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Coal India Management Trainee (CIL MT)

Practice Questions Study Guide Cheat Sheet Flashcards

4.4 Algebra, Mensuration, Geometry & Data Interpretation (DI)

Q: If α and β are the real roots of the quadratic equation 2x² - 6x + 3 = 0, what is the exact value of the algebraic expression (α/β) + (β/α) + 3(α + β)?

13.0. For 2x^2 - 6x + 3 = 0, we have a = 2, b = -6, c = 3. Sum of roots: \alpha + \beta = -\frac{b}{a} = -\frac{-6}{2} = 3. Product of roots: \alpha \beta = \frac{c}{a} = \frac{3}{2} = 1.5. Simplify the fractional term: \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{(\alpha+\beta)^2 - 2\alpha\beta}{\alpha\beta} = \frac{3^2 - 2(1.5)}{1.5} = \frac{9 - 3}{1.5} = \frac{6}{1.5} = 4 Evaluate the complete expression: \left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right) + 3(\alpha + \beta) = 4 + 3(3) = 4 + 9 = 13.0.

Q: A solid metallic right circular cylinder of base radius 6 cm and height 24 cm is melted down and recast into identical small solid spheres, each of radius 3 cm. Assuming zero loss of metal during the recasting process, how many such spherical balls can be formed?

Use the volume conservation principle (V_{\text{cylinder}} = N \times V_{\text{sphere}}): Volume of cylinder: V_{\text{cyl}} = \pi r^2 h = \pi \times (6)^2 \times 24 = \pi \times 36 \times 24 = 864\pi\text{ cm}^3 Volume of one sphere: V_{\text{sph}} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi \times (3)^3 = \frac{4}{3}\pi \times 27 = 36\pi\text{ cm}^3 Total number of spheres (N): N = \frac{V_{\text{cyl}}}{V_{\text{sph}}} = \frac{864\pi}{36\pi} = 24.

Q: In a Coal India dispatch pie chart representing total subsidiary production of 180 million tonnes (MT), the central sector angle allocated to South Eastern Coalfields Limited (SECL) is 108°, while Northern Coalfields Limited (NCL) is allocated 72°. What is the absolute difference in coal production between SECL and NCL, and what percentage of total production does SECL represent?

Difference: 18 MT; SECL Share: 30.0%. 1. In a pie chart, 360^{\circ} corresponds to total production (180 MT). \text{Tonnage per degree} = \frac{180\text{ MT}}{360^{\circ}} = 0.5\text{ MT per degree} Difference in sector angles between SECL and NCL: \Delta \theta = 108^{\circ} - 72^{\circ} = 36^{\circ} Absolute difference in production: \text{Difference} = 36^{\circ} \times 0.5\text{ MT/deg} = 18\text{ MT} Percentage share of SECL: \text{SECL Share} = \left(\frac{108^{\circ}}{360^{\circ}}\right) \times 100\% = \frac{3}{10} \times 100\% = 30.0\%.

Key Takeaways

Quadratic roots are governed by discriminant $\Delta = b^2 - 4ac$ , sum $\alpha + \beta = -\frac{b}{a}$ , and product $\alpha\beta = \frac{c}{a}$ , allowing quick equation construction and sign analysis.
Symmetric algebraic identities and reciprocal powers ( $x + \frac{1}{x} = k \implies x^2 + \frac{1}{x^2} = k^2 - 2, x^3 + \frac{1}{x^3} = k^3 - 3k$ ) simplify high-degree polynomials in competitive tests.
2D and 3D mensuration relies on fundamental volume and surface area relationships (cylinders $\pi r^2 h$ , cones $\frac{1}{3}\pi r^2 h$ , spheres $\frac{4}{3}\pi r^3$ ), governed by the principle of volume conservation during metal/material recasting.
Data Interpretation (DI) requires mastery of tabular data, bar graphs, line charts, and pie chart angle conversions ( $1\% = 3.6^{\circ}$ ), emphasizing percentage growth, year-on-year analysis, and ratio evaluation.
Speed in DI is achieved by calculating approximate percentages, using common denominators, and avoiding complete manual arithmetic when options have distinct divergence.

Last updated: August 2026

Algebra, Mensuration, Geometry & Data Interpretation (DI)

Quantitative Aptitude in CIL MT Paper-I culminates in advanced algebraic relationships, 2D/3D mensuration geometry, and Data Interpretation (DI). Engineers are expected to interpret graphical outputs, calculate geometrical material clearances, and manipulate polynomial equations with precision.

1. Algebraic Identities & Quadratic Polynomials

Core Polynomial Identities

Standard Binomial & Trinomial Expansions:
$(a \pm b)^2 = a^2 \pm 2ab + b^2$ $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$ $a^2 - b^2 = (a - b)(a + b)$ $(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3 = a^3 \pm b^3 \pm 3ab(a \pm b)$ $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
The Conditional Cubic Identity:
$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$ $= \frac{1}{2}(a + b + c)\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right]$ $\text{If } a + b + c = 0, \quad \text{then } a^3 + b^3 + c^3 = 3abc$
Reciprocal Powers ( $x + 1/x = k$ ):
- $x^2 + \frac{1}{x^2} = k^2 - 2$
- $x^3 + \frac{1}{x^3} = k^3 - 3k$
- $x^4 + \frac{1}{x^4} = (k^2 - 2)^2 - 2$
- If $x - \frac{1}{x} = k \implies x^2 + \frac{1}{x^2} = k^2 + 2, \quad x^3 - \frac{1}{x^3} = k^3 + 3k$

Quadratic Equations & Nature of Roots

For $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$ :

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Sum of Roots: $\alpha + \beta = -\frac{b}{a}$
Product of Roots: $\alpha \beta = \frac{c}{a}$
Discriminant Analysis ( $\Delta = b^2 - 4ac$ ):
- $\Delta > 0$ : Two distinct real roots (rational if $\Delta$ is a perfect square).
- $\Delta = 0$ : Two equal real roots ( $\alpha = \beta = -b/2a$ ).
- $\Delta < 0$ : Complex conjugate roots.

2. 2D Mensuration & Plane Geometry

Shape	Perimeter / Circumference	Area ( $A$ )	Special Properties
Scalene Triangle	$P = a + b + c$	$A = \sqrt{s(s-a)(s-b)(s-c)}$	Semi-perimeter $s = (a+b+c)/2$ (Heron's Formula)
Equilateral Triangle	$P = 3a$	$A = \frac{\sqrt{3}}{4}a^2$	Height $h = \frac{\sqrt{3}}{2}a$ , Inradius $r = \frac{a}{2\sqrt{3}}$ , Circumradius $R = \frac{a}{\sqrt{3}}$
Right Triangle	$P = a + b + h$	$A = \frac{1}{2} \times \text{base} \times \text{height}$	Hypotenuse $h = \sqrt{a^2 + b^2}$
Rectangle	$P = 2(l + w)$	$A = l \times w$	Diagonal $d = \sqrt{l^2 + w^2}$
Rhombus	$P = 4a$	$A = \frac{1}{2} d_1 d_2$	Diagonals bisect at $90^{\circ}$ ; $\text{Side } a = \frac{1}{2}\sqrt{d_1^2 + d_2^2}$
Trapezium	$P = a + b + c + d$	$A = \frac{1}{2}(a + b) \times h$	$a, b$ are parallel sides, $h$ is perpendicular distance
Circle	$C = 2\pi r$	$A = \pi r^2$	Sector Area $= \frac{\theta}{360^{\circ}}\pi r^2$ , Arc Length $= \frac{\theta}{360^{\circ}}2\pi r$
Annulus (Ring)	—	$A = \pi(R^2 - r^2)$	Width $w = R - r$

3. 3D Mensuration: Solids & Volume Conservation

Solid Object	Volume ( $V$ )	Curved / Lateral Surface Area ( $CSA$ )	Total Surface Area ( $TSA$ )
Cuboid	$V = l \times w \times h$	$LSA = 2h(l + w)$	$TSA = 2(lw + wh + hl)$ , Diagonal $= \sqrt{l^2 + w^2 + h^2}$
Cube	$V = a^3$	$LSA = 4a^2$	$TSA = 6a^2$ , Diagonal $= a\sqrt{3}$
Right Cylinder	$V = \pi r^2 h$	$CSA = 2\pi rh$	$TSA = 2\pi r(h + r)$
Hollow Cylinder	$V = \pi(R^2 - r^2)h$	$CSA = 2\pi(R + r)h$	$TSA = 2\pi(R + r)h + 2\pi(R^2 - r^2)$
Right Cone	$V = \frac{1}{3}\pi r^2 h$	$CSA = \pi r l$	$TSA = \pi r(l + r)$ , Slant Height $l = \sqrt{r^2 + h^2}$
Sphere	$V = \frac{4}{3}\pi r^3$	$CSA = 4\pi r^2$	$TSA = 4\pi r^2$
Hemisphere	$V = \frac{2}{3}\pi r^3$	$CSA = 2\pi r^2$	$TSA = 3\pi r^2$

Volume Conservation in Recasting: When a solid metal ingot or billet is melted down and recast into $N$ identical smaller bodies: $V_{\text{original}} = N \times V_{\text{new}}$

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Data Interpretation Taxonomy and Metric Computation

4. Data Interpretation (DI) Analytical Frameworks

Data Interpretation in CIL MT tests rapid data extraction and percentage comparisons from four primary display formats:

1. Table Charts

Data is arranged in row-by-column grids. Typical questions evaluate:

Row/Column Sums: Rapid grouping of values by round tens or hundreds.
Ratio Analysis: Expressing output of one division relative to another without high-precision decimals.

2. Bar Charts & Stacked Bars

Bars represent discrete categories (e.g., subsidiary-wise annual production in Million Tonnes). Visual height corresponds directly to magnitude.

3. Pie Charts (Angular vs. Percentage Mapping)

A full circular pie chart represents $100\%$ of data distributed across $360^{\circ}$ :

100\% = 360^{\circ} \iff 1\% = 3.6^{\circ} \iff 1^{\circ} = \frac{10}{36}\% = \frac{5}{18}\%

\text{Value of Sector} = \left(\frac{\theta^{\circ}}{360^{\circ}}\right) \times \text{Total Value}

4. Line Graphs & Trend Metrics

Used for continuous time series (monthly coal dispatches, overburden removal over years).

Percentage Increase / Decrease: $\Delta\% = \left(\frac{\text{Value}_{\text{current}} - \text{Value}_{\text{base}}}{\text{Value}_{\text{base}}}\right) \times 100\%$
Compound Annual Growth Rate (CAGR) Approximation: $\text{CAGR} = \left(\frac{V_{\text{final}}}{V_{\text{initial}}}\right)^{1/n} - 1$

Worked Example: Subsidiary Production Table

Consider annual coal dispatch data across five CIL subsidiaries:

Subsidiary	2024 Dispatch (MT)	2025 Dispatch (MT)	Growth (MT)	Growth Rate (%)
ECL	50.0	55.0	$+5.0$	$+10.00\%$
BCCL	40.0	46.0	$+6.0$	$+15.00\%$
CCL	60.0	72.0	$+12.0$	$+20.00\%$
NCL	120.0	132.0	$+12.0$	$+10.00\%$
SECL	150.0	165.0	$+15.0$	$+10.00\%$
Total	420.0	470.0	+50.0	+11.90%

CCL achieved the highest individual growth rate ( $20.00\%$ ).
SECL contributed the largest absolute incremental tonnage ( $15.0\text{ MT}$ ).
Total CIL dispatch grew by $\frac{50}{420} \times 100\% = 11.90\%$ .

Test Your Knowledge

If α and β are the real roots of the quadratic equation 2x² - 6x + 3 = 0, what is the exact value of the algebraic expression (α/β) + (β/α) + 3(α + β)?

10.0

11.5

13.0

15.0

Test Your Knowledge

A solid metallic right circular cylinder of base radius 6 cm and height 24 cm is melted down and recast into identical small solid spheres, each of radius 3 cm. Assuming zero loss of metal during the recasting process, how many such spherical balls can be formed?

Test Your Knowledge

In a Coal India dispatch pie chart representing total subsidiary production of 180 million tonnes (MT), the central sector angle allocated to South Eastern Coalfields Limited (SECL) is 108°, while Northern Coalfields Limited (NCL) is allocated 72°. What is the absolute difference in coal production between SECL and NCL, and what percentage of total production does SECL represent?