WBJEE Mathematics - Algebra Syllabus

Mathematics - Algebra syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics - Algebra in WBJEE, not a summary of it.

1. 1

   Number Systems and Complex Numbers

   3 topics

   • Complex Numbers
     • Algebra of complex numbers, modulus and argument
     • Conjugate, polar form and Euler representation
     • De Moivre's theorem and nth roots of unity
     • Cube roots of unity and their properties
   • Logarithms and Indices
     • Laws of logarithms and change of base
     • Logarithmic and exponential inequalities
   • Sets, Relations and Functions
     • Operations on sets and Venn diagrams
     • Types of relations and equivalence relations
     • Mappings and composition of functions
2. 2

   Equations and Series

   3 topics

   • Quadratic Equations
     • Nature of roots and discriminant
     • Relation between roots and coefficients
     • Formation of equations from given roots
     • Common roots and sign of a quadratic expression
   • Sequences and Series
     • Arithmetic, geometric and harmonic progressions
     • Arithmetic, geometric and harmonic means
     • Sum of special series and infinite GP
   • Binomial Theorem
     • Binomial theorem for positive integral index
     • General and middle terms
     • Properties of binomial coefficients
3. 3

   Combinatorics, Matrices and Determinants

   3 topics

   • Permutations and Combinations
     • Fundamental principle of counting
     • Permutations of distinct and identical objects
     • Combinations and their properties
     • Circular permutations and applications
   • Matrices
     • Types of matrices and operations
     • Transpose, symmetric and skew-symmetric matrices
     • Inverse of a matrix
   • Determinants
     • Determinants up to third order and properties
     • Solution of linear equations by Cramer's rule
     • Consistency of systems of equations
4. 4

   Probability and Statistics

   3 topics

   • Probability
     • Classical and axiomatic definitions
     • Addition and multiplication theorems
     • Conditional probability and Bayes' theorem
     • Random variable and binomial distribution
   • Statistics
     • Measures of central tendency
     • Variance and standard deviation
   • Mathematical Reasoning
     • Statements and logical connectives
     • Tautology and contradiction

Mathematics - Algebra flashcards for WBJEE

18 of 50 cards from the Mathematics - Algebra deck — real questions with worked answers.

1. What is the standard (rectangular) form of a complex number, and what do its parts represent?

   A complex number is written $z = x + iy$, where $x = \operatorname{Re}(z)$ is the real part, $y = \operatorname{Im}(z)$ is the imaginary part, and $i = \sqrt{-1}$ with $i^{2} = -1$.

2. State the rule for adding and multiplying two complex numbers $z_1 = a + ib$ and $z_2 = c + id$.

   Addition: $z_1 + z_2 = (a+c) + i(b+d)$. Multiplication: $z_1 z_2 = (ac - bd) + i(ad + bc)$.

3. How is the modulus $|z|$ of a complex number $z = x + iy$ defined?

   $|z| = \sqrt{x^{2} + y^{2}}$, the distance of the point $(x,y)$ from the origin in the Argand plane.

4. Define the argument (amplitude) of a complex number $z = x + iy$.

   $\arg(z) = \theta$ where $\tan\theta = \dfrac{y}{x}$; it is the angle the line joining the origin to $(x,y)$ makes with the positive real axis. The principal value lies in $(-\pi, \pi]$.

5. What is the conjugate of $z = x + iy$, and what is the product $z\bar{z}$?

   The conjugate is $\bar{z} = x - iy$, and $z\bar{z} = x^{2} + y^{2} = |z|^{2}$.

6. Express $\dfrac{1}{z}$ in terms of the conjugate and modulus of $z$.

   $\dfrac{1}{z} = \dfrac{\bar{z}}{|z|^{2}}$, valid for $z \neq 0$.

7. Write the polar (trigonometric) form of a complex number with modulus $r$ and argument $\theta$.

   $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ and $\theta = \arg(z)$.

8. State Euler's representation of a complex number.

   $z = r e^{i\theta} = r(\cos\theta + i\sin\theta)$, using Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$.

9. How do moduli and arguments combine under multiplication of two complex numbers?

   $|z_1 z_2| = |z_1||z_2|$ and $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$ (modulo $2\pi$).

10. How do moduli and arguments combine under division of two complex numbers?

    $\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}$ and $\arg\!\left(\dfrac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$.

11. State De Moivre's theorem for an integer power $n$.

    $(\cos\theta + i\sin\theta)^{n} = \cos n\theta + i\sin n\theta$ for all integers $n$.

12. Using De Moivre's theorem, write the formula for the $n$th roots of a complex number $r(\cos\theta + i\sin\theta)$.

    $z_k = r^{1/n}\left[\cos\!\dfrac{\theta + 2k\pi}{n} + i\sin\!\dfrac{\theta + 2k\pi}{n}\right]$, for $k = 0, 1, \dots, n-1$, giving $n$ distinct roots.

13. What are the $n$th roots of unity, and what general form do they take?

    They are the $n$ solutions of $z^{n} = 1$, given by $z_k = e^{i 2k\pi/n} = \cos\dfrac{2k\pi}{n} + i\sin\dfrac{2k\pi}{n}$ for $k = 0, 1, \dots, n-1$.

14. What is the sum of all $n$th roots of unity?

    The sum is $0$ (for $n \geq 2$). Their product is $(-1)^{n+1}$.

15. Geometrically, how are the $n$th roots of unity arranged in the Argand plane?

    They lie on the unit circle $|z| = 1$ as vertices of a regular $n$-gon, equally spaced by angle $\dfrac{2\pi}{n}$, with one vertex at $1$.

16. Define the non-real cube roots of unity and the symbol $\omega$.

    The cube roots of unity are $1, \omega, \omega^{2}$ where $\omega = \dfrac{-1 + i\sqrt{3}}{2} = e^{i 2\pi/3}$ and $\omega^{2} = \dfrac{-1 - i\sqrt{3}}{2}$.

17. State the two key identities satisfied by the cube roots of unity.

    $1 + \omega + \omega^{2} = 0$ and $\omega^{3} = 1$ (so $\omega^{2} = \bar{\omega}$ and $\omega^{4} = \omega$).

18. Factorize $a^{3} - b^{3}$ and $a^{3} + b^{3}$ using cube roots of unity.

    $a^{3} - b^{3} = (a-b)(a-\omega b)(a-\omega^{2}b)$ and $a^{3} + b^{3} = (a+b)(a+\omega b)(a+\omega^{2}b)$.

See more Mathematics - Algebra flashcards →

Planning Mathematics - Algebra for WBJEE

Mathematics - Algebra (WBJEE) FAQ