WBJEE Mathematics - Algebra Syllabus

State the product and quotient laws of logarithms.

$\log_a(mn) = \log_a m + \log_a n$ and $\log_a\!\left(\dfrac{m}{n}\right) = \log_a m - \log_a n$.

State the power law of logarithms.

$\log_a(m^{p}) = p\,\log_a m$. In particular $\log_a \sqrt[n]{m} = \dfrac{1}{n}\log_a m$.

State the change-of-base formula for logarithms.

$\log_a x = \dfrac{\log_b x}{\log_b a}$. A useful special case: $\log_a b = \dfrac{1}{\log_b a}$.

Evaluate $\log_a 1$, $\log_a a$, and $a^{\log_a x}$.

$\log_a 1 = 0$, $\log_a a = 1$, and $a^{\log_a x} = x$.

List the basic laws of indices (exponents) for $a^{m}$ and $a^{n}$.

$a^{m}\cdot a^{n} = a^{m+n}$, $\dfrac{a^{m}}{a^{n}} = a^{m-n}$, $(a^{m})^{n} = a^{mn}$, $a^{0} = 1$, and $a^{-m} = \dfrac{1}{a^{m}}$.

How does the direction of a logarithmic inequality $\log_a x > \log_a y$ depend on the base?

If $a > 1$: $\log_a x > \log_a y \iff x > y$ (preserved). If $0 < a < 1$: $\log_a x > \log_a y \iff x < y$ (reversed). In both cases $x, y > 0$.

Solve the type of inequality $a^{x} > a^{y}$ for the two cases of base $a$.

If $a > 1$: $a^{x} > a^{y} \iff x > y$. If $0 < a < 1$: $a^{x} > a^{y} \iff x < y$ (inequality reverses).

Define the union, intersection, and difference of two sets $A$ and $B$.

Union $A \cup B = \{x : x \in A \text{ or } x \in B\}$; Intersection $A \cap B = \{x : x \in A \text{ and } x \in B\}$; Difference $A - B = \{x : x \in A \text{ and } x \notin B\}$.

State De Morgan's laws for sets.

$(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

State the inclusion-exclusion formula for the cardinality of a union of two sets.

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

How many subsets and how many proper subsets does a set with $n$ elements have?

It has $2^{n}$ subsets (the power set) and $2^{n} - 1$ proper subsets.

Define a relation from set $A$ to set $B$.

A relation $R$ from $A$ to $B$ is any subset of the Cartesian product $A \times B$; i.e., a set of ordered pairs $(a,b)$ with $a \in A,\ b \in B$.

Define reflexive, symmetric, and transitive relations on a set $A$.

Reflexive: $(a,a) \in R$ for all $a \in A$. Symmetric: $(a,b) \in R \Rightarrow (b,a) \in R$. Transitive: $(a,b) \in R$ and $(b,c) \in R \Rightarrow (a,c) \in R$.

What is an equivalence relation?

A relation that is reflexive, symmetric, and transitive simultaneously. It partitions the set into disjoint equivalence classes.

Distinguish injective (one-one), surjective (onto), and bijective functions.

Injective: distinct inputs give distinct outputs ($f(x_1)=f(x_2)\Rightarrow x_1=x_2$). Surjective: every element of the codomain is an image. Bijective: both injective and surjective.

How many functions, and how many one-one functions, exist from a set of $m$ elements to a set of $n$ elements?

Total functions: $n^{m}$. One-one functions (when $n \geq m$): $\dfrac{n!}{(n-m)!} = {}^{n}P_m$.

Define the composition of two functions $f$ and $g$.

$(g \circ f)(x) = g(f(x))$, defined when the range of $f$ lies in the domain of $g$. Composition is associative but generally not commutative.

State the general quadratic equation and the quadratic formula for its roots.

For $ax^{2} + bx + c = 0$ with $a \neq 0$, the roots are $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

Define the discriminant of a quadratic and how it determines the nature of the roots (real coefficients).

$\Delta = b^{2} - 4ac$. If $\Delta > 0$: two distinct real roots; $\Delta = 0$: two equal (repeated) real roots; $\Delta < 0$: two complex conjugate roots.

For real-coefficient quadratics, when are the roots rational versus irrational (given $\Delta > 0$)?

If $a,b,c$ are rational and $\Delta$ is a perfect square, the roots are rational; if $\Delta > 0$ but not a perfect square, the roots are irrational and occur as conjugate surds $p \pm \sqrt{q}$.

State the relations between the roots $\alpha, \beta$ and the coefficients of $ax^{2} + bx + c = 0$.

Sum: $\alpha + \beta = -\dfrac{b}{a}$; Product: $\alpha\beta = \dfrac{c}{a}$.

How do you form a quadratic equation given its roots $\alpha$ and $\beta$?

$x^{2} - (\alpha + \beta)x + \alpha\beta = 0$, i.e. $x^{2} - (\text{sum})x + (\text{product}) = 0$.

Express $\alpha^{2} + \beta^{2}$ and $(\alpha - \beta)^{2}$ in terms of the sum and product of roots.

$\alpha^{2} + \beta^{2} = (\alpha+\beta)^{2} - 2\alpha\beta$ and $(\alpha - \beta)^{2} = (\alpha+\beta)^{2} - 4\alpha\beta$.

What condition makes two quadratics $a_1x^{2}+b_1x+c_1=0$ and $a_2x^{2}+b_2x+c_2=0$ have both roots common?

They are proportional: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$.

State the condition for two quadratics to have exactly one common root.

Eliminating gives $(c_1 a_2 - c_2 a_1)^{2} = (b_1 c_2 - b_2 c_1)(a_1 b_2 - a_2 b_1)$, and the common root is $x = \dfrac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}$.

For the quadratic expression $f(x) = ax^{2} + bx + c$ with $\Delta < 0$, what is the sign of $f(x)$?

$f(x)$ has the same sign as $a$ for all real $x$ (it never changes sign, since there are no real roots).

For $f(x) = ax^{2} + bx + c$ with real roots $\alpha < \beta$, describe the sign of $f(x)$ across the real line.

$f(x)$ has the same sign as $a$ for $x < \alpha$ or $x > \beta$ (outside the roots), and the opposite sign of $a$ for $\alpha < x < \beta$ (between the roots).

State the formulas for the $n$th term and sum of the first $n$ terms of an arithmetic progression (AP).

$n$th term: $T_n = a + (n-1)d$. Sum: $S_n = \dfrac{n}{2}[2a + (n-1)d] = \dfrac{n}{2}(a + l)$, where $l$ is the last term.

State the $n$th term and sum formulas for a geometric progression (GP).

$n$th term: $T_n = ar^{n-1}$. Sum of $n$ terms: $S_n = a\dfrac{r^{n}-1}{r-1}$ (for $r \neq 1$). Sum to infinity: $S_\infty = \dfrac{a}{1-r}$ for $|r| < 1$.

State the relationship between the AM, GM, and HM of two positive numbers, and the AM-GM inequality.

For positive $a,b$: $\text{AM} = \dfrac{a+b}{2}$, $\text{GM} = \sqrt{ab}$, $\text{HM} = \dfrac{2ab}{a+b}$. They satisfy $\text{AM} \geq \text{GM} \geq \text{HM}$ and $\text{GM}^{2} = \text{AM}\times\text{HM}$.