Inter 2nd Year

Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 4 Theory of Equations to solve questions creatively.

Intermediate 2nd Year Maths 2A Theory of Equations Formulas

→ If n is a non-negative integer and a₀, a₁, a₂, ……….. a_n are real or complex numbers and a₀ ≠ 0, then the expression f(x) = a₀xⁿ + a₁x^{n – 1} + a₂x^{n – 2} + ……. + a_n is called a polynomial in x of degree n.

→ f(x) = a₀xⁿ + a₁x^{n – 1} + a₂x^{n – 2} + ……. + a_n = 0 is called a polynomial equation in x of degree n (a₀ ≠ 0). Every non-constant polynomial equation has atleast one root.

→ If f(α) = 0 then α is called a root of the equation f(x) = 0.

→ If f(α) = 0 then (x – α) is a factor of f(x).

Relation between roots and coefficients of an equation:
→ If α β γ are the roots of x³ + p₁x² + p₂x + p₃ = 0 then sum of the roots s₁ = α + β + γ = – p₁.
Sum of the products of two roots taken at a time s₂ = αβ + βγ + γα = p₂.
Product of all the roots, s₃ = αβγ = – p₃.

→ If α, β, γ, δ are the roots of x⁴ + p₁x³ + p₂x² + p₃x + p₄ = 0 then sum of the roots s₁ = α + β + γ + δ = – p₁.
Sum of the products of roots taken two at a time
s₂ = αβ + αγ + αδ + βγ + βδ + γδ = p₂.
Sum of the products of roots taken three at a time .
s₃ = αβγ + βγδ + γδα + δαβ = – p₃.
Product of the roots, s₄ = αβγδ = p₄.

→ For a cubic equation, when the roots are

• In A.P., then they are taken as a – d, a, a + d.
• In G.P., then they are taken as \(\frac{a}{r}\), a, ar.
• In H.P., then they are taken as \(\frac{1}{a-d}, \frac{1}{a^{\prime}}, \frac{1}{a+d}\).

→ For a bi quadratic equation, if the roots are

• In A.P., then they are taken as a – 3d, a – d, a + d, a + 3d.
• In C.P., then they are taken as \(\frac{1}{a-3 d}, \frac{1}{a-d}, \frac{1}{a+d}, \frac{1}{a+3 d}\).

→ In an equation with real coefficients, imaginary roots occur in conjugate pairs.

→ In an equation with rational coefficients, irrational roots occur in pairs of conjugate surds.

→ The equation whose roots are those of the equation f(x) = 0 with contrary signs is f(- x) = 0.

→ The equation whose roots are multiplied by kfa 0) of those of the proposed equation f(x) = 0 is f \(\left(\frac{x}{k}\right)\) = 0.

→ The equation whose roots are reciprocals of the roots of f(x) = 0 is f \(\left(\frac{1}{x}\right)\) = 0.

→ The equation whose roots are exceed by h than those of f(x) = 0 is f(x – h) = 0.

→ The equation whose roots are diminished by h than those of f(x) 0 is f(x + h) = 0.

→ The equation whose roots are the square of the roots of f(x) = 0 is obtained by eliminating square root from f(√x) = 0.

→ If f(x) = p₀xⁿ + p₁x^{n – 1} + p₂x^{n – 2} + ……. + p_n = 0 then to eliminate the second term,
f(x) = 0 can be transformed to f(x + h) = 0 where h = \(\frac{-p_{1}}{n \cdot p_{0}}\).

→ If an equation is unaltered by changing x into \(\frac{1}{x}\) then it is a reciprocal equation.

→ A reciprocal equation f(x) = p₀xⁿ + p₁x^{n – 1} + …… + pⁿ = 0 is said to be a reciprocal equation of first class p_i = p_{n – i} for all i.

→ A reciprocal equation f(x) = p₀xⁿ + p₁x^{n – 1} + …… + pⁿ = 0 is said to be a reciprocal equation of second class if p_i = – p_{n – i} for all i.

→ For an odd degree reciprocal equation of class one, – 1 is a root and for an odd degree reciprocal equation of class two, 1 is a root.

→ For an even degree reciprocal equation of class two, 1 and – 1 are roots.

→ If f(x) = 0 is an equation of degree ‘n’ then to eliminate r^th term, f(x) = 0 can be transformed to f(x + h) = 0 where h is a constant such that f^{(n – r + 1)}(h) = 0 i.e.,(n – r + 1)^th derivative of f(h) is zero.

→ Every n^th degree equation has exactly n roots real or imaginary.

→ Relation between, roots and coefficients of an equation.

(i) If α, β, γ are the roots of x³ + p₁x² + p₂x + p₃ = 0 the sum of the roots s₁ = α + β + γ = -p₁.
Sum of the products of two roots taken at a time s₂ = αβ + βγ + γα = -p₂
Product of all the roots, s₃ = αβγ= – p₃.

(ii) If α, β, γ, δ are the roots of x⁴ + p₁x³ + p₂x² + p₃x + p₄ = 0 then

• Sum of the roots s₁ = a+P+y+S = -p₁.
  s₂ = αβ + αγ + αδ + βα + βδ + γδ = p₂.
• Sum of the products of roots taken three at a time
  s₃ = αβγ + βγδ + γδα + δαβ = – p₃.
• Product of the roots, s₄ = αβγδ = p₄

→ For the equation xⁿ + p₁x^n-1 + p₂x^n-2 + ……… + p_n = 0

• Σ α₂ = p₁² – 2p₂
• Σ α₃ = -p₁³ + 3p₁p₂ – 3p₃
• Σ α₄ =p₁⁴ – 4p₁²p² + 2p₂² + 4p₁p₃ – 4p₄
• Σ α₂β = 3p₃ – p₁p₂
• Σ α₂βγ = p₁p₃ – 4p₄

Note: For the equation x³ + p₁x² + p₂x + p₃ = 0 Σα₂β₂ — p₂ -2p₁p₃

→ To remove the second term from a nth degree equation, the roots must be diminished by \(\frac{-\mathrm{a}_{1}}{\mathrm{na}_{0}}\) and the resultant equation will not contain the term with x^n-1.

→ If α₁ , α₂ ………………. , α_n are the roots of f(x) = 0, the equation

• Whose roots are \(\) is f\(\left(\frac{1}{x}\right)\) = 0
• Whose roots are kα₁, kα₂ …,kα_n is f\(\left(\frac{x}{h}\right)\) = 0
• Whose roots are α₁ – h, α₂ – h, …. α_n – h is f(x + h) = 0.
• Whose roots are α₁ + h, α₂ + h, ………….. α_n + h is f(x – h) = 0
• Whose roots are α₁², α₂²…. α₁² is f (f√y) = 0

→ In any equation with rational coefficients, irrational roots occur in conjugate pairs.

→ In any equation with real coefficients, complex roots occur in conjugate pairs.

→ If α is r – multiple root of f(x) = 0, then a is a (r – 1) – multiple root of f¹(x) = 0 and (r-2) – Multiple root of f ¹¹(x) = 0 and non multiple root of f^r-1(x) =0.

→ If f(x) = xⁿ + p₁x^n-1 + …………. + p_n-1x + p_n and f(a) and f(b) are of opposite sign, then at least
one real root of f(x) =0 lies between a and b.

(a) For a cubic equation, when the roots are

• In A.P., then they are taken as a – d, a, a + d
• In G.P., then are taken as \(\frac{a}{r}\), a, ar
• In H.P., then they are taken as \(\frac{1}{a-d}, \frac{1}{a}, \frac{1}{a+d}\)

(b) For a bi quadratic equation, if the roots are

• In A.P., then they are taken as a – 3d, a + d, a + 3d
• In G.P., then they are \(\frac{a}{d^{3}}, \frac{a}{d}\), ad, ad³
• In H.P., then they are taken as \(\frac{1}{a-3 d}, \frac{1}{a-d}, \frac{1}{a+d}, \frac{1}{a+3 d}\)

→ It an equation is unaltered by changing x into \(\frac{1}{x}\), then it is a reciprocal equation.

• A reciprocal equation f (x) = p₀xⁿ + p₁x^n-1 + ……………….. + p_n = 0 is said to be a reciprocal equation of first class p_i = p_n-i for all i.
• A reciprocal equation f (x) = p₀xⁿ + p₁x^n-1 + ……………….. + p_n = 0 = 0 is said to be a reciprocal equation of second class p_i = p_n-i for all i.
• For an odd degree reciprocal equation of class one, -1 is a root and for an odd degree reciprocal equation of class two, 1 is a root.
• For an even degree reciprocal equation of class two, 1 and -1 are roots.

→ If f(x) = 0 is an equation of degree ‘n’ then to eliminate rth term, .f(x) = 0 can be transformed to f(x+h) = 0 where h is a constant such that f^(n-r+1)(h) =0 i.e., (n – r + 1)^th derivative of f(h) is zero.

Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 3 Quadratic Expressions to solve questions creatively.

Intermediate 2nd Year Maths 2A Quadratic Expressions Formulas

→ If a, b, c are real or complex numbers and a ≠ 0, then the expression ax² + bx + c is called a quadratic expression in the variable x.
Eg: 4x² – 2x + 3

→ If a, b, c are real or complex numbers and a ≠ 0, then ax² + bx + c = 0 is called a quadratic equation in x.
Eg: 2x² – 5x + 6 = 0

→ A complex number α is said to be a root or solution of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0

→ The roots of ax² + bx + c = 0 are \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

→ If α, β are roots of ax² + bx + c = 0, then α + β = \(\frac{-b}{a}\) and αβ = \(\frac{C}{a}\)

→ The equation of whose roots are α, β is x² – (α + β)x + αβ = 0

→ Nature of the roots: ∆ = b² – 4ac is called the discriminant of the quadratic equation ax² + bx + c = 0. Let α, β be the roots of the quadratic equation ax² + bx + c = 0
Case 1: If a, b, c are real numbers, then

• ∆ = 0 ⇔ α = β = \(\frac{-b}{2 a}\) (a repeated root or double root)
• ∆ > 0 ⇔ α and β are real and distinct.
• ∆ < 0, ⇔ α and β are non- real complex numbers conjugate to each other.

Case 2: If a, b, c are rational numbers, then

• ∆ = 0 ⇔ α and β are rational and equal (ei) α = \(\frac{-b}{2 a}\), a double root or a repeated root.
• ∆ > 0 and is a square of a rational number ⇔ α and β are rational and distinct.
• ∆ > 0 but not a square of a rational number ⇔ α and β are conjugate surds.
• ∆ < 0, ⇔ α and β are non- real ⇔ α and β are non-real con conjugate complex numbers.

→ Let a, b and c are rational numbers, α and β be the roots of the equations ax² + bx + c = 0. Then

• α, β are equal rational numbers if ∆ = 0.
• α, β are distinct rational numbers if ∆ is the square of a non zero rational numbers.
• α, β are conjugate surds if ∆ > 0 and ∆ is not the square of a nonzero square of a rational number.

→ If a₁x² + b₁x + c₁ = 0, a₂x² + b₂x + c₂ = 0 have two same roots, then \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

→ If α, β are roots of ax² + bx + c = 0,

• the equation whose roots are \(\frac{1}{\alpha}, \frac{1}{\beta}\) is f \(\left(\frac{1}{x}\right)\) = 0. If c ≠ 0 (ie) αβ ≠ 0
• the equation whose roots are α + k, β + k is f(x – k) = 0
• the equation whose roots are kα and kβ is f\(\left(\frac{x}{k}\right)\) = 0
• the equation whose roots are equal but opposite in sign is f(-x) = 0
  (ie) the equation whose roots are – α, – β is f(-x) = 0.

→ If the roots of ax² + bx + c = 0 are complex roots then for x ∈ R, ax² + bx + c and ‘a’ have the same sign.

→ If α and β (α < β) are the roots of ax² + bx + c = 0 then

• ax² + bx + c and ‘a’ are of opposite sign when α < x < β
• ax² + bx + c and ‘a’ are of the same sign if x < α or x > β.

→ Let f(x) = ax² + bx + c be a quadratic function

• If a > 0 then f(x) has minimum value at x = \(\frac{-b}{2 a}\) and the minimum value is given by \(\frac{4 a c-b^{2}}{4 a}\)
• If a < 0 then f(x) has maximum value at x = \(\frac{-b}{2 a}\) and the maximum value is given by \(\frac{4 a c-b^{2}}{4 a}\)

→ A necessary and sufficient condition for the quadratic equation a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 to have a common root is (c₁a₂ – c₂a₁)² = (a₁b₂ – a₂b₁) (b₁c₂ – b₂c₁).

→ If a₁b₂ – a₂b₁ = 0 then common root of a₁x² + b₁x + c₁ = 0, a₂x² + b₂x + c₂ = 0 is \(\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\).

→ The standard form of a quadratic ax² + bx + c = 0 where a, b, c ∈ R and a ≠ 0

→ The roots of ax² + bx + c = 0 are \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

→ For the equation ax² + bx + c = 0, sum of the roots = \(-\frac{b}{a}\), product of the roots = \(\frac{c}{a}\).

→ If the roots of a quadratic are known, the equation is x² – (sum of the roots)x +(product of the roots)= 0

→ “Irrational roots” of a quadratic equation with “rational coefficients” occur in conjugate pairs. If p + √q is a root of ax² + bx + c = 0, then p – √q is also a root of the equation.

→ “Imaginary” or “Complex Roots” of a quadratic equation with “real coefficients” occur in conjugate pairs. If p + iq is a root of ax² + bx + c = 0. Then p – iq is also a root of the equation.

→ Nature of the roots of ax² + bx + c = 0

Nature of the Roots	Condition
Imagine	b2 – 4ac < 0
Equal	b2 – 4ac = 0
Real	b2 – 4ac ≥ 0
Real and different	b2 – 4ac > 0
Rational	b2 – 4ac is a perfect square a, b, c being rational
Equal in magnitude and opposite in sign	b = 0
Reciprocal to each other	c = a
Both positive	b has a sign opposite to that of a and c
Both negative	a, b, c all have same sign
Opposite sign	a, c are of opposite sign

→ Two equations a₁x² + b₁x + c₁ = 0, a₂x² + b₂x + c₂ = 0 have exactly the same roots if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

→ The equations a₁x² + b₁x + c₁ = 0, a2x² + b₂x + c₂ = 0 have a common root, if (c₁a₂ – c₂a₁)² = (a₁b₂ – a₂b₁)(b₁c₂ – b₂c₁) and the common root is \(\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\) if a₁b₂ ≠ a₂b₁

→ If f(x) = 0 is a quadratic equation, then the equation whose roots are

• The reciprocals of the roots of f(x) = 0 is f\(\left(\frac{1}{x}\right)\) = 0
• The roots of f(x) = 0, each ‘increased’ by k is f(x – k) = 0
• The roots of f(x) = 0, each ‘diminished’ by k is f(x + k) = 0
• The roots of f(x) = 0 with sign changed is f(-x) = 0
• The roots of f(x) = 0 each multiplied by k(≠0) is f\(\left(\frac{x}{k}\right)\) = 0

→ Sign of the expression ax² + bx + c = 0

• The sign of the expression ax² + bx + c is same as that of ‘a’ for all values of x if b² – 4ac ≤ 0 i.e. if the roots of ax² + bx + c = 0 are imaginary or equal.
• If the roots of the equation ax² + bx + c = 0 are real and different i.e b² – 4ac > 0, the sign of the expression is same as that of ‘a’ if x does not lie between the two roots of the equation and opposite to that of ‘a’ if x lies between the roots of the equation.

→ The expression ax² + bx + c is positive for all real values of x if b² – 4ac < 0 and a > 0.

→ The expression ax² + bx + c has a maximum value when ‘a’ is negative and x = –\(\frac{\mathrm{b}}{2 \mathrm{a}}\). Maximum value of the expression = \(\frac{4 a c-b^{2}}{4 a}\)

→ The expression ax² + bx + c has a maximum value when ‘a’ is positive and x = –\(\frac{\mathrm{b}}{2 \mathrm{a}}\). Minimum value of the expression = \(\frac{4 a c-b^{2}}{4 a}\)

Theorem 1:
If the roots of ax² + bx + c = 0 are imaginary, then for x ∈ R , ax² + bx + c and a have the same sign.
Proof:
The root are imaginary
b² – 4ac < 0 4ac – b² > 0
\(\frac{a x^{2}+b x+c}{a}=x^{2}+\frac{b}{a} x+\frac{c}{a}=\left(x+\frac{b}{2 a}\right)^{2}+\frac{c}{a}-\frac{b^{2}}{4 a^{2}}=\left(x+\frac{b}{2 a}\right)^{2}+\frac{4 a c-b^{2}}{4 a^{2}}\)
∴ For x ∈ R, ax² + bx + c = 0 and a have the same sign.

Theorem 2.
If the roots of ax² + bx + c = 0 are real and equal to α = \(\frac{-b}{2 a}\), then α ≠ x ∈ R ax² + bx + c and a will have same sign.
Proof:
The roots of ax² + bx + c = 0 are real and equal
⇒ b² = 4ac ⇒ 4ac – b² = 0
\(\frac{a x^{2}+b x+c}{a}\) = x + \(\frac{b}{a}\)x + \(\frac{c}{a}\)
= \(\left(x+\frac{b}{2 a}\right)^{2}+\frac{c}{a}-\frac{b^{2}}{4 a^{2}}\)
= \(\left(x+\frac{b}{2 a}\right)^{2}+\frac{4 a c-b^{2}}{4 a^{2}}\)
= \(\left(x+\frac{b}{2 a}\right)^{2}\) > 0 for x ≠ \(\frac{-b}{2 a}\) = α
For α ≠ x ∈ R, ax² + bx + c and a have the same sign.

Theorem 3.
Let be the real roots of ax² + bx + c = 0 and α < β. Then
(i) x ∈ R, α < x< β ax² + bx + c and a have the opposite signs
(ii) x ∈ R, x < α or x> β ax² + bx + c and a have the same sign.
Proof:
α, β are the roots of ax² + bx + c = 0
Therefore, ax² + bx + c = a(x – α)(x – β)
\(\frac{a x^{2}+b x+c}{a}\) = (x – α)(x – β)

(i) Suppose x ∈ R, α < x < β
⇒ x < α < β then x – α < 0, x – β < 0 ⇒ (x – α)(x – β) > 0 ⇒ \(\frac{a x^{2}+b x+c}{a}\) > 0
⇒ ax² + bx + c, a have a same sign

(ii) Suppose x ∈ R, x > β, x > β > α then x – α > 0, x – β > 0
⇒ (x – α)(x – β) > 0 ⇒ \(\frac{a x^{2}+b x+c}{a}\) > 0
⇒ ax² + bx + c, a have same sign
∴ x ∈ R, x < α or x > β ⇒ ax² + bx + c and a have the same sign.

Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 2 De Moivre’s Theorem to solve questions creatively.

Intermediate 2nd Year Maths 2A De Moivre’s Theorem Formulas

Statement:
→ If ‘n’ is an integer, then (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
If n’ is a rational number, then one of the values of
(cos θ + i sin θ)ⁿ is cos nθ + i sin nθ

n^th roots of unity:
→ n^th roots of unity are {1, ω, ω² …….. ω^{n – 1}}.
Where ω = \(\left[\cos \frac{2 k \pi}{n}+i \sin \frac{2 k \pi}{n}\right]\) k = 0, 1, 2 ……. (n – 1).
If ω is a n^th root of unity, then

• ωⁿ = 1
• 1 + ω + ω² + ………… + ω^{n – 1} = 0

Cube roots of unity:
→ 1, ω, ω² are cube roots of unity when

• ω³ = 1
• 1 + ω + ω² = 0
• ω = \(\frac{-1+i \sqrt{3}}{2}\), ω² = \(\frac{-1-i \sqrt{3}}{2}\)
• Fourth roots of unity roots are 1, – 1, i, – i

→ If Z₀ = r₀ cis θ₀ ≠ 0, then the n^th roots of Z₀ are α_k = (r₀)^1/n cis\(\left(\frac{2 k \pi+\theta_{0}}{n}\right)\) where k = 0, 1, 2, ……… (n – 1)

→ If n is any integer, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

→ If n is any fraction, one of the values of (cosθ + i sinθ)ⁿ is cos nθ + i sin nθ.

→ (sinθ + i cosθ)ⁿ = cos(\(\frac{n \pi}{2}\) – nθ) + i sin(\(\frac{n \pi}{2}\) – nθ)

→ If x = cosθ + i sinθ, then x + \(\frac{1}{x}\) = 2 cosθ, x – \(\frac{1}{x}\) = 2i sinθ

→ xⁿ + \(\frac{1}{x^{n}}\) = 2cos nθ, xⁿ – \(\frac{1}{x^{n}}\) = 2i sin nθ

→ The nth roots of a complex number form a G.P. with common ratio cis\(\frac{2 \pi}{n}\) which is denoted by ω.

→ The points representing n^th roots of a complex number in the Argand diagram are concyclic.

→ The points representing n^th roots of a complex number in the Argand diagram form a regular polygon of n sides.

→ The points representing the cube roots of a complex number in the Argand diagram form an equilateral triangle.

→ The points representing the fourth roots of complex number in the Argand diagram form a square.

→ The nth roots of unity are 1, w, w²,………. , w^n-1 where w = cis\(\frac{2 \pi}{n}\)

→ The sum of the nth roots of unity is zero (or) the sum of the nth roots of any complex number is zero.

→ The cube roots of unity are 1, ω, ω² where ω = cis\(\frac{2 \pi}{3}\), ω² = cis\(\frac{4 \pi}{3}\) or
ω = \(\frac{-1+i \sqrt{3}}{2}\)
ω² = \(\frac{-1-i \sqrt{3}}{2}\)
1 + ω + ω² = 0
ω³ = 1

→ The product of the n^th roots of unity is (-1)^n-1 .

→ The product of the n^th roots of a complex number Z is Z(-1)^n-1 .

→ ω, ω² are the roots of the equation x² + x + 1 = 0

Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 1 Complex Numbers to solve questions creatively.

Intermediate 2nd Year Maths 2A Complex Numbers Formulas

Definition of a complex number:
→ A number of the type z = x + yi, where x, y ∈ R and i = √- 1 i.e., i² = – 1 is called a complex number ‘x’ is called real part of z, and ‘y’ is called imaginary part of z. We write x = Re(z) and y = Im(z). A number z = x + yi is said to be purely real iff y = 0 (x ≠ 0) and purely imaginary iff x = 0 (y ≠ 0)
A complex number a + ib is an ordered pair of real numbers. It is denoted by (a, b), a ∈ R, b ∈ R.

→ Two complex numbers z₁ = (a, b), z₂ = (c, d) are said to be equal iff a = c and b = d.

→ If z₁ = (a, b), z₂ = (c, d) then

• z₁ + z₁ = (a + c, b + d)
• z₁ – z₂ = (a – c, b – d)
• z₁. Z₂ = (ac – bd, ad + bc) and
• \(\frac{z_{1}}{z_{2}}=\left(\frac{a c+b d}{c^{2}+d^{2}}, \frac{b c-a d}{c^{2}+d^{2}}\right)\)

Modulus and Amplitude:
→ The modulus of a complex number z = x + iy is defined as a non-negative real number r = \(\sqrt{x^{2}+y^{2}}\). It is denoted by |z|.

→ Any real number θ satisfying the equation cos θ = \(\frac{x}{r}\), sin θ = \(\frac{y}{r}\) is called an amplitude or argument of z. The unique argument θ of z satisfying – π < θ ≤ π is called the principal argument of z and is denoted by Arg z.

→ The polar form or modulus amplitude form of the complex number
z = x + iy is r(cos θ + i sin θ)

Conjugate of a complex number:
→ If z = x + iy i.e., x, y ∈ R, then the complex number x – iy is called conjugate of z and is written as z̅. The sum and product of a complex number and its Conjugate is always purely real.

Some properties of modulus, amplitude and conjugate:

• (z̅) = z
• z + z̅ = 2 Re (z) and z – z̅ = 2 Im (z)
• \(\overline{Z_{1} Z_{2}}\) = \(\overline{Z_{1}}\) \(\overline{Z_{2}}\)
• \(\) and \(\) (z₂ ≠ 0) and |z₁z₂| = |z₁| |z₂|
• zz̅ = |z|²
• |z| = |z̅|;
• |z₁ + z₂| ≤ |z₁| + |z₂|, |z₁ – z₂| ≤ |z₁| + |z₂|
• |z₁ – z₂| > | |z₁| – |z₂| |
  In (vii) and (viii) equality holds iff amp (z₁) – amp (z₂) is an integral multiple of 2π.
• amp (z₁ z₂) = amp (z₁) + amp (z₂) + nπ, for some n ∈ {- 1, 0, 1}
• amp \(\left(\frac{Z_{1}}{Z_{2}}\right)\) = amp (z₁) – amp (z₂) + nπ, for some n ∈ {- 1, 0, 1}
• \(\frac{1}{{cis} \alpha}\) = cis(- α)
• cis α cis β = cis (α + β)
• \(\frac{{cis} \alpha}{{cis} \beta}\) = cis (α – β)

De-Moivre’s theorem:

• If n is any integer, then (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
• If n = \(\frac{p}{q}\), where p and q are integers having no common factor and q > 1, then cos nθ + i sin nθ is one of the q values of (cos θ + i sin θ)ⁿ
• If z₀ = r₀ cis θ₀ ≠ 0, then the n^th roots of z₀ are α_k = r₀^1/n cis \(\left(\frac{2 k \pi+\theta_{0}}{n}\right)\),
  k = 0, 1, 2, 3 … (n-1)

Cube roots of unity:

• The cube roots of unity are 1, ω = \(\frac{-1+\sqrt{3 i}}{2}\) and ω² = \(\frac{-1-\sqrt{3 i}}{2}\)
• 1 + ω + ω² = 0 and w³ = 1; 1 + ω = – ω ², 1 + ω² = – ω, ω + ω² = – 1
• Either of the two non-real cube roots of unity is square of the other.
• For either of the two non – real cube roots a.and of unity α + β = – 1, αβ = – 1, α² = β, β² = α and α³ = β³ = 1
• (-1)^1/3 = – 1, – ω – ω²
• The n^th roots of unity are cis\(\left(\frac{2 k \pi}{n}\right)\), k = 0, 1, 2, ….. (n – 1)

Formulae:

→ Modulus of Z = \(\sqrt{x^{2}+y^{2}}\)

→ If \(\sqrt{a+i b}\) = (x + iy), then x = \(\sqrt{\frac{\sqrt{a^{2}+b^{2}}+a}{2}}\) and y = \(\sqrt{\frac{\sqrt{a^{2}+b^{2}}-a}{2}}\)

→ Conjugate of a + ib = a – ib

→ Conjugate of a – ib = a + ib

→ Any number of the form x + iy where x, y ∈ R and i² = -1 is called a Complex Number.

→ In the complex number x + iy, x is called the real part and y is called the imaginary part of the complex number.

→ A complex number is said to be purely imaginary if its real part is zero and is said to be purely real if its imaginary part is zero.
(a) Two complex numbers are said to be equal if their real parts are equal and their imaginary parts are equal.
(b) In the set of complex numbers, there is no meaning to the phrases one complex is greater than or less than another i.e. If two complex numbers are not equal, we say they are unequal.
(c) a+ ib > c + id is meaningful only when b = 0, d = 0.

→ Two complex numbers are conjugate if their sum and product are both real. They are of the form a + ib, a – ib.

→ cisθ₁ cisθ₂ = cis(θ₁ + θ₂), \(\frac{\operatorname{cis} \theta_{1}}{\operatorname{cis} \theta_{2}}\) = cis(θ₁ – θ₂), \(\frac{1}{\cos \theta+i \sin \theta}\) = cosθ – isinθ

→ \(\frac{a_{1}+i b_{1}}{a_{2}+i b_{2}}=\frac{\left(a_{1} a_{2}+b_{1} b_{2}\right)+i\left(a_{2} b_{1}-a_{1} b_{2}\right)}{a_{2}^{2}+b_{2}^{2}}\)

→ \(\frac{1+i}{1-i}\) = i, \(\frac{1-i}{1+i}\) = i

→ \(\sqrt{x^{2}+y^{2}}\) is called the modulus of the complex number x + iy and is denoted by r or |x + iy|

→ Any value of 0 obtained from the equations cos θ = \(\frac{x}{r}\), sin θ = \(\frac{y}{r}\) is called an amplitude of the complex number.

→ The amplitude lying between -π and π is called the principal amplitude of the complex number. Rule for choosing the principal amplitude.

→ If θ is the principal amplitude, then -π < 0 < π

→If α is the principle amplitude of a complex number, general amplitude = 2nπ + α where n ∈ Z.

• Amp (Z₁ Z₂) = Amp Z₁ + AmpZ₂
• Amp z + Amp z̅ = 2π (when z is a negative real number) = 0 (otherwise)

→ r(cos θ + i sin θ) is the modulus amplitude form of x + iy.

→ If the amplitude of a complex number is \(\frac{\pi}{2}\), its real part is zero.

→ If the amplitude of a complex number is \(\frac{\pi}{4}\), its real part is equal to its imaginary part.

Students can go through AP Inter 2nd Year Zoology Notes Lesson 5(a) Human Reproductive System will help students in revising the entire concepts quickly.

AP Inter 2nd Year Zoology Notes Lesson 5(a) Human Reproductive System

→ The sex organs of the Male Reproductive System in the pelvic region are a pair of testes, accessory ducts, glands and external genitalia.

→ Testes are suspended within a pouch called scrotum.

→ Scrotum maintains temperature of 2-2.5°C, necessary for spermatogenesis.

→ Scrotal sac is connected to abdominal cavity through ‘inguinal canal’.

→ Testis is held in position in the scrotum by the ‘gubernaculum’.

→ Testis is enclosed in a fibrous envelope called ‘tunica albuginea’.

→ A pouch of serous membrane ’tunica vaginalis’ covers testis.

→ Seminiferous tubules are present in the lobules of testis.

→ Spermatogonial mother cells of seminiferous tubules play an important role in spermatogenesis.

→ ‘Nourishing cells’ or ‘sertoli cells’ provide nutrition to spermatozoa.

→ Leydig cells produce ‘Testosterone’ hormone, that controls secondary sexual characters and spermatogenesis.

→ Epididymis is a storage organ of sperms and gives the sperms time to mature.

→ Urethra is the shared terminal duct of the reproductive arid urinary systems. It opens into penis through a opening called ‘urethral meatus’.

→ Penis is the copulatory organ of male and the enlarged, bulbous end of penis is called glans penis, and is covered by a foreskin or prepuce.

→ Seminal vesicles produce seminal fluid which contains nutrients for the nourishment of sperms.

→ Prostate gland’s secretion activates the spermotozoa and provides nutrition to them.

→ The secretion of bulbourethral glands acts as a flushing agent that washes out the acidic urinary residues that remain in urethra, before the semen is ejaculated.

→ The Female reproductive system consists of a pair of ovaries, a pair of oviducts, uterus, vagina and external genitalia.

→ Ovaries are enveloped by outer germinal epithelium and inner tunica albuginea.

→ The ovarian stroma is divided into an outer cortex and inner medulla. Cortex is more dense due to ovarian follicles and medulla is abundant with blood and lymphatic vessels and nerve fibres.

→ Fallopian tube is the site of fertilization which conducts ovum or zygote towards the uterus by peristalsis.

→ Uterus is a muscular pear-shaped structure and carries foetus, it is also known as womb.

→ Uterus opens into the vagina is called the cervix. The cavity of cervix is called cervical canal.

→ Bartholin’s glands lubricate vagina and are homologous to the bulbourethral glands of the male reproductive system.

→ Skene’s glands secrete a lubricating fluid when stimulated and are homologus to the prostate gland.

→ The nourishment of newborn baby is provided by milk produced from mother’s mammary glands.

→ Gametogenesis in. male is called Spermatogenesis and that in a female is called Oogenesis.

→ Spermatozoa is composed of head, neck middle piece and tail.

→ The cap like structure on the head of spermatozoa is called acrosome.

→ The middle piece of sperm contains mitochondria that produce energy for the movement of tail.

→ Ovum is produced from modified secondary follicle called Graafian follicle.

→ Ovum is sorrounded by a membrane called Zona Pellucida.

→ Corpus luteum secretes Progesterone for the maintenance of pregnancy.

→ The menstrual flow results due to breakdown of endometrial lining of the uterus and its blood vessels.

→ During follicular phase, primary follicles mature into Graafian follicles and produce gonadotropins LH & FSH.

→ Rupture of Graafian follicle by LH results in the release of ovum.

→ The enzyme hyaluronidase from acrosome of a sperm dissolves hyaluronic acid of the follicle cells thus making easy penetration of sperm into a ovum.

→ The nuclear union of sperm and ovum results in the formation of synkaryon or zygotic nucleus.

→ Implantation of blastocyte occurs on the 6th day of fertilization.

→ The extra embryonic or foetal membranes are chorion, amnion, allantois and yolk sac.

→ The sclerotome of enlbryo forms vertebral column.
The myotome fonns voluntary muscles.
The dermotome forms dermis of skin and other connective tissues.

→ The notochord and neural tube are formed by the converge and involution of chorda mesodermal cells through the Hensen’s node.

→ The placenta of human is called chorioallantoic placenta as allantois fuses with the chorion in the process of vascuralisation.

→ Placenta is deciduous, because during parturition placenta is Cast off causing extensive haemorrhage and there by bleeding.

→ Progesterone secreted by placenta is essential for the maintenance of preganancy after 4th month.

→ Human Chorionic Gonadotropin (HCG) presence is used as a test in the detection of pregnancy.

→ The gestation period in humans is 266 days or 38 weeks.

→ Most of the major organs are formed by the end of first trimester and by the end of nine months of pregnancy, the foetus is fully developed.

→ Oxytocin acts on the uterine muscle during parturition and causes stronger uterine contractions.

→ Mammary glands produce milk after the birth of pregnancy, called lactation.

→ Breast feeding is recommended by doctors for bringing up a healthy baby.

→ The primary aim of life is procreation (reproduction)

→ Sexual reproduction starts with formation and fusion of male and female gametes.

→ The process of formation of gametes is known as gametogenesis.

→ Fertilisation is the complete and permanent fusion of two gametes from parents to form a diploid zygote. This process is also called syngamy. [IPE]

→ The Testes produce sperms and male hormones (Testosterone) called androgens. [IPE]

→ The ovary produces ova and female hormones estrogen and progesterone. [IPE]

→ Human female sex organs present in the pelvic region are as follows: [IPE]

• Ovaries
• Fallopian tubes
• Uterus
• Vagina
• Vulva

→ Human male sex organs present in the pelvic region are as follows: [IPE]

1. Testes
2. Epididymis
3. Vasa deferentia
4. Urethra
5. Penis
6. Accessory glands

→ Seminal plasma in humans is rich in fructose, calcium [2009, 2010 PMT]

→ Sertoli cells are found in seminiferous tubules [2010 PMT]

→ The signal for parturition originate from placenta as well as fully dev eloped foetus. [2010, 12]

→ Menstrual flow occurs due to lack of progesterone. [NEET-2013]

→ The main function of mammalian corpus luteum is to produce progesterone. [NEET-2014]

→ Capacitation refers to changes in the sperm before fertilisation. [NEET-2015]

→ Hysterectomy is surgical removal of uterus [NEET-2015]

→ In human females, meiosis-II in the gamete is not completed until fertilisation. [NEET-2015]

→ Hormones like hCG, hPL, estrogen, progesterone are produced by placenta. [NEET-2016]

→ Capacitation occurs in female reproductive tract. [NEET-2017]

→ Colostrum, the yellowish fluid secreted by mother is very essential to impart immunity to the new bom infants because it contains Immunoglobulin A. [NEET-2019]

→ Receptors for sperm binding in mammals are present on zona peliucida. [NEET-2021]