10. Integral Calculus (Definite & Indefinite)
Key Formulas:
- ∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C
- King's Property: ∫[a to b] f(x) dx = ∫[a to b] f(a + b - x) dx
Example 1: Evaluate ∫[0 to π/2] [sin x / (sin x + cos x)] dx
- Step 1: Apply King's Property: Replace x with π/2 - x.
- Step 2: I = ∫[0 to π/2] [cos x / (cos x + sin x)] dx.
- Step 3: Add both integrals: 2I = ∫[0 to π/2] 1 dx = π/2.
- Final Result: I = π/4
Example 2: Evaluate ∫ e^x [(1 + sin x) / (1 + cos x)] dx
- Step 1: Simplify inside: [1 + 2 sin(x/2) cos(x/2)] / [2 cos²(x/2)] = (1/2) sec²(x/2) + tan(x/2).
- Step 2: Let f(x) = tan(x/2). Then f'(x) = (1/2) sec²(x/2).
- Step 3: Matches e^x [f(x) + f'(x)] form.
- Final Result: e^x tan(x/2) + C
Homework Problems
1. ∫[-2 to 2] x⁵ dx
Answer: 0
2. ∫ e^x [1/x - 1/x²] dx
Answer: e^x/x + C
3. Area between y = x² and y = x
Answer: 1/6 sq units
4. ∫[0 to 1] dx / (1 + x²)
Answer: π/4
11. Differential Equations
Key Formulas:
- First Order Linear DE: dy/dx + P y = Q -> IF = e^(∫ P dx)
- Solution: y × IF = ∫ (Q × IF) dx + C
Example 1: Solve dy/dx + y tan x = sec x
- Step 1: P = tan x, Q = sec x.
- Step 2: IF = e^(∫ tan x dx) = e^(ln sec x) = sec x.
- Step 3: Solution: y sec x = ∫ (sec x × sec x) dx = ∫ sec² x dx = tan x + C.
- Final Result: y sec x = tan x + C
Example 2: Solve dy/dx = (x + y)/x
- Step 1: Substitute y = v x -> dy/dx = v + x dv/dx.
- Step 2: v + x dv/dx = (x + vx)/x = 1 + v -> x dv/dx = 1.
- Step 3: Separate variables: dv = dx/x -> v = ln|x| + C.
- Step 4: Replace v = y/x: y/x = ln|x| + C.
- Final Result: y = x ln|x| + C x
Homework Problems
1. IF of dy/dx + 3y = e^x
Answer: e^(3x)
2. Order and Degree of d²y/dx² + (dy/dx)³ = 0
Answer: Order=2, Degree=1
3. Solution of dy/dx = y/x
Answer: y = C x
4. Solution of dy/dx = e^(x-y)
Answer: e^y = e^x + C
12. Straight Lines & Pair of Lines
Key Formulas:
- Perpendicular Distance: d = |ax1 + by1 + c| / sqrt(a² + b²)
- Foot of Perpendicular: (x - x1)/a = (y - y1)/b = -(ax1 + by1 + c)/(a² + b²)
Example 1: Find foot of perpendicular from (0, 0) on line 3x + 4y - 25 = 0
- Step 1: Apply foot formula: (h - 0)/3 = (k - 0)/4 = -(-25)/(3² + 4²).
- Step 2: h/3 = k/4 = 25/25 = 1.
- Step 3: h = 3, k = 4.
- Final Result: Foot = (3, 4)
Example 2: Find angle between pair of lines 2x² - 5xy + 2y² = 0
- Step 1: Compare with ax² + 2hxy + by² = 0: a = 2, b = 2, h = -5/2.
- Step 2: tan θ = 2 sqrt(h² - ab) / |a + b|.
- Step 3: tan θ = 2 sqrt(25/4 - 4) / |2 + 2| = 2 sqrt(9/4) / 4 = 3/4.
- Final Result: θ = tan⁻¹(3/4)
Homework Problems
1. Distance from (1, 1) to 3x + 4y + 3 = 0
Answer: 2
2. Distance between 2x + 3y = 6 and 2x + 3y = -7
Answer: 13 / sqrt(13) = sqrt(13)
3. Angle between lines y = sqrt(3)x and y = x
Answer: 15°
4. Condition for pair of lines ax² + 2hxy + by² = 0 to be perpendicular
Answer: a + b = 0
13. Conic Sections
Tangency Conditions:
- Parabola y² = 4ax: c = a/m
- Ellipse x²/a² + y²/b² = 1: c² = a² m² + b²
- Hyperbola x²/a² - y²/b² = 1: c² = a² m² - b²
Example 1: Find 'c' if y = 3x + c is tangent to y² = 12x
- Step 1: Parabola y² = 12x -> 4a = 12 -> a = 3. Slope m = 3.
- Step 2: Apply tangency condition c = a / m.
- Step 3: c = 3 / 3 = 1.
- Final Result: c = 1
Example 2: Find eccentricity of hyperbola x²/16 - y²/9 = 1
- Step 1: a² = 16, b² = 9. Formula: e = sqrt(1 + b²/a²).
- Step 2: e = sqrt(1 + 9/16) = sqrt(25/16) = 5/4.
- Final Result: 5/4 = 1.25
Homework Problems
1. Focus of y² = 16x
Answer: (4, 0)
2. Eccentricity of rectangular hyperbola
Answer: sqrt(2)
3. Director circle of x² + y² = 25
Answer: x² + y² = 50
4. Latus rectum length of x²/25 + y²/9 = 1
Answer: 2b²/a = 18/5
14. Vector Algebra
Key Formulas:
- Scalar Triple Product [a b c] = a · (b × c)
- Projection of a on b = (a · b) / |b|
Example 1: Projection of a = i + 3j + 7k on b = 7i - j + 8k
- Step 1: a · b = (1)(7) + (3)(-1) + (7)(8) = 7 - 3 + 56 = 60.
- Step 2: |b| = sqrt(7² + (-1)² + 8²) = sqrt(49 + 1 + 64) = sqrt(114).
- Step 3: Projection = 60 / sqrt(114).
- Final Result: 60 / sqrt(114)
Example 2: Find 'x' if vectors i + j + k, 2i + 3j + k, x i + j + 2k are coplanar
- Step 1: Coplanar means [a b c] = det|[1,1,1],[2,3,1],[x,1,2]| = 0.
- Step 2: Expand: 1(6 - 1) - 1(4 - x) + 1(2 - 3x) = 0.
- Step 3: 5 - 4 + x + 2 - 3x = 0 -> 3 - 2x = 0 -> x = 3/2.
- Final Result: x = 3/2
Homework Problems
1. Value of i · (j × k)
Answer: 1
2. If a ⊥ b, then a · b = ?
Answer: 0
3. Area of triangle with sides a = i+j, b = j+k
Answer: sqrt(3)/2
4. Angle between i + j and j + k
Answer: 60°
15. Three-Dimensional Geometry
Key Formulas:
- Shortest Distance between Skew Lines: d = |(a2 - a1) · (b1 × b2)| / |b1 × b2|
- Direction Cosines: l² + m² + n² = 1
Example 1: Find third direction cosine if l = 1/2, m = 1/2
- Step 1: l² + m² + n² = 1.
- Step 2: (1/2)² + (1/2)² + n² = 1 -> 1/4 + 1/4 + n² = 1.
- Step 3: n² = 1/2 -> n = 1/sqrt(2).
- Final Result: 1/sqrt(2)
Example 2: Find distance of point (2, 3, 4) from origin
- Step 1: d = sqrt(2² + 3² + 4²) = sqrt(4 + 9 + 16) = sqrt(29).
- Final Result: sqrt(29)
Homework Problems
1. Distance between (1, 2, 3) and (4, 6, 3)
Answer: 5
2. Midpoint of (2, 4, 6) and (8, 10, 12)
Answer: (5, 7, 9)
3. If line makes 90°, 60° with X, Y axes, angle with Z?
Answer: 30°
4. Perpendicular distance of (3, 4, 5) from Y-axis
Answer: sqrt(3²+5²) = sqrt(34)
16. Probability & Statistics
Key Formulas:
- Bayes' Theorem: P(A_i|B) = [P(A_i) P(B|A_i)] / Σ [P(A_k) P(B|A_k)]
- Var(aX + b) = a² Var(X)
Example 1: Variance of first 5 natural numbers (1, 2, 3, 4, 5)
- Step 1: Mean = (1+2+3+4+5)/5 = 3.
- Step 2: Mean of squares = (1 + 4 + 9 + 16 + 25)/5 = 55/5 = 11.
- Step 3: Var = 11 - 3² = 11 - 9 = 2.
- Final Result: 2
Example 2: Find P(A|B) if P(A) = 0.6, P(B) = 0.4, P(A ∩ B) = 0.2
- Step 1: Formula P(A|B) = P(A ∩ B) / P(B).
- Step 2: P(A|B) = 0.2 / 0.4 = 1/2 = 0.5.
- Final Result: 0.5
Homework Problems
1. P(A ∪ B) if P(A)=0.5, P(B)=0.4, P(A∩B)=0.2
Answer: 0.7
2. Probability of getting sum 7 on 2 dice
Answer: 6/36 = 1/6
3. If SD(X) = 3, find Var(X)
Answer: 9
4. Probability of a leap year having 53 Sundays
Answer: 2/7