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
(d) cyclic quadrilateral
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.
Since all the elements of the second row of the determinant is 0, so the value of determinant is zero.
(b) 3ω (ω − 1)
(d) 3ω (1 − ω)
Correct Option: (b)
Applying C1→ C1 + C2 + C3 , then
(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)
10z2 -4iz -m = 0
By using quadratic formula, we get
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).
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
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, f(ω2) = 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
According to the properties of cube root of unity:
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
Correct option: (c)
(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
Correct option: (b)