CAT (Common Admission Test) Quantitative Aptitude (QA) - Algebra Syllabus

Quantitative Aptitude (QA) - Algebra syllabus: all 5 chapters

Expand any chapter to see its topics and sub-topics.

1. 1

   Linear and Quadratic Equations

   3 topics

   • Linear Equations
     • Equations in one and two variables
     • Consistency and unique/infinite/no solutions
     • Integer solutions and word problems
   • Quadratic Equations
     • Nature of roots and discriminant
     • Sum and product of roots
     • Maxima and minima of a quadratic
   • Higher Degree Equations
     • Cubic equations and Vieta's relations
     • Descartes' rule of signs
2. 2

   Inequalities and Modulus

   3 topics

   • Linear and Quadratic Inequalities
     • Wavy curve method
     • Sign analysis of rational expressions
   • Modulus Functions
     • Properties of absolute value
     • Equations and inequalities involving modulus
   • Inequalities of Means
     • AM-GM-HM inequality
     • Cauchy-Schwarz applications
3. 3

   Functions and Graphs

   3 topics

   • Function Basics
     • Domain, range and co-domain
     • Even, odd and periodic functions
     • Composite and inverse functions
   • Graph Transformations
     • Shifting, stretching and reflection
     • Graphs of modulus and greatest integer functions
   • Maxima and Minima
     • Maxima/minima using AM-GM
     • Quadratic and constraint-based optimisation
4. 4

   Logarithms, Indices and Surds

   3 topics

   • Laws of Indices and Surds
     • Rationalisation of surds
     • Comparison of surd magnitudes
   • Logarithm Properties
     • Change of base and laws of logs
     • Characteristic and mantissa
   • Logarithmic Equations and Inequalities
     • Solving log equations
     • Inequalities with variable base
5. 5

   Sequences and Series

   3 topics

   • Arithmetic Progression
     • nth term and sum of n terms
     • Arithmetic mean and inserting means
   • Geometric Progression
     • Sum to n terms and infinite GP
     • Geometric mean
   • Harmonic and Special Series
     • Harmonic progression and AGP
     • Sum of squares and cubes of natural numbers
     • Telescoping series

CAT (Common Admission Test) Quantitative Aptitude (QA) - Algebra flashcards — 50 questions with answers

1. What is the standard form of a linear equation in one variable, and how many solutions does it have?

   ax + b = 0 (a ≠ 0). It has exactly one solution: x = -b/a.

2. For a system of two linear equations a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, what condition gives a unique solution?

   a₁/a₂ ≠ b₁/b₂. The lines intersect at exactly one point.

3. For two linear equations, what condition gives infinitely many solutions, and what gives no solution?

   Infinitely many (coincident lines): a₁/a₂ = b₁/b₂ = c₁/c₂. No solution (parallel lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

4. What is the standard form of a quadratic equation and the quadratic formula for its roots?

   ax² + bx + c = 0 (a ≠ 0). Roots: x = [-b ± √(b² - 4ac)] / (2a).

5. In a quadratic ax²+bx+c=0, what is the discriminant and what does it determine?

   D = b² - 4ac. It determines the nature of roots: D>0 two distinct real roots, D=0 two equal real roots, D<0 two complex conjugate roots.

6. For ax²+bx+c=0 with roots α and β, what are the sum and product of roots?

   Sum: α + β = -b/a. Product: α·β = c/a.

7. How do you form a quadratic equation given its roots α and β?

   x² - (α + β)x + (α·β) = 0, i.e., x² - (sum of roots)x + (product of roots) = 0.

8. For ax²+bx+c=0, when are the roots reciprocals of each other, and when are they equal in magnitude but opposite in sign?

   Reciprocal roots: a = c (product = 1). Equal magnitude/opposite sign: b = 0 (sum = 0).

9. What does it mean for a quadratic with real coefficients to have a complex root, and what is the other root?

   Complex roots occur only in conjugate pairs. If a + bi is a root, then a - bi is also a root.

10. For ax²+bx+c=0, what is the value of α² + β² in terms of coefficients?

    α² + β² = (α+β)² - 2αβ = (b²/a²) - 2c/a = (b² - 2ac)/a².

11. State the Factor Theorem and the Remainder Theorem for polynomials.

    Remainder Theorem: when polynomial P(x) is divided by (x - a), the remainder is P(a). Factor Theorem: (x - a) is a factor of P(x) iff P(a) = 0.

12. For a cubic equation ax³+bx²+cx+d=0 with roots α, β, γ, what are the Vieta relations?

    α+β+γ = -b/a; αβ+βγ+γα = c/a; αβγ = -d/a.

13. How many roots does a polynomial equation of degree n have, and what is Descartes' Rule of Signs?

    It has exactly n roots (counting multiplicity, complex included). Descartes' Rule: the number of positive real roots equals the number of sign changes in P(x), or less by an even number.

14. When solving a linear inequality, what operation reverses the inequality sign?

    Multiplying or dividing both sides by a negative number reverses the inequality sign (e.g., -2x > 4 → x < -2).

15. How do you solve a quadratic inequality like ax²+bx+c > 0 (a>0) using its roots α<β?

    Expression is positive outside the roots: x < α or x > β. It is negative between the roots: α < x < β. (Reverse the intervals if a < 0.)

16. Define |x| (modulus/absolute value) and give the solutions of |x| = a for a > 0.

    |x| = x if x ≥ 0, and -x if x < 0. For |x| = a (a>0): x = a or x = -a.

17. Solve the inequalities |x| < a and |x| > a (a > 0).

    |x| < a means -a < x < a. |x| > a means x < -a or x > a.

18. State the triangle inequality and reverse triangle inequality for moduli.

    Triangle: |x + y| ≤ |x| + |y|. Reverse: |x - y| ≥ ||x| - |y||.

19. What does |x - a| represent geometrically on the number line?

    The distance between x and a. So |x - a| < r means x lies within distance r of a (i.e., a - r < x < a + r).

20. State the AM-GM-HM inequality for positive reals and the condition for equality.

    AM ≥ GM ≥ HM. For positive numbers, arithmetic mean ≥ geometric mean ≥ harmonic mean. Equality holds iff all numbers are equal.

CAT (Common Admission Test) Quantitative Aptitude (QA) - Algebra FAQ