Math Formula Sheet AIEEE

QUADRATIC EQUATION & EXPRESSION

1. Quadratic expression :

A polynomial of degree two of the form ax2 _{+ bx + c, a ≠ 0 is} called a quadratic expression in x.

2. Quadratic equation :

An equation ax2 _{+ bx + c = 0, a ≠ 0, a, b, c ∈ R has two and} only two roots, given by

α = − +b b − ac a 2 ₄ 2 and β = − −b b − ac a 2 ₄ 2 3. Nature of roots :

Nature of the roots of the given equation depends upon the nature of its discriminant D i.e. b2 _{– 4ac.}

Suppose a, b, c ∈ R, a ≠ 0 then

(i) If D > 0 ⇒ roots are real and distinct (unequal) (ii) If D = 0 ⇒ roots are real and equal (Coincident) (iii) If D < 0 ⇒ roots are imaginary and unequal i.e.

non real complex numbers. Suppose a, b, c ∈ Q a ≠ 0 then

(i) If D > 0 and D is a perfect square ⇒ roots are rational & unequal

(ii) If D > 0 and D is not a perfect square ⇒roots are irrational and unequal.

For a quadratic equation their will exist exactly 2 roots real or imaginary. If the equation ax2 _{+ bx + c = 0 is satisfied for} more than 2 distinct values of x, then it will be an identity & will be satisfied by all x. Also in this case a = b = c = 0.

4. Conjugate roots :

Irrational roots and complex roots occur in conjugate pairs i.e.

if one root α + iβ, then other root α – iβ

if one root α + β, then other root α – β

5. Sum of roots :

S = α + β = −_ab = −_{Coefficient of x}Coefficient of x2 Product of roots :

P = αβ = c_a = _{Coefficient of x}constant term2

6. Formation of an equation with given roots : x2 _{– Sx + P = 0}

⇒ x2 _{– (Sum of roots) x + Product of roots = 0} 7. Roots under particular cases :

For the equation ax2 _{+ bx + c = 0, a ≠ 0}

(i) If b = 0 ⇒ roots are of equal magnitude but of opposite sign.

(ii) If c = 0 ⇒ one root is zero and other is –b/a (iii) If b = c = 0 ⇒ both roots are zero

(v) If a > 0, c < 0 or a < 0, c > 0 ⇒ roots are of opposite signs

(vi) If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 ⇒ both roots are –ve

(vii) If a > 0, b < 0 , c > 0 or a < 0, b > 0, c < 0 ⇒ both roots are +ve.

8. Symmetric function of the roots :

If roots of quadratic equation ax2 _{+ bx + c, a ≠ 0 are α and β,} then (i) (α – β) = (α β+ )2−₄αβ _{= ±} b ac a 2₋₄ (ii) α2 _{+ β}2 _{= (α + β)}2 _{– 2αβ =} b ac a 2 2 2 − (iii) α2 _{– β}2 _{= (α + β) (α β+ )}2_{−4αβ =} −b b − ac a 2 2 4 (iv) α3 _{+ β}3 _{= (α + β)}3 _{– 3(α + β) αβ =} −b b − ac a ( 2 ) 3 3 (v) α3 _{– β}3 _{= (α – β) [α}2_{+ β}2 _{– αβ]} = (α β+ )2 −₄αβ_[α2_{+ β}2 _{– αβ]} = (b ac b) ac a 2 2 3 4 − − (vi) α4 _{+ β}4 _{= (α}2 _{+ β}2₎2 _{– 2α}2_β2 ={(α + β)2 _–2αβ}2 _{– 2α}2_β2 ₌ b ac a 2 2 2 2 −

F

HG

I

KJ

– 2 2 2 c a (vii) α4 _{– β}4 _=(α2 _{+ β}2_{) (α}2 _{– β}2_{) =} −b b − ac b − ac a ( 2 ) 2 4 2 4 (viii) α2 _{+ αβ + β}2 _{=(α + β)}2 _{– αβ =} b ac a 2 2 + (ix) α_β + _αβ = α _αββ 2₊ 2 = (α β+ _αβ) − αβ 2 ₂ (x)

F

HG

α_β

I

KJ

2 +

F

HG

_αβ

I

KJ

2 = α_{α β}β 4 4 2 2 + = [(b ac) a c ] a c 2 2 2 2 2 2 2 2 − −

9. Condition for common roots :

The equations a1 x2 + b1 x + c1 = 0 and a2x2 + b2x + c2 = 0 have

(i) One common root if b c_{c a}1 2 b c_{c a}2 1 1 2 2 1 − − = c a c a a b1 21 2 a b2 12 1 − −

(ii) Both roots common if a_a1 2 =

b b12 =

c c12

10. Maximum and Minimum value of quadratic expression : In a quadratic expression ax2 _{+ bx + c = a} x b a D a +

F

HG

I

KJ

−

L

N

MM

2 4

O

_Q

PP

2 2 , Where D = b2 _{– 4ac}

(i) If a > 0, quadratic expression has minimum value 4 4 2 ac b a −

at x = −₂b_a and there is no maximum value. (ii) If a < 0, quadratic expression has maximum value

4 4 2 ac b a −

at x = −₂b_a and there is no minimum value.

11. Location of roots :

Let f(x) = ax2_{+ bx + c, a ≠ 0 then w.r.to f(x) = 0} (i) If k lies between the roots then a.f(k) < 0

(necessary & sufficient) (ii) If between k1 & k2 their is exactly one root of k1, k2

themselves are not roots

f(k1) . f(k2) < 0 (necessary & sufficient) (iii) If both the roots are less than a number k

D ≥ 0, a.f(k) > 0, −₂b_a < k (necessary & sufficient) (iv) If both the roots are greater than k

D ≥ 0, a.f(k) > 0, −₂b_a > k (necessary & sufficient)

(v) If both the roots lies in the interval (k1, k2) D ≥ 0, a.f(k1) > 0, a.f(k2) > 0, k1 <

−b a 2 < k2 (vi) If k1, k2 lies between the roots

a.f(k1) < 0, a.f(k2) < 0

(vii) λ will be the repeated root of f(x) = 0 if f(λ) = 0 and f'(λ) = 0

12. For cubic equation ax3 _{+ bx}2 _{+ cx + d = 0 :}

We have α + β + γ = −_ab, αβ + βγ + γα = c_a and αβγ = −_ad where α, β, γ are its roots.

13. For biquadratic equation ax4 _{+ bx}3 _{+ cx}2 _{+ dx + e = 0 :} We have α + β + γ + δ = –b

a, αβγ + βγδ + γδα + γδβ =

−d a

COMPLEX NUMBER

1. Complex Number :

A number of the form z = x + iy (x, y ∈ R, i = −1) is called a complex number, where x is called a real part i.e. x = Re(z) and y is called an imaginary part i.e. y = Im(z).

Modulus |z| = x2₊y2_,

amplitude or amp(z) = arg(z) = θ = tan–1 y

x.

(i) Polar representation :

x = r cosθ, y = r sinθ, r = |z| = x2 ₊y2 (ii) Exponential form :

z = reiθ _{, where r = |z|, θ = amp.(z)}

(iii) Vector representation :

P(x, y) then its vector representation is z = OP→ 2. Integral Power of lota :

i = −1, i2 _{= –1, i}3 _{= –i , i}4 _{= 1}

Hence i4n+1 _{= i, i}4n+2 _{= –1, i}4n+3 _{= –i, i}4n _{or i}4(n+1) _{= 1} 3. Complex conjugate of z :

If z = x + iy, then _z = x – iy is called complex conjugate of z

* _z is the mirror image of z in the real axis. * |z| = |_z|

* z + _z = 2Re(z) = purely real

* z – _z = 2i Im(z) = purely imaginary * z_z = |z|2 * z₁ +z₂+ +.... z_n = _z1 + z2 + ... + zn * z₁ −z₂ = _z1 – z2 * z z1 2 = z1z2 * _zz1 2

F

HG

I

KJ

= z_z1 2

F

HG

I

KJ

(provided z2 ≠ 0) *

e j

zn = (_z)n *

c h

z = z * If α = f(z), then α = f(_z)

Where α = f(z) is a function in a complex variable with real coefficients.

* z + z = 0 or z = –z ⇒ z = 0 or z is purely imaginary * z = z ⇒ z is purely real

4. Modulus of a complex number :

Magnitude of a complex number z is denoted as |z| and is defined as |z| = (Re( ))z 2₊(Im( ))z 2 _{, |z| ≥ 0} (i) zz = |z|2 _{= |} z|2 (ii) z–1 ₌ z z | |2 (iii) |z1 ± z2|2 = |z1|2 + |z2|2 ± 2 Re (z1 z2)

(iv) |z1 + z2|2 + |z1 – z2|2 = 2 [|z1|2 + |z2|2] (v) |z1 ± z2| ≤ |z1| + |z2|

(vi) |z1 ± z2| ≥ |z1| – |z2|

5. Argument of a complex number :

Argument of a complex number z is the ∠ made by its radius vector with +ve direction of real axis.

arg z = θ, z ∈ 1st _quad. = π – θ, z ∈ 2nd _quad. = –θ, z ∈ 3rd _quad. = θ – π, z ∈ 4th _quad. (i) arg (any real + ve no.) = 0 (ii) arg (any real – ve no.) = π

(iii) arg (z – z) = ± π/2

(iv) arg (z1.z2) = arg z1 + arg z2 + 2 kπ (v) arg z_z1

2

F

HG

I

KJ

= arg z1 – arg z2 + 2 kπ

(vi) arg (_z) = –arg z = arg

F

HG

1_z

I

KJ

, if z is non real = arg z, if z is real

(vii) arg (– z) = arg z + π, arg z ∈ (–π, 0] = arg z – π, arg z ∈ (0, π] (viii) arg (zn_{) = n arg z + 2 kπ}

(ix) arg z + arg _z = 0

argument function behaves like log function.

6. Square root of a complex no.

a ib+ = ±

L

NMM

| |z a i z a₂+ + | |₂−

O

QPP

, for b > 0

= ±

L

_NMM

| |z a i z a₂+ − | |₂−

O

_QPP

, for b < 0 7. De-Moiver's Theorem :

It states that if n is rational number, then (cosθ + isinθ)n _{= cosθ + isin nθ}

and (cosθ + isinθ)–n _{= cos nθ – i sin nθ} 8. Euler's formulae as z = reiθ_{, where}

eiθ _{= cosθ + isinθ and e}–iθ _{= cosθ – i sinθ} ∴ eiθ _{+ e}–iθ _{= 2cosθ and e}iθ _{– e}–iθ _{= 2 isinθ}

9. nth _{roots of complex number z}1/n = r1/n cos 2m sin 2 n i m n π θ+ π θ

F

HG

I

KJ

+

F

_HG

+

I

_KJ

L

NM

O

QP

, where m = 0, 1, 2, ...(n – 1)

(i) Sum of all roots of z1/n _{is always equal to zero} (ii) Product of all roots of z1/n _{= (–1)}n–1 _z

10. Cube root of unity :

cube roots of unity are 1, ω, ω2 _where

ω = − +1 3

2

11. Some important result : If z = cosθ + isinθ (i) z + 1_z = 2cosθ (ii) z – 1_z = 2 isinθ (iii) zn + 1 zn = 2cosnθ

(iv) If x = cosα + isinα , y = cos β + i sin β & z = cosγ + isinγ

and given x + y + z = 0, then

(a) 1_x + 1_y + 1_z = 0 (b) yz + zx + xy = 0 (c) x2 _{+ y}2 _{+ z}2 _{= 0 (d) x}3 _{+ y}3 _{+ z}3 _{= 3xyz} 12. Equation of Circle :

* |z – z1| = r represents a circle with centre z1 and radius r.

* |z| = r represents circle with centre at origin. * |z – z1| < r and |z – z1| > r represents interior and

exterior of circle |z – z1| = r.

* zz + az + az + b = 0 represents a general circle where a ∈ c and b ∈ R.

* Let |z| = r be the given circle, then equation of tangent at the point z1 is zz1 + zz1 = 2r2

* diametric form of circle : arg

F

HG

z z_{z z}−₋ 1

I

KJ

2 = ± π 2, or z z_{z z}−₋ 1 2 + z z z z − − 1₂ = 0 or z− z1₂+z2 = |z1₂−z2| or |z – z1|2 + |z – z2|2 = |z1 – z2|2

Where z1, z2 are end points of diameter and z is any point on circle.

13. Some important points :

(i) Distance formula PQ = |z2 – z1| (ii) Section formula

For internal division = m z1 2_m m z_m2 1

1 2

+ +

MATHS FORMULA - POCKET BOOK MATHS FORMULA - P For external division = m z1 2_m m z_m2 1

1 2

− −

(iii) Equation of straight line.

* Parametric form z = tz1 + (1 – t)z2 where t ∈ R

* Non parametric form

z z z z z z 1 1 1 1 1 2 2 = 0.

* Three points z1, z2, z3 are collinear if

z z z z z z 1 1 2 2 3 3 1 1 1 = 0

(iv) The complex equation z z_{z z−}− 1

2 = k represents a circle if k ≠ 1 and a straight line if k = 1.

(v) The triangle whose vertices are the points represented by complex numbers z1, z2, z3 is equilateral if

1

2 3

z −z + z₃1−z₁ + z₁1−z₂ = 0

i.e. if z12 + z22 + z32 = z1z2 + z2z3 + z1z3.

(vi) |z – z1| = |z – z2| = λ, represents an ellipse if |z1 – z2| < λ, having the points z1 and z2 as its foci and if |z1 – z2| = λ, then z lies on a line segment connecting z1 & z2

(vii) |z – z1| ~ |z – z2| = λ represents a hyperbola if |z1 – z2| > λ, having the points z1 and z2 as its foci, and if |z1 – z2| = λ, then z lies on the line passing through z1 and z2 excluding the points between z1 & z2. (viii) If four points z1, z2, z3, z4 are concyclic,

then _zz1 z_z2 1 4 − −

F

HG

I

KJ

zz33 zz42 − −

F

HG

I

KJ

is purely real.

(ix) If three complex numbers are in A.P., then they lie on a straight line in the complex plane.

(x) If z1, z2, z3 be the vertices of an equilateral triangle and z0 be the circumcentre,

then z12 + z22 + z32 = 3z02.

(xi) If z1, z3, z3 ... zn be the vertices of a regular polygon of n sides & z0 be its centroid, then z12 + z22 + ... + zn2= nz02.

(xii) If z1, z2, z3 be the vertices of a triangle, then the triangle is equilateral

iff (z1 – z2)2 + (z2 – z3)2 + (z3 – z1)2 = 0.

(xiii) If z1, z2, z3 are the vertices of an isosceles triangle, right angled at z2,

then z12 + z22+ z32 = 2z2 (z1 + z3).

(xiv) z1, z2, z3. z4 are vertices of a parallelogram then z1+ z3 = z2 + z4

PERMUTATION & COMBINATION

1. Factorial notation

-The continuous product of first n natural numbers is called factorial

i.e. n or n! = 1. 2. 3...(n – 1).n n! = n(n – 1)! = n(n – 1)(n – 2)! & so on or n (n – 1)... (n – r + 1) = _{(n r}n₋!_)! Here 0! = 1 and (–n)! = meaningless. 2. Fundamental principle of counting

-(i) Addition rule : If there are two operations such that they can be done independently in m and n ways respectively, then either (any one) of these two operations can be done by (m + n) ways.

Addition ⇒ OR (or) Option

(ii) Multiplication rule : Let there are two tasks of an operation and if these two tasks can be performed in m and n different number of ways respectively, then the two tasks together can be done in m × n ways. Multiplication ⇒ And (or) Condition

(iii) Bijection Rule : Number of favourable cases = Total number of cases

– Unfavourable number of cases. 3. Permutations (Arrangement of objects)

-(i) The number of permutations of n different things taken r at a time is n_p r = n n r ! ( − )!

(ii) The number of permutations of n dissimilar things taken all at a time is n_p

n = n!

(iii) The number of permutations of n distinct objects taken r at a time, when repetition of objects is allowed is nr_.

(iv) If out of n objects, 'a' are alike of one kind, 'b' are alike of second kind and 'c' are alike of third kind and the rest distinct, then the number of ways of permuting the n objects is _{a b c! ! !}n!

4. Restricted Permutations

-(i) The number of permutations of n dissimilar things taken r at a time, when m particular things always occupy definite places = n–m_p

r–m

(ii) The number of permutations of n different things taken r at a time, when m particular things are always to be excluded (included)

= n–m_P

r (n–mCr–m × r!) 5. Circular Permutations

-When clockwise & anticlockwise orders are treated as different.

(i) The number of circular permutations of n different things taken r at a time nPr

r

(ii) The number of circular permutations of n different things taken altogether nPn

n = (n – 1)!

When clockwise & anticlockwise orders are treated as same.

(i) The number of circular permutations of n different things taken r at a time nPr

r 2

(ii) The number of circular permutations of n different things taken all together nPn

n 2 =

1

6. Combination (selection of objects)

-The number of combinations of n different things taken r at a time is denoted by n_C r or C (n, r) n_C r = n r n r ! !( − )! = n r P r! (i) n_C r = nCn–r (ii) n_C r + nCr–1 = n+1Cr (iii) n_C r = nCs ⇒ r = s or r + s = n (iv) n_C 0 = nCn = 1 (v) n_C 1 = nCn–1 = n (vi) n_C r = n r n–1Cr–1 (vii) n_C r = 1 r (n – r + 1) nCr–1 7. Restricted combinations

-The number of combinations of n distinct objects taken r at a time, when k particular objects are always to be

(i) included is n–k_C r–k (ii) excluded is n–k_C

r

(iii) included and s particular things are to be excluded is n–k–s_C

r–k

8. Total number of combinations in different cases -(i) The number of selections of n identical objects, taken

at least one = n

(ii) The number of selections from n different objects, taken at least one

= n_C

1 + nC2 + nC3 + ... + nCn = 2n – 1

(iii) The number of selections of r objects out of n iden-tical objects is 1.

(iv) Total number of selections of zero or more objects from n identical objects is n + 1.

(v) Total number of selections of zero or more objects out of n different objects

= n_C

0 + nC1 + nC2 + nC3 + ... + nCn = 2n (vi) The total number of selections of at least one out of

a1 + a2 + ... + an objects where a1 are alike (of one kind), a2 are alike (of second kind), ... an are alike (of nth _{kind) is}

[(a1 + 1) (a2 + 1) (a3 + 1) + ... + (an + 1)] – 1 (vii) The number of selections taking atleast one out of a1 + a2 + a3 + ... + an + k objects when a1 are alike (of one kind), a2 are alike (of second kind), ... an are alike (of kth kind) and k are distinct is [(a1 + 1) (a2 + 1) (a3 + 1) ... (an + 1)] 2k – 1 9. Division and distribution

-(i) The number of ways in which (m + n + p) different objects can be divided into there groups containing m, n, & p different objects respectively is (m n p_{m n p}+ +_{! ! !})! (ii) The total number of ways in which n different objects

are to be divided into r groups of group sizes n1, n2, n3, ... nr respectively such that size of no two groups is same is _{n n} n _n

r

!

! !... !

1 2 .

(iii) The total number of ways in which n different objects are to be divided into groups such that k1 groups have group size n1, k2 groups have group size n2 and so on, kr groups have group size nr, is given as

n

n k n k n k k k k

!

(iv) The total number of ways in which n different objects are divided into k groups of fixed group size and are distributed among k persons (one group to each) is given as

(number of ways of group formation) × k!

10. Selection of light objects and multinomial theorem -(i) The coefficient of xn _{in the expansion of (1 – x}–r_{) is}

equal to n + r – 1_C r – 1

(ii) The number of solution of the equation x1 + x2 + ... + xr = n, n ∈ N under the condition n1 ≤ x1 ≤ n'1, n2 ≤ x2 ≤ n'2 , ... nr ≤ xr ≤ n'r

where all x'is are integers is given as Coefficient of xn _is

xn1 +xn1 1+ +xn1 xn2 +xn2 1+ +xn2 xnr +xnr 1+ +xnr

LNM

e

+ _... '

j e

+ _... '

j e

_... + _... '

j

OQP

11. Derangement Theorem

-(i) If n things are arranged in a row, then the number of ways in which they can be rearranged so that no one of them occupies the place assigned to it is

= n!

NM

L

1 1−_1!+_2!1 −_3!1 + _4!1 −.... ( )+ −1 1n_n!

O

QP

(ii) If n things are arranged at n places then the number of ways to rearrange exactly r things at right places is = n_r!

L

NM

1 1−_1!+₂1_!−₃1_!+₄1_!+.... ( )+ −1n r− _{(n r−}1 _)!

O

QP

12. Some Important results

-(a) Number of total different straight lines formed by joining the n points on a plane of which m(<n) are collinear is n_C

2 – mC2 + 1.

(b) Number of total triangles formed by joining the n points on a plane of which m(< n) are collinear is n_C

3 – mC3.

(c) Number of diagonals in a polygon of n sides is n_C

2 – n.

(d) If m parallel lines in a plane are intersected by a family of other n parallel lines. Then total number of parallelogram so formed is m_C

2 × nC2.

(e) Given n points on the circumference of a circle, then number of straight lines n_C

2 number of triangles n_C

3 number of quadrilaterals n_C

4

(f) If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. Then the number of part into which these lines divide the plane is = 1 + Σn

(g) Number of rectangles of any size in a square of n × n is r r n 3 1 =

∑

and number of squares of any size is r

r n 2 1 =

∑

.

(h) Number of rectangles of any size in a rectangle of n × p is np₄ (n + 1) (p + 1) and number of squares of any size is r n =

∑

1 (n + 1 – r) (p + 1 – r).

PROBABILITY

1. Mathematical definition of probability : Probability of an event

= No of favourable cases to event A. _{Total no of cases.} Note : (i) 0 ≤ P (A) ≤ 1

(ii) Probability of an impossible event is zero (iii) Probability of a sure event is one.

(iv) P(A) + P(Not A) = 1 i.e. P(A) + P(A) = 1 2. Odds for an event :

If P(A) = m_n and P(A) = n m−_n

Then odds in favour of A = P A_{P A}( )_{( )} = _{n m}m₋ and odds in against of A = P A_P( )_(A) = n m_m−

3. Set theoretical notation of probability and some impor-tant results :

(i) P(A + B) = 1 – P(_{A B}) (ii) P(A/B) = P AB_{P B}(_{( )})

(iii) P(A + B) = P(AB) + P(AB) + P(AB) (iv) A ⊂ B ⇒ P(A) ≤ P(B)

(v) P(_AB) = P(B) – P(AB)

(vi) P(AB) ≤ P(A) P(B) ≤ P(A + B) ≤ P(A) + P(B) (vii) P(Exactly one event) = P(AB) + P(AB) (viii) P(_A + _B) = 1 – P(AB) = P(A) + P(B) – 2P(AB)

= P(A + B) – P(AB)

(ix) P(neither A nor B) = P (_{A B}) = 1 – P(A + B)

(x) When a coin is tossed n times or n coins are tossed once, the probability of each simple event is ₂1n. (xi) When a dice is rolled n times or n dice are rolled once,

the probability of each simple event is ₆1n.

(xii) When n cards are drawn (1 ≤ n ≤ 52) from well shuffled deck of 52 cards, the probability of each simple event is 521_C

n .

(xiii) If n cards are drawn one after the other with replace-ment, the probability of each simple event is _{( )52}1 n. (xiv) P(none) = 1 – P (atleast one)

(xv) Playing cards :

(a) Total cards : 52 (26 red, 26 black)

(b) Four suits : Heart, diamond, spade, club (13 cards each)

(c) Court (face) cards : 12 (4 kings, 4 queens, 4 jacks)

(xvi) Probability regarding n letters and their envelopes : If n letters corresponding to n envelopes are placed in the envelopes at random, then

(a) Probability that all the letters are in right enve-lopes = _n!1

(b) Probability that all letters are not in right enve-lopes = 1 – _n!1

(c) Probability that no letters are in right envelope = _2!1 – _3!1 + _4!1 .... + (–1)n1

n!

(d) Probability that exactly r letters are in right envelopes = _r!1

NM

L

₂1_!−₃1_!+ ₄1_!+... ( )+ −1n r− _{(n r−}1 _)!

O

QP

4. Addition Theorem of Probability :

(i) When events are mutually exclusive i.e. n (A ∩ B) = 0 ⇒ P(A ∩ B) = 0

∴ P(A ∪ B) = P(A) + P(B)

(ii) When events are not mutually exclusive i.e. P(A ∩ B) ≠ 0

∴ P(A ∪ B) = P(A) + P(B) – P(A ∩ B) or P(A + B) = P(A) + P(B) – P(AB)

(iii) When events are independent i.e. P(A ∩ B) = P(A) P (B)

∴ P(A + B) = P(A) + P(B) – P(A) P(B)

5. Conditional probability :

P(A/B) = Probability of occurrence of A, given that B has already happened = P A B(_{P B( )}∩ )

P(B/A) = Probability of occurrence of B, given that A has already happened = P A B(_{P A( )}∩ )

Note : If the outcomes of the experiment are equally likely, then P(A/B) = No of sample pts in A B. _{No of pts in B. .} . ∩ .

(i) If A and B are independent event, then P(A/B) = P(A) and P(B/A) = P(B)

(ii) Multiplication Theorem :

P(A ∩ B) = P(A/B). P(B), P(B) ≠ 0 or P(A ∩ B) = P(B/A) P(A), P(A) ≠ 0 Generalized :

P(E1 ∩ E2 ∩ E3 ∩ ... ∩ En)

= P(E1) P(E2/E1) P(E3/E1 ∩ E2) P(E4/E1 ∩ E2 ∩ E3) ... If events are independent, then

P(E1 ∩ E2 ∩ E3 ∩ ... ∩ En) = P(E1) P(E2) ... P(En) 6. Probability of at least one of the n Independent events :

If P1, P2, ... Pn are the probabilities of n independent events A, A2, .... An then the probability of happening of at least one of these event is.

1 – [(1 – P1) (1 – P2)...(1 – Pn)]

7. Total Probability :

Let A1, A2, ... An are n mutually exclusive & set of exhaustive events and event A can occur through any one of these events, then probability of occurence of A

P(A) = P(A ∩ A1) + P(A ∩ A2) + ... + P(A ∩ An) = r n =

∑

1 P(Ar) P(A/Ar) 8. Baye's Rule :

Let A1, A2, A3 be any three mutually exclusive & exhaustive events (i.e. A1∪ A2 ∪ A3 = sample space & A1 ∩ A2 ∩A3 = φ) an sample space S and B is any other event on sample space then, P(Ai/B) = P B A P A P B A P A P B A P Ai i P B A P A ( / ) ( ) ( / 1) ( )1 + ( / 2) ( )2 + ( / 3) ( )3 , i = 1, 2, 3 9. Probability distribution :

(i) If a random variable x assumes values x1, x2, ...xn with probabilities P1, P2, ... Pn respectively then (a) P1 + P2 + P3 + ... + Pn = 1

(b) mean E(x) = Σ Pixi (c) Variance = Σx2_P

i – (mean)2 = Σ (x2) – (E(x))2 (ii) Binomial distribution : If an experiment is repeated n

times, the successive trials being independent of one another, then the probability of

-r success is n_C r Pr qn–r atleast r success is k r n =

∑

n_C k Pk qn–k

where p is probability of success in a single trial, q = 1 – p

(a) mean E(x) = np (b) E (x2_{) = npq + n}2 _p2

(c) Variance E(x2_{) – (E(x))}2 _{= npq} (d) Standard deviation = npq

10. Truth of the statement :

(i) If two persons A and B speaks truth with the probabil-ity p1 & p2 respectively and if they agree on a state-ment, then the probability that they are speaking truth will be given by

p p

p p_{1 2} +(1−1 2p₁) (1−p₂).

(ii) If A and B both assert that an event has occurred, probability of occurrence of which is α then the prob-ability that event has occurred.

Given that the probability of A & B speaking truth is p1, p2.

α

α α

p p

p p_{1 2} +(1− ) (1 21−p₁) (1−p₂)

(iii) If in the second part the probability that their lies (jhuth) coincides is β then from above case required probability will be

α

α α β

p p

PROGRESSION AND SERIES

1. Arithmetic Progression (A.P.) :

(a) General A.P. — a, a + d, a + 2d, ... , a + (n – 1) d where a is the first term and d is the common difference (b) General (nth_{) term of an A.P. —}

Tn = a + (n – 1)d [nth term from the beginning] If an A.P. having m terms, then nth _{term from end} = a + (m – n)d

(c) Sum of n terms of an A.P. — Sn =

n

2 [2a + (n – 1)d] = n

2[a + Tn]

Note : If sum of n terms i.e. Sn is given then Tn = Sn – Sn–1 where Sn–1 is sum of (n – 1) terms.

(d) Supposition of terms in A.P. — (i) Three terms as a - d, a, a + d

(ii) Four terms as a – 3d, a – d, a + d, a + 3d (iii) Five terms as a – 2d, a – d, a, a + d, a + 2d (e) Arithmetic mean (A.M.) :

(i) A.M. of n numbers A1, A2, ... An is defined as

A.M. = A1+A2+..._n +An = Σ_nAi = Sum of numbers_n (ii) For an A.P., A.M. of the terms taken symmetrically

from the beginning and from the end will always be constant and will be equal to middle term or A.M. of middle term.

(iii) If A is the A.M. between two given nos. a and b, then

A = a b+

2 i.e. 2A = a + b

(iv) If A1, A2,... An are n A.M's between a and b, then A1 = a + d, A2 = a + 2d,... An = a + nd, where d = b a_n−₊₁

(v) Sum of n A.M's inserted between a and b is n₂ (a + b) (vi) Any term of an A.P. (except first term) is equal to the half of the sum of term equidistant from the term i.e. an =

1

2(an–r + an+r), r < n 2. Geometric Progression (G.P.)

(a) General G.P. — a, ar, ar2 _{, ...}

where a is the first term and r is the common ratio (b) General (nth_{) term of a G.P. — T}

n = arn–1

If a G.P. having m terms then nth _{term from end = ar}m–n (c) Sum of n terms of a G.P. — Sn = a r r n (1 ) 1 − − = a T r r n − − 1 , r < 1 = a r(_rn₋−₁1) = T r a_rn₋−₁ , r > 1 (d) Sum of an infinite G.P. — S∞ = a r 1− , |r|<1

(e) Supposition of terms in G.P. — (i) Three terms as a_r , a, ar (ii) Four terms as _ra3 ,

a

r , ar, ar3 (iii) Five terms as _ra2 , a_r , a, ar, ar2

(f) Geometric Mean (G.M.) —

(i) Geometrical mean of n numbers x1, x2, ... xn is defined as

G.M. = (x1 x2 ... xn)1/n.

(i) If G is the G.M. between two given numbers a and b, then

G2 _{= ab ⇒ G =} ab

(ii) If G1, G2, ... Gn are n G.M's between a and b, then G1 = ar, G2 = ar2,... Gn = arn, where r = b a

F

HG

I

KJ

1/n+1 (iii) Product of the n G.M.'s inserted between a & b is

(ab)n/2

3. Arithmetico - Geometric Progression (A.G.P.) : (a) General form — a, (a + d)r, (a + 2d) r2_{, ...} (b) General (nth_{) term — T}

n= [a + (n – 1) d] rn–1 (c) Sum of n terms of an A.G.P — Sn =

a r 1− + r. d r r n ( ) ( ) 1 1 1 2 − − −

(d) Sum of infinite terms of an A.G.P. S∞ = a r 1− + dr r (1₋ )2 4. Sum standard results :

(a) Σn = 1 + 2 + 3 + ... + n = n n( ₂+1) (b) Σn2 _{= 1}2 _{+ 2}2 _{+ 3}2 _{+ ... + n}2 ₌ n n( +1 2)( n+1) 6 (c) Σn3 _{= 1}3 _{+ 2}3 _{+ 3}3 _{+ .... + n}3 ₌

L

n n( + )

NM

2 1

O

QP

2 (d) Σa = a + a + .... + (n times) = na (e) Σ(2n – 1) = 1 + 3 + 5 + .... (2n – 1) = n2 (f) Σ2n = 2 + 4 + 6 + .... + 2n = n (n + 1) 5. Harmonic Progression (H.P) (a) General H.P. — 1_a, _{a d}1₊ , _a+1_2d +... (b) General (nth _{term) of a H.P. — T} n = _{a n+(}1_−1)d = _{n term coresponding to A P}th 1 _{. .} (c) Harmonic Mean (H.M.)

(i) If H is the H.M. between a and b, then H = _{a b}2ab₊ (ii) If H1, H2,...,Hn are n H.M's between a and b,

then H1 = ab n bn a ( + ) + 1 , ..., Hn = ab n na b ( + ) + 1

or first find n A.M.'s between 1_a & 1_b, then their reciprocal will be required H.M's.

6. Relation Between A.M., G.M. and H.M. (i) AH = G2

(ii) A ≥ G ≥ H

(iii) If A and G are A.M. and G.M. respectively between two +ve numbers, then these numbers are

BINOMIAL THEOREM

1. Binomial Theorem for any +ve integral index :

(x + a)n ₌ n_C 0 xn + nC1 xn–1 a + nC2 xn–2 a2 + ... + n_C r xn–r ar + .... + nCn an = r n =

∑

0 n_C r xn–r ar

(i) General term - Tr+1 = nCr xn–rar is the (r + 1)th term from beginning.

(ii) (m + 1)th _{term from the end = (n – m + 1)}th _from begin-ning = Tn–m+1

(iii) middle term

(a) If n is even then middle term = n th

2+1

F

HG

I

KJ

term (b) If n is odd then middle term = n

th +

F

HG

21

I

KJ

and n th +

F

HG

23

I

KJ

term

Binomial coefficient of middle term is the greatest bino-mial coefficient.

2. To determine a particular term in the given expasion : Let the given expansion be x _x

n α β ±

F

HG

1

I

KJ

, if xn _{occurs in} Tr+1 (r + 1)th term then r is given by n α – r (α + β) = m and for x0_{, n α – r (α + β) = 0}

3. Properties of Binomial coefficients :

For the sake of convenience the coefficients n_C

0, nC1, n_C

2... nCr...nCn are usually denoted by C0, C1,... Cr ... Cn respectively. * C0 + C1 + C2 + ... + Cn = 2n * C0 – C1 + C2 – C3 + ... + Cn = 0 * C0 + C2 + C4 + ... = C1 + C3 + C5 + .... = 2n–1 * n r C = n_r n−1Cr−1 = n_r n_r−−1₁ n−2C_r−2 and so on ... * 2nCn r+ = 2n n r n r ! ! ! − +

c h c h

* n r C + nC_r−1 = n r C +1 * C1 + 2C2 + 3C3 + ... + nCn = n.2n–1 * C1 – 2C2 + 3C3 ... = 0 * C0 + 2C1 + 3C2 + ...+ (n + 1)Cn = (n + 2)2n–1 * C02 + C12 + C22 + ... + Cn2 = 2 2 n n

c h

c h

! ! = 2n_C n * C02 – C12 + C22 – C32 + ... = ₁ /20, _, /2 if n is odd C if n is ev en n _n n −

RS|

T|c h

Note : 2 1 0 n+ _{C +} 2 1 1 n+ _{C + .... +} 2n 1 n C + ₌ 2 1 1 n n C + + + 2 1 2 n n C + + + ... 2n+1C2n+1 = 22n * C0 + C1 2 + C2 3 + ... + C n+n1 = 2 1 1 1 n n + ₋ + * C0 – C1 2 + C2 3 – C3 4 .... + ( )− + 1 1 n n C n = 1 1 n+ 4. Greatest term :

(i) If (n_{x a}+₊1)a ∈ Z (integer) then the expansion has two greatest terms. These are kth _{and (k + 1)}th _{where x & a} are +ve real nos.

(ii) If (n_{x a}+₊1)a ∉ Z then the expansion has only one

great-est term. This is (k + 1)th _{term k =} (n )a x a

+ +

L

NM

1

O

QP

,

{[.] denotes greatest integer less than or equal to x} 5. Multinomial Theorem : (i) (x + a)n ₌ r n = ∑ 0 n_C r xn–r ar, n ∈N = r n = ∑ 0 n n r) r ! ( − ! ! xn–r ar = r s n+ =∑ n s r ! ! ! xs ar, (ii) (x + y + z)n ₌ r s t n+ + =∑ n s r t ! ! ! ! xr ys zt Generalized (x1 + x2 +... xk)n = r r1+ +2 r nk= ∑ ... n r r r_k ! ! !.... ! 1 2 x x x r r k rk 1 21 2...

6. Total no. of terms in the expansion (x1 + x2 +... xn)m is m+n–1_C

TRIGONOMETRIC RATIO AND IDENTITIES

1. Some important results : (i) Arc length AB = rθ

Area of circular sector = 1

2r2θ

(ii) For a regular polygon of side a and number of sides n (a) Internal angle of polygon = (n – 2) π

n

(b) Sum of all internal angles = (n – 2) π

(c) Radius of incircle of this polygon r = a

2 cot

π

n

(d) Radius of circumcircle of this polygon R = ₂a cosec_nπ

(e) Area of the polygon = 1₄ na2 _cot π n

F

HG

I

KJ

(f) Area of triangle = 1₄a2 _cos π

n (g) Area of incircle = π ₂a _n 2 cotπ

F

HG

I

KJ

(h) Area of circumcircle = π ₂a ec_n 2 cos π

F

HG

I

KJ

2. Relation between system of measurement of angles :

D 90 = G 100 = 2C π & π radian = 1800 3. Trigonometric identities : (i) sin2_{θ + cos}2_{θ = 1} (ii) cosec2_{θ – cot}2_{θ = 1} (iii) sec2_{θ – tan}2_{θ = 1} 4. Sign convention :

y

II quadrant I quadrant sin & cosec All +ve

are +ve

x' O x

III quadrant IV quadrant tan & cot cos & sec

are +ve are +ve

y'

5. T-ratios of allied angles : The signs of trigonometrical ratio in different quadrant.

Allied∠ of (–θ) 900 _{± θ 180}0 _{± θ 270}0 _{± θ 360}0 _{± θ} T-ratios

sinθ –sinθ cosθ _msinθ –cosθ ±sinθ

cosθ cosθ _msinθ –cosθ ±sinθ cosθ

tanθ –tanθ _mcotθ ±tanθ _mcotθ ±tanθ

cotθ –cotθ _mtanθ ±cotθ _mtanθ ±cotθ

secθ secθ mcosecθ –secθ ±cosecθ secθ

cosecθ –cosecθ secθ _mcosecθ –sec θ ±cosecθ

6. Sum & differences of angles of t-ratios : (i) sin(A ± B) = sinA cosB ± cosA sinB (ii) cos(A ± B) = cosA cosB ± sinA sinB (iii) tan (A ± B) = tan tan

tan tan

A B

A B

±

(iv) cot (A ± B) = cot cot_cotA_B±cotBm 1_A

(v) sin(A + B) sin(A – B) = sin2_{A – sin}2_{B = cos}2 _{B – cos}2 _A (vi) cos(A + B) cos (A – B) = cos2_{A – sin}2_{B = cos}2_{B – sin}2_A (vii) tan(A + B + C)

= ₁tan_{−tan tan}A+_AtanB_B+₋tan_{tan tan}C_B−tan tan tan_CA_{−tan tan}B_C C_A = S1 _SS3 2 1 − − Generalized tan (A + B + C + ... ) = S1_S S3_S S5_S S7_S 2 4 6 8 1 − + − + − + − + − ... ... Where S1 = Σ tan A S2 = Σ tan A tan B,

S3 = Σtan A tan B tan C & so on (viii) sin (A + B + C) = Σ sin A cos B cos C – Π sin A

= Π cos A (Numerator of tan (A + B + C)) (ix) cos (A + B + C) = Π cos A – Σ sin A sin B cos C

= Π cos A (Denominator of tan (A + B + C)) for a triangle A + B + C = π

Σ tan A = Π tan A

Σ sin A = Σ sin A cos B cos C 1 + Π cos A = Σ sin A sin B cos C (viii) sin750 ₌ 3 1 2 2 + = cos150 (ix) cos750 ₌ 3 1 2 2 − = sin150 (x) tan750 _{= 2 + 3 = cot15}0 (xi) cot750 _{= 2 – = tan15}0

7. Formulaes for product into sum or difference and vice-versa :

(i) 2sinA cosB = sin(A + B) + sin(A – B) (ii) 2cosA sinB = sin(A + B) – sin(A – B) (iii) 2cosA cosB = cos(A + B) + cos(A – B) (iv) 2sinA sinB = cos(A – B) – cos(A + B) (v) sinC + sinD = 2sin

F

HG

C D₂+

I

KJ

cos

F

HG

C D₂−

I

KJ

(vi) sinC – sinD = 2cos

F

HG

C D+₂

I

KJ

sin

F

HG

C D₂−

I

KJ

(vii) cosC + cosD = 2cos

F

HG

C D₂+

I

KJ

cos

F

HG

C D₂−

I

KJ

(viii) cosC – cosD = 2sin

F

HG

C D₂+

I

KJ

sin

F

HG

D C₂−

I

KJ

(ix) tanA + tanB = _{cos cos}sin(_AA B+ _B)

8. T-ratios of multiple and submultiple angles : (i) sin2A = 2sinA cosA = ₁2 2

tan tan A

A

+

= (sin A + cos A)2 _{– 1 = 1 – (sin A – cos A)}2

⇒ sinA = 2sinA/2 cosA/2 = ₁2 2 2₂

tan /

tan /

A A

+

(ii) cos2A = cos2_{A – sin}2_{A = 2cos}2_{A – 1} = 1 – 2sin2_{A =} 1 1 2 2 − + tan tan A A

(iii) tan2A = ₁2 2 tan tan A A − ⇒ tanA = 2 2 1 2 2 tan / tan / A A −

(iv) sin3θ = 3sinθ – 4sin3_{θ = 4sin(60}0 _{– θ) sin(60}0 _{+ θ) sinθ} = sin θ (2 cos θ – 1) (2 cos θ + 1)

(v) cos3θ = 4cos3_{θ – 3cosθ}

= 4cos(600 _{– θ) cos(60}0 _{+ θ) cosθ} = cos θ (1 – 2 sin θ) (1 + 2 sin θ) (vi) tan3A = 3_{1 3} 3 2 tan tan tan A A A − −

= tan(600 _{– A) tan(60}0 _{+ A)tanA} (vii) sinA/2 = 1−cos A₂

(viii) cosA/2 = 1+cos A₂

(ix) tanA/2 = 1 1 − + cos cos A A = 1−cos sin A A , A ≠ (2n + 1)π

9. Maximum and minimum value of the expression : acosθ + bsinθ

Maximum (greatest) Value = _a2_+b2 Minimum (Least) value = – _a2_+b2 10. Conditional trigonometric identities :

If A, B, C are angles of triangle i.e. A + B + C = π, then (i) sin2A + sin2B + sin2C = 4sinA sinB sinC

i.e. Σ sin 2A = 4 Π (sin A)

(ii) cos2A + cos2B + cos2C = –1– 4cosA cosB cosC (iii) sinA + sinB + sinC = 4cosA/2 cosB/2 cosC/2 (iv) cosA + cosB + cosC = 1 + 4 sinA/2 sinB/2 sinC/2 (v) sin2_{A + sin}2_{B + sin}2_{C = 1 – 2sinA sinB cosC} (vi) cos2_{A + cos}2_{B + cos}2_{C = 1 – 2cosA cosB cosC} (vii) tanA + tanB + tanC = tanA tanB tanC

(viii) cotB cotC + cotC cotA + cotA cotB = 1 (ix) Σ tan A/2 tan B/2 = 1

(x) Σcot A cot B = 1 (xi) Σcot A/2 = Π cot A/2 11. Some useful series :

(i) sinα + sin(α + β) + sin(α + 2β) + .... + to nterms

= sin sin sin α β β β +

F

_HG

−

I

_KJ

L

NM

n21

O

QP

L

NM

n2

O

QP

2 , β≠ 2nπ

(ii) cosα + cos(α + β) + cos(α + 2β) + .... + to nterms

= cos sin sin α β β β +

F

_HG

−

I

_KJ

L

NM

n21

O

QP

n2 2 β≠ 2nπ

(iii) cosα .cos2α . cos22_{α ....cos(2}n–1 _{α) =} sin

sin 2 2 n n α α, α ≠ nπ = 1 , α = 2kπ = –1 , α = (2k+1)π

TRIGONOMETRIC EQUATIONS 1. General solution of the equations of the form

(i) sinθ = 0 ⇒ θ = nπ, n ∈ I (ii) cosθ = 0 ⇒ θ = (2n + 1) ₂π, n ∈ I (iii) tanθ = 0 ⇒ θ = nπ, n ∈ I (iv) sinθ = 1 ⇒ θ = 2nπ + π₂ (v) cosθ = 1 ⇒ θ = 2πn (vi) sinθ = –1 ⇒ θ = 2nπ – π₂ or 2nπ + 3₂π (vii) cosθ = –1 ⇒ θ = (2n + 1)π

(viii) sinθ = sinα ⇒ θ = nπ + (–1)n_α (ix) cosθ = cosα ⇒ θ = 2nπ ± α

(x) tanθ = tanα⇒ θ = nπ + α

(xi) sin2_{θ = sin}2_{α ⇒ θ = nπ ± α} (xii) cos2_{θ = cos}2_{α ⇒ θ = nπ ± α} (xiii) tan2_{θ = tan}2_{α ⇒ θ = nπ ± α}

2. For general solution of the equation of the form

a cosθ + bsinθ = c, where c ≤ a2₊b2 , divide both side by

a2₊b2

and put _a2a_+b2 = cosα, b

a2₊b2 = sinα.

Thus the equation reduces to form cos(θ – α) = _a2c_+b2 = cosβ(say) now solve using above formula 3. Some important points :

(i) If while solving an equation, we have to square it, then the roots found after squaring must be checked wheather they satisfy the original equation or not. (ii) If two equations are given then find the common

val-ues of θ between 0 & 2π and then add 2nπ to this common solution (value).

INVERSE TRIGONOMETRIC FUNCTIONS

1. If y = sin x, then x = sin–1 _{y, similarly for other inverse} T-functions.

2. Domain and Range of Inverse T-functions : Function Domain (D) Range (R) sin–1 _{x – 1 ≤ x ≤ 1 –} π 2 ≤ θ ≤ π 2 cos–1 _{x – 1 ≤ x ≤ 1 0 ≤ θ ≤ π} tan–1 _{x – ∞ < x < ∞ –} π 2 < θ < π 2 cot–1 _{x – ∞ < x < ∞ 0 < θ < π} sec–1 _{x x ≤ – 1, x ≥ 1 0 ≤ θ ≤ π, θ ≠} π 2 cosec–1 _{x x ≤ – 1, x ≥ 1 –} π 2 ≤ θ ≤ π 2, θ ≠ 0

3. Properties of Inverse T-functions : (i) sin–1 _{(sin θ) = θ provided –} π

2 ≤ θ ≤

π

2 cos–1 _{(cos θ) = θ provided θ ≤ θ ≤ π} tan–1 _{(tan θ) = θ provided –} π

2 < θ <

π

2

cot–1 _{(cot θ) = θ provided 0 < θ < π} sec–1 _{(sec θ) = θ provided 0 ≤ θ <} π

2 or

π

2 < θ ≤ π

cosec–1 _{(cosec θ) = θ provided –} π

2 ≤ θ < 0

or 0 < θ ≤ π₂

(ii) sin (sin–1 _{x) = x provided – 1 ≤ x ≤ 1} cos (cos–1 _{x) = x provided – 1 ≤ x ≤ 1} tan (tan–1 _{x) = x provided – ∞ < x < ∞} cot (cot–1 _{x) = x provided – ∞ < x < ∞}

sec (sec–1 _{x) = x provided – ∞ < x ≤ – 1 or 1 ≤ x < ∞} cosec (cosec–1 _{x) = x provided – ∞ < x ≤ – 1}

or 1 ≤ x < ∞

(iii) sin–1 _{(– x) = – sin}–1 _x, cos–1 _{(– x) = π – cos}–1 _x tan–1 _{(– x) = – tan}–1 _x cot–1 _{(– x) = π – cot}–1 _x cosec–1 _{(– x) = – cosec}–1 _x sec–1 _{(– x) = π – sec}–1 _x

(iv) sin–1 _{x + cos}–1 _{x =} π

2, ∀ x ∈ [– 1, 1]

tan–1 _{x + cot}–1 _{x =} π

2, ∀ x ∈ R

sec–1 _{x + cosec}–1 _{x =} π

4. Value of one inverse function in terms of another inverse function :

(i) sin–1 _{x = cos}–1

1_−x2 = tan–1 x x 1₋ 2 = cot –1 1−x2 x = sec–1 1 1_−x2 = cosec –1 1 x , 0 ≤ x ≤ 1

(ii) cos–1 _{x = sin}–1

1_−x2 _{= tan}–1 1−x2 x = cot –1 x x 1− 2 = sec–1 1 x = cosec–1 1 1−x2 , 0 ≤ x ≤ 1

(iii) tan–1 _{x = sin}–1 x

x 1+ 2 = cos–1 1 1+x2 = cot–1 1 x = sec–1 1_+x2 _{= cosec}–1 1+x2 x , x ≥ 0 (iv) sin–1 1 x

F

HG

I

KJ

= cosec–1 _{x, ∀ x ∈ (– ∞, 1] ∪ [1, ∞)} (v) cos–1 1 x

F

HG

I

KJ

= sec–1 _{x, ∀ x ∈ (– ∞, 1] ∪ [1, ∞)} (vi) tan–1 1 x

F

HG

I

KJ

= cot_cot − − > − + <

RS

T|

1 1 0 0 x for x x for x π

5. Formulae for sum and difference of inverse trigonomet-ric function :

(i) tan–1_{x + tan}–1_{y = tan}–1 x y xy

+ −

F

HG

1

I

KJ

; if x > 0, y > 0, xy < 1 (ii) tan–1_{x + tan}–1_{y = π + tan}–1 x y

xy

+ −

F

HG

1

I

KJ

; if x > 0, y > 0, xy > 1 (iii) tan–1_{x – tan}–1_{y = tan}–1 x y

xy

− +

F

HG

1

I

KJ

; if xy > –1 (iv) tan–1_{x – tan}–1_{y = π + tan}–1 x y

xy

− +

F

HG

1

I

KJ

; if x > 0, y < 0, xy < –1 (v) tan–1_{x + tan}–1_{y + tan}–1_{z = tan}–1 x y z xyz

xy yz zx

+ + −

− − −

F

HG

1

I

KJ

(vi) sin–1_{x ± sin}–1_{y = sin}–1

LNM

x 1−y2 ±y 1−x2

OQP

_; if x,y ≥ 0 & x2 _{+ y}2 _{≤ 1}

(vii) sin–1_{x ± sin}–1_{y = π – sin}–1

LNM

x 1−y2 ±y 1−x2

OQP

_; if x,y ≥ 0 & x2 _{+ y}2 _{> 1}

(viii) cos–1_{x ± cos}–1_{y = cos}–1

LNM

xym 1−x2 1−y2

OQP

_; if x,y > 0 & x2 _{+ y}2 _{≤ 1}

(ix) cos–1_{x ± cos}–1_{y = π – cos}–1

LNM

xym 1−x2 1−y2

OQP

_; if x,y > 0 & x2 _{+ y}2 _{> 1}

6. Inverse trigonometric ratios of multiple angles (i) 2sin–1_{x = sin}–1_(2x

1_−x2 _{), if –1 ≤ x ≤ 1} (ii) 2cos–1_{x = cos}–1_(2x2 _{–1), if –1 ≤ x ≤ 1} (iii) 2tan–1_{x = tan}–1 2

1 2 x x −

F

HG

I

KJ

= sin–1 2 1 2 x x +

F

HG

I

KJ

= cos–1 1 1 2 2 − +

F

HG

xx

I

KJ

(iv) 3sin–1_{x = sin}–1_{(3x – 4x}3₎

(v) 3cos–1_{x = cos}–1_(4x3 _{– 3x)} (vi) 3tan–1_{x = tan}–1 3

1 3 3 2 x x x − −

F

HG

I

KJ

PROPERTIES & SOLUTION OF TRIANGLE

Properties of triangle :

1. A triangle has three sides and three angles. In any ∆ABC, we write BC = a, AB = c, AC = b

B B C C A A a b c

and ∠BAC = ∠A, ∠ABC = ∠B, ∠ACB = ∠C

2. In ∆∆∆∆∆ABC : (i) A + B + C = π (ii) a + b > c, b +c > a, c + a > b (iii) a > 0, b > 0, c > 0 3. Sine formula : a A sin = b B sin = c C sin = k(say)

or sinA_a = sinB_b = sinC_c = k (say)

4. Cosine formula : cos A = b2+₂c_bc2 −a2 cos B = c a b ac 2 2 2 2 + − cos C = a b c ab 2 2 2 2 + −

5. Projection formula : a = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A 6. Napier's Analogies : tan A B₂− = a b_{a b}−₊ cot C₂ tan B C₂− = b c_{b c}−₊ cot A₂ tan C A−₂ = c a_{c a}−₊ cot B₂

7. Half angled formula - In any ∆∆∆∆∆ABC : (a) sin A₂ = (s b s c) ( ) bc − − sin B 2 = (s c s a) ( ) ca − − sin C₂ = (s a s b− ) (_ab − ) where 2s = a + b + c (b) cos A 2 = s s a bc ( − ) cos B₂ = s s b(_ca− ) cos C₂ = s s c(_ab− ) (c) tan A₂ = (s b s c−_{s s a(}) (_{− )}− ) tan B₂ = (s c s a−_{s s b(}) (_{− )}− ) tan C₂ = (s b s a−_{s s c(}) (_{− )}− ) 8. ∆∆∆∆∆, Area of triangle :

(i) ∆ = 1₂ab sin C = 1₂ bc sin A = 1₂ ca sin B

(ii) ∆ = s s a s b s c( − ) ( − ) ( − ) 9. tan A 2 tan B 2 = s c s − tan B 2 tan C 2 = s a s − tan C 2 tan A 2 = s b s −

10. Circumcircle of triangle and its radius : (i) R = _2sina _A = _2sinb _B = _2sinc _C

11. Incircle of a triangle and its radius : (iii) r = ∆_s

(iv) r = (s – a) tan A₂ = (s – b) tan B₂ = (s – c) tan C₂

(v) r = 4R sin A₂ sin B₂ sin C₂

(vi) cos A + cos B + cos C = 1 + _Rr

(vii) r = a B C A sin sin cos 2 2 2 = b A C B sin sin cos 2 2 2 = c B A C sin sin cos 2 2 2

12. The radii of the escribed circles are given by : (i) r1 = ∆ s a− , r2 = ∆ s b− , r3 = ∆ s c− (ii) r1 = s tan A 2, r2 = s tan B 2, r3 = s tan C 2 (iii) r1 = 4R sin A 2 cos B 2 cos C 2, r2 = 4R cos A 2 sin B 2 cos C 2, r3 = 4R cos A 2 cos B 2 sin C 2 (iv) r1 + r2 + r3 – r = 4R (v) 1 1 r + 1 2 r + 1 3 r = 1_r (vi) 1 12 r + 1 22 r + 1 32 r + 1 2 r = a2 b2 c2 2 + + ∆ (vii) _bc1 + _ca1 + _ab1 = _2Rr1 (viii) r1r2 + r2r3 + r3r1 = s2

(ix) ∆ = 2R2 _{sin A sin B sin C = 4Rr cos} A

2 cos B 2 cos C 2 (x) r1 = a B C A cos cos cos 2 2 2 , r2 = b C A B cos cos cos 2 2 2 , r3 = c A B C cos cos cos 2 2 2

(ii) d = h (cotα – cotβ)

α β

d

h

HEIGHT AND DISTANCE

1. Angle of elevation and depression :

If an observer is at O and object is at P then ∠XOP is called angle of elevation of P as seen from O.

If an observer is at P and object is at O, then ∠QPO is called angle of depression of O as seen from P.

2. Some useful result :

(i) In any triangle ABC if AD : DB = m : n

∠ACD = α , ∠BCD = β & ∠BDC = θ

then (m + n) cotθ = m cotα – ncot β

C B B A A D m _n α β θ

POINT

1. Distance formula :

Distance between two points P(x1, y1) and Q(x2, y2) is given by d(P, Q) = PQ

= (x₂ x₁)2 (y y ) 2 1 2

− + −

= (Difference of x coordinate)2+(Difference of ycoordinate)2

Note : (i) d(P, Q) ≥ 0

(ii) d(P, Q) = 0 ⇔ P = Q (iii) d(P, Q) = d(Q, P)

(iv) Distance of a point (x, y) from origin (0, 0) = _x2 _+y2 2. Use of Distance Formula :

(a) In Triangle :

Calculate AB, BC, CA

(i) If AB = BC = CA, then ∆ is equilateral.

(ii) If any two sides are equal then ∆ is isosceles. (iii) If sum of square of any two sides is equal to

the third, then ∆ is right triangle.

(iv) Sum of any two equal to left third they do not form a triangle

i.e. AB = BC + CA or BC = AC + AB

or AC = AB + BC. Here points are collinear. (b) In Parallelogram :

Calculate AB, BC, CD and AD.

(i) If AB = CD, AD = BC, then ABCD is a parallelo-gram.

(ii) If AB = CD, AD = BC and AC = BD, then ABCD is a rectangle.

(iii) If AB = BC = CD = AD, then ABCD is a rhombus. (iv) If AB = BC = CD = AD and AC = BD, then ABCD

is a square.

(C) For circumcentre of a triangle :

Circumcentre of a triangle is equidistant from vertices i.e. PA = PB = PC.

Here P is circumcentre and PA is radius.

(i) Circumcentre of an acute angled triangle is in-side the triangle.

(ii) Circumcentre of a right triangle is mid point of the hypotenuse.

(iii) Circumcentre of an obtuse angled triangle is outside the triangle.

3. Section formula : (i) Internally : AP BP = m n = λ, Here λ > 0 A (x , y )1 1 P B (x , y )2 2 m n P

HG

F

mx_{m n}2₊+nx1, my_{m n}2 +₊ny1

I

KJ

(ii) Externally : AP BP = m n = λ _{A(x , y )} 1 1 B(x , y )2 2 P n m P

HG

F

mx_{m n}2₋−nx1, my_{m n}2₋−ny1

I

KJ

(iii) Coordinates of mid point of PQ are

HG

F

x1+₂x y2, 1₂+y2

I

KJ

(iv) The line ax + by + c = 0 divides the line joining the

points

(x1, y1) & (x2, y2) in the ratio = –

( ) ( ) ax by c ax21 by2 c + + + +

(v) For parallelogram – midpoint of diagonal AC = mid point of diagonal BD

(vi) Coordinates of centroid G

HG

F

x1+x₃2 +x3,y1+y₃2+y3

I

KJ

(vii) Coordinates of incentre I

ax bx cx a b c ay by cy a b c 1+ 2+ 3 1 2 3 + + + + + +

F

HG

,

I

KJ

(viii) Coordinates of orthocentre are obtained by solving the equation of any two altitudes.

4. Area of Triangle :

The area of triangle ABC with vertices A(x1, y1), B(x2, y2) and C(x3, y3). ∆ = 1₂ x y x y x y 1 1 2 2 3 3 1 1 1 (Determinant method) = 1₂ x y x y x y x y 1 1 2 2 3 3 1 1 = 1₂ [x1y2 + x2y3 + x3y1 – x2y1 – x3y2 – x1y3] [Stair method] Note :

(i) Three points A, B, C are collinear if area of triangle is zero.

(ii) If in a triangle point arrange in anticlockwise then value of ∆ be +ve and if in clockwise then ∆ will be –ve.

5. Area of Polygon :

Area of polygon having vertices (x1, y1), (x2, y2), (x3, y3) ... (xn, yn) is given by area = 1₂ x y x y x y x y x y n n 1 1 2 2 3 3 1 1 M M

. Points must be taken in order.

6. Rotational Transformation :

If coordinates of any point P(x, y) with reference to new axis will be (x', y') then

x

B

y

B

x'→ cosθ sinθ

y'→–sinθ cosθ

7. Some important points :

(i) Three pts. A, B, C are collinear, if area of triangle is zero

(ii) Centroid G of ∆ABC divides the median AD or BE or CF in the ratio 2 : 1

(iii) In an equilateral triangle, orthocentre, centroid, circumcentre, incentre coincide.

(iv) Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining orthocentre and circumcentre in the ratio 2 : 1

(v) Area of triangle formed by coordinate axes & the line ax + by + c = 0 is ₂c_ab2 .

STRAIGHT LINE

1. Slope of a Line : The tangent of the angle that a line makes with +ve direction of the x-axis in the anticlockwise sense is called slope or gradient of the line and is generally denoted by m. Thus m = tan θ.

(i) Slope of line || to x-axis is m = 0

(ii) Slope of line || to y-axis is m = ∞ (not defined) (iii) Slope of the line equally inclined with the axes is 1

or – 1

(iv) Slope of the line through the points A(x1, y1) and B(x2, y2) is

y y

x2₂ x1₁

−

− .

(v) Slope of the line ax + by + c = 0, b ≠ 0 is – _ba (vi) Slope of two parallel lines are equal.

(vii) If m1 & m2 are slopes of two ⊥ lines then m1m2 = – 1. 2. Standard form of the equation of a line :

(i) Equation of x-axis is y = 0 (ii) Equation of y-axis is x = 0

(iii) Equation of a straight line || to x-axis at a distance b from it is y = b

(iv) Equation of a straight line || to y-axis at a distance a from it is x = a

(v) Slope form : Equation of a line through the origin and having slope m is y = mx.

(vi) Slope Intercept form : Equation of a line with slope m and making an intercept c on the y-axis is y = mx + c. (vii) Point slope form : Equation of a line with slope m

and passing through the point (x1, y1) is y– y1 = m(x – x1)

(viii) Two point form : Equation of a line passing through the points (x1, y1) & (x2, y2) is

y y y y − − 1 2 1 = x x x x − − 1 2 1

(ix) Intercept form : Equation of a line making intercepts a and b respectively on x-axis and y-axis is x_a + _by = 1. (x) Parametric or distance or symmetrical form of the line : Equation of a line passing through (x1, y1) and making an angle θ, 0 ≤ θ ≤ π, θ ≠ ₂π with the +ve direction of x-axis is

x x− ₁

cosθ =

y y− ₁

sinθ = r

⇒ x = x1 + r cos θ, y = y1 + r sin θ

Where r is the distance of any point P(x, y) on the line from the point (x1, y1)

(xi) Normal or perpendicular form : Equation of a line such that the length of the perpendicular from the origin on it is p and the angle which the perpendicular makes with the +ve direction of x-axis is α, is

x cos α + y sin α = p. 3. Angle between two lines :

(i) Two lines a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0 are (a) Parallel if a_a1 2 = b b12 ≠ c c12 (b) Perpendicular if a1a2 + b1b2 = 0 (c) Identical or coincident if a_a1 2 = b b12 = c c12 (d) If not above three, then θ = tan–1 a b a b

a a b b

2 1 1 2 1 2 1 2

− −

(ii) Two lines y = m1 x + c and y = m2 x + c are (a) Parallel if m1 = m2

(b) Perpendicular if m1m2 = –1

(c) If not above two, then θ = tan–1 m m

m m 1 2 1 2 1 − +

4. Position of a point with respect to a straight line : The line L(xi, yi) i = 1, 2 will be of same sign or of opposite sign according to the point A(x1, y1) & B (x2, y2) lie on same side or on opposite side of L (x, y) respectively.

5. Equation of a line parallel (or perpendicular) to the line ax + by + c = 0 is ax + by + c' = 0 (or bx – ay + λ = 0) 6. Equation of st. lines through (x1,y1) making an angle ααααα

with y = mx + c is y – y1 = m m ±tan tan α α 1 m (x – x1)

7. length of perpendicular from (x1, y1) on ax + by + c = 0 is |ax1_a2by_b12 c|

+ +

+

8. Distance between two parallel lines ax + by + ci = 0, i = 1, 2 is |c_a12 c_b22|

− +

9. Condition of concurrency for three straight lines

Li≡≡≡≡≡ ai x + bi y + ci = 0, i = 1, 2, 3 is a b c a b c a b c 1 1 1 2 2 2 3 3 3 = 0

10. Equation of bisectors of angles between two lines : a x b y c a b 1 1 1 1 2 1 2 + + + = ± a x b y c a b 2 2 2 2 2 2 2 + + +

11. Family of straight lines :

The general equation of family of straight line will be written in one parameter

The equation of straight line which passes through point of intersection of two given lines L1 and L2 can be taken as

λ

12. Homogeneous equation : If y = m1x and y = m2x be the two equations

represented by ax2 _{+ 2hxy + by}2 _{= 0 , then m}

1 + m2 = –2h/b and m1m2 = a/b

13. General equation of second degree :

ax2 _{+ 2hxy + by}2 _{+ 2gx + 2fy + c = 0 represent a pair of} straight line if ∆≡

a h g h b f

g f c = 0

If y = m1x + c & y = m2x + c represents two straight lines then m1 + m2 =

−2h

b , m1m2 =

a b.

14. Angle between pair of straight lines : The angle between the lines represented by

ax2 _{+ 2hxy + by}2 _{+ 2gx + 2fy + c = 0 or ax}2 _{+ 2hxy + by}2 _{= 0} is tanθ = 2 2 h ab a b − + ( )

(i) The two lines given by ax2 _{+ 2hxy + by}2 _{= 0 are} (a) Parallel and coincident iff h2 _{– ab = 0}

(b) Perpendicular iff a + b = 0

(ii) The two line given by ax2 _{+ 2hxy + by}2 _{+ 2gx + 2fy + c} = 0 are

(a) Parallel if h2 _{– ab = 0 & af}2 _{= bg}2 (b) Perpendicular iff a + b = 0 (c) Coincident iff g2 _{– ac = 0}

13. Combined equation of angle bisector of the angle between the lines ax2 _{+ 2hxy + by}2 _{= 0 is}

x y a b 2₋ 2 − = xy h

CIRCLE

1. General equation of a circle : x2 _{+ y}2 _{+ 2gx + 2fy + c = 0} where g, f and c are constants

(i) Centre of the cirle is (–g, –f) i.e.

F

HG

−1₂coeff of x. ,−₂1coeff of y.

I

KJ

(ii) Radius is g2₊f2₋c

2. Central (Centre radius) form of a circle :

(i) (x – h)2 _{+ (y – k)}2 _{= r}2 _{, where (h, k) is circle centre and} r is the radius.

(ii) x2 _{+ y}2 _{= r}2 _{, where (0, 0) origin is circle centre and r is} the radius.

3. Diameter form : If (x1, y1) and (x2, y2) are end pts. of a diameter of a circle, then its equation is

(x – x1) (x – x2) + (y – y1) (y – y2) = 0 4. Parametric equations :

(i) The parametric equations of the circle x2 _{+ y}2 _{= r}2 _are x = rcosθ, y = r sinθ ,

where point θ ≡ (r cos θ, r sin θ) (ii) The parametric equations of the circle

(x – h)2 _{+ (y – k)}2 _{= r}2 _{are x = h + rcosθ, y = k + rsinθ} (iii) The parametric equations of the circle

x2 _{+ y}2 _{+ 2gx + 2fy + c = 0 are}

x = –g + g2₊f2₋c_{cosθ, y = –f + g}2_+f2_{−c sinθ} (iv) For circle x2 _{+ y}2 _{= a}2_{, equation of chord joining θ}

1 & θ2 is xcosθ1 θ2 2 + + ysinθ1 θ2 2 + = r cosθ1 θ2 2 − .

5. Concentric circles : Two circles having same centre C(h, k) but different radii r1 & r2 respectively are called concentric circles.

6. Position of a point w.r.t. a circle : A point (x1, y1 ) lies outside, on or inside a circle

S ≡ x2 _{+ y}2 _{+ 2gx + 2fy + c = 0 according as} S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c is +ve, zero or –ve 7. Chord length (length of intercept) = 2 r2₋p2

8. Intercepts made on coordinate axes by the circle : (i) x axis = 2 g2₋c

(ii) y axis = 2 _f2_−c 9. Length of tangent = S1

10. Length of the intercept made by line : y = mx + c with the circle x2 _{+ y}2 _{= a}2 _is 2 a m c m 2 2 2 2 1 1 ( + )− + or (1 + m2) |x1 – x2| where |x1 – x2| = difference of roots i.e.

D a

11. Condition of Tangency : Circle x2 _{+ y}2 _{= a}2 _{will touch the} line y = mx + c if c = ±a _1+m2