aws - 02 maths

IIT-JEE Achiever 2012-2014 / Mathematics / AWS-02 1

MATHEMATICSMATHEMATICSMATHEMATICSMATHEMATICSClassClassClassClass IIIIIIII IIT-JEEIIT-JEEIIT-JEEIIT-JEE AchieverAchieverAchieverAchiever (2012(2012(2012(2012 ---- 2014)2014)2014)2014)

AdditionalAdditionalAdditionalAdditionalWorkWorkWorkWork SessionSessionSessionSession

WorksheetWorksheetWorksheetWorksheet ---- 02020202

TopicTopicTopicTopic ComplexComplexComplexComplex NumbersNumbersNumbersNumbers

Date:Date:Date:Date: July,July,July,July, 2013201320132013

Multiple choice questions with one correct alternative

1. If ( 1) is a cube root of unity and (1 + )7 = A + B then A and B are respectively

(A) 0, 1 (B) 1, 1 (C) 1, 0 (D) 1, 1

2. If )z1(/)zww( is purely real where, w = + i, 0 and z 1, then the set of the values of z is

(A) {z : |z| = 1} (B) }zz:z{ = (C) {z : z 1} (D) {z : |z| = 1, z 1}

3. If z is a complex number satisfying 3|6i6z| , the minimum and maximum values of arg(z)

are

(A)3

2and

3

(B)

8

3and

8

(C)

12

5and

12

(D) none of these

4. 1, z1, z2, z3,zn 1 are nth roots of unity then the value of1n21 z3

1...

z3

1

z3

1

++

+

is equal to

(A)2

1

13

3nn

1n

+

(B) 113

3nn

1n

(C) 113

3nn

1n

+

(D) none of these

5. Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose

vertices are the roots of the equation 350zzzz 33 =+ is

(A) 48 (B) 32 (C) 40 (D) 80

6. Let z and be two complex numbers such that 2|iz||iz|and1||,1|z| == then z equals

(A) 1 or i (B) i or i (C) 1 or 1 (D) i or 1

7. A man walks a distance of 3 units from the origin towards the north-east (N 45 E) direction. Form there,

he walks a distance of 4 units towards the north-west (N 45 W) direction to reach a point P. Then the

position of P in the Argand plane is

(A) i4e3 4/i + (B) 4/ie)i43( (C) 4/ie)i34( + (D) 4/ie)i43( +

8. Let z1 = a + ib, z2 = c + id, |z1| = 1 = |z2| and 0)zzRe( 21 = . If a, b, c, d R, then which of the following

is not true?

(A) a2 + c2 = 1 (B) b2 + d2 = 1 (C) ac + bd = 0 (D) ac + bd = 1

Multiple choice questions with one or more than one correct alternative/s

9. If (1 + x)n = C0 + C1x ++Cnxn, where n is a positive integer, then

(A)4

ncos2...CCC 2/n420

=+ (B)

4

nsin2....CCC 2/n531

=+


(C)4

ncos22...CCC 2/)2n(2n840

+=+++ (D) none of these

10. x11 + 1 can be expressed as

(A) =

+

+

5

1r

2 122

)1r2(cosx2x)1x( (B)

=

+

++

4

0r

2 122

)1r2(cosx2x)1x(

(C) =

+

++

5

1r

2 122

)1r2(cosx2x)1x( (D)

=

+

+++

4

0r

2 122

)1r2(cosx2x)1x(

11. If cos + cos + cos = sin + sin + sin = 0, then

(A) cos (3) + cos (3) + cos (3) = 3cos ( + + )

(B) sin (3) + sin (3) + sin (3) = 3sin ( + + )

(C) sin (2) + sin (2) + sin (2) = 0

(D) cos (2) + cos (2) + cos (2) = 0.

12. If the imaginary part of the complex number (z 1) (cos i sin ) + (z 1)1 (cos + i sin ) is zero

then

(A) | z 1 | = 1 (B) arg (z 1) =

(C)

=2

)1zarg( (D) 2|1z| =

13. If x, y, a, b are real numbers such thatb

y

a

xpandiba)iyx( 5

1

=+=+ , then

(A) a b is a factor of p (B) a + b is a factor of p

(C) a + ib is a factor of p (D) a ib is a factor of p

Read the passage given below and answer questions 14 and 15 by choosing the

correct alternative/s

Given that z = ( )21

2t3i)t2( ++ , where t is a real number and t2 < 3. Using this information answer the

following:

14.1z

1z

+is

(A) 3 (B) independent of t (C) depends on t (D) 2 + t

15. Locus of z

(A) is a circle of radius 3 (B) is an ellipse with major axis 32

(C) has (2, 0) as its centre (D) has (0, 2) as its centre

Read the passage given below and answer questions 16 to 18 by choosing the

correct alternative

Let A, B, C be three sets of complex numbers as defined below :

A = {z : Im z 1}

B = {z : |z 2 i| = 3}

}2)z)i1Re((:z{C ==

16. The number of elements in the set A B C is


(A) 0 (B) 1 (C) 2 (D)

17. Let z be any point in A B C. Then, |z + 1 i|2 + |z 5 i|2 lies between

(A) 25 and 29 (B) 30 and 34 (C) 35 and 39 (D) 40 and 44

18. Let z be any point in A B C and let w be any point satisfying |w 2 i| < 3. Then, |z| |w| + 3 lies

between

(A) 6 and 3 (B) 3 and 6 (C) 6 and 6 (D) 3 and 9

Numerical Problems

19. If z 0 and 2 + cos + i sin =z

3, then find the value of ( ) 2|z|zz2 +

20. If |z i| 2 and z0 = 5 + 3i find the maximum value of |iz + z0|.

* * *

MultiplechoicequestionswithoneormorethanonReadthepassagegivenbelowandanswerquestionsReadthepassagegivenbelowandanswerquestionsNumericalProblems

aws - 02 maths

Documents