Class 11 Maths — Paper 1 ➕

A∪B = {1,2,3,4,5,6,7}
A∩B = {3,4,5}
A−B = {1,2} (elements in A but not in B)

n(M∪S) = n(M) + n(S) − n(M∩S) = 35 + 25 − 10 = 50 students

A function f from set A to set B is a rule that assigns to each element of A exactly one element of B.
Domain: set of all input values (set A)
Range: set of all output values (subset of B)
Co-domain: the full set B

Function undefined when denominator = 0.
x²−4 = 0 → x = ±2
Domain = R − {−2, 2} = all real numbers except ±2

f(0) = 0 + 0 + 2 = 2
f(1) = 1 + 3 + 2 = 6
f(−1) = 1 − 3 + 2 = 0

A complex number is of the form z = a + bi where a, b are real numbers and i = √(−1).
a = real part, b = imaginary part.
Example: z = 3 + 4i (a=3, b=4)

Addition: (3+4i)+(2−3i) = 5+i

Multiplication: (3+4i)(2−3i) = 6−9i+8i−12i²
= 6−i−12(−1) = 6−i+12 = 18−i

|z| = √(1²+1²) = √2 ≈ 1.414
arg(z) = tan⁻¹(b/a) = tan⁻¹(1/1) = 45° = π/4

Conjugate of 5−3i = 5+3i
(5−3i)(5+3i) = 5²+3² = 25+9 = 34
(Product of complex number and its conjugate = square of modulus)

150° = 150 × π/180 = 5π/6 radians
3π/4 = 3π/4 × 180/π = 135°

In right triangle with hypotenuse h, opposite o, adjacent a:
sinθ = o/h, cosθ = a/h
sin²θ + cos²θ = o²/h² + a²/h² = (o²+a²)/h² = h²/h² = 1 ✓ (by Pythagoras)

sin(A+B) = sinA cosB + cosA sinB

sin 75° = sin(45°+30°) = sin45°cos30° + cos45°sin30°
= (1/√2)(√3/2) + (1/√2)(1/2)
= √3/(2√2) + 1/(2√2) = (√3+1)/(2√2) = (√6+√2)/4

cosθ = √(1−sin²θ) = √(1−9/25) = √(16/25) = 4/5
tanθ = sinθ/cosθ = (3/5)/(4/5) = 3/4

|4 5|
det(A) = (2×5) − (3×4) = 10 − 12 = −2

D = |2 1| = (2)(−1)−(1)(1) = −2−1 = −3
|1 −1|
Dx = |5 1| = (5)(−1)−(1)(1) = −5−1 = −6
|1 −1|
Dy = |2 5| = (2)(1)−(5)(1) = 2−5 = −3
|1 1|
x = Dx/D = −6/−3 = 2
y = Dy/D = −3/−3 = 1

1. If two rows (or columns) are identical, determinant = 0.
2. If any row (or column) is all zeros, determinant = 0.
3. Interchanging two rows (or columns) changes sign of determinant.
4. Multiplying a row by k multiplies determinant by k.

Locus: path traced by a point satisfying a given condition.

Let P(x,y) be equidistant from A(3,0) and B(−3,0).
PA = PB → √((x−3)²+y²) = √((x+3)²+y²)
Squaring: x²−6x+9+y² = x²+6x+9+y²
−12x = 0 → x = 0 (the y-axis)

d = √((x₂−x₁)²+(y₂−y₁)²) = √((−1−2)²+(7−3)²)
= √(9+16) = √25 = 5 units

Using point-slope form: y−y₁ = m(x−x₁)
y−3 = 4(x−2)
y−3 = 4x−8
4x−y−5 = 0

Standard form: (x−h)² + (y−k)² = r²
Centre = (h,k), radius = r

General form: x² + y² + 2gx + 2fy + c = 0
Centre = (−g, −f), radius = √(g²+f²−c)

Compare with x²+y²+2gx+2fy+c=0:
2g=−6 → g=−3, 2f=4 → f=2, c=−12
Centre = (−g,−f) = (3,−2)
r = √(9+4+12) = √25 = 5

(x−2)² + (y+3)² = 25
Expanding: x²−4x+4+y²+6y+9 = 25
x²+y²−4x+6y−12 = 0

Conic sections are curves obtained by intersection of a plane with a double cone.
Types:
1. Circle: plane parallel to base
2. Parabola: plane parallel to slant side
3. Ellipse: plane cuts at an angle
4. Hyperbola: plane parallel to axis

Standard form: y² = 4ax (opens right)
For y²=8x: 4a=8 → a=2
Focus = (2,0)
Directrix: x = −2
Axis: y=0 (x-axis)

Standard form: x²/a² + y²/b² = 1 (a > b)
For x²/25 + y²/16 = 1: a²=25 → a=5, b²=16 → b=4
c = √(a²−b²) = √(25−16) = √9 = 3
Foci = (±3, 0), Length of major axis = 10