JEE Main logo
JEE Main Test Series Complex Numbers and Quadratic Equations, Previous Year Questions with Solutions

JEE Main Complex Numbers and Quadratic Equations is an important chapter. Getting a detailed understanding about the kinds of questions asked from this chapter will help students predict the difficulty level of the questions asked in JEE Main and JEE Advanced exams. Some of the important topics included in this chapter include algebra of complex numbers, modulus and argument, complex conjugate, properties of complex numbers, square root of complex numbers, etc.

Complex numbers and quadratic equations are one of the important chapters that must be included in the preparation of competitive entrance exams like JEE Main. Each year, JEE aspirants can expect 2-4 questions either asked directly and indirectly in JEE Main Question Paper

Previous Year Questions

JEE Main Test series - Previous Year Asked Questions and Solutions

Doing previous year’s question papers help all JEE aspirants in getting a detailed understanding of different style of questions, which all the candidates are supposed to attempt.

Some of the questions asked in last year JEE Main papers are mentioned below:

Question- The number of real values of x for which the equality ∣∣3x2+12x+6∣∣=5x+16 looks good would be [AMU 1999]

A) 4

B) 3

C) 2

D) 1

The correct Answer is C

Question: What will the number of roots of the equation log(−2x) =2log(x+1) are [AMU 2001]

A) 3

B) 2

C) 1

D) None of these

The correct Answer is B.

Question : If (1 + i) (1 + 2i) (1 + 3i) ….. (1 + ni) = a + ib, then 2 * 5 * 10….(1 + n2) would be equal to?

Solution:

 (1 + i) (1 + 2i) (1 + 3i) ….. (1 + ni) = a + ib …..(i)

(1 − i) (1 − 2i) (1 − 3i) ….. (1 − ni) = a − ib …..(ii)

So on multiplying both (i) and (ii), the answer would be 2 * 5 * 10 ….. (1 + n2) = a2 + b2

Question: If two roots of the equation x3−3x+2=0 are same, then the roots will be [MP PET 1985]

A) 2, 2, 3

B) 1, 1, -2

C) - 2, 3, 3

D) -2, -2, 1

The correct option is B

Question : If z is a complex number, then what will be the lowest value of |z| + |z − 1|?

A. 1

B. 2

C. 4

D. 5

Solution:

Here |−z|=|z| and |z1 + z2| ≤ |z1| + |z2|; Now |z| + |z − 1| = |z| + |1 − z| ≥ |z + (1 − z)|

= |1|.Therefore the, minimum value would be 1. The correct option is A.

Question : What will be complex number z that satisfies the equations |\frac{z-12}{z-8i}| = \frac{5}{3}∣z−8iz−12∣=35, |\frac{z-4}{z-8}| = 1∣z−8z−4∣=1

A. 1

B. 2

C. 4

D. 6

Solution:

If we put x = 6 in (i), the answer we will get is y2 − 25y + 136 = 0

y = 17, 8

The correct option is 6. Hence, z = 6 + 17i or z = 6 + 8i

Question : In case cosα + cos β + cos γ = sin α + sin β + sin γ = 0 then find out the value of cos 3α + cos 3β + cos 3γ?

A. 3 cos (α + β + γ).

B. 4 cos (α + β + γ).

C. 5 cos (α + β + γ).

D. 8 cos (α + β + γ).

Solution:

The correct option is A. 3 cos (α + β + γ).

Question : If the cube roots of unity are 1, ω, ω2, then what will be the roots of the equation (x − 1)3 + 8 = 0.

A. x is equal to −1, 1 − 2ω, 1 − 2ω2

B. x = −4, 1 − 2ω, 1 − 2ω2

C. x = −7, 1 − 2ω, 1 − 2ω2

D. x = −9, 1 − 2ω, 1 − 2ω2

Solution:

(x − 1)3 = −8 ⇒ x − 1 = (−8)1/3

x − 1 = −2, −2ω, −2ω2

The correct option is A. x = −1, 1 − 2ω, 1 − 2ω2

Question: If x is equal to a + b, y is equal to aα + bβ and z = aβ + bα, where α and β are considered the complex cube roots of unity, then look at the value of xyz?

A. a3 + b3

B. 2 a3 + b3

C. = 3 a3 + b3

D. = 4 a3 + b3

Solution:

The correct options is A. (a + b) (a2− ab + b2) would be equal to a3 + b3

Quick Formulas

Quick Formulas for Revision

Organic Chemistry is a sub-division of Chemistry that deals with carbon compounds. Learning formulas and preparing short notes is vital for all the JEE aspirants as it helps them come to the answers quickly. Some of the formulas that are important for you to learn in general organic chemistry include:

Complex number: A complex number is defined as the sum of a real number and an imaginary number. For instance, 5 + 2i is a complex number where 5 would be the real part while 2 is the imaginary part.

Algebra : Algebra of complex numbers include addition, subtraction, multiplication and division of complex numbers. For instance, adding 3 + 4i and 9 + 8i is (3+4i) + (9+4i) would be equal to 12 + 12i.

Conjugate of complex numbers: the basic thing about conjugate is that they are different by the sign of imaginary part, if the imaginary part of the number is +ve then it’s conjugate imaginary part becomes -ve.

For example

z = 2 + 2i

z= 2 - 2i

The discriminant of quadratic equations: Discriminant of quadratic equations is explained by D = b2 - 4ac.

If we put + and - one after another, we will get get two required solutions of the equation, if D is lesser than zero, then both roots will be different. In case D is equal to 0, then roots are same and if D<0 then the roots are complex and they exist in pairs of conjugate numbers.

Any equation which is in the form a1x(n-1) + a2x(n-2) + a3x(n-3) + … + an = 0 is a polynomial equation of degree n.

Equality of Complex Number

Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, iff x1 = x2 and y1 = y2 i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2)

Note: Order relation “more than’’ and “less than” are not considered for complex number.

While revising the concepts, all the candidates must understand the derivation of formulas and try to use them on their own. Having a complete understanding of exact formula is helpful in coming to a solution.

The above mentioned formulae play an important role in getting a good score in the exam, candidates must make a list of formula so that they can quickly revise all of them before exams or anytime when you required to revise the chapter. Complex numbers and Quadratic equations are among the most important topics asked in IIT JEE Maths. So give you best while preparing for Complex numbers and quadratic equations that consists of real numbers and the imaginary part usually represented as 'i'.

Comments



No Comments To Show