1st Class of MATHS - Banking Exams Quantitative Aptitude

Essential Mathematics for Competitive Exams

Method of Interval and Inequalities

The method of interval is a systematic approach to solve inequalities, especially those involving polynomials and rational expressions. The key is to factorize the expression, analyze the sign changes across intervals, and apply the correct logic based on the inequality sign and the parity of powers.

• Key Steps:
  • Factorize the expression and ensure the coefficient of $x$ is positive.
  • Set each factor to zero to find critical points.
  • Arrange critical points in increasing order on a number line.
  • Start with a positive sign…

Essential Mathematics for Competitive Exams

Method of Interval and Inequalities

The method of interval is a systematic approach to solve inequalities, especially those involving polynomials and rational expressions. The key is to factorize the expression, analyze the sign changes across intervals, and apply the correct logic based on the inequality sign and the parity of powers.

• Key Steps:
  • Factorize the expression and ensure the coefficient of $x$ is positive.
  • Set each factor to zero to find critical points.
  • Arrange critical points in increasing order on a number line.
  • Start with a positive sign on the rightmost interval.
  • Alternate signs at each critical point if the power is odd; keep the sign if even.
  • For $>$ (greater than), select intervals with positive sign; for $<$, select negative.
• Shortcuts:
  • For quadratic inequalities: If $>$, answer is outside the roots; if $<$, answer is between the roots.
  • If a factor is always positive and nonzero, it can be omitted from the inequality.

Quadratic Equations: Roots, Nature, and Properties

Quadratic equations are central to algebra, with their roots' nature and sign determined by discriminant and coefficients. The sum and product of roots provide direct information about their sign and possible values.

• Roots:
  • Sum: $-b/a$
  • Product: $c/a$
• Nature:
  • $D > 0$: Real and distinct roots
  • $D = 0$: Real and equal roots
  • $D < 0$: Complex conjugate roots
• Sign of Roots:
  • Both positive: Sum $> 0$, Product $> 0$
  • Both negative: Sum $< 0$, Product $> 0$
  • One positive, one negative: Product $< 0$
• Special Cases:
  • If sum of coefficients is zero, $x=1$ is a root.
  • If more than two roots, all coefficients must be zero (identity).

Logarithmic and Exponential Functions

Logarithmic and exponential functions are interconnected, with specific domain and range conditions. Their properties are crucial for solving equations and inequalities.

• Domain of $\log_a x$:
  • $x > 0$, $a > 0$, $a \neq 1$
• Range: All real numbers
• Key Properties:
  • $\log_a (mn) = \log_a m + \log_a n$ (if coefficients are 1 and bases are same)
  • $\log_a (m/n) = \log_a m - \log_a n$
  • $k \log_a m = \log_a m^k$
  • Base change: $\log_a b = \frac{\log_c b}{\log_c a}$
  • $a^{\log_a b} = b$
  • If base and argument are interchanged in reciprocal, $\log_a b = 1/\log_b a$
• Exponential Function $y = a^x$:
  • Domain: All real $x$
  • Range: $(0, \infty)$
  • $a > 0$, $a \neq 1$

Logarithmic and Exponential Inequalities

Solving inequalities involving logarithms and exponentials requires careful attention to base and argument conditions, as well as the direction of the inequality.

• For $\log_a x > \log_a y$:
  • If $a > 1$, $x > y$ and $x, y > 0$
  • If $0 < a < 1$, $x < y$ and $x, y > 0$
• For $\log_a x > k$:
  • If $a > 1$, $x > a^k$
  • If $0 < a < 1$, $x < a^k$ and $x > 0$
• For $a^x > a^y$:
  • If $a > 1$, $x > y$
  • If $0 < a < 1$, $x < y$

Modulus and Its Inequalities

The modulus function $|x|$ is used to express the absolute value of $x$. Inequalities involving modulus have direct interval interpretations.

• Key Results:
  • $|x| = a \implies x = \pm a$
  • $|x| > a \implies x < -a$ or $x > a$
  • $|x| < a \implies -a < x < a$
  • For $|x - a| < b$, $a - b < x < a + b$
• If modulus is outside a variable, do not apply direct formula; use case analysis.

Trigonometric Functions: Angles, Quadrants, and Reduction

Trigonometric functions are defined for all real angles, with their signs and values depending on the quadrant. Reduction formulas and quadrant analysis are essential for simplification.

• Angle Measures:
  • $180^\circ = \pi$ radians
  • $1^\circ = \pi/180$ radians
  • To convert: Degrees to radians: multiply by $\pi/180$; radians to degrees: multiply by $180/\pi$
• Quadrant Signs:
  • I: All positive
  • II: $\sin$, $\csc$ positive
  • III: $\tan$, $\cot$ positive
  • IV: $\cos$, $\sec$ positive
• Reduction:
  • $\sin(n\pi \pm \theta) = \pm \sin\theta$
  • $\cos(n\pi \pm \theta) = \mp \cos\theta$
  • For odd multiples of $\pi/2$, sine and cosine convert into each other with sign changes.

Trigonometric Identities and Formulas

A set of standard trigonometric identities is frequently used for simplification and solving equations.

• Sum and Difference:
  • $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
  • $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
  • $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
• Double and Triple Angle:
  • $\sin 2A = 2\sin A \cos A$
  • $\cos 2A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$
  • $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$
  • $\sin 3A = 3\sin A - 4\sin^3 A$
  • $\cos 3A = 4\cos^3 A - 3\cos A$
• Product to Sum:
  • $2\sin A \cos B = \sin(A + B) + \sin(A - B)$
  • $2\cos A \cos B = \cos(A + B) + \cos(A - B)$
  • $2\sin A \sin B = \cos(A - B) - \cos(A + B)$

Trigonometric Equations and Solution Methods

Solving trigonometric equations involves using identities and considering the periodicity and range of the functions.

• Standard Solutions:
  • $\sin x = \sin \alpha \implies x = n\pi + (-1)^n \alpha$
  • $\cos x = \cos \alpha \implies x = 2n\pi \pm \alpha$
  • $\tan x = \tan \alpha \implies x = n\pi + \alpha$
  • For squared equations: $x = n\pi \pm \alpha$
• Counting Solutions:
  • Consider the interval and periodicity to count the number of solutions.

Range and Graphs of Functions

The range of a function is determined by analyzing its algebraic form, often using vertex form for quadratics or by considering the domain of the inner function for composite cases.

• Quadratic $y = ax^2 + bx + c$:
  • If $a > 0$, minimum at $x = -b/2a$, value $y_{min} = -D/4a$
  • If $a < 0$, maximum at $x = -b/2a$, value $y_{max} = -D/4a$
  • For restricted domain $[p, q]$, check if $-b/2a$ lies within $[p, q]$; otherwise, range is between $f(p)$ and $f(q)$.
• For $a\sin x + b\cos x + c$:
  • Range: $[c - \sqrt{a^2 + b^2}, c + \sqrt{a^2 + b^2}]$
• For rational functions (linear/linear):
  • Range: All real except the ratio of leading coefficients.

Special Cases and Common Pitfalls

Certain cases require special attention, such as always positive/negative quadratics, modulus with variable outside, and domain restrictions in logarithmic and trigonometric equations.

• Always Positive/Negative Quadratic:
  • If $a > 0$ and $D < 0$, $ax^2 + bx + c > 0$ for all real $x$.
  • If $a < 0$ and $D < 0$, $ax^2 + bx + c < 0$ for all real $x$.
• Modulus with Variable Outside:
  • Do not apply direct modulus formulas; use case analysis.
• Logarithmic Equations:
  • Always check that the argument and base conditions are satisfied after solving.
• Trigonometric Equations:
  • Consider all possible solutions within the given interval, accounting for periodicity.

Functional Equations and Advanced Results

Some functional equations have standard forms and solutions, especially for logarithmic and exponential types.

• If $f(xy) = f(x) + f(y)$ for all $x, y > 0$ and $f(x) \neq 0$, then $f(x) = k \ln x$
• If $f(x + y) = f(x)f(y)$ for all real $x, y$, then $f(x) = a^x$

These forms are used to identify and solve functional equations in competitive exams.