CAT Algebra and Functions

Algebra is the engine of the Quantitative Ability section in the CAT exam, powering a significant portion of its questions. Success here hinges not on rote memorization, but on a deep, intuitive understanding of how equations, functions, and inequalities behave. Mastering these concepts allows you to deconstruct complex word problems, spot hidden patterns, and execute efficient calculations—skills critical for tackling CAT's challenging time constraints.

Foundational Equations: Linear and Quadratic

Linear equations in one variable form the simplest algebraic relationships, expressed as $ax+b=0$, where $a\ne 0$. The solution is straightforward: $x=-b/a$. In CAT, their power is often applied within word problems involving ages, ratios, or speed-distance-time, where you must first construct the equation from the narrative. For example: "Five years ago, A's age was twice B's age. Ten years later, A's age will be 1.5 times B's age." Setting up and solving the linear equations is the key.

Quadratic equations, of the standard form $a{x}^2+bx+c=0$ ($a\ne 0$), are a CAT staple. You must be fluent in both solving them and analyzing their properties.

• Solving: Factorization, using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, and completing the square.
• Analyzing the Discriminant ($D=b^2-4ac$): This determines the nature of the roots. If $D>0$, roots are real and distinct. If $D=0$, roots are real and equal. If $D<0$, roots are imaginary (a rare but possible CAT scenario).
• Root Properties: For a quadratic $a{x}^2+bx+c=0$ with roots $\alpha$ and $\beta$, the sum of roots $\alpha +\beta =-b/a$ and the product $\alpha \beta =c/a$. These relationships are frequently used to solve problems without explicitly finding the roots.

A typical CAT twist: Given a condition on the roots (e.g., one root is square of the other), use these properties to form a new equation or find a parameter.

Mastering Inequalities and Absolute Values

Inequalities test your understanding of number line logic. The rules for manipulation are similar to equations, with one critical exception: multiplying or dividing by a negative number reverses the inequality sign.

CAT often combines inequalities with absolute value, defined as $|x|=x$ if $x\ge 0$, and $|x|=-x$ if $x<0$. Solving $|expression|\le k$ (where $k\ge 0$) translates to $-k\le expression\le k$. For $|expression|\ge k$, it translates to $expression\le -k$ or $expression\ge k$.

Graphical interpretation is a powerful tool. The solution to an inequality like $x^2-5x+6<0$ is found by identifying where the corresponding quadratic parabola lies below the x-axis. Factor to $(x-2)(x-3)<0$; the parabola opens upward, so it is negative between the roots. Thus, the solution is $2<x<3$.

Functions: Behavior, Types, and Composition

A function $f:A\to B$ is a rule that assigns exactly one output in set $B$ to every input from set $A$. Key concepts include:

• Domain: All possible input values ($A$).
• Range: All possible output values (subset of $B$).
• CAT-relevant Function Types:
• Polynomial Functions: Like $f(x)=a_n{x}^n+...+a_{1}x+a_0$. Quadratics are a special case.
• Modulus Function: $f(x)=|x|$, which creates a V-shaped graph.
• Greatest Integer Function (GIF): $f(x)=[x]$, denoting the greatest integer less than or equal to $x$.

A common question type involves composite functions, where the output of one function becomes the input of another: $(f\circ g)(x)=f(g(x))$. Work from the inside out. Understanding that functions are like machines helps; if $f(x)=x^2+1$ and $g(x)=2x$, then $f(g(x))=(2x{)}^2+1=4x^2+1$.

Logarithms: The Exponent Unpackers

Logarithms are the inverse operations of exponentiation. The statement $a^b=c$ is equivalent to ${\log }_{a}c=b$ (where $a>0,a\ne 1,c>0$). They are indispensable for solving equations where the variable is in an exponent.

You must internalize these core laws:

1. Product Rule: ${\log }_a(mn)={\log }_{a}m+{\log }_{a}n$
2. Quotient Rule: ${\log }_a(m/n)={\log }_{a}m-{\log }_{a}n$
3. Power Rule: ${\log }_a(m^n)=n{\log }_{a}m$
4. Base Change Rule: ${\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}$

CAT questions often involve simplifying expressions using these laws or solving logarithmic equations. A crucial step is always checking the domain—the arguments of all log functions must remain positive throughout the solution process.

Sequences and Series: Recognizing Patterns

CAT tests your ability to identify and work with progressions, which are ordered lists of numbers following a specific rule.

• Arithmetic Progression (AP): Consecutive terms have a constant difference $d$. The $n^{th}$ term is $T_n=a+(n-1)d$. The sum of the first $n$ terms is $S_n=\frac{n}{2}[2a+(n-1)d]$.
• Geometric Progression (GP): Consecutive terms have a constant ratio $r$. The $n^{th}$ term is $T_n=a{r}^{n-1}$. The sum of the first $n$ terms is $S_n=a\frac{(1-r^n)}{(1-r)}$ for $r\ne 1$.

For an infinite GP with $|r|<1$, the sum converges to $S_{\infty }=\frac{a}{1-r}$. Word problems involving installments, growth/decay, or recurring decimals often cloak a GP within them. The key is to extract the first term ($a$) and the common ratio ($r$) from the description.

Common Pitfalls

1. Ignoring Domain Restrictions: This is most fatal with logarithms (argument must be > 0) and denominators (cannot be zero). Always state conditions at the start of your solution. For inequalities, multiplying by an expression without knowing its sign can lead to an incorrect reversal of the inequality.

1. Misapplying Root Properties: A common error is assuming that if the sum of roots is $S$ and product is $P$, the quadratic is always $x^2-Sx+P=0$. This is only true when the leading coefficient $a=1$. The correct general form is $a{x}^2-aSx+aP=0$.

1. Confusing Function Composition Order: Remember, $(f\circ g)(x)$ is not the same as $(g\circ f)(x)$. Function composition is generally not commutative. Always apply the inner function first.

1. Arithmetic Errors in Progressions: When finding the number of terms $(n)$ in an AP using $T_n=a+(n-1)d$, solving for $n$ gives $n=\frac{T_n-a}{d}+1$. Forgetting the "$+1$" is a frequent oversight. Similarly, in GP, ensure you use the correct power: the $n^{th}$ term involves $r^{n-1}$, not $r^n$.

Summary

• Core Tools: Proficiency in solving linear and quadratic equations, along with analyzing roots via the discriminant and sum/product formulas, is non-negotiable for CAT QA.
• Inequality Logic: Solve by algebraic manipulation (mind the sign reversal) or graphical interpretation; absolute value equations must be broken into cases based on the definition.
• Functions as Machines: Understand domain, range, and how to navigate composite functions. Recognize the graph shapes of key functions like modulus and quadratic.
• Logarithm Laws: Use product, quotient, and power rules fluently to simplify or solve equations, while vigilantly enforcing domain conditions.
• Pattern Recognition: AP and GP questions are often embedded in word problems. Correctly identify the first term and common difference/ratio to apply the $n^{th}$ term and sum formulas efficiently.
• Strategic Focus: CAT algebra tests conceptual clarity and application speed. Practice moving seamlessly between pure algebra and its manifestation in complex word problems.