2}=64$ $\Rightarrow C...
Read More →Ram put some buttons on the table.
Question: Ram put some buttons on the table. There were 4 blue, 7 red, 3 black and 6 white buttons in all. All of a sudden, a cat jumped on the table and knocked out one button on the floor. What is the probability that the button on the floor is blue? (a) 7/20 (b) 3/5 (c) 1/5 (d) Solution: (c) 1/5 Probability = Number of blue buttons on the table / Total number of buttons on the table = 4/20 [divide both numerator and denominator by 4] = 1/5...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: The area of a trapezium is 180 cm2and its height is 9 cm. If one of the parallel sides is longer than the other by 6 cm, the length of the longer of the parallel sides is (a) 17 cm (b) 23 cm (c) 18 cm (d) 24 cm Solution: (b) $23 \mathrm{~cm}$ Let the length of the parallel sides be $x \mathrm{~cm}$ and $(x+6) \mathrm{cm}$, respectively. Then, area of the trapezium $=\left\{\frac{1}{2} \times(x+x+6) \times 9\right\} \mathrm{cm}^{2}$ $=\left\{\frac{1}{2} \tim...
Read More →Listed below are the temperature in oC for 10 days.
Question: Listed below are the temperature inoC for 10 days. 6, 8, 0, 3, 2, 0, 1, 5, 4, 4 What is the range of the data? Solution: (a) 8 (b) 13oC (c) 10oC (d) 12oC (b) 13oC The difference between the lowest and the highest observation in a given data is called its Range. Then, Range = Highest observation Lowest observation = 5 (-8) = 5 + 8 = 13oC...
Read More →Solve this
Question: If $\frac{z-1}{z+1}$ is purely imaginary and $z=-1$, show that $|z|=1$ Solution: Let z= a + ib Now, $\frac{z-1}{z+1}=\frac{a+i b-1}{a+i b+1}$ $=\frac{(a-1)+i b}{(a+1)+i b}$ $\Rightarrow \frac{(a-1)+i b}{(a+1)+i b} \times \frac{(a+1)-i b}{(a+1)-i b}$ $=\frac{a^{2}+a-i a b-a-1+i b+i a b+i b-i^{2} b^{2}}{(a+1)^{2}+b^{2}}$ $=\frac{a^{2}+-1+i b+i b+b^{2}}{(a+1)^{2}+b^{2}}=\frac{a^{2}+b^{2}-1+2 i b}{(a+1)^{2}+b^{2}}$ Given that $\frac{z-1}{z+1}$ is purely imaginary $\Rightarrow$ real part $=...
Read More →Which of the following is a reasonable
Question: Which of the following is a reasonable conclusion for the given data? (a) (1/20)thstudent voted for blue colour (b) Green is the least popular colour (c) The number of students who voted for red colour is two times the number of students who voted for yellow colour (d) Number of students liking together yellow and green colour is approximately the same as those for red colour. Solution: (d) Number of students liking together yellow and green colour is approximately the same as those fo...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: The parallel sides of a trapezium are in the ratio 3 : 4 and the perpendicular distance between them is 12 cm. If the area of the trapezium is 630 cm2, then its shorter of the parallel sides is (a) 45 cm (b) 42 cm (c) 60 cm (d) 36 cm Solution: (a) 45 cm Let the length of the parallel sides be $3 \mathrm{x} \mathrm{cm}$ and $4 \mathrm{x} \mathrm{cm}$, respectively. Then, area of the trapezium $=\left\{\frac{1}{2} \times(3 x+4 x) \times 12\right\} \mathrm{cm}...
Read More →Solve this
Question: If $z^{2}+|z|^{2}=0$, show that $z$ is purely imaginary. Solution: Let $z=a+i b$ $\left.\Rightarrow|z|=\sqrt{(} a^{2}+b^{2}\right)$ Now, $z^{2}+|z|^{2}=0$ $\Rightarrow(a+i b)^{2}+a^{2}+b^{2}=0$ $\Rightarrow a^{2}+2 a b i+i^{2} b^{2}+a^{2}+b^{2}=0$ $\Rightarrow a^{2}+2 a b i-b^{2}+a^{2}+b^{2}=0$ $\Rightarrow 2 a^{2}+2 a b i=0$ $\Rightarrow 2 a(a+i b)=0$ Either a = 0 or z = 0 Since $z \neq 0$ $a=0 \Rightarrow z$ is purely imaginary....
Read More →If 400 students voted in all,
Question: If 400 students voted in all, then how many did vote Others colour as their favorite? (a) 6 (b) 20 (c) 24 (d) 40 Solution: (c) 24 From the pie chart, 6% out of 400 students voted for others, So, = (6%/100%) 400 = 0.06 400 = 24...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: The lengths of the parallel sides of a trapezium are 19 cm and 13 cm and its area is 128 cm2. The distance between the parallel sides is (a) 9 cm (b) 7 cm (c) 8 cm (d) 12.5 cm Solution: (c) 8 cm Let the distance between the parallel sides be $\mathrm{x} \mathrm{cm}$. Then, area of the trapezium $=\left\{\frac{1}{2} \times(19+13) \times \mathrm{x}\right\} \mathrm{cm}^{2}$ $=\left(\frac{1}{2} \times 32 \times x\right) \mathrm{cm}^{2}$ $=16 x \mathrm{~cm}^{2}$...
Read More →Which colour received 1/5
Question: Which colour received 1/5 of the votes? (a) Red (b) Blue (c) Green (d) Yellow Solution: (c) Green = 1/5 100 = 0.2 100 = 20%...
Read More →Find real values of x and y for which
Question: Find real values of x and y for which $\left(x^{4}+2 x i\right)-\left(3 x^{2}+i y\right)=(3-5 i)+(1+2 i y)$ Solution: We have, $\left(x^{4}+2 x i\right)-\left(3 x^{2}+i y\right)=(3-5 i)+(1+2 i y)$ $\Rightarrow x^{4}+2 x i-3 x^{2}+i y=3-5 i+1+2 i y$ $\Rightarrow\left(x^{4}-3 x^{2}\right)+i(2 x-y)=4+i(2 y-5)$ On equating real and imaginary parts, we get $x^{4}-3 x^{2}=4$ and $2 x-y=2 y-5$ $\Rightarrow x^{4}-3 x^{2}-4=0$ eq(i) and $2 x-y-2 y+5=0$ eq(ii) Now from eq (i), $x^{4}-3 x^{2}-4=0...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: The parallel sides of a trapezium measure 14 cm and 18 cm and the distance between them is 9 cm. The area of the trapezium is (a) 96 cm2 (b) 144 cm2 (c) 189 cm2 (d) 207 cm2 Solution: (b) $144 \mathrm{~cm}^{2}$ Area of the trapezium $=\left\{\frac{1}{2} \times(14+18) \times 9\right\} \mathrm{cm}^{2}$ $=\left(\frac{1}{2} \times 32 \times 9\right) \mathrm{cm}^{2}$ $=144 \mathrm{~cm}^{2}$...
Read More →What is the probability of choosing
Question: What is the probability of choosing a vowel from the alphabets? (a) 21/26 (b) 5/26 (c) 1/26 (d) 3/26 Solution: (b) 5/26 Probability = Number of Vowels/ Total number of alphabets = 5/26...
Read More →Which of the following is not a random
Question: Which of the following is not a random experiment? (a) Tossing a coin (b) Rolling a dice (c) Choosing a card from a deck of 52 cards (d) Throwing a stone from a roof of a building Solution: (d) Throwing a stone from a roof of a building. There is only one output that is the stone will fall down therefore it is not a random experiment....
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