### Math Assignment Class XII Ch -09 | Differential Equations

Math Assignment  Class XII  Ch - 09   Differential Equations Extra questions of chapter 09 Differential Equations, class XII  with answers and  hints to the difficult questions, strictly according to the CBSE Board syllabus. Important and useful math. assignment for the students of class XII

## Math Assignment  Class XI  Ch - 05COMPLEX NUMBERS

Extra questions of chapter 05 COMPLEX NUMBERS class 11  with answers and  hints to the difficult questions, strictly according to the CBSE & DAV Board syllabus. Important and useful math. assignment for the students of class 11

## COMPLEX NUMBERS (XI)

Strictly according to the CBSE and DAV Board

Express the following complex number in a + ib form

Question1:  i+ i10 + i11 + i12.

Question 2:  (7 - 2i) - (4 + i) + (- 3 + 5i)

Question 3: (2 + 3i) (2 - 3i)(1 + i2)

Question 4:

$\frac{\left ( \sqrt{2}+i\sqrt{3} \right )+\left ( \sqrt{2}-i\sqrt{3} \right )}{\left ( \sqrt{3}+i\sqrt{2} \right )+\left ( \sqrt{3}-i\sqrt{2} \right )}$

Answer: $\sqrt{6}/3+0i$

Question 5:

$\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$

Question 6:  If 3 + yi - 2i = x - i. Find y.

Answer: x = 3, y = -1

Question 7: Find the modulus of (1 - i)10.

Question 8: Find the conjugate of  $\frac{(3-2i)(2+3i)}{(1+2i)(2-i)}$

Answer: $\frac{63}{25}+\frac{16}{25}i$

Question 9:

$If\: \: \left ( \frac{1+i}{1-i} \right )^{3}-\left ( \frac{1-i}{1+i} \right )^{3}=x+iy,\: \: find\: \: x,\: y$

Answer: x = 0, y = -2

Question 10:

$Evaluate:\: \: 2x^{3}+2x^{2}-7x+72,\: \: when\: x=\frac{3-5i}{2}$

Solution Hint

$x=\frac{3-5i}{2}$

2x = 3 - 5i

2x - 3 = - 5i

Squaring on both side and arranging all terms on the one side of equal we get

2x2 - 6x + 17 = 0 ...... (i)

Now divide the given cubic equation by eqn. (i) we get quotient = x+4 and remainder = 4

2x3 + 2x2 - 7x + 72 = (2x2 - 6x + 17)(x + 4) + 4

= 0 × (x + 4) + 4

= 4
Question 11: Evaluate: x4 + 4x3 + 6x2 + 4x + 9,   where  x = -1+ i√ 2

Question 12:

If z = 2 - 3i, then show that z2 - 4z + 13 = 0. Hence, find the value of 4z- 3z+ 2z + 170.

Question 13:

Find the real values of x and y if   $\frac{x+iy}{1+i}$  is the conjugate of  5 - i

Answer: x = 4, y = 6

Question 14: Evaluate: $\sqrt{-16}+\sqrt{-25}-\sqrt{-36}+\sqrt{-625}$

Question 15: Find two real numbers x and y if (x – iy) (3 + 5i) is the conjugate of  – 6 – 24i.

Question 16: Write the conjugate of complex number  in the form a + ib.    $\frac{1}{1+i}$

Answer: $\frac{1+i}{2}$

Question 17: If x + iy=   $\frac{(a^{2}+1)^{2}}{2a-i}$ ,  then find the value of x2 + y2.

Answer:  x2 + y2 =   $\frac{(a^{2}+1)^{4}}{4a^{2}+1}$

Question 18: If (x - iy)1/3 = a +ib,  x, y, a ∊ R, then  show that    $\frac{x}{a}-\frac{y}{b}=-2(a^{2}+b^{2})$

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Question 19:

Express the complex numbers in standard form and hence convert it in polar form.

$Z=\frac{1-i}{1+i}-\frac{1+i}{1-i}$

Answer: Z = 0 - 2i

Polar form is :  $2\left [ cos\left ( -\frac{\pi }{2} \right )+isin\left ( -\frac{\pi }{2} \right ) \right ]$

Question 20:

Convert the complex number 2 - 2i into polar form. Also write its argument.

Question 21: Find the square root of the complex number 4 - 4√3i

Question 22:  Find the square root of  Z =  $\frac{-22+19i}{2+i}$

Question 23: Convert the complex number  $3\left [ cos\frac{5\pi }{3}-isin\frac{\pi }{6} \right ]$    into polar form.

Question 24: Find principal argument of the complex number $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$

Answer:  $\theta =\pi /4$

THANKS FOR YPUR VISIT