NDA / CDS / AFCAT · Mathematics · Chapter 2

Calculus, Coordinate Geometry and Vectors

Straight lines, circles and conics, 3D geometry, limits, derivatives, maxima-minima, integration, area under curves, differential equations and vector algebra.

In exams: NDA: 45–55 Qs 23 min

1. Chapter at a Glance

Coordinate geometry (2D and 3D), calculus and vector algebra contribute about 45–55 questions to NDA Paper I. Calculus alone (limits, continuity, differentiation, applications, integration, area and differential equations) usually gives 25–30 questions — the single biggest block in the paper. These topics reward practice: once you know the standard forms, most questions take under a minute. CDS and AFCAT do not test calculus, but they use the straight line, distance and section formulae in coordinate-based arithmetic questions, so the first two sections are useful for every candidate.

Priority order for NDA
  • First master differentiation and integration formulas — they appear in many questions at once (tangents, maxima, area, differential equations).
  • Next learn the standard equations of conics (circle, parabola, ellipse, hyperbola) and their key parameters.
  • Then 3D geometry: direction cosines, angle between lines, equation of a plane, distance of a point from a plane.
  • Vectors: dot product, cross product and their geometric meanings — often 6–8 easy questions.

2. Straight Lines (2D)

  • Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²].
  • Section formula (internal, ratio m : n): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2).
  • Centroid of a triangle = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).
  • Slope m = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines: m₁ = m₂. Perpendicular lines: m₁m₂ = −1.
  • Forms of a line: slope-intercept y = mx + c; point-slope y − y₁ = m(x − x₁); intercept form x/a + y/b = 1; general form ax + by + c = 0 (slope −a/b).
  • Angle between two lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.
  • Perpendicular distance of (x₁, y₁) from ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²). Distance between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ − c₂|/√(a² + b²).

3. Circles and Conic Sections

A circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

ConicStandard equationKey facts
Parabolay² = 4axFocus (a, 0); directrix x = −a; latus rectum 4a; eccentricity e = 1
Ellipsex²/a² + y²/b² = 1 (a > b)b² = a²(1 − e²); foci (±ae, 0); latus rectum 2b²/a; 0 < e < 1
Hyperbolax²/a² − y²/b² = 1b² = a²(e² − 1); foci (±ae, 0); latus rectum 2b²/a; e > 1
Rectangular hyperbolaxy = c² or x² − y² = a²e = √2
Eccentricity decides the conic

e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola, e > 1 hyperbola. For x²/25 + y²/16 = 1: a = 5, b = 4, so 16 = 25(1 − e²) ⇒ e = 3/5.

4. Three-Dimensional Geometry

  • Distance between two points in space: √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
  • Direction cosines (l, m, n) satisfy l² + m² + n² = 1. If direction ratios are (a, b, c), then l = a/√(a² + b² + c²), and so on.
  • Angle between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂): cos θ = l₁l₂ + m₁m₂ + n₁n₂. Perpendicular if a₁a₂ + b₁b₂ + c₁c₂ = 0.
  • Equation of a plane: ax + by + cz + d = 0, where (a, b, c) are the direction ratios of the normal. Intercept form: x/a + y/b + z/c = 1.
  • Distance of (x₁, y₁, z₁) from the plane ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).
  • A line making angles α, β, γ with the axes satisfies cos²α + cos²β + cos²γ = 1 (and sin²α + sin²β + sin²γ = 2).

5. Limits and Continuity

  • Standard limits: lim(x→0) sin x / x = 1; lim(x→0) (1 − cos x)/x² = 1/2; lim(x→0) (eˣ − 1)/x = 1; lim(x→0) (aˣ − 1)/x = ln a; lim(x→0) ln(1 + x)/x = 1; lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹; lim(x→0) (1 + x)^(1/x) = e.
  • For 0/0 or ∞/∞ forms, L'Hôpital's rule: differentiate numerator and denominator separately.
  • A function is continuous at a if the left limit = right limit = f(a). It is differentiable at a if left and right derivatives are equal. Differentiable ⇒ continuous, but not conversely: |x| is continuous at 0 but not differentiable there.

6. Differentiation

f(x)f′(x)f(x)f′(x)
xⁿnxⁿ⁻¹eˣeˣ
sin xcos xaˣaˣ ln a
cos x−sin xln x1/x
tan xsec²xsin⁻¹x1/√(1 − x²)
sec xsec x tan xtan⁻¹x1/(1 + x²)
cot x−cosec²x√x1/(2√x)
  • Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v²; chain rule dy/dx = (dy/du)(du/dx).
  • Parametric: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt).
  • Logarithmic differentiation for forms like xˣ: if y = xˣ, ln y = x ln x, so dy/dx = xˣ(1 + ln x).

7. Applications of Derivatives

  • Slope of the tangent at (x₁, y₁) = dy/dx at that point; the normal has slope −1/(dy/dx).
  • f is increasing where f′(x) > 0 and decreasing where f′(x) < 0.
  • Maxima and minima: solve f′(x) = 0; if f″(x) < 0 it is a local maximum, if f″(x) > 0 a local minimum.
  • Rate of change: if a quantity depends on time, dV/dt = (dV/dr)(dr/dt). Example: for a sphere V = (4/3)πr³, dV/dt = 4πr²(dr/dt).
  • Classic results: of all rectangles with a given perimeter, the square has the maximum area; the product of two positive numbers with a fixed sum is maximum when they are equal.
Local maximum: f′ = 0, f″ < 0 Local minimum: f′ = 0, f″ > 0 xy
At both turning points the tangent is horizontal (f′(x) = 0). The sign of the second derivative tells which is which.

8. Integration

∫ f(x) dxResult (+ C)∫ f(x) dxResult (+ C)
xⁿ (n ≠ −1)xⁿ⁺¹/(n + 1)1/xln |x|
eˣeˣaˣaˣ/ln a
sin x−cos xcos xsin x
sec²xtan xcosec²x−cot x
1/(1 + x²)tan⁻¹x1/√(1 − x²)sin⁻¹x
1/(a² + x²)(1/a) tan⁻¹(x/a)1/(x² − a²)(1/2a) ln |(x − a)/(x + a)|
  • Integration by parts: ∫u v dx = u∫v dx − ∫[u′ ∫v dx] dx. Choose u by the ILATE order (Inverse, Log, Algebraic, Trigonometric, Exponential).
  • ∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C — a frequently tested shortcut.
  • Definite integral properties: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx; ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even and 0 if f is odd.
  • Using the first property: ∫₀^(π/2) sinⁿx/(sinⁿx + cosⁿx) dx = π/4 for every n.
  • Area between y = f(x), the x-axis and x = a to x = b is ∫ₐᵇ |f(x)| dx. Area enclosed by y² = 4ax and its latus rectum = 8a²/3; area of the ellipse x²/a² + y²/b² = 1 is πab.

9. Differential Equations

  • Order = highest derivative present; degree = power of the highest derivative after making the equation free of radicals and fractions in derivatives.
  • Variable separable: write all x terms with dx and all y terms with dy, then integrate.
  • Linear equation dy/dx + Py = Q: integrating factor IF = e^(∫P dx); solution y·IF = ∫Q·IF dx + C.
  • Number of arbitrary constants in the general solution = order of the equation; the family y = mx + c (two constants) gives a second-order equation y″ = 0.

10. Vector Algebra

  • For a = a₁i + a₂j + a₃k, |a| = √(a₁² + a₂² + a₃²); unit vector â = a/|a|.
  • Dot product a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. a·b = 0 ⇔ perpendicular. Projection of a on b = (a·b)/|b|.
  • Cross product |a × b| = |a||b| sin θ; direction perpendicular to both (right-hand rule). a × b = 0 ⇔ parallel. a × b = −(b × a).
  • Area of a parallelogram with adjacent sides a, b = |a × b|; area of a triangle = ½|a × b|.
  • Scalar triple product [a b c] = a·(b × c) gives the volume of the parallelepiped; it is zero when the three vectors are coplanar.
  • i·i = j·j = k·k = 1; i × j = k, j × k = i, k × i = j.

11. Worked Examples — Step-by-Step NDA Calculus and Coordinate Problems

Formulas alone do not win marks; the order in which you apply them does. The examples below follow the pattern of recent NDA papers. Read the question, decide the method in one line, and only then calculate.

Example 1 — Limits using standard forms

Find lim (x → 0) (e³ˣ − 1)/x.
Method: use lim (eᵏˣ − 1)/x = k. Answer: 3. Direct substitution gives 0/0, so a standard limit or L'Hospital's rule is needed: differentiating top and bottom gives 3e³ˣ/1 → 3.

Example 2 — Continuity with an unknown constant

f(x) = (x² − 9)/(x − 3) for x ≠ 3 and f(3) = k. For what k is f continuous at 3?
Method: the limit must equal the value. lim (x → 3) (x − 3)(x + 3)/(x − 3) = 6, so k = 6.

Example 3 — Rate of change

The radius of a circular oil patch grows at 2 cm/s. How fast is the area growing when the radius is 10 cm?
Method: A = πr², so dA/dt = 2πr × dr/dt = 2π × 10 × 2 = 40π cm²/s.

Example 4 — Maximum area with a fixed perimeter

A rectangular camp area is to be fenced with 200 m of wire. Find the maximum area.
Method: 2(l + b) = 200 ⇒ b = 100 − l, A = l(100 − l). dA/dl = 100 − 2l = 0 ⇒ l = 50 m, b = 50 m. Maximum area = 2,500 m². A square always gives the largest area for a given perimeter.

Example 5 — Area between two curves

Find the area enclosed by y = x² and y = 2x.
Method: intersections at x = 0 and x = 2. Area = ∫₀² (2x − x²) dx = [x² − x³/3]₀² = 4 − 8/3 = 4/3 square units.

Example 6 — Locus and the circle

A point moves so that its distance from (0, 0) is always twice its distance from (3, 0). Find the locus.
Method: x² + y² = 4[(x − 3)² + y²] ⇒ 3x² + 3y² − 24x + 36 = 0 ⇒ x² + y² − 8x + 12 = 0, a circle with centre (4, 0) and radius 2.

Example 7 — Angle between vectors

Find the angle between a = i + j and b = i + k.
Method: a · b = 1, |a| = |b| = √2, so cos θ = 1/2 and θ = 60°.

Choosing the method quickly
  • 0/0 or ∞/∞ form → factorise, use a standard limit, or apply L'Hospital's rule.
  • Words like maximum, minimum, greatest, least → set the first derivative to zero and check the sign of the second derivative.
  • Area enclosed by curves → find the intersection points first; they are the limits of integration.
  • Locus questions → write the condition with the distance formula, then simplify and identify the conic.

12. Master Formula Table

TopicFormulaNote
Point–line distance|ax₁ + by₁ + c|/√(a² + b²)Same idea for point–plane in 3D
Angle between linestan θ = |(m₁ − m₂)/(1 + m₁m₂)|⊥ when m₁m₂ = −1
CircleCentre (−g, −f), r = √(g² + f² − c)Coefficients of x², y² must be 1
Parabola y² = 4axFocus (a, 0), latus rectum 4ae = 1
Ellipseb² = a²(1 − e²)Area πab
Hyperbolab² = a²(e² − 1)e > 1
Direction cosinesl² + m² + n² = 1cos²α + cos²β + cos²γ = 1
Standard limitssin x/x → 1; (eˣ − 1)/x → 1; (1 + x)^(1/x) → ex → 0
Chain ruledy/dx = (dy/du)(du/dx)Parametric: (dy/dt)/(dx/dt)
Maxima / minimaf′ = 0; f″ < 0 max, f″ > 0 minSecond-derivative test
By parts∫uv = u∫v − ∫(u′∫v)ILATE
King's property∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dxGives π/4 type answers
Even/odd∫₋ₐᵃ odd = 0; even = 2∫₀ᵃCheck symmetry first
Linear DEIF = e^(∫P dx)y·IF = ∫Q·IF dx + C
Dot / crossa·b = |a||b|cos θ; |a × b| = |a||b|sin θ0 ⇒ ⊥ / parallel
Triangle area (vectors)½|a × b|Parallelogram: |a × b|
Triple product[a b c] = volume0 ⇒ coplanar

13. Master Summary Table

Sub-topicFrequent question typeTypical NDA QsBest approach
Straight linesSlope, perpendicularity, distance from a line4–6Convert to y = mx + c
Circles and conicsCentre-radius, eccentricity, focus, latus rectum5–7Bring to standard form first
3D geometryDirection cosines, angle between lines, plane distance5–7Work with direction ratios
Limits, continuityStandard limits, L'Hôpital, piecewise continuity4–6Identify the form first
DifferentiationChain, parametric, implicit, logarithmic5–7Write intermediate variable
ApplicationsTangent slope, increasing intervals, max/min4–6Second-derivative test
Integration and areaStandard integrals, definite-integral properties, areas8–10Look for symmetry/King's rule
Differential equationsOrder-degree, variable separable, linear3–5Find IF quickly
VectorsDot/cross product, projection, area, coplanarity5–8Use component form

14. Mind Map

Calculus, Coordinate Geometry and Vectors
2D geometry
  • Lines, slopes
  • Circles
  • Parabola, ellipse, hyperbola
3D geometry
  • Direction cosines
  • Lines and planes
  • Distances, angles
Differential calculus
  • Limits, continuity
  • Derivatives
  • Tangents, max/min
Integral calculus
  • Indefinite integrals
  • Definite integrals
  • Area, differential equations
Vectors
  • Dot product
  • Cross product
  • Triple product

15. Tips and Tricks

Speed techniques
  • Check by differentiating: in an indefinite-integral question, differentiate each option — the one that returns the integrand is correct.
  • Odd function test: for ∫₋ₐᵃ, test f(−x) = −f(x) first; the answer may simply be 0.
  • Limits: for x → 0 in trigonometric limits, replace sin x by x and 1 − cos x by x²/2 (small-angle approximations).
  • Conics: always divide to make the right-hand side 1 before reading a and b.
  • Vectors: for perpendicularity, write the dot product and set it to zero; for parallel vectors, compare component ratios.
  • Differential equations: if an option is a solution, substitute it into the equation — quicker than solving.
  • Calculus questions are long; aim for 50–60 seconds each and skip any that need more than two lines of work in the first round.
Traps to avoid
  • Forgetting the chain rule factor: d/dx sin(3x) = 3 cos 3x, not cos 3x.
  • Writing ∫1/x dx = ln x instead of ln |x|.
  • Mixing the ellipse relation b² = a²(1 − e²) with the hyperbola relation b² = a²(e² − 1).
  • Assuming every continuous function is differentiable.

16. Important One-Liners

  • Two lines are perpendicular if the product of their slopes is −1.
  • The slope of a line parallel to the x-axis is 0; a line parallel to the y-axis has undefined slope.
  • The centroid divides each median in the ratio 2 : 1.
  • Eccentricity of a circle is 0 and of a parabola is 1.
  • The length of the latus rectum of y² = 4ax is 4a.
  • Direction cosines of the x-axis are (1, 0, 0).
  • |x| is continuous everywhere but not differentiable at x = 0.
  • Derivative of a constant is 0; derivative of eˣ is eˣ.
  • At a local maximum the first derivative is zero and the second derivative is negative.
  • The integral of an odd function over [−a, a] is zero.
  • ∫₀^(π/2) sin x dx = ∫₀^(π/2) cos x dx = 1.
  • Area of the ellipse x²/a² + y²/b² = 1 is πab.
  • Order of a differential equation equals the number of arbitrary constants in its general solution.
  • The dot product of perpendicular vectors is zero; the cross product of parallel vectors is the zero vector.
  • i × i = 0 and i × j = k.
  • Three vectors are coplanar if their scalar triple product is zero.

17. PYQ-Based Questions with Detailed Answers

Q1NDA pattern

What is the distance of the point (2, 3) from the line 3x + 4y − 3 = 0?

  1. 3
  2. 4
  3. 15
  4. 3/5
Answer and explanation

Distance = |3(2) + 4(3) − 3|/√(9 + 16) = |6 + 12 − 3|/5 = 15/5 = 3. The option 15 forgets to divide by √(a² + b²) = 5.

Answer: (A) 3

Q2NDA pattern

What is the radius of the circle x² + y² − 6x + 8y = 0?

  1. 3
  2. 4
  3. 5
  4. 10
Answer and explanation

Here 2g = −6 ⇒ g = −3, 2f = 8 ⇒ f = 4, c = 0. Centre (3, −4), radius = √(9 + 16 − 0) = 5. The circle passes through the origin because c = 0.

Answer: (C) 5

Q3NDA pattern

What is the eccentricity of the ellipse 9x² + 25y² = 225?

  1. 3/5
  2. 4/5
  3. 5/4
  4. 3/4
Answer and explanation

Dividing by 225: x²/25 + y²/9 = 1, so a = 5 and b = 3. b² = a²(1 − e²) ⇒ 9 = 25(1 − e²) ⇒ e² = 16/25 ⇒ e = 4/5. The value 5/4 is greater than 1 and cannot be the eccentricity of an ellipse.

Answer: (B) 4/5

Q4NDA pattern

A line makes angles of 45° and 60° with the x-axis and y-axis respectively. What angle does it make with the z-axis?

  1. 30°
  2. 45°
  3. 60°
  4. 90°
Answer and explanation

cos²45° + cos²60° + cos²γ = 1 ⇒ 1/2 + 1/4 + cos²γ = 1 ⇒ cos²γ = 1/4 ⇒ cos γ = 1/2 ⇒ γ = 60° (taking the acute angle).

Answer: (C) 60°

Q5NDA pattern

What is lim(x→0) (sin 5x)/(sin 3x)?

  1. 0
  2. 1
  3. 3/5
  4. 5/3
Answer and explanation

Write it as [(sin 5x)/5x] × [3x/(sin 3x)] × (5/3). Each bracket tends to 1, so the limit is 5/3. Small-angle shortcut: sin 5x ≈ 5x and sin 3x ≈ 3x near 0.

Answer: (D) 5/3

Q6NDA pattern

If y = xˣ, what is dy/dx?

  1. x·xˣ⁻¹
  2. xˣ ln x
  3. xˣ(1 + ln x)
  4. xˣ(1 − ln x)
Answer and explanation

Take logs: ln y = x ln x. Differentiate: (1/y) dy/dx = ln x + 1. So dy/dx = xˣ(1 + ln x). Option (A) wrongly treats the power as a constant; (B) treats the base as constant — the correct answer combines both effects.

Answer: (C) xˣ(1 + ln x)

Q7NDA pattern

What is the slope of the tangent to the curve y = x³ − 2x at x = 1?

  1. −1
  2. 0
  3. 1
  4. 3
Answer and explanation

dy/dx = 3x² − 2. At x = 1, slope = 3 − 2 = 1. The value −1 is the y-coordinate of the point (1, −1), not the slope.

Answer: (C) 1

Q8NDA pattern

The maximum value of f(x) = x(12 − x) for real x is:

  1. 24
  2. 36
  3. 48
  4. 144
Answer and explanation

f(x) = 12x − x², f′(x) = 12 − 2x = 0 ⇒ x = 6; f″(x) = −2 < 0, so it is a maximum. f(6) = 6 × 6 = 36. Shortcut: two numbers with a fixed sum (x and 12 − x) have the greatest product when they are equal.

Answer: (B) 36

Q9NDA pattern

What is ∫ eˣ (sin x + cos x) dx?

  1. eˣ cos x + C
  2. eˣ sin x + C
  3. −eˣ cos x + C
  4. eˣ (sin x − cos x) + C
Answer and explanation

The integrand has the form eˣ[f(x) + f′(x)] with f(x) = sin x (since f′(x) = cos x). So the integral is eˣ sin x + C. Check by differentiating: d/dx(eˣ sin x) = eˣ sin x + eˣ cos x ✓.

Answer: (B) eˣ sin x + C

Q10NDA pattern

What is the value of ∫ from −1 to 1 of x³ cos x dx?

  1. 0
  2. 1
  3. 2 cos 1
  4. 2 sin 1
Answer and explanation

x³ is odd and cos x is even, so x³ cos x is odd. The integral of an odd function over a symmetric interval [−a, a] is 0. No integration is needed.

Answer: (A) 0

Q11NDA pattern

What are the order and degree of the differential equation (d²y/dx²)³ + (dy/dx)⁴ + y = 0?

  1. Order 2, degree 3
  2. Order 2, degree 4
  3. Order 1, degree 4
  4. Order 3, degree 2
Answer and explanation

The highest derivative is d²y/dx², so the order is 2. Its power is 3, so the degree is 3. The power 4 belongs to the first derivative, which does not decide the degree.

Answer: (A) Order 2, degree 3

Q12NDA pattern

For what value of λ are the vectors 2i + λj + k and i − 2j + 3k perpendicular?

  1. 3/2
  2. 5/2
  3. −5/2
  4. 1/2
Answer and explanation

Dot product = 2(1) + λ(−2) + 1(3) = 5 − 2λ = 0 ⇒ λ = 5/2.

Answer: (B) 5/2

Q13NDA pattern — tough

What is the area of the triangle whose two sides are represented by the vectors a = i + j and b = j + k?

  1. √3/2
  2. √3
  3. 3/2
  4. 1/2
Answer and explanation

a × b = (i + j) × (j + k) = i×j + i×k + j×j + j×k = k − j + 0 + i = i − j + k. |a × b| = √3. Area of the triangle = ½ × √3 = √3/2. √3 is the parallelogram area.

Answer: (A) √3/2

Q14NDA pattern — tough

What is the value of ∫ from 0 to π/2 of √(sin x)/(√(sin x) + √(cos x)) dx?

  1. 0
  2. π/4
  3. π/2
  4. π
Answer and explanation

Let I be the integral. Replacing x by π/2 − x swaps sin and cos, giving I = ∫ √(cos x)/(√(cos x) + √(sin x)) dx. Adding, 2I = ∫₀^(π/2) 1 dx = π/2, so I = π/4.

Answer: (B) π/4

Q15NDA pattern

The value of lim (x → 0) (e³ˣ − 1)/x is:

  1. 3
  2. 1
  3. 0
  4. e³
Answer and explanation

Using the standard result lim (eᵏˣ − 1)/x = k, the limit is 3. By L'Hospital's rule: differentiate to get 3e³ˣ, which tends to 3 as x → 0.

Answer: (A) 3

Q16NDA pattern

A rectangular area is to be fenced with 200 m of wire. The maximum area that can be enclosed is:

  1. 2,500 m²
  2. 2,000 m²
  3. 10,000 m²
  4. 1,250 m²
Answer and explanation

With b = 100 − l, area A = l(100 − l). Setting dA/dl = 100 − 2l = 0 gives l = 50 m, and the second derivative −2 confirms a maximum. The figure is a 50 m square with area 2,500 m².

Answer: (A) 2,500 m²

Q17NDA pattern

The area enclosed between the curves y = x² and y = 2x is:

  1. 4/3 square units
  2. 8/3 square units
  3. 4 square units
  4. 2/3 square units
Answer and explanation

The curves meet at x = 0 and x = 2. Between them y = 2x lies above y = x², so the area is ∫₀² (2x − x²) dx = 4 − 8/3 = 4/3.

Answer: (A) 4/3 square units

18. Quick Revision Summary

Revise in five minutes
  • Lines: slope, perpendicular condition m₁m₂ = −1, point–line distance formula.
  • Conics: standard forms, eccentricity (0, <1, 1, >1), focus and latus rectum.
  • 3D: l² + m² + n² = 1; cos θ = l₁l₂ + m₁m₂ + n₁n₂; point–plane distance.
  • Limits: sin x/x → 1, (eˣ − 1)/x → 1, (1 + x)^(1/x) → e; L'Hôpital for 0/0.
  • Derivatives: standard table, chain, product, quotient, parametric, logarithmic.
  • Applications: tangent slope = dy/dx; maxima where f′ = 0 and f″ < 0.
  • Integrals: standard table, by parts (ILATE), eˣ[f + f′] shortcut, King's property, odd-even rule.
  • Differential equations: order-degree, variable separable, IF = e^(∫P dx).
  • Vectors: dot = 0 ⇒ ⊥, cross = 0 ⇒ parallel, ½|a × b| = triangle area, [a b c] = 0 ⇒ coplanar.
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