separate into real and imaginary parts arc[cos(3i/4)]

 separation into real and imaginary parts

separate into real and imaginary parts arc[cos(3i/4)]

separate into real and imaginary parts cos (inverse) (3i/4)

 

let arc[cos(3i/4)] = x + iy --------------------------(0)

therefore (3i/4) =cos(x+iy)

cos(x+iy) = (3i/4) 

cosx cos(iy) - sinx sin(iy) =0 + (3i/4) 

cosx coshy - i sinx sinhy =  0 +(3i/4) 

 

using real parts 

cosx coshy = 0----------------------(1)

 

using imaginary parts

-sinx sinhy = (3/4) --------------------(2)

using (1)

cosx =0 or coshy = 0

so 

cosx = 0

or x =(pi/2)---------------------------(3)

 

substituting in (2)

-sinx sinhy = (3/4)

-sin(pi/2)  sinhy = (3/4)

- (1) * sinhy =(3/4)

sinhy = (-3/4)

y = sinh inverse (-3/4)

using sinh inverse (u) = ln[u + sqrt{(u^2) +1} ]

y =ln[(-3/4) + sqrt{ [(-3/4)^2] +1 }]

y = ln[ (-3/4) + sqrt {25/16} ]

y = ln[(-3/4) +(5/4)]

y =ln[2/4] 

y = ln[1/2]

y = ln[2^(-1)]

y = { - ln(2) } ----------------------------(4)

real part =x =pi/2

imaginary part =y = { - ln(2) } or ln[1/2]

 

use eqn(3) and eqn(4) in eqn(0)

arc[cos(3i/4)] = x + iy

 

arc[cos(3i/4)] = (pi/2)  - i ln2

or

 arc[cos(3i/4)] =(pi/2) +i ln(1/2)

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 

solution  

 

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution  

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

 

Solve the following pair of linear equations:

 px + qy = p – q 

 qx – py = p + q

solution

 

(ii) ax + by = c
     bx + ay = 1 + c

solution  

 

(iii) 

(x/a) -(y/b) = 0

ax +by = (a^2)  + (b^2)

solution  

 

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

 

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution   

 ABCD is a cyclic quadrilateral  Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

 

exercise 3.6

2

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. 

solution

 

 (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

 

(iii) 

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

 

solution

 

exercise 3.5

4

A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs.1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs.1180 as hostel charges. Find the fixed charges and the
cost of food per day.

solution

 

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction. 

 solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution 

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution 

 (v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
 

8x + 5y = 9
3x + 2y = 4

solution

 

2

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution 

 

(ii) For which value of k will the following pair of linear equations have no solution?
 

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

  solution

 

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution 

(ii) 

2x + y = 5
3x + 2y = 8

solution 

 

 

(iii) 

3x – 5y = 20
6x – 10y = 40

solution 

 

(iv) 

x – 3y – 7 = 0
3x – 3y – 15 = 0

 solution

exercise 3.4

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. 

solution

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

solution

 

 
 
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