NCERT Solutions Class 11 Maths - All 16 Chapters

All 16 Chapters

Sets

Venn diagrams, Operations on sets

› 2

Relations and Functions

Domain, Range, Types of functions

› 3

Trigonometric Functions

Ratios, Identities, Graphs

› 4

Principle of Mathematical Induction

Proof by induction

› 5

Complex Numbers and Quadratic Equations

Argand plane, Modulus, Argument

› 6

Linear Inequalities

Solution on number line

› 7

Permutations and Combinations

nPr, nCr, Counting

› 8

Binomial Theorem

General term, Middle term

› 9

Sequences and Series

AP, GP, HP, AGP

› 10

Straight Lines

Slope, Equations of line, Angle between lines

› 11

Conic Sections

Circle, Parabola, Ellipse, Hyperbola

› 12

Introduction to 3D Geometry

Distance, Section formula

› 13

Limits and Derivatives

First principles, Algebra of limits

› 14

Mathematical Reasoning

Statements, Implications, Contradiction

› 15

Statistics

Mean, Variance, Standard deviation

› 16

Probability

Sample space, Events, Axiomatic approach

›

Chapter 1: Sets

Q1. If U = {1,2,3,4,5,6,7,8,9,10} and A = {2,4,6,8,10}, find A'.

Solution

A' = U - A = {1, 3, 5, 7, 9}

Q2. Let A = {1,2,3}, B = {3,4,5}. Find A union B, A intersection B.

Solution

A ∪ B = {1,2,3,4,5}
A ∩ B = {3}

Q3. Verify De Morgan's law for A = {1,2,3,4} and B = {3,4,5,6} with U = {1,...,8}.

Solution

(A ∪ B)' = {1,2,5,6,7,8}' = {7,8}
A' ∩ B' = {5,6,7,8} ∩ {1,2,7,8} = {7,8}
Both are equal. ✓

Q4. In a class of 50 students, 30 take Maths, 25 take Physics, 10 take both. How many take neither?

Solution

n(M ∪ P) = 30 + 25 - 10 = 45
Neither = 50 - 45 = 5

Chapter 2: Relations and Functions

Q1. Find the domain and range of f(x) = √(x-2).

Solution

Domain: x - 2 ≥ 0 ⇒ x ≥ 2, i.e., [2, ∞)
Range: [0, ∞)

Q2. Check if f(x) = x² is one-one from R to R.

Solution

f(1) = f(-1) = 1 but 1 ≠ -1. Not one-one since different inputs give same output.

Q3. Let f(x) = 2x + 3. Find f(f(x)).

Solution

f(f(x)) = f(2x+3) = 2(2x+3) + 3 = 4x + 9

Q4. Find gof if f(x) = x+1 and g(x) = x².

Solution

gof(x) = g(f(x)) = g(x+1) = (x+1)² = x² + 2x + 1

Chapter 3: Trigonometric Functions

Key Identities:
• sin²x + cos²x = 1
• 1 + tan²x = sec²x
• 1 + cot²x = cosec²x
• sin(A±B) = sinA cosB ± cosA sinB

Q1. Find the value of sin 75°.

Solution

sin 75° = sin(45° + 30°)
= sin45²cos30 + cos45²sin30
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4

Q2. Prove: (sinx + cosx)² = 1 + sin2x.

Solution

LHS = sin²x + cos²x + 2sinx cosx = 1 + sin2x = RHS ✓

Q3. Solve: sin x = 1/2 for x in [0, 2π].

Solution

x = π/6 or x = 5π/6

Chapter 4: Principle of Mathematical Induction

Q1. Prove by induction: 1 + 2 + 3 + ... + n = n(n+1)/2.

Solution

Base case (n=1): 1 = 1(2)/2 = 1 ✓
Assume for n=k: 1+2+...+k = k(k+1)/2
Prove for n=k+1:
1+2+...+k+(k+1) = k(k+1)/2 + (k+1) = (k+1)(k+2)/2 ✓

Chapter 5: Complex Numbers and Quadratic Equations

Q1. Find the modulus and argument of z = 1 + i.

Solution

|z| = √(1+1) = √2
arg(z) = tan⁻²(1/1) = π/4

Q2. Solve: x² + 4 = 0.

Solution

x² = -4 = 4i²
x = ±2i

Q3. Express (-1 + i√3) in polar form.

Solution

r = √(1+3) = 2
θ = π - π/3 = 2π/3
z = 2(cos 2π/3 + i sin 2π/3) = 2e^i2π/3

Chapter 6: Linear Inequalities

Q1. Solve: 3x - 7 < 5x + 1.

Solution

-7 - 1 < 5x - 3x
-8 < 2x
x > -4
Solution set: (-4, ∞)

Q2. Solve: |2x - 3| ≤ 5.

Solution

-5 ≤ 2x - 3 ≤ 5
-2 ≤ 2x ≤ 8
-1 ≤ x ≤ 4
Solution: [-1, 4]

Chapter 7: Permutations and Combinations

Q1. Find 10P3.

Solution

10P3 = 10!/(10-3)! = 10 × 9 × 8 = 720

Q2. Find 12C4.

Solution

12C4 = 12!/(4!8!) = (12×11×10×9)/(4×3×2×1) = 495

Q3. In how many ways can 5 boys and 3 girls be seated alternately?

Solution

Arrange 5 boys: 5!
Place girls in gaps: 6P3
Total = 5! × 6P3 = 120 × 120 = 14400

Chapter 8: Binomial Theorem

(a+b)ⁿ = ∫_r=0ⁿ nCr a^n-rb^r

Q1. Find the 4th term in the expansion of (x + 2)²¹.

Solution

T₄ = 22C3 · x¹⁹ · 2² = 1540 · x¹⁹ µ 4 = 6160x¹⁹

Q2. Find the middle term in (x + y)²¹.

Solution

n = 20 (even), so middle term = T₁₁ = 20C10 · x²¹¹

Chapter 9: Sequences and Series

Q1. Find the sum of first 20 terms of the AP: 3, 7, 11, 15, ...

Solution

a = 3, d = 4, n = 20
S₂₀ = 20/2[2(3) + 19(4)] = 10[6 + 76] = 820

Q2. Find the sum of GP: 2, 6, 18, 54, ... to 6 terms.

Solution

a = 2, r = 3, n = 6
S₆ = 2(3² - 1)/(3 - 1) = 2(728)/2 = 728

Q3. Find the AM between 5 and 15.

Solution

AM = (5 + 15)/2 = 10

Q4. Find the 10th term of GP: 3, 6, 12, 24, ...

Solution

a = 3, r = 2
T₁₀ = 3 × 2² = 3 × 512 = 1536

Chapter 10: Straight Lines

Q1. Find the equation of line passing through (2, 3) with slope 4.

Solution

y - 3 = 4(x - 2)
y - 3 = 4x - 8
4x - y - 5 = 0

Q2. Find the angle between 2x + 3y = 5 and x - y = 2.

Solution

m₁ = -2/3, m₂ = 1
tanθ = |(1 - (-2/3))/(1 + 1(-2/3))| = |(5/3)/(1/3)| = 5
θ = tan⁻²(5)

Chapter 11: Conic Sections

Key Formulas:
• Circle: (x-h)² + (y-k)² = r²
• Parabola: y² = 4ax, Focus (a,0)
• Ellipse: x²/a² + y²/b² = 1
• Hyperbola: x²/a² - y²/b² = 1

Q1. Find the equation of circle with centre (2, -3) and radius 5.

Solution

(x - 2)² + (y + 3)² = 25

Q2. Find the focus of y² = 12x.

Solution

4a = 12 ⇒ a = 3
Focus: (3, 0)

Chapter 12: Introduction to 3D Geometry

Q1. Find the distance between (1, 2, 3) and (4, 6, 8).

Solution

d = √[(4-1)² + (6-2)² + (8-3)²]
= √(9 + 16 + 25) = √50 = 5√2

Q2. Find the coordinates of the point dividing the join of (1, -2, 3) and (4, 5, -3) in ratio 2:1.

Solution

x = (2×4 + 1×1)/3 = 3
y = (2×5 + 1×(-2))/3 = 8/3
z = (2×(-3) + 1×3)/3 = -1
Point: (3, 8/3, -1)

Chapter 13: Limits and Derivatives

Key Formulas:
• lim_x→a (xⁿ - aⁿ)/(x - a) = na^n-1
• d/dx(xⁿ) = nx^n-1
• d/dx(sin x) = cos x
• d/dx(e^x) = e^x

Q1. Find lim_x→2 (x² - 4)/(x - 2).

Solution

(x² - 4)/(x - 2) = (x-2)(x+2)/(x-2) = x+2
lim = 2 + 2 = 4

Q2. Find the derivative of x² + sin x.

Solution

d/dx(x² + sin x) = 2x + cos x

Q3. Find lim_x→0 sin x / x.

Solution

Standard limit: lim_x→0 sin x / x = 1

Chapter 14: Mathematical Reasoning

Q1. Write the negation of: "All students are勤劳."

Solution

Negation: "Some students are not勤劳."

Q2. Check if p → q is true when p is false and q is true.

Solution

In implication, p → q is true whenever p is false.
So T → T is TRUE.

Chapter 15: Statistics

Q1. Find the mean of: 4, 7, 8, 11, 12, 16, 20.

Solution

Mean = (4+7+8+11+12+16+20)/7 = 78/7 = 11.14

Q2. Find the variance of: 2, 4, 6, 8, 10.

Solution

Mean = 6
Variance = [(2-6)²+(4-6)²+(6-6)²+(8-6)²+(10-6)²]/5
= [16+4+0+4+16]/5 = 40/5 = 8

Chapter 16: Probability

Q1. A die is thrown. Find the probability of getting a number less than 5.

Solution

Favorable outcomes: {1,2,3,4} = 4
P = 4/6 = 2/3

Q2. Two coins are tossed. Find P(at least one head).

Solution

Sample space: {HH, HT, TH, TT}
At least one head: {HH, HT, TH} = 3
P = 3/4

Q3. A card is drawn from a deck of 52. Find P(king or queen).

Solution

P(king) = 4/52, P(queen) = 4/52
P(king or queen) = 8/52 = 2/13

NCERT Solutions Class 11 Mathematics

All 16 Chapters

Sets

Relations and Functions

Trigonometric Functions

Principle of Mathematical Induction

Complex Numbers and Quadratic Equations

Linear Inequalities

Permutations and Combinations

Binomial Theorem

Sequences and Series

Straight Lines

Conic Sections

Introduction to 3D Geometry

Limits and Derivatives

Mathematical Reasoning

Statistics

Probability

Chapter 1: Sets

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to 3D Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability