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IIT JEE Main Solved Practice Paper Set 1.1: Complex Numbers

Jan 9, 2017 16:32 IST

    Find Solved Practice Paper for IIT JEE Main in this article. This practice paper consists of 10 questions. These questions are from the chapter complex numbers of Mathematics. We have tried to include all important concepts in these questions. Also, detailed solution of each question has been provided. All questions are very important from examination point of view.

    Q1. If z1, z2, z3 ,. z4 are the four complex numbers represented by the vertices of a quadrilateral taken in order such that z1 – z4 =  z2  − z3

           complex number, iit jee, upsee, wbjee

          (a) rhombus                                        

          (b) square

          (c) rectangle                                       

          (d) cyclic quadrilateral

     complex number, iit jee, upsee, wbjee

    Since two angle of the quadrilateral is 90° and the sum of opposite angles of a quadrilateral is 180° so, the quadrilateral ABCD is cyclic quadrilateral.

    UPSEE 2017 Solved Sample Paper Set-1

     complex number, iit jee, upsee, wbjee

    Since all the elements of the second row of the determinant is 0, so the value of determinant is zero.

     complex number, iit jee, upsee, wbjee

     complex number, iit jee, upsee, wbjee

          (a) 3ω

          (b) 3ω (ω − 1)

          (c) 3ω2                                  

          (d) 3ω (1 − ω)

    Correct Option: (b)

    Sol.

    Applying C1C1 + C2 + C3 , then

     complex number, iit jee, upsee, wbjee

     complex number, iit jee, upsee, wbjee

    JEE Main Mathematics Solved Sample Paper Set-VII

     complex number, iit jee, upsee, wbjee

                    (a) Roots are purely imaginary for all negative realvalues of m

                    (b) Roots are not real for all complex numbers m

                    (c) Roots are purely irnaginary for all positive real values of m

                    (d) None of the above

    Correct Option: (a)

    Sol.                 

    Given:

    10z2 -4iz -m = 0

    By using quadratic formula, we get

     complex number, iit jee, upsee, wbjee

    If m is negative, then D will also be negative.

    So, the roots of z will be purely imaginary.

    Hence, the correct option is (a).

    JEE Main Physics Solved Sample Paper Set-VII

    Q6.   Common roots of the equations z3 +2z2 + 2z + 1 = 0 and z1985 + z100 + 1 = 0 are

             (a) ω, ω2         

             (b) ω, ω3

             (c) ω2, ω3        

             (d) None of these

    Sol.

    Correct Option: (a)

    We have equation,

    z3 +2z2 + 2z + 1 = 0

    (z + 1) (z2 + z + 1) = 0

    Roots of the above equations are −1,  ω and ω2

    Let   f(z) = z1985 + z100 + 1

    Putting z = −1, ω and ω2 respectively, we get

    f(−1) = (−1)1985 + (−1)100 + 1 ≠ 0

    Therefore, −1 is not a root of the equation

    Again, f(ω) = ω1985 + ω100+ 1

                       = (ω3)661 ω2 + (ω3)33 ω+1

                      = ω2 +ω + 1 = 0

    Therefore, ω is a root of the equation f(z) = 0.

    Similarly, f2) = 0

    Hence, ω and ω2 are the common roots.

    Q7.    The value of the expression 1×(2 - ω) (2 - ω2) + 2(3 - ω) (3 - ω2)+........+ (n - 1) (n - ω) (n - ω2), where ω is an imaginary cube root of unity is

     complex number, iit jee, upsee, wbjee

    According to the properties of cube root of unity:

     complex number, iit jee, upsee, wbjee

     complex number, iit jee, upsee, wbjee

    Q8.      Let z be a complex number such that the imaginary part of z is non zero and

                a = z2 + z + 1 is real. Then a cannot take the value

                (a) - 1

                (b) 1/3

                (c) 1/2

                (d) 3/4

     complex number, iit jee, upsee, wbjee

     complex number, iit jee, upsee, wbjee

    Correct option: (c)

    Sol.

    We have,

     complex number, iit jee, upsee, wbjee

     complex number, iit jee, upsee, wbjee

                (a) is equal to 5/2

                (b) lies in the interval (1, 2)

                (c) is strictly greater than 5/2

                (d) is strictly greater than 3/2 but less than 5/2

    Sol.

    Correct option: (b)

     complex number, iit jee, upsee, wbjee

    JEE Main Chemistry Solved Sample Paper Set-VII

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