Find the modulus of each of the following:
Question: Find the modulus of each of the following: $(1+2 i)(i-1)$ Solution: Given: z = (1 + 2i)(i 1) Firstly, we calculate the (1 + 2i)(i 1) and then find the modulus So, we open the brackets, $1(\mathrm{i}-1)+2 \mathrm{i}(\mathrm{i}-1)$ $=1(i)+(1)(-1)+2 i(i)+2 i(-1)$ $=i-1+2 i^{2}-2 i$ $=-i-1+2(-1)\left[\because i^{2}=-1\right]$ $=-i-1-2$ $=-i-3$ Now, we have to find the modulus of (-3 - i) So, $|z|=|-3-i|=|-3+(-1) i|=\sqrt{(-3)^{2}+(-1)^{2}}=\sqrt{9+1}=\sqrt{10}$...
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: 5 Solution: Given: z = 5 The above equation can be re written as z = 5 + 0i Now, we have to find the modulus of (5 + 0i) So, $|z|=|5+0 i|=\sqrt{(5)^{2}+(0)^{2}}=5$...
Read More →Every whole number is a rational number.
Question: Every whole number is a rational number. Solution: True Every whole number can be written in the form of $-\frac{p}{q}$, where $p, q$ are integers and $q \neq 0$. Hence, every whole number is a rational number....
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: $\frac{(2-i)(1+i)}{(1+i)}$ Solution: Given: $\frac{(2-i)(1+i)}{(1+i)}$ Firstly, we calculate $\frac{(2-i)(1+i)}{(1+i)}$ and then find its modulus $\frac{(2-i)(1+i)}{(1+i)}=\frac{2(1)+2(i)+(-i)(1)+(-i)(i)}{(1+i)}$ $=\frac{2+2 i-i-i^{2}}{1+i}$ $=\frac{2+i-(-1)}{1+i}\left[\because i^{2}=-1\right]$ $=\frac{3+i}{1+i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 1 + i $=\frac{3+i}{1+i} \times \frac{1-i}{1-i}$ $=\frac{...
Read More →Every whole number is an integer.
Question: Every whole number is an integer. Solution: True. Every whole number is an integer but, every integer is not whole number....
Read More →0 is the smallest rational number
Question: 0 is the smallest rational number Solution: False. Negative rational number below 0 is infinite. So, the smallest rational number does not exist....
Read More →If x and y are two rational numbers
Question: If x and y are two rational numbers such that x y, then x y is always a positive rational number. Solution: True. Let x = 4, y = 2 Then, = x y = 4 2 = 2...
Read More →The negative of a negative rational number
Question: The negative of a negative rational number is a positive rational number. Solution: True. Example, let us take - is a negative rational number. Then negative of negative rational number = (-) = (positive rational number)...
Read More →In the adjacent figure, ABCD is a rectangle.
Question: In the adjacent figure,ABCDis a rectangle. IfBMandDNare perpendiculars fromBandDonAC, prove that∆BMC∆DNA.Is it true thatBM=DN? Solution: Refer to the figure given in the book. In $\Delta B M C$ and $\Delta D N A$ : $\angle D N A=\angle B M C=90^{\circ}$ $\angle B C M=\angle D A N \quad$ (alternate angles) $B C=D A \quad$ (opposite sides) By AAS congruency criteria : $\Delta B M C \cong \Delta D N A$ Yes, it is true that $B M$ is equal to $D N$. (by corresponding parts of congruent tria...
Read More →The rational numbers can be
Question: The rational numbers can be represented on the number line. Solution: True....
Read More →Every integer is a rational number.
Question: Every integer is a rational number. Solution: True. In integer denominator remain 1. So, every integer is a rational number....
Read More →Every fraction is a rational number.
Question: Every fraction is a rational number. Solution: True. Because rational numbers can be expressed in the p/q form and fraction is also a part of whole which can be expressed in the form of p/q....
Read More →The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.
Question: The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides. Solution: Let the lengths of two sides of the parallelogram be $x \mathrm{~cm}$ and $(x+10) \mathrm{cm}$, respectively. Then, its perimeter $=2[x+(x+10)] \mathrm{cm}$ $=2[x+x+10] \mathrm{cm}$ $=2[2 x+10] \mathrm{cm}$ $=4 x+20 \mathrm{~cm}$ $4 x+20=140$ $\Rightarrow 4 x=140-20$ $\Rightarrow 4 x=120$ $\Rightarrow x=\frac{120}{4}$ $\Rightarrow x=30$ Leng...
Read More →The rational numbers
Question: The rational numbers and 1 are on the opposite sides of zero on the number line. Solution: True. is positive rational number so it is lies to the right side of 0 on the number line. -1 is negative rational number so it is lies to the left side of 0 on the number line....
Read More →The rational number -8/-3 lies neither
Question: The rational number -8/-3 lies neither to the right nor to the left of zero on the number line. Solution: False. -8/-3 is written as 8/3 it is a positive rational number. So it is lies to the right side of 0 on the number line....
Read More →The rational number 7/-4
Question: The rational number 7/-4 is lies to the right side zero on the number line. Solution: False. The given rational number is negative so it is lies to the left side of 0 on the number line....
Read More →The rational number 57/23
Question: The rational number 57/23 lies to the left of zero on the number line. Solution: False. The given rational number is positive so it is lies to the right side of 0 on the number line....
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: $\frac{(3+2 i)^{2}}{(4-3 i)}$ Solution: Given: $\frac{(3+2 i)^{2}}{(4-3 i)}$ Firstly, we calculate $\frac{(3+2 i)^{2}}{(4-3 i)}$ and then find its modulus $\frac{(3+2 i)^{2}}{(4-3 i)}=\frac{9+4 i^{2}+12 i}{(4-3 i)}\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=\frac{9+4(-1)+12 i}{4-3 i}\left[\because \mathrm{i}^{2}=-1\right]$ $=\frac{5+12 i}{4-3 i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 4 + 3i $=\frac...
Read More →The reciprocal of
Question: The reciprocal of x-1is 1/x. Solution: False. X-1= 1/x Then, reciprocal of 1/x = x/1 = x...
Read More →Two sides of a parallelogram are in the ratio 5 : 3.
Question: Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides. Solution: Let the lengths of two sides of the parallelogram be $5 x \mathrm{~cm}$ and $3 x \mathrm{~cm}$, respectively. Then, its perimeter $=2(5 x+3 x) \mathrm{cm}$ $=16 x \mathrm{~cm}$ $\therefore 16 x=64$ $\Rightarrow x=\frac{64}{16}$ $\Rightarrow x=4$ Other side $\Rightarrow(3 \times 4) \mathrm{cm}=12 \mathrm{~cm}$...
Read More →There are countless rational
Question: There are countless rational numbers between 5/6 and 8/9. Solution: True....
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: 3i Solution: Given: $z=3 i$ The above equation can be re written as $z=0+3 i$ Now, we have to find the modulus of (0 + 3i) So, $|z|=|0+3 i|=\sqrt{(0)^{2}+(3)^{2}}=\sqrt{9}=3$ Hence, the modulus of (3i) is 3...
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: $(7+24 i)$ Solution: Given: $z=(7+24 i)$ Now, we have to find the modulus of $(7+24 \mathrm{i})$ So, $|z|=|7+24 i|=\sqrt{(7)^{2}+(24)^{2}}=\sqrt{49+576}=\sqrt{625}=25$ Hence, the modulus of (7 + 24i) is 25...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when If $x=a(\theta-\sin \theta)$ and $y=a(1+\cos \theta)$, find $\frac{d y}{d x}$ at $\theta=\frac{\pi}{3}$ Solution: Here, $x=(\theta-\sin \theta)$ and $y=a(1+\cos \theta)$ then, $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}(1-\cos \theta)$ $\frac{\mathrm{dy}}{\mathrm{d} \theta}=\mathrm{a}(-\sin \theta)$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{dx}}{\mathrm{d} \theta}}=\frac{\mathrm{a}(-\si...
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: $(-3-4 i)$ Solution: Given: $z=(-3-4 i)$ Now, we have to find the modulus of $(-3-4 i)$ So, $|z|=|-3-4 i|=\sqrt{(-3)^{2}+(-4)^{2}}=\sqrt{9+16}=\sqrt{25}=5$ Hence, the modulus of $(-3-4 i)$ is 5...
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