Mathematics (Paper I — MCQ, part of 20 marks with Statistics)¶
Exam reality: MCQs, no calculator, −20% negative marking. Arithmetic/DI items are the reliable marks; calculus/transform items are recognition-level — know definitions, standard results and properties, not long computation.
1. Arithmetic Reasoning (highest-yield)¶
Percentage¶
\[
x\% \text{ of } y = \frac{xy}{100} \qquad \text{\% change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%
\]
- Successive changes of \(a\%\) then \(b\%\): net \(= a + b + \frac{ab}{100}\%\)
- Fraction equivalents (memorize): \(\frac{1}{8} = 12.5\%\), \(\frac{1}{6} = 16.\overline{6}\%\), \(\frac{1}{3} = 33.\overline{3}\%\), \(\frac{3}{8} = 37.5\%\), \(\frac{5}{8} = 62.5\%\)
Ratio and Proportion¶
- \(a : b = ka : kb\); divide amount \(M\) in ratio \(a:b:c\) → shares \(\frac{aM}{a+b+c}\), etc.
- Proportion: \(a:b :: c:d \iff ad = bc\)
- Mixture (alligation): ratio of cheap : dear \(= (d - m) : (m - c)\) where \(m\) = mean price
Average¶
\[
\text{Average} = \frac{\text{sum}}{\text{count}} \qquad \text{new member changes average by } \frac{x_{new} - \bar{x}_{old}}{n+1}
\]
- Average of first \(n\) naturals \(= \frac{n+1}{2}\); consecutive numbers → average = middle value
Profit and Loss¶
\[
\text{Profit\%} = \frac{SP - CP}{CP} \times 100 \qquad SP = CP\left(1 \pm \frac{g}{100}\right)
\]
- Discount is always on marked price: \(SP = MP(1 - d\%)\)
- Two successive discounts \(a\%, b\%\) ≡ single \(a + b - \frac{ab}{100}\%\)
- Sold two items at same price, one +x% and one −x% → overall loss \(= \frac{x^2}{100}\%\)
Time and Work¶
- Rate = \(\frac{1}{\text{days}}\); A + B together: \(\frac{1}{T} = \frac{1}{a} + \frac{1}{b} \Rightarrow T = \frac{ab}{a+b}\)
- \(M_1D_1H_1W_2 = M_2D_2H_2W_1\) (men-days-hours-work)
- Pipes: inlet positive, outlet negative rates
Time, Speed, Distance (supports DI)¶
- \(d = vt\); average speed for equal distances \(= \frac{2v_1v_2}{v_1+v_2}\)
- km/h → m/s: multiply by \(\frac{5}{18}\)
- Trains: crossing pole → own length; crossing platform → sum of lengths; opposite directions → speeds add
Interest (bridges to Financial Management)¶
\[
SI = \frac{PRT}{100} \qquad A = P\left(1 + \frac{r}{100}\right)^t \quad (CI)
\]
- CI − SI for 2 years \(= P\left(\frac{r}{100}\right)^2\)
Data Interpretation & Verification¶
- Read table/bar/pie/line data; compute %, ratio, average, growth rate
- Pie chart: value \(= \frac{\text{sector}°}{360°} \times\) total
- Verification items: check unit consistency, totals matching 100%, impossible percentages — eliminate options first (negative marking)
2. Function and Limit¶
Function Facts¶
- Odd function: \(f(-x) = -f(x)\) (e.g., \(x^3, \sin x, \tan x\)) — graph symmetric about origin
- Even function: \(f(-x) = f(x)\) (e.g., \(x^2, \cos x, |x|\)) — symmetric about y-axis
- odd × odd = even · even × even = even · odd × even = odd
- \(\int_{-a}^{a}(\text{odd}) = 0\); \(\int_{-a}^{a}(\text{even}) = 2\int_0^a\)
Standard Limits (recognition)¶
\[
\lim_{x\to 0}\frac{\sin x}{x} = 1 \qquad \lim_{x\to 0}\frac{1-\cos x}{x^2} = \frac{1}{2} \qquad \lim_{x\to 0}\frac{e^x - 1}{x} = 1
\]
\[
\lim_{x\to 0}\frac{\ln(1+x)}{x} = 1 \qquad \lim_{x\to\infty}\left(1 + \frac{1}{x}\right)^x = e \qquad \lim_{x\to a}\frac{x^n - a^n}{x - a} = na^{n-1}
\]
- L'Hôpital for \(\frac{0}{0}, \frac{\infty}{\infty}\): differentiate top and bottom.
3. Differentiation, Maxima and Minima¶
Standard Derivatives¶
| \(f(x)\) | \(f'(x)\) | \(f(x)\) | \(f'(x)\) | |
|---|---|---|---|---|
| \(x^n\) | \(nx^{n-1}\) | \(\tan x\) | \(\sec^2 x\) | |
| \(e^x\) | \(e^x\) | \(\ln x\) | \(1/x\) | |
| \(\sin x\) | \(\cos x\) | \(a^x\) | \(a^x\ln a\) | |
| \(\cos x\) | \(-\sin x\) | \(\sin^{-1}x\) | \(\frac{1}{\sqrt{1-x^2}}\) |
Rules: product \((uv)' = u'v + uv'\); quotient \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\); chain \(\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}\)
Maxima/Minima Procedure¶
- \(f'(x) = 0\) → critical points
- \(f''(x) < 0\) → maximum; \(f''(x) > 0\) → minimum; \(= 0\) → higher test/inflection
Classic results: \(x + \frac{1}{x}\) has min 2 (\(x>0\)); rectangle of given perimeter with max area = square; \(\max(a\sin x + b\cos x) = \sqrt{a^2+b^2}\)
4. Integration¶
| Integral | Result | Integral | Result | |
|---|---|---|---|---|
| \(\int x^n dx\) | \(\frac{x^{n+1}}{n+1}\) | \(\int \sec^2x\,dx\) | \(\tan x\) | |
| \(\int \frac{dx}{x}\) | \(\ln\|x\|\) | \(\int \frac{dx}{a^2+x^2}\) | \(\frac{1}{a}\tan^{-1}\frac{x}{a}\) | |
| \(\int e^x dx\) | \(e^x\) | \(\int \frac{dx}{\sqrt{a^2-x^2}}\) | \(\sin^{-1}\frac{x}{a}\) | |
| \(\int \sin x\,dx\) | \(-\cos x\) | \(\int_0^\infty e^{-ax}dx\) | \(\frac{1}{a}\) |
- By parts: \(\int u\,dv = uv - \int v\,du\) (choose \(u\) by LIATE)
- Definite: \(\int_0^{\pi/2}\sin^n = \int_0^{\pi/2}\cos^n\) (Wallis recognition)
5. Coordinate Geometry (equation recognition)¶
| Curve | Standard Equation | Key Facts |
|---|---|---|
| Straight line | \(y = mx + c\); \(ax+by+c=0\) | slope \(m=\tan\theta\); parallel: \(m_1 = m_2\); perpendicular: \(m_1m_2 = -1\); distance of point: \(\frac{\|ax_1+by_1+c\|}{\sqrt{a^2+b^2}}\) |
| Circle | \((x-h)^2 + (y-k)^2 = r^2\) | general: \(x^2+y^2+2gx+2fy+c=0\), center \((-g,-f)\), \(r=\sqrt{g^2+f^2-c}\) |
| Parabola | \(y^2 = 4ax\) | focus \((a,0)\), directrix \(x=-a\), \(e = 1\) |
| Ellipse | \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) | \(e<1\), foci \((\pm ae, 0)\), \(b^2 = a^2(1-e^2)\) |
| Hyperbola | \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) | \(e>1\), \(b^2 = a^2(e^2-1)\) |
| Sphere | \((x-a)^2+(y-b)^2+(z-c)^2=R^2\) | center + radius |
| Cylinder | \(x^2+y^2=a^2\) (axis = z) | one variable missing → axis along it |
| Cone | \(x^2+y^2=z^2\tan^2\alpha\) | homogeneous 2nd degree through origin |
MCQ trick: identify conic from general equation \(Ax^2 + Bxy + Cy^2 + \ldots\): \(B^2-4AC < 0\) ellipse/circle, \(=0\) parabola, \(>0\) hyperbola.
6. Differential Equations¶
- Order = highest derivative; degree = power of that derivative (after clearing radicals)
- Linear DE: \(\frac{dy}{dx} + P(x)y = Q(x)\) → integrating factor \(IF = e^{\int P\,dx}\), solution \(y \cdot IF = \int Q \cdot IF\,dx\)
- \(\frac{dy}{dx} = ky \Rightarrow y = Ce^{kx}\) (growth/decay)
- 2nd order: \(y'' + \omega^2 y = 0 \Rightarrow y = A\cos\omega x + B\sin\omega x\) (SHM)
7. Series and Transforms (recognition level — links to Signal Analysis)¶
Taylor / Maclaurin Series¶
\[
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots
\]
\[
e^x = 1 + x + \frac{x^2}{2!} + \cdots \quad \sin x = x - \frac{x^3}{3!} + \cdots \quad \cos x = 1 - \frac{x^2}{2!} + \cdots \quad \frac{1}{1-x} = 1 + x + x^2 + \cdots
\]
Fourier Series (periodic signals)¶
\[
f(t) = a_0 + \sum_{n=1}^{\infty}\left(a_n\cos n\omega_0 t + b_n\sin n\omega_0 t\right)
\]
- Even function → only cosine terms (\(b_n = 0\)); odd → only sine terms
- Half-wave symmetry → no even harmonics
Fourier Transform / Integral (aperiodic signals)¶
\[
X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt
\]
Pairs to recognize: \(\delta(t) \leftrightarrow 1\); \(e^{-at}u(t) \leftrightarrow \frac{1}{a+j\omega}\); convolution ↔ multiplication.
Laplace Transform¶
| \(f(t)\) | \(F(s)\) | \(f(t)\) | \(F(s)\) | |
|---|---|---|---|---|
| \(1\) | \(\frac{1}{s}\) | \(e^{-at}\) | \(\frac{1}{s+a}\) | |
| \(t\) | \(\frac{1}{s^2}\) | \(\sin\omega t\) | \(\frac{\omega}{s^2+\omega^2}\) | |
| \(t^n\) | \(\frac{n!}{s^{n+1}}\) | \(\cos\omega t\) | \(\frac{s}{s^2+\omega^2}\) |
Properties: \(\mathcal{L}\{f'\} = sF(s) - f(0)\); final value \(\lim_{t\to\infty}f = \lim_{s\to 0}sF(s)\).
Z-Transform (discrete)¶
\[
X(z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n}
\]
\(\delta[n] \leftrightarrow 1\); \(u[n] \leftrightarrow \frac{1}{1-z^{-1}}\); \(a^nu[n] \leftrightarrow \frac{1}{1-az^{-1}}\), ROC \(|z|>|a|\); stability ⇔ ROC includes unit circle.
Compare-table (favorite MCQ + Paper II question): FT for spectra of aperiodic continuous signals · LT for continuous system analysis with initial conditions (\(s\)-domain) · ZT for discrete-time systems (\(z\)-domain). (Details: transforms note.)
8. Speed Tactics for the Math MCQs¶
- Attempt arithmetic/DI first — guaranteed marks; calculus items only if the result is a memorized standard.
- Eliminate by properties (odd/even, sign, units, limiting cases) before calculating.
- Options far apart → estimate; options close → compute exactly.
- Skip if 2 options can't be eliminated (−20% rule).
- No calculator: keep fraction↔percent table and squares up to 25, cubes up to 12 memorized.
Key Exam Points — Mathematics
- 20 marks total shared with Statistics; arithmetic + DI are the reliable scores.
- Successive % change \(a + b + \frac{ab}{100}\) and same-price-sale loss \(\frac{x^2}{100}\%\) are recurring MCQ patterns.
- Conic discriminant \(B^2 - 4AC\) classifies the curve instantly.
- Know the FT/LT/ZT definition + 3 pairs each — doubles as Paper II material.
- \(\frac{ab}{a+b}\) for two workers/pipes; \(\frac{2v_1v_2}{v_1+v_2}\) for average speed.