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Have you ever faced a situation where solving an equation led to the square root of a negative number? Do you know how engineers, physicists, and mathematicians deal with this situation? Yes, this is where Complex Numbers come in! This chapter discusses the idea of Complex numbers and quadratic equations. A complex number has both real and imaginary parts. The imaginary parts are represented by i(iota). Students will carry out basic algebraic operations on complex numbers in the same way as on real numbers, such as addition, subtraction, multiplication, and division. The quadratic equations, or second-degree order equations, educate one on how to determine the roots of these equations through the discriminant and quadratic formulas. This article on NCERT solutions for class 11 maths will help students to understand all the concepts more thoroughly.
The NCERT solutions for Class 11 complex numbers and quadratic equations deliver step-by-step processes to solve exercises and simplify complicated matters. In addition to textbook exercises, students are recommended to solve Class 11 Maths Chapter 4 Question Answer to reinforce learning and ensure they are well prepared for exams. Through the use of resources such as the NCERT Exemplar solutions for Class 11 Maths, students can be able to solve more complex problems and enhance their knowledge of both complex numbers and quadratic equations. Students need to refer to the NCERT Class 10 Maths books to gain more knowledge. With consistent practice and a clear understanding of these concepts, students can approach both theoretical and applied mathematics confidently. For syllabus, notes, and PDF, refer to this link: NCERT.
Class 10 Maths chapter 4 solutions Exercise: 4.1 Page number: 82-83 Total questions: 14 |
Question 1: Express the given complex number in the form a+ib: (5i)(-3i/5)?
Answer:
Given $a+i b:(5 i) \times\left(-\frac{3}{5} i\right)$
$(5 i) \times\left(-\frac{3}{5} i\right)=-\left(5 \times \frac{3}{5}\right)\left(i^2\right)$
$\begin{aligned} & =-3\left(i^2\right) \\ & =3 .\end{aligned}$
Question 2: Express each of the complex number in the form $a+ib$ .
Answer:
We know that $i^4 = 1$
Now, we will reduce $i^9+i^{19}$ into
$i^9+i^{19}=\left(i^4\right)^2 \cdot i+\left(i^4\right)^3 \cdot i^3$
$= (1)^2.i+(1)^3.(-i)$ $(\because i^4 = 1 , i^3 = -i\ and \ i^2 = -1)$
$=i-i = 0$
Now, in the form of $a+ib$ we can write it as
$o+io$
Therefore, the answer is $o+io$
Question 3: Express each of the complex number in the form a+ib.
$i^{-39}$
Answer:
We know that $i^4 = 1$
Now, we will reduce $i^{-39}$ into
$i^{-39}$ $= (i^{4})^{-9}.i^{-3}$
$= (1)^{-9}.(-i)^{-1}$ $(\because i^4 = 1 , i^3 = -i)$
$= \frac{1}{-i}$
$= \frac{1}{-i} \times \frac{i}{i}$
$= \frac{i}{-i^2}$ $(\because i^2 = -1)$
$= \frac{i}{-(-1)}$
$=i$
Now, in the form of $a+ib$ we can write it as
$o+i1$
Therefore, the answer is $o+i1$
Question 4: Express each of the complex number in the form a+ib.
$3(7+7i)+i(7+7i)$
Answer:
Given problem is
$3(7+7i)+i(7+7i)$
Now, we will reduce it into
$3(7+7i)+i(7+7i)$ $= 21+21i+7i+7i^2$
$= 21+21i+7i+7(-1)$ $(\because i^2 = -1)$
$= 21+21i+7i-7$
$=14+28i$
Therefore, the answer is $14+i28$
Question 5: Express each of the complex number in the form $a+ib$ .
$(1-i)-(-1+6i)$
Answer:
Given problem is
$(1-i)-(-1+6i)$
Now, we will reduce it into
$(1-i)-(-1+6i)$$=1-i+1-6i$
$= 2-7i$
Therefore, the answer is $2-7i$
Question 6: Express each of the complex number in the form $a+ib$ .
$\left ( \frac{1}{5}+i\frac{2}{5} \right )-\left ( 4+i\frac{5}{2} \right )$
Answer:
Given problem is
$\left ( \frac{1}{5}+i\frac{2}{5} \right )-\left ( 4+i\frac{5}{2} \right )$
Now, we will reduce it into
$\left ( \frac{1}{5}+i\frac{2}{5} \right )-\left ( 4+i\frac{5}{2} \right ) = \frac{1}{5}+i\frac{2}{5}-4-i\frac{5}{2}$
$= \frac{1-20}{5}+i\frac{(4-25)}{10}$
$= -\frac{19}{5}-i\frac{21}{10}$
Therefore, the answer is $-\frac{19}{5}-i\frac{21}{10}$
Question 7: Express each of the complex number in the form $a+ib$ .
$\left [ \left ( \frac{1}{3}+i\frac{7}{3} \right )+\left ( 4+i\frac{1}{3} \right ) \right ]-\left ( -\frac{4}{3}+i \right )$
Answer:
Given problem is
$\left [ \left ( \frac{1}{3}+i\frac{7}{3} \right )+\left ( 4+i\frac{1}{3} \right ) \right ]-\left ( -\frac{4}{3}+i \right )$
Now, we will reduce it into
$\left [ \left ( \frac{1}{3}+i\frac{7}{3} \right )+\left ( 4+i\frac{1}{3} \right ) \right ]-\left ( -\frac{4}{3}+i \right ) = \frac{1}{3}+i\frac{7}{3} + 4+i\frac{1}{3} + \frac{4}{3}-i$
$=\frac{1+4+12}{3}+i\frac{(7+1-3)}{3}$
$=\frac{17}{3}+i\frac{5}{3}$
Therefore, the answer is $\frac{17}{3}+i\frac{5}{3}$
Question 8: Express each of the complex number in the form $a+ib$ .
$(1-i)^4$
Answer:
The given problem is
$(1-i)^4$
Now, we will reduce it into
$(1-i)^4 = ((1-i)^2)^2$
$= (1^2+i^2-2.1.i)^2$ $(using \ (a-b)^2= a^2+b^2-2ab)$
$=(1-1-2i)^2$ $(\because i^2 = -1)$
$= (-2i)^2$
$= 4i^2$
$= -4$
Therefore, the answer is $-4+i0$
Question 9: Express each of the complex number in the form $a+ib$ .
$\left ( \frac{1}{3}+3i \right )^3$
Answer:
Given problem is
$\left ( \frac{1}{3}+3i \right )^3$
Now, we will reduce it into
$\left ( \frac{1}{3}+3i \right )^3=\left ( \frac{1}{3} \right )^3+(3i)^3+3.\left ( \frac{1}{3} \right )^2.3i+3.\frac{1}{3}.(3i)^2$ $(using \ (a+b)^3=a^3+b^3+3a^2b+3ab^2)$
$= \frac{1}{27}+27i^3+i + 9i^2$
$= \frac{1}{27}+27(-i)+i + 9(-1)$ $(\because i^3=-i \ and \ i^2 = -1)$
$=\frac{1}{27}-27i+i-9$
$=\frac{1-243}{27}-26i$
$=-\frac{242}{27}-26i$
Therefore, the answer is
$-\frac{242}{27}-26i$
Question 10: Express each of the complex number in the form $a+ib$ .
$\left ( -2-\frac{1}{3}i \right )^3$
Answer:
Given problem is
$\left ( -2-\frac{1}{3}i \right )^3$
Now, we will reduce it into
$\left ( -2-\frac{1}{3}i \right )^3=-\left ( (2)^3+\left ( \frac{1}{3}i \right )^3 +3.(2)^2\frac{1}{3}i+3.\left ( \frac{1}{3}i \right )^2.2 \right )$ $(using \ (a+b)^3=a^3+b^3+3a^2b+3ab^2)$
$=-\left ( 8+\frac{1}{27}i^3+3.4.\frac{1}{3}i+3.\frac{1}{9}i^2.2 \right )$
$=-\left ( 8+\frac{1}{27}(-i)+4i+\frac{2}{3}(-1) \right )$ $(\because i^3=-i \ and \ i^2 = -1)$
$=-\left ( 8-\frac{1}{27}i+4i-\frac{2}{3} \right )$
$=-\left ( \frac{(-1+108)}{27}i+\frac{24-2}{3} \right )$
$=-\frac{22}{3}-i\frac{107}{27}$
Therefore, the answer is $-\frac{22}{3}-i\frac{107}{27}$
Question 11: Find the multiplicative inverse of each of the complex numbers.
$4-3i$
Answer:
Let $z = 4-3i$
Then,
$\bar z = 4+ 3i$
And
$|z|^2 = 4^2+(-3)^2 = 16+9 =25$
Now, the multiplicative inverse is given by
$z^{-1}= \frac{\bar z}{|z|^2}= \frac{4+3i}{25}= \frac{4}{25}+i\frac{3}{25}$
Therefore, the multiplicative inverse is
$\frac{4}{25}+i\frac{3}{25}$
Question 12: Find the multiplicative inverse of each of the complex numbers.
$\sqrt{5}+3i$
Answer:
Let $z = \sqrt{5}+3i$
Then,
$\bar z = \sqrt{5}-3i$
And
$|z|^2 = (\sqrt5)^2+(3)^2 = 5+9 =14$
Now, the multiplicative inverse is given by
$z^{-1}= \frac{\bar z}{|z|^2}= \frac{\sqrt5-3i}{14}= \frac{\sqrt5}{14}-i\frac{3}{14}$
Therefore, the multiplicative inverse is $\frac{\sqrt5}{14}-i\frac{3}{14}$
Question 13: Find the multiplicative inverse of each of the complex numbers.
$-i$
Answer:
Let $z = -i$
Then,
$\bar z = i$
And
$|z|^2 = (0)^2+(1)^2 = 0+1 =1$
Now, the multiplicative inverse is given by
$z^{-1}= \frac{\bar z}{|z|^2}= \frac{i}{1}= 0+i$
Therefore, the multiplicative inverse is $0+i1$
Question 14: Express the following expression in the form of $a+ib:$
$\frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})}$
Answer:
Given problem is
$\frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})}$
Now, we will reduce it into
$\frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})} = \frac{3^2- (\sqrt5i)^2}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})}$ $(using \ (a-b)(a+b)=a^2-b^2)$
$=\frac{9-5i^2}{\sqrt3+\sqrt2i-\sqrt3+\sqrt2i}$
$=\frac{9-5(-1)}{2\sqrt2i}$ $(\because i^2 = -1)$
$=\frac{14}{2\sqrt2i}\times \frac{\sqrt2i}{\sqrt2i}$
$=\frac{7\sqrt2i}{2i^2}$
$=-\frac{7\sqrt2i}{2}$
Therefore, the answer is $(0-i\frac{7\sqrt2}{2})$
Class 10 Maths chapter 4 solutions Miscellaneous Exercise Page number: 85-86 Total questions: 14 |
Question 1: Evaluate $\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^3$ .
Answer:
The given problem is
$\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^3$
Now, we will reduce it into
$\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^3=\left[\left(i^4\right)^4 \cdot i^2+\frac{1}{\left(i^4\right)^6 \cdot i}\right]^3$
$=\left [ 1^4.(-1)+\frac{1}{1^6.i} \right ]^3$ $(\because i^4 = 1, i^2 = -1 )$
$= \left [ -1+\frac{1}{i} \right ]^3$
$= \left [ -1+\frac{1}{i} \times \frac{i}{i}\right ]^3$
$= \left [ -1+\frac{i}{i^2} \right ]^3$
$= \left [ -1+\frac{i}{-1} \right ]^3 = \left [ -1-i \right ]^3$
Now,
$-(1+i)^3=-(1^3+i^3+3.1^2.i+3.1.i^2)$ $(using \ (a+b)^3=a^3+b^3+3.a^2.b+3.a.b^2)$
$= -(1 - i +3i+3(-1))$ $(\because i^3=-i , i^2 = -1)$
$= -(1 - i +3i-3)= -(-2+2i)$
$=2-2i$
Therefore, answer is $2-2i$
Answer:
Let two complex numbers are
$z_1=x_1+iy_1$
$z_2=x_2+iy_2$
Now,
$z_1.z_2=(x_1+iy_1).(x_2+iy_2)$
$=x_1x_2+ix_1y_2+iy_1x_2+i^2y_1y_2$
$=x_1x_2+ix_1y_2+iy_1x_2-y_1y_2$ $(\because i^2 = -1)$
$=x_1x_2-y_1y_2+i(x_1y_2+y_1x_2)$
$Re(z_1z_2)= x_1x_2-y_1y_2$
$=Re(z_1z_2)-Im(z_1z_2)$
Hence proved
Answer:
Given problem is
$\small \left ( \frac{1}{1-4i}-\frac{2}{1+i} \right )\left ( \frac{3-4i}{5+i} \right )$
Now, we will reduce it into
$\small \left ( \frac{1}{1-4i}-\frac{2}{1+i} \right )\left ( \frac{3-4i}{5+i} \right ) = \left ( \frac{(1+i)-2(1-4i)}{(1+i)(1-4i)} \right )\left ( \frac{3-4i}{5+i} \right )$
$=\left ( \frac{1+i-2+8i}{1-4i+i-4i^2} \right )\left ( \frac{3-4i}{5+i} \right )$
$=\left ( \frac{-1+9i}{1-3i-4(-1)} \right )\left ( \frac{3-4i}{5+i} \right )$
$=\left ( \frac{-1+9i}{5-3i} \right )\left ( \frac{3-4i}{5+i} \right )$
$=\left ( \frac{-3+4i+27i-36i^2}{25+5i-15i-3i^2} \right )= \left ( \frac{-3+31i+36}{25-10i+3} \right )= \frac{33+31i}{28-10i}= \frac{33+31i}{2(14-5i)}$
Now, multiply numerator an denominator by $(14+5i)$
$\Rightarrow \frac{33+31i}{2(14-5i)}\times \frac{14+5i}{14+5i}$
$\Rightarrow \frac{462+165i+434i+155i^2}{2(14^2-(5i)^2)}$ $(using \ (a-b)(a+b)=a^2-b^2)$
$\Rightarrow \frac{462+599i-155}{2(196-25i^2)}$
$\Rightarrow \frac{307+599i}{2(196+25)}= \frac{307+599i}{2\times 221}= \frac{307+599i}{442}= \frac{307}{442}+i\frac{599}{442}$
Therefore, answer is $\frac{307}{442}+i\frac{599}{442}$
Question 4: If $\small x-iy=\sqrt{\frac{a-ib}{c-id}}$ , prove that $\small (x^2+y^2)^2=\frac{a^2+b^2}{c^2+d^2}.$
Answer:
the given problem is
$\small x-iy=\sqrt{\frac{a-ib}{c-id}}$
Now, multiply the numerator and denominator by
$\sqrt{c+id}$
$x-iy = \sqrt{\frac{a-ib}{c-id}\times \frac{c+id}{c+id}}$
$= \sqrt{\frac{(ac+bd)+i(ad-bc)}{c^2-i^2d^2}}= \sqrt{\frac{(ac+bd)+i(ad-bc)}{c^2+d^2}}$
Now, square both the sides
$(x-iy)^2=\left ( \sqrt{\frac{(ac+bd)+i(ad-bc)}{c^2+d^2}} \right )^2$
$=\frac{(ac+bd)+i(ad-bc)}{c^2+d^2}$
$x^2-y^2-2ixy=\frac{(ac+bd)+i(ad-bc)}{c^2+d^2}$
On comparing the real and imaginary part, we obtain
$x^2-y^2 = \frac{ac+bd}{c^2+d^2} \ \ and \ \ -2xy = \frac{ad-bc}{c^2+d^2} \ \ \ -(i)$
Now,
$(x^2+y^2)^2= (x^2-y^2)^2+4x^2y^2$
$= \left ( \frac{ac+bd}{c^2+d^2} \right )^2+\left ( \frac{ad-bc}{c^2+d^2} \right )^2 \ \ \ \ (using \ (i))$
$=\frac{a^2c^2+b^2d^2+2acbd+a^2d^2+b^2c^2-2adbc}{(c^2+d^2)^2}$
$=\frac{a^2c^2+b^2d^2+a^2d^2+b^2c^2}{(c^2+d^2)^2}$
$=\frac{a^2(c^2+d^2)+b^2(c^2+d^2)}{(c^2+d^2)^2}$
$=\frac{(a^2+b^2)(c^2+d^2)}{(c^2+d^2)^2}$
$=\frac{(a^2+b^2)}{(c^2+d^2)}$
Hence proved.
Question 5: If $\small z_1=2-i, z_2=1+i$ , find $\small \left |\frac{z_1+z_2+1}{z_1-z_2+1} \right |$ .
Answer:
It is given that
$\small z_1=2-i, z_2=1+i$
Then,
$\small \left |\frac{z_1+z_2+1}{z_1-z_2+1} \right | =\left | \frac{2-i+1+i+1}{2-i-1-i+1} \right | = \left | \frac{4}{2(1-i)} \right |= \left | \frac{2}{(1-i)} \right |$
Now, multiply the numerator and denominator by $1+i$
$\Rightarrow \left | \frac{2}{(1-i)} \times \frac{1+i}{1+i} \right |=\left |\frac{2(1+i)}{1^2-i^2} \right |=\left | \frac{2(1+i)}{1+1} \right |= \left| 1+i \right |$
Now,
$|1+i| = \sqrt{1^2+1^2}=\sqrt{1+1}=\sqrt{2}$
Therefore, the value of
$\small \left |\frac{z_1+z_2+1}{z_1-z_2+1} \right |$ is $\sqrt{2}$
Question 6: If $\small a+ib=\frac{(x+i)^2}{2x^2+1}$ , prove that $\small a^2+b^2=\frac{(x^2+1)^2}{(2x^2+1)^2}$ .
Answer:
It is given that
$\small a+ib=\frac{(x+i)^2}{2x^2+1}$
Now, we will reduce it into
$\small a+ib=\frac{(x+i)^2}{2x^2+1} = \frac{x^2+i^2+2xi}{2x^2+1}=\frac{x^2-1+2xi}{2x^2+1}=\frac{x^2-1}{2x^2+1}+i\frac{2x}{2x^2+1}$
On comparing real and imaginary part. we will get
$a=\frac{x^2-1}{2x^2+1}\ and \ b=\frac{2x}{2x^2+1}$
Now,
$a^2+b^2=\left ( \frac{x^2-1}{2x^2+1} \right )^2+\left ( \frac{2x}{2x^2+1} \right )^2$
$= \frac{x^4+1-2x^2+4x^2}{(2x^2+1)^2}$
$= \frac{x^4+1+2x^2}{(2x^2+1)^2}$
$= \frac{(x^2+1)^2}{(2x^2+1)^2}$
Hence proved
Question 7 (i): Let $\small z_1=2-i,z_2=-2+i.$ Find
$\small Re\left ( \frac{z_1z_2}{\bar{z_1}} \right )$
Answer:
It is given that
$\small z_1=2-i \ and \ z_2=-2+i$
Now,
$z_1z_2= (2-i)(-2+i)= -4+2i+2i-i^2=-4+4i+1= -3+4i$
And
$\bar z_1 = 2+i$
Now,
$\frac{z_1z_2}{\bar z_1}= \frac{-3+4i}{2+i}= \frac{-3+4i}{2+i}\times \frac{2-i}{2-i}= \frac{-6+3i+8i-4i^2}{2^2-i^2}= \frac{-6+11i+4}{4+1}$ $= \frac{-2+11i}{5}= -\frac{2}{5}+i\frac{11}{5}$
Now,
$Re\left ( \frac{z_1z_2}{z_1} \right )= -\frac{2}{5}$
Therefore, the answer is
$-\frac{2}{5}$
Question 7 (ii): Let $\small z_1=2-i,z_2=-2+i.$ Find
$\small Im\left ( \frac{1}{z_1\bar{z_1}} \right )$
Answer:
It is given that
$z_1= 2-i$
Therefore,
$\bar z_1= 2+i$
NOw,
$z_1\bar z_1= (2-i)(2+i)= 2^2-i^2=4+1=5$ $(using \ (a-b)(a+b)= a^2-b^2)$
Now,
$\frac{1}{z_1\bar z_1}= \frac{1}{5}$
Therefore,
$Im\left ( \frac{1}{z_1\bar z_1} \right )= 0$
Therefore, the answer is 0.
Question 8: Find the real numbers x and y if $\small (x-iy)(3+5i)$ is the conjugate of $\small -6-24i$.
Answer:
Let
$z = \small (x-iy)(3+5i) = 3x+5xi-3yi-5yi^2= 3x+5y+i(5x-3y)$
Therefore,
$\bar z = (3x+5y)-i(5x-3y) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(i)$
Now, it is given that
$\bar z = -6-24i \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(ii)$
Comparing (i) and (ii), we will get
$(3x+5y)-i(5x-3y) = -6-24i$
On comparing the real and imaginary part, we will get
$3x+5y=-6 \ \ \ and \ \ \ 5x-3y = 24$
On solving these we will get
$x = 3 \ \ \ and \ \ \ y =- 3$
Therefore, the values of x and y are 3 and -3 respectively
Question 9: Find the modulus of $\small \frac{1+i}{1-i}-\frac{1-i}{1+i}$ .
Answer:
Let
$z =\small \frac{1+i}{1-i}-\frac{1-i}{1+i}$
Now, we will reduce it into
$z =\small \frac{1+i}{1-i}-\frac{1-i}{1+i} = \frac{(1+i)^2-(1-i)^2}{(1+i)(1-i)}= \frac{1^2+i^2+2i-1^2-i^2+2i}{1^2-i^2}$ $= \frac{4i}{1+1}= \frac{4i}{2}=2i$
Now,
$r\cos\theta = 0 \ \ and \ \ r\sin \theta = 2$
square and add both the sides. we will get,
$r^2\cos^2\theta+r^2\sin^2 \theta = 0^2+2^2$
$r^2(\cos^2\theta+\sin^2 \theta) = 4$
$r^2 = 4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\because \cos^2\theta+\sin^2 \theta = 1)$
$r = 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\because r > 0)$
Therefore, the modulus of
$\small \frac{1+i}{1-i}-\frac{1-i}{1+i}$ is 2
Question 10: If $\small (x+iy)^3=u+iv$ , then show that $\small \frac{u}{x}+\frac{v}{y}=4 (x^2-y^2).$
Answer:
it is given that
$\small (x+iy)^3=u+iv$
Now, expand the Left-hand side
$x^3+(iy)^3+3.(x)^2.iy+3.x.(iy)^2= u + iv$
$x^3+i^3y^3+3x^2iy+3xi^2y^2= u + iv$
$x^3-iy^3+3x^2iy-3xy^2= u + iv$ $(\because i^3 = -i \ \ and \ \ i^2 = -1)$
$x^3-3xy^2+i(3x^2y-y^3)= u + iv$
On comparing real and imaginary part. we will get,
$u = x^3-3xy^2 \ \ \ and \ \ \ v = 3x^2y-y^3$
Now,
$\frac{u}{x}+\frac{v}{y}= \frac{x(x^2-3y^2)}{x}+\frac{y(3x^2-y^2)}{y}$
$= x^2-3y^2+3x^2-y^2$
$= 4x^2-4y^2$
$= 4(x^2-y^2)$
Hence proved
Answer:
Let
$\alpha = a+ib$ and $\beta = x+iy$
It is given that
$\small |\beta|=1\Rightarrow \sqrt{x^2+y^2} = 1\Rightarrow x^2+y^2 = 1$
and
$\small \bar \alpha = a-ib$
Now,
$\small \left | \frac{\beta -\alpha }{1-\bar{\alpha }\beta } \right | = \left | \frac{(x+iy)-(a+ib)}{1-(a-ib)(x+iy)} \right | = \left | \frac{(x-a)+i(y-b)}{1-(ax+iay-ibx-i^2yb)} \right |$
$\small = \left | \frac{(x-a)+i(y-b)}{(1-ax-yb)-i(bx-ay)} \right |$
$\small = \frac{\sqrt{(x-a)^2+(y-b)^2}}{\sqrt{(1-ax-yb)^2+(bx-ay)^2}}$
$\small = \frac{\sqrt{x^2+a^2-2xa+y^2+b^2-yb}}{\sqrt{1+a^2x^2+b^2y^2-2ax+2abxy-by+b^2x^2+a^2y^2-2abxy}}$
$\small = \frac{\sqrt{(x^2+y^2)+a^2-2xa+b^2-yb}}{\sqrt{1+a^2(x^2+y^2)+b^2(x^2+y^2)-2ax+2abxy-by-2abxy}}$
$\small = \frac{\sqrt{1+a^2-2xa+b^2-yb}}{\sqrt{1+a^2+b^2-2ax-by}}$ $\small (\because x^2+y^2 = 1 \ given)$
$\small =1$
Therefore, value of $\small \left | \frac{\beta -\alpha }{1-\bar{\alpha }\beta } \right |$ is 1
Question 12: Find the number of non-zero integral solutions of the equation $\small |1-i|^x=2^x$.
Answer:
Given problem is
$\small |1-i|^x=2^x$
Now,
$( \sqrt{1^2+(-1)^2 })^x=2^x$
$( \sqrt{1+1 })^x=2^x$
$\left ( \sqrt{2 }\right )^x=2^x$
$2^{\frac{x}{2}}= 2^x$
$\frac{x}{2}=x$
$\frac{x}{2}=0$
x = 0 is the only possible solution to the given problem
Therefore, there are 0 number of non-zero integral solutions of the equation $\small |1-i|^x=2^x$
Question 13: If $\small (a+ib)(c+id)(e+if)(g+ih)=A+iB,$ then show that $\small (a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2)=A^2+B^2$
Answer:
It is given that
$\small (a+ib)(c+id)(e+if)(g+ih)=A+iB,$
Now, take mod on both sides
$\left | (a+ib)(c+id)(e+if)(g+ih) \right |= \left | A+iB \right |$
$|(a+ib)||(c+id)||(e+if)||(g+ih)|= \left | A+iB \right |$ $(\because |z_1z_2|=|z_1||z_2|)$
$(\sqrt{a^2+b^2})(\sqrt{c^2+d^2})(\sqrt{e^2+f^2})(\sqrt{g^2+h^2})= (\sqrt{A^2+B^2})$
Square both sides, we will get
$({a^2+b^2})({c^2+d^2})({e^2+f^2})({g^2+h^2})= (A^2+B^2)$
Hence proved
Question 14: If $\small \left ( \frac{1+i}{1-i} \right )^m=1,$ then find the least positive integral value of $\small m$ .
Answer:
Let
$z = \left ( \frac{1+i}{1-i} \right )^m$
Now, multiply both numerator and denominator by $(1+i)$
We will get,
$z = \left ( \frac{1+i}{1-i}\times \frac{1+i}{1+i} \right )^m$
$= \left ( \frac{(1+i)^2}{1^2-i^2} \right )^m$
$= \left ( \frac{1^2+i^2+2i}{1+1} \right )^m$
$= \left ( \frac{1-1+2i}{2} \right )^m$ $(\because i^2 = -1)$
$= \left ( \frac{2i}{2} \right )^m$
$= i^m$
We know that $i^4 = 1$
Therefore, the least positive integral value of $\small m$ is 4.
Also, read,
Question:
If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\frac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then n is equal to ______
Solution:
We have,
$\begin{aligned}
& \alpha=\omega \\
& \therefore\left(\omega^{\mathrm{k}}+\frac{1}{\omega^{\mathrm{k}}}\right)^2=\omega^{2 \mathrm{k}}+\frac{1}{\omega^{2 \mathrm{k}}}+2 \\
& =\omega^{2 \mathrm{k}}+\omega^{\mathrm{k}}+2 \quad \because \omega^{3 \mathrm{k}}=1
\end{aligned}$
$\begin{aligned}
& \therefore \sum_{k=1}^n\left(\omega^{2 k}+\omega^k+2\right)=20 \\
& \Rightarrow\left(\omega^2+\omega^4+\omega^6+\ldots+\omega^{2 n}\right)+\left(\omega+\omega^2+\omega^3+\ldots+\right. \\
& \left.\omega^n\right)+2 n=20
\end{aligned}$
Now if $n=3 m, \quad m \in I$
Then $0+0+2 \mathrm{n}=20 \Rightarrow \mathrm{n}=10$ (not satisfy) if $n=3 m+1$, then
$\begin{aligned}
& \omega^2+\omega+2 n=20 \\
& -1+2 n=20 \Rightarrow n=\frac{21}{2} \text { (not possible) } \\
& \text { if } n=3 m+2, \\
& \left(\omega^8+\omega^{10}\right)+\left(\omega^4+\omega^5\right)+2 n=20 \\
& \Rightarrow\left(\omega^2+\omega\right)+\left(\omega+\omega^2\right)+2 n=20 \\
& 2 n=22
\end{aligned}$
$\begin{aligned} & \mathrm{n}=11 \text { satisfy } \mathrm{n}=3 \mathrm{~m}+2 \\ & \therefore \mathrm{n}=11\end{aligned}$
Hence, the correct answer is 11.
The topics discussed in the NCERT Solutions for class 11, chapter 4, Complex Numbers and Quadratic Equations are:
A complex number is expressed as a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.
Imaginary Numbers: The square root of a negative real number is called an imaginary number, represented as √ − 1 = i.
Equality of Complex Number: Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal if x1 = x2 and y1 = y2.
Addition: (z1 + z2) = (x1 + x2) + i(y1 + y2)
Subtraction: (z1 - z2) = (x1 - x2) + i(y1 - y2)
Multiplication: (z1 * z2) = (x1x2 - y1y2) + i(x1y2 + x2y1)
Division: (z1 / z2) = [(x1x2 + y1y2) + i(x2y1 - x1y2)] / (x22 + y22), where z2 ≠ 0.
Conjugate of Complex Number: The conjugate of a complex number z = x + iy is represented as z¯ = x - iy.
Modulus of a Complex Number: |z| = √(x2 + y2)
Argument of a Complex Number: The angle made by the line joining the point z to the origin, with the positive X-axis in an anti-clockwise sense is called the argument (arg) of the complex number.
When x > 0 and y > 0 ⇒ arg(z) = θ
When x < 0 and y > 0 ⇒ arg(z) = π - θ
When x < 0 and y < 0 ⇒ arg(z) = -(π - θ)
When x > 0 and y < 0 ⇒ arg(z) = -θ
Polar Form of a Complex Number: z = |z| (cosθ + isinθ), where θ = arg(z).
The general polar form of z is z = |z| [cos(2nπ + θ) + isin(2nπ + θ)], where n is an integer.
Here are some approaches that students can use to solve complex numbers and quadratic equation problems.
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The links below allow students to access all the Maths solutions from the NCERT book.
Also, read,
Given below are the subject-wise NCERT solutions of class 11 NCERT:
NCERT solutions for class 11 biology |
NCERT solutions for class 11 maths |
NCERT solutions for class 11 chemistry |
NCERT solutions for class 11 physics |
Here are some useful links for NCERT books and the NCERT syllabus for class 11:
Addition/Subtraction: Combine real & imaginary parts separately.
Multiplication: Distribute like algebra, using $i^2=-1$.
Division: Multiply by the conjugate to simplify.
Modulus: Distance from origin, $|z|=\sqrt{a^2+b^2}$.
Conjugate: Flip the sign of $i$, useful in division & simplifying expressions.
If the discriminant $b^2-4 a c<0$, use $i$ to express the square root:
$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$
Definition: If $z=a+b i$, its conjugate is $\bar{z}=a-b i$.
Its Uses:
Expressed as:
$
z=r(\cos \theta+i \sin \theta) \quad \text { or } \quad z=r e^{i \theta}
$
Where:
- $r=|z|=\sqrt{a^2+b^2}$ (modulus)
- $\theta=\tan ^{-1}(b / a)$ (argument)
Useful for multiplication, division, and powers of complex numbers.
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