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1.COMPLEX NUMBERS
1. DEFINITION : Complex numbers are definited as expressions of the form a + ib where a, b ∈ R &
i = − 1. It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z). www.MathsBySuhag.com , www.TekoClasses.com EVERY COMPLEX NUMBER CAN BE REGARDED AS
Purely real Purely imaginary Imaginary if b = 0 if a = 0 if b ≠ 0 Note : (a) The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complete Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C. (b) Zero is both purely real as well as purely imaginary but not imaginary.
(c) i = − 1 is called the imaginary unit. Also i² = − l ; i^3 = −i ; i^4 = 1 etc.
(d) (^) a b = (^) a b only if atleast one of either a or b is non-negative.www. Maths By Suhag .com
2. CONJUGATE COMPLEX : If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z. i.e. z = a − ib. Note that : www.MathsBySuhag.com , www.TekoClasses.com (i) z + z = 2 Re(z) (ii) z − z = 2i Im(z) (iii) z z = a² + b² which is real (iv) If z lies in the 1st^ quadrant then (^) z lies in the 4th^ quadrant and − (^) z lies in the 2nd^ quadrant. 3. ALGEBRAIC OPERATIONS : The algebraic operations on complex numbers are similiar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative. e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless. However in real numbers if a^2 + b^2 = 0 then a = 0 = b but in complex numbers, z 12 + z 22 = 0 does not imply z 1 = z 2 = 0.www.MathsBySuhag.com , www.TekoClasses.com 4. EQUALITY IN COMPLEX NUMBER : Two complex numbers z 1 = a 1 + ib 1 & z 2 = a 2 + ib 2 are equal if and only if their real & imaginary parts coincide. 5. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS: (a) Cartesian Form (Geometric Representation) : Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x, y). length OP is called modulus of the complex number denoted by z & θ is called the argument or amplitude eg. z = (^) x 2 +y^2
θ = tan−^1
y x (angle made by OP with positive x−axis)
NOTE :(i) z is always non negative. Unlike real numbers z =
z if z z if z
> − <
0 0
is not correct
(ii) Argument of a complex number is a many valued function. If θ is the argument of a complex number then 2 nπ + θ ; n ∈ I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nπ. (iii) The unique value of θ such that – π < θ ≤ π is called the principal value of the argument. (iv) Unless otherwise stated, amp z implies principal value of the argument. (v) By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus.www.MathsBySuhag.com , www.TekoClasses.com
(vi) There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers. (b) Trignometric / Polar Representation : z = r (cos θ + i sin θ) where | z | = r ; arg z = θ ; z = r (cos θ − i sin θ) Note: cos θ + i sin θ is also written as CiS θ.
Also cos x = 2
e ix^ + e−ix & sin x = 2
e ix^ − e−ix are known as Euler's identities.
(c) Exponential Representation : z = reiθ^ ; | z | = r ; arg z = θ ; z = re−^ iθ
6. IMPORTANT PROPERTIES OF CONJUGATE / MODULI / AMPLITUDE : If z , z 1 , z 2 ∈ C then ;
(a) z + z = 2 Re (z) ; z − z = 2 i Im (z) ; (z )= z ; z 1 +^ z 2 = z 1 + z 2 ;
z 1 − z 2 = z 1 − z^2 ; z 1 z 2 = z 1. z 2
1 z
z
2
1 z
z ; z 2 ≠ 0
(b) | z | ≥ 0 ; | z | ≥ Re (z) ; | z | ≥ Im (z) ; | z | = | z | = | – z | ; z (^) z = | z |^2 ;
| z 1 z 2 | = | z 1 |. | z 2 | ; 2
1 z
z
|z |
|z |
2
(^1) , z 2 ≠^ 0 ,^ | z
n (^) | = | z |n (^) ;
| z 1 + z 2 |^2 + | z 1 – z 2 |^2 = 2 [ | z 1 |^2 + |z 2 |^2 ]www.MathsBySuhag.com , www.TekoClasses.com
z 1 − z 2 ≤ z 1 + z 2 ≤ z 1 + z 2 [ TRIANGLE INEQUALITY ] (c) (i) amp (z 1. z 2 ) = amp z 1 + amp z 2 + 2 kπ. k ∈ I
(ii) amp z z
1 2
= amp z 1 − amp z 2 + 2 kπ ; k ∈ I
(iii) amp(zn) = n amp(z) + 2kπ. where proper value of k must be chosen so that RHS lies in (− π , π ]. (7) VECTORIAL REPRESENTATION OF A COMPLEX : Every complex number can be considered as if it is the position vector of that point. If the point P
represents the complex number z then,
→ OP (^) = z &
→ OP (^) = z.
NOTE :(i) If
→ OP = z = r^ ei^ θ^ then^
→ OQ = z 1 = r ei (θ^ +^ φ)^ = z. e iφ. If
→ OP and^
→ OQ are of unequal magnitude then φ
Λ Λ OQ = OP ei (ii) If A, B, C & D are four points representing the complex numbers z 1 , z 2 , z 3 & z 4 then
AB CD if 2 1
4 3 z z
z z −
is purely real ;
AB ⊥ CD if 2 1
4 3 z z
z z −
is purely imaginary ]
(iii) If z 1 , z 2 , z 3 are the vertices of an equilateral triangle where z 0 is its circumcentre then (a) z 12 + z 22 + z 32 − z 1 z 2 − z 2 z 3 − z 3 z 1 = 0 (b) z 12 + z 22 + z 32 = 3 z 02
8. DEMOIVRE’S THEOREM : Statement : cos n θ + i sin n θ is the value or one of the values of (cos θ + i sin θ)n^ ¥ n ∈ Q. The theorem is very useful in determining the roots of any complex quantity Note : Continued product of the roots of a complex quantity should be determined using theory of equations.
root must be the conjugate of it i.e. β = p − q & vice versa.
4. A quadratic equation whose roots are α & β is (x − α)(x − β) = 0 i.e. x^2 − (α + β) x + α β = 0 i.e. x^2 − (sum of roots) x + product of roots = 0. 5. Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity. 6. Consider the quadratic expression , y = ax² + bx + c , a ≠ 0 & a , b , c ∈ R then (i) The graph between x , y is always a parabola. If a > 0 then the shape of the parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards. (ii) ∀ x ∈ R , y > 0 only if a > 0 & b² − 4ac < 0 (figure 3). (iii) ∀ x ∈ R , y < 0 only if a < 0 & b² − 4ac < 0 (figure 6). Carefully go through the 6 different shapes of the parabola given below.
7. SOLUTION OF QUADRATIC INEQUALITIES:
ax^2 + bx + c > 0 (a ≠ 0). (i) If D > 0, then the equation ax^2 + bx + c = 0 has two different roots x 1 < x 2. Then a > 0 x ∈ (−∞, x 1 ) ∪ (x 2 , ∞) a < 0 x ∈ (x 1 , x 2 )www.MathsBySuhag.com , www.TekoClasses.com (ii) If D = 0, then roots are equal, i.e. x 1 = x 2. In that case a > 0 x ∈ (−∞, x 1 ) ∪ (x 1 , ∞) a < 0 x ∈ φ
(iii) Inequalities of the form P x Q x
( ) ( )
0 can be quickly solved using the method of intervals.
8.MAXIMUM & MINIMUM VALUE of y = ax² + bx + c occurs at x = − (b/2a) according as ; a < 0 or
a > 0. y ∈ 4 4
ac b^2 a
− ∞
, if a > 0 & y ∈ − ∞^
−
, 4 4
ac b^2 a
if a < 0.
9.COMMON ROOTS OF 2 QUADRATIC EQUATIONS [ONLY ONE COMMON ROOT] : Let α be the common root of ax² + bx + c = 0 & a′x^2 + b′x + c′ = 0 Thereforea α² + bα + c = 0 ;
a′α² + b′α + c′ = 0. By Cramer’s Rule ab ab
bc bc ac a c
2 ′− ′
α
′− ′
α (^) Therefore, α =
ac a c
bc b c ab a b
ca c a ′− ′
.So the condition for a common root is (ca′ − c′a)² = (ab′ − a′b)(bc′ − b′c).
10. The condition that a quadratic function f (x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c may be resolved into two linear factors is that ;
abc + 2 fgh − af^2 − bg^2 − ch^2 = 0 OR
a h g h b f g f c
11. THEORY OF EQUATIONS : If α 1 , α 2 , α 3 , ......αn are the roots of the equation;
f(x) = a 0 xn^ + a 1 xn-1^ + a 2 xn-2^ + .... + an-1x + an = 0 where a 0 , a 1 , .... an are all real & a 0 ≠ 0 then, α 1 = − a a
1 0
α 1 α 2 = +
a a
2 0
, α 1 α 2 α 3 = −
a a
3 0
, ....., α 1 α 2 α 3 ........αn = (−1)n^
a a
n 0 Note : (i) If α is a root of the equation f(x) = 0, then the polynomial f(x) is exactly divisible by (x − α) or (x − α) is a factor of f(x) and conversely. (ii) Every equation of nth degree (n ≥ 1) has exactly n roots & if the equation has more than n roots, it is an identity. (iii) If the coefficients of the equation f(x) = 0 are all real and α + iβ is its root, then α − iβ is also a root. i.e. imaginary roots occur in conjugate pairs. (iv) If the coefficients in the equation are all rational & α + β^ is one of its roots, then α − β^ is also a root where α, β ∈ Q & β is not a perfect square. (v) If there be any two real numbers 'a' & 'b' such that f(a) & f(b) are of opposite signs, then f(x) = 0 must have atleast one real root between 'a' and 'b'. (vi) Every eqtion f(x) = 0 of degree odd has atleast one real root of a sign opposite to that of its last term.
12. LOCATION OF ROOTS : www.MathsBySuhag.com , www.TekoClasses.com Let f (x) = ax^2 + bx + c, where a > 0 & a, b, c ∈ R. (i) Conditions for both the roots of f (x) = 0 to be greater than a specified number ‘d’ are b^2 − 4ac ≥ 0; f (d) > 0 & (− b/2a) > d. (ii) Conditions for both roots of f (x) = 0 to lie on either side of the number ‘d’ (in other words the number ‘d’ lies between the roots of f (x) = 0) is f (d) < 0. (iii) Conditions for exactly one root of f (x) = 0 to lie in the interval (d , e) i.e. d < x < e are b^2 − 4ac > 0 & f (d). f (e) < 0. (iv) Conditions that both roots of f (x) = 0 to be confined between the numbers p & q are (p < q). b^2 − 4ac ≥ 0; f (p) > 0; f (q) > 0 & p < (− b/2a) < q. 13. LOGARITHMIC INEQUALITIES (i) For a > 1 the inequality 0 < x < y & loga x < loga y are equivalent. (ii) For 0 < a < 1 the inequality 0 < x < y & loga x > loga y are equivalent. (iii) If a > 1 then loga x < p 0 < x < ap (iv) If a > 1 then logax > p x > ap (v) If 0 < a < 1 then loga x < p x > ap (vi) If 0 < a < 1 then logax > p 0 < x < ap www.MathsBySuhag.com , www.TekoClasses.com 3. Sequence & Progression(AP, GP, HP, AGP, Spl. Series) DEFINITION : A sequence is a set of terms in a definite order with a rule for obtaining the terms. e.g. 1 , 1/2 , 1/3 , ....... , 1/n , ........ is a sequence. AN ARITHMETIC PROGRESSION (AP) : AP is a sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference. If a is the first term & d the common difference, then AP can be written as a, a + d, a + 2 d, ....... a + (n – 1)d, ........ nth^ term of this AP tn = a +
(n – 1)d, where d = an – an-1. The sum of the first n terms of the AP is given by ; Sn = n 2 [2 a + (n – 1)d] = n 2 [a
- l ]. where l is the last term. NOTES :(i) If each term of an A.P. is increased, decreased, multiplied or divided by the same non zero number, then the resulting sequence is also an AP. (ii) Three numbers in AP can be taken as a – d , a , a + d ; four numbers in AP can be taken as a – 3d, a – d, a + d, a + 3d ; five numbers in AP are a – 2d , a – d , a, a + d, a + 2d & six terms in AP are a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d etc. (iii) The common difference can be zero, positive or negative. (iv) The sum of the two terms of an AP equidistant from the beginning & end is constant and equal to the sum of first & last terms.
Fig. 1 Fig. 2 y y y
O x^ O x^ O x
Roots are real & Roots are Roots are complex
a > 0 a > 0
a > 0
x 1 x 2
Fig. 4 Fig. 5 y y y
O
O O
x
x x a < 0 a < 0 (^) a < 0
Roots are real & Roots are Roots are complex
x 1 x 2
(v) Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it. (vi) tr = Sr − Sr− 1 (vii) If a , b , c are in AP 2 b = a + c. GEOMETRIC PROGRESSION (GP) : GP is a sequence of numbers whose first term is non zero & each of the succeeding terms is equal to the proceeding terms multiplied by a constant. Thus in a GP the ratio of successive terms is constant. This constant factor is called the COMMON RATIO of the series & is obtained by dividing any term by that which immediately proceeds it. Therefore a, ar, ar^2 , ar^3 , ar^4 , ...... is a GP with a as the first term & r as common ratio.www.MathsBySuhag.com , www.TekoClasses.com
(i) nth^ term = a rn –1^ (ii) Sum of the Ist^ n terms i.e. Sn =
a r( ) r
n (^) −
−
1 1
, if r ≠ 1.
(iii) Sum of an infinite GP when r < 1 when n → ∞ rn^ → 0 if r < 1 therefore,
S∞ = (|r|^1 ) 1 r
a < −
(iv) If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP. (v) Any 3 consecutive terms of a GP can be taken as a/r, a, ar ; any 4 consecutive terms of a GP can be taken as a/r^3 , a/r, ar, ar^3 & so on. (vi) If a, b, c are in GP b^2 = ac.www.MathsBySuhag.com , www.TekoClasses.com HARMONIC PROGRESSION (HP) : A sequence is said to HP if the reciprocals of its terms are in AP.If the sequence a 1 , a 2 , a 3 , .... , an is an HP then 1/a 1 , 1/a 2 , .... , 1/an is an AP & converse. Here we do not have the formula for the sum of the n terms of an HP. For HP whose
first term is a & second term is b, the nth^ term is tn = a b b + ( n − 1 ) ( a −b)
If a, b, c are in HP b = 2ac a + c
or
a c
a b b c
− −. MEANS ARITHMETIC MEAN : If three terms are in AP then the middle term is called the AM between the other two, so if a, b, c are in AP, b is AM of a & c. AM for any n positive number a 1 , a 2 , ... , an is ;
A =
a a a a n
1 +^2 +^3 +^ .....^ + n.www.MathsBySuhag.com , www.TekoClasses.com
n - ARITHMETIC MEANS BETWEEN TWO NUMBERS : If a, b are any two given numbers & a, A 1 , A 2 , .... , An, b are in AP then A 1 , A 2 , ... An are the n AM’s between a & b.
A 1 = a + b a n
−
( b a) n
−
, ...... , AAn = a +
n b a n
( − )
- 1 = a + d , = a + 2 d, ...... , An = a + nd ,
where d = b a n
−
NOTE : Sum of n AM’s inserted between a & b is equal to n times the single AM between a & b i.e.
r
n
=
1
Ar = nA where A is the single AM between a & b.
GEOMETRIC MEANS : If a, b, c are in GP, b is the GM between a & c. b² = ac, therefore b = a c ; a > 0, c > 0.
n-GEOMETRIC MEANS BETWEEN a, b : If a, b are two given numbers & a, G 1 , G 2 , ..... , Gn, b are in GP. Then G 1 , G 2 , G 3 , ...., Gn are n GMs between a & b.
G 1 = a(b/a)1/n+1, G 2 = a(b/a)2/n+1, ...... , Gn = a(b/a)n/n+ = ar , = ar² , ...... = arn, where r = (b/a)1/n+ NOTE : The product of n GMs between a & b is equal to the nth^ power of the single GM between a & b
i.e. (^) rπ
n = 1 Gr^ = (G)
n (^) where G is the single GM between a & b.
HARMONIC MEAN : If a, b, c are in HP, b is the HM between a & c, then b = 2ac/[a + c]. THEOREM : If A, G, H are respectively AM, GM, HM between a & b both being unequal & positive then, (i) G² = AH (ii) A > G > H (G > 0). Note that A, G, H constitute a GP. ARITHMETICO-GEOMETRIC SERIES : A series each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series. e.g. 1 + 3x + 5x^2 + 7x^3 + ..... Here 1, 3, 5, .... are in AP & 1, x, x^2 , x^3 ..... are in GP. www.MathsBySuhag.com , www.TekoClasses.com Standart appearance of an Arithmetico-Geometric Series is Let Sn = a + (a + d) r + (a + 2 d) r² + ..... + [a + (n − 1)d] rn−^1 SUM TO INFINITY :
If r < 1 & n → ∞ then Limit n → ∞ rn^ = 0. S∞ =
a r
d r 1 − (^1) r^2
−
SIGMA NOTATIONS THEOREMS :
(i) r
n
=
1
(ar ± br) = r
n
=
1
ar ± r
n
=
1
br. (ii) r
n
=
1
k ar = k r
n
=
1
ar. (iii) r
n
=
1
k = nk ; where k is a constant.
RESULTS (i) r
n
1
r =
n (n + 1 ) 2 (sum of the first n natural nos.)
(ii) r
n
1
r² = n (n + 1 ) ( 2 n+ 1 ) 6 (sum of the squares of the first n natural numbers)
(iii) r
n
1
r^3 = n^ n
2 12 4
( + ) r r
n
=
(^1)
2 (sum of the cubes of the first n natural numbers)
(iv) r
n
=
1
r^4 =
n 30 (n +^ 1)^ (2n^ +^ 1)^ (3n²^ +^ 3n^ −^ 1)www.MathsBySuhag.com , www.TekoClasses.com METHOD OF DIFFERENCE : If T 1 , T 2 , T 3 , ...... , Tn are the terms of a sequence then some times the terms T 2 − T 1 , T 3 − T 2 , ....... constitute an AP/GP. nth^ term of the series is determined & the sum to n terms of the sequence can easily be obtained. Remember that to find the sum of n terms of a series each term of which is composed of r factors in AP, the first factors of several terms being in the same AP, we “write down the nth term, affix the next factor at the end, divide by the number of factors thus increased and by the common difference and add a constant. Determine the value of the constant by applying the initial conditions”.
4.PERMUTATION AND COMBINATION DEFINITIONS :1. PERMUTATION : Each of the arrangements in a definite order which can be made by taking some or all of a number of things is called a PERMUTATION. 2.COMBINATION : Each of the groups or selections which can be made by taking some or all of a number of things without reference to the order of the things in each group is called a COMBINATION. FUNDAMENTAL PRINCIPLE OF COUNTING : If an event can occur in ‘m’ different ways, following which another event can occur in ‘ n ’ different ways, then the total number of different ways of simultaneous occurrence of both events in a definite order is m × n. This can be extended to any number of events. RESULTS :(i) A Useful Notation : n! = n (n − 1) (n − 2)......... 3. 2. 1 ; n! = n. (n − 1) !0! = 1! = 1 ; (2n)! = 2n. n! [1. 3. 5. 7...(2n − 1)] Note that factorials of negative integers are not defined. (ii) If nPr denotes the number of permutations of n different things, taking r at a time, then
(iv) Equation of a straight line passsing through (x 1 , y 1 ) & (x 2 , y 2 ) is x y 1
x y 1
x y 1
2 2
4. MINORS : The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands For example, the
minor of a 1 in (Key Concept 2) is
b c b c
2 2 3 3
& the minor of b 2 is
a c a c
1 1 3 3
. Hence a determinant of order two will have “4 minors” & a determinant of order three will have “9 minors”. 5. COFACTOR : If Mij represents the minor of some typical element then the cofactor is defined as : Cij = (−1)i+j^. Mij ; Where i & j denotes the row & column in which the particular element lies. Note that the value of a determinant of order three in terms of ‘Minor’ & ‘Cofactor’ can be written as : D = a 11 M 11 − a 12 M 12 + a 13 M 13 OR D = a 11 C 11 + a 12 C 12 + a 13 C 13 & so on ....... 6. PROPERTIES OF DETERMINANTS : P −−−− 1 : The value of a determinant remains unaltered , if the rows & columns are inter changed. e.g.
if D =
a b c a b c a b c
a a a b b b c c c
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
= (^) = D′ D & D′ are transpose of each other. If D′ = − D then it
is SKEW SYMMETRIC determinant but D′ = D 2 D = 0 D = 0 Skew symmetric determinant of third order has the value zero .www.MathsBySuhag.com , www.TekoClasses.com P −−−− 2 : If any two rows (or columns) of a determinant be interchanged , the value of determinant is changed in sign only. e.g.
Let D =
a b c a b c a b c
1 1 1 2 2 2 3 3 3
& D′ =
a b c a b c a b c
2 2 2 1 1 1 3 3 3
Then D′ = − D.
P −−−− 3 : If a determinant has any two rows (or columns) identical , then its value is zero. e. g.
Let D =
a b c a b c a b c
1 1 1 1 1 1 3 3 3
then it can be verified that D = 0.
P −−−− 4 : If all the elements of any row (or column) be multiplied by the same number , then the determinant is multiplied by that number.
e.g. If D =
a b c a b c a b c
1 1 1 2 2 3 3 3
2 and D′^ =
Ka Kb Kc a b c a b c
1 1 1 2 2 3 3 3
2 Then^ D′= KD
P −−−− 5 : If each element of any row (or column) can be expressed as a sum of two terms t hen t he determinant can be expressed as the sum of two determinants. e.g. a x b y c z a b c a b c
a b c a b c a b c
x y z a b c a b c
1 1 1 2 2 2 3 3 3
1 1 1 2 2 2 3 3 3
2 2 2 3 3 3
= +
P −−−− 6 : The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column) .e.g. Let D
a b c a b c a b c
1 1 1 2 2 2 3 3 3
and D′ =
a ma b m b c m c a b c a n a b n b c n c
1 2 1 2 1 2 2 2 2 3 1 3 1 3 1
. Then D′ = D.
Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must remain unchanged. P −−−− 7 : If by putting x = a the value of a determinant vanishes then (x − a) is a factor of the determinant.
7.MULTIPLICATION OF TWO DETERMINANTS :(i)
a b a b x
l m l m
a l b l a m b m a l b l a m b m
1 1 2 2
1 1 2 2
1 1 1 2 1 1 1 2 2 1 2 2 2 1 2 2
=
Similarly two determinants of order three are multiplied.
(ii) If D =
a b c a b c a b c
1 1 1 2 2 2 3 3 3
≠ 0 then , D² =
A B C A B C A B C
1 1 1 2 2 2 3 3 3
where AAi,Bi,Ci are cofactors
PROOF : Consider
a b c a b c a b c
1 1 1 2 2 2 3 3 3
×
A A A B B B C C C
1 2 3 1 2 3 1 2 3
D D D
0 0 0 0 0 0
Note : a 1 A 2 + b 1 B 2 + c 1 C 2 = 0 etc. therefore
, D x
A A A B B B C C C
1 2 3 1 2 3 1 2 3
= D^3
A A A B B B C C C
1 2 3 1 2 3 1 2 3
= D² OR
A B C A B C CA B C
1 1 1 2 2 2 3 3 3
= D²
8. SYSTEM OF LINEAR EQUATION (IN TWO VARIABLES) :
(i) Consistent Equations : Definite & unique solution. [ intersecting lines ] (ii) Inconsistent Equation : No solution. [ Parallel line ] (iii) Dependent equation : Infinite solutions. [ Identical lines ] Let a 1 x + b 1 y + c 1 = 0 & a 2 x + b 2 y + c 2 = 0 then : a a
b b
c c
1 2
1 2
1 2
= ≠ (^) Given equations are inconsistent& a a
b b
c c
1 2
1 2
1 2
= = (^) Given equations are dependent
9. CRAMER'S RULE :[ SIMULTANEOUS EQUATIONS INVOLVING THREE UNKNOWNS ] Let ,a 1 x + b 1 y + c 1 z = d 1 ...(I) ; a 2 x + b 2 y + c 2 z = d 2 ...(II) ; a 3 x + b 3 y + c 3 z = d 3 ...(III)
Then , x =
D D
1 , Y = D
D
2 , Z = D
D
Where D =
a b c a b c a b c
1 1 1 2 2 2 3 3 3
; D 1 =
d b c d b c d b c
1 1 1 2 2 2 3 3 3
; D 2 =
a d c a d c a d c
1 1 1 2 2 2 3 3 3
& D 3 =
a b d a b d a b d
1 1 1 2 2 2 3 3 3 NOTE : (a) If D ≠ 0 and alteast one of D 1 , D 2 , D 3 ≠ 0 , then the given system of equations are consistent and have unique non trivial solution. (b) If D ≠ 0 & D 1 = D 2 = D 3 = 0 , then the given system of equations are consistent and have trivial solution only .www.MathsBySuhag.com , www.TekoClasses.com (c) If D = D 1 = D 2 = D 3 = 0 , then the given system of equations are
consistentand have infinite solutions. In case
3 3 3 3
2 2 2 2
1 1 1 1
a x by c z d
a x b y cz d
ax by cz d represents these parallel
planes then also D = D 1 = D 2 = D 3 = 0 but the system is inconsistent. (d) If D = 0 but atleast one of D 1 , D 2 , D 3 is not zero then the equations are inco nsist ent and have no solution.
10. If x , y , z are not all zero , the condition for a 1 x + b 1 y + c 1 z = 0 ; a 2 x + b 2 y + c 2 z = 0 & a 3 x + b 3 y
- c 3 z = 0 to be consistent in x , y , z is that
a b c a b c a b c
1 1 1 2 2 2 3 3 3
= 0.Remember that if
a given system of linear equations have Only Zero Solution for all its variables then the given equations are said to have TRIVIAL SOLUTION
6.MATRICES USEFUL IN STUDY OF SCIENCE, ECONOMICS AND ENGINEERING
1. Definition : Rectangular array of m n numbers. Unlike determinants it has no value.
A =
a a a a a a
a a a
n n
m m m n
11 12 1 21 22 2
1 2
...... ...... : : : : ......
or
a a a a a a
a a a
n n
m m mn
11 12 1 21 22 2
1 2
...... ...... : : : : ......
Abbreviated as : A = (^) [ a^ i j ] 1 ≤ i ≤ m ; 1 ≤ j ≤ n, i denotes
the row andj denotes the column is called a matrix of order m × n.
2. Special Type Of Matrices : (a) Row Matrix : A = [ a 11 , a 12 , ...... a1n ] having one row. (1 × n) matrix. (or row vectors)
(b) Column Matrix : A =
a a
a (^) m
11 21
1
:
having one column. (m × 1) matrix (or column vectors)
(c) Zero or Null Matrix : (A = Om ×××× n) An m × n matrix all whose entries are zero.
A =
0 0 0 0 0 0
is a 3 × 2 null matrix & B =
0 0 0 0 0 0 0 0 0
is 3 × 3 null matrix
(d) Horizontal Matrix : A matrix of order m × n is a horizontal matrix if n > m. (^)
(e) Verical Matrix : A matrix of order m × n is a vertical matrix if m > n.
(f) Square Matrix : (Order n) If number of row = number of column a square matrix. Note
(i) In a square matrix the pair of elements aij & aj i are called Conjugate Elements .e.g.
a a a a
11 12 21 22
(ii) The elements a 11 , a 22 , a 33 , ...... ann are called Diagonal Elements. The line along which the diagonal elements lie is called " Principal or Leading " diagonal. The qty Σ ai i = trace of the matrice written as , i.e. tr A Triangular Matrix Diagonal Matrix denote as ddia (d 1 , d 2 , ....., dn) all elements except the leading diagonal are zero diagonal Matrix Unit or Identity Matrix Note: Min. number of zeros in a diagonal matrix of order n = n(n – 1) "It is to be noted that with square matrix there is a corresponding determinant formed by the elements of A in the same order."
3. Equality Of Matrices : Let A = [a (^) i j ] & B = [b (^) i j ] are equal if , (i) both have the same order. (ii) ai j = b (^) i j for each pair of i & j.
4.Algebra Of Matrices :Addition : A + B = (^) [ a^ i j +^ bi j] where A & B are of the same type. (order)
(a) Addition of matrices is commutative. i.e. A + B = B + A, A = m × n; B = m × n (b)Matrix addition is associative. (A + B) + C = A + (B + C) Note : A , B & C are of the same type. (c) Additive inverse. If A + B = O = B + A A = m × n
5. Multiplication Of A Matrix By A Scalar : IfA =
c a b
b c a
a b c ;k A =
kc ka k b
kb kc ka
ka kb kc
6.Multiplication Of Matrices : (Row by Column) AB exists if , A = m × n & B = n × p 2 × 3 3 × 3 AB exists , but BA does not AB ≠ BA
Note : In the product AB ,
A prefactor B post factor
=
A = (a 1 , a 2 , ...... an) &B =
n
2
1
b
b
b
1 × n n × 1 AA
B = [a 1 b 1 + a 2 b 2 + ...... + an bn] If A = (^) [ ai j ] m × n & B = (^) [ bi j ] n × p
matrix , then (A B)i j = r
n
=
1
ai r. br j Properties Of Matrix Multiplication :
1. Matrix multiplication is not commutative.
AAAA = (^)
; B =
; AB =
;BA =
ABAB ≠ BA (in general)
2. AB =
ABAB = O (^) / A = O or B = O
Note: If A and B are two non- zero matrices such that AB = O then A and B are called the divisors of zero. Also if [AB] = O | AB | | A | | B | = 0 | A | = 0 or | B | = 0 but not the converse. If A and B are two matrices such that (i) AB = BA A and B commute each other (ii) AB = – BA A and B anti commute each other
3. Matrix Multiplication Is Associative : If A , B & C are conformable for the product AB & BC, then (A. B). C = A. (B. C) 4. Distributivity : A B C A B A C A B C A C BC
( ) ( )
Provided A, B & C are conformable for respective products
5. POSITIVE INTEGRAL POWERS OF A SQUARE MATRIX : For a square matrix A , A^2 A = (A A) A = A (A A) = A^3. Note that for a unit matrix I of any order , Im^ = I for all m ∈ N. 6. MATRIX POLYNOMIAL : If f (x) = a 0 xn^ + a 1 xn – 1^ + a 2 xn – 2^ + ......... + anx^0 then we define a matrix polynomial f (A) = a 0 An^ + a 1 An– (^1) + a 2 A
n–2 (^) + ..... + a nI
n (^) where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomial f (x). DEFINITIONS : (a) Idempotent Matrix : A square matrix is idempotent provided A^2 = A. Note that An^ = A (^) ∀ n > 2 , n ∈ N. (b) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m ∈^ N, if Am^ = O , Am–1^ ≠ O. (c) Periodic Matrix : A square matrix is which satisfies the relation AK+1^ = A, for some positive integer K, is a periodic matrix. The period of the matrix is the least value of K for which this holds true. Note that period of an idempotent matrix is 1. (d) Involutary Matrix : If A^2 = I , the matrix is said to be an involutary matrix. Note that A = A–1^ for an involutary matrix.
7. The Transpose Of A Matrix : (Changing rows & columns) Let A be any matrix. Then , A = ai j of order m × n AT^ or A′ = [ aj i ] for 1 ≤ i ≤ n & 1 ≤ j ≤ m of order n × m Properties of Transpose : If AT^ & BT^ denote the transpose of A and B , (a) (A ± B)T^ = AT^ ± BT^ ; note that A & B have the same order. IMP. (b) (A B)T^ = BT^ AT^ A & B are conformable for matrix product AB. (c) (AT)T^ = A (d) (k A)T^ = k AT^ k is a scalar. General : (A 1 , A 2 , ...... An)T^ = AnT^ , ....... , AT 2 , A 1 T (reversal law for transpose) 8. Symmetric & Skew Symmetric Matrix : A square matrix A = (^) [ a^ i j ] is said to be , symmetric if , ai j = aj i ∀ i & j (conjugate elements are equal) (Note A = AT)
Note: Max. number of distinct entries in a symmetric matrix of order n is 2
n (n+ 1 ) . and skew symmetric if , ai j = − aj i ∀ i & j (the pair of conjugate elements are additive inverse of each other) (Note A = –AT^ ) Hence If A is skew symmetric, then ai i = − ai i ai i = 0 ∀ i Thus the digaonal elements of a skew symmetric matrix are all zero , but not the converse. Properties Of Symmetric & Skew Matrix : P −−−− 1 A is symmetric if AT^ = A A is skew symmetric if AT^ = − A P −−−− 2 A + ATis a symmetric matrix A − AT^ is a skew symmetric matrix. Consider (A + AT)T^ = AT^ + (AT)T^ = AT^ + A = A + AT^ A + AT^ is symmetric. Similarly we can prove that A − AT
(vi) EQUALLY LIKELY EVENTS : Events are said to be EQUALLY LIKELY when each event is as likely to occur as any other event. (vii) EXHAUSTIVE EVENTS : Events A,B,C ........ L are said to be EXHAUSTIVE EVENTS if no event outside this set can result as an outcome of an experiment. For example, if A & B are two events defined on a sample space S, then A & B are exhaustive A ∪ B = S P (A ∪ B) = 1. (viii) CLASSICAL DEF. OF PROBABILITY : If n represents the total number of equally likely , mutually exclusive and exhaustive outcomes of an experiment and m of them are favourable to the happening of the event A, then the probability of happening of the event A is given by P(A) = m/n.
Note : (1) 0 ≤ P(A) ≤ 1 (2) P(A) + P( A ) = 1, Where A = Not A ..
(3) If x cases are favourable to A & y cases are favourable to (^) A then P(A) = x ( x + y)
and P( (^) A ) = y ( x + y) We say that ODDS IN FAVOUR OF A are x: y & odds against A are y : x
Comparative study of Equally likely , Mutually Exclusive and Exhaustive events. Experiment Events E/L M/E Exhaustive
- Throwing of a die A : throwing an odd face {1, 3, 5} No Yes No B : throwing a composite face {4,. 6}
- A ball is drawn from E 1 : getting a W ball an urn containing 2W, E 2 : getting a R ball No Yes Yes 3R and 4G balls E 3 : getting a G ball
- Throwing a pair of A : throwing a doublet dice {11, 22, 33, 44, 55, 66} B : throwing a total of 10 or Yes No No more {46, 64, 55, 56, 65, 66}
- From a well shuffled E 1 : getting a heart pack of cards a card is E 2 : getting a spade Yes Yes Yes drawn E 3 : getting a diamond E 4 : getting a club
- From a well shuffled A = getting a heart pack of cards a card is B = getting a face card No No No drawn
RESULT −−−− 2 www.MathsBySuhag.com , www.TekoClasses.com AUB = A+ B = A or B denotes occurence of at least A or B. For 2 events A & B : (See fig.1) (i) P(A∪B) = P(A) + P(B) − P(A∩B) =
P(A. B ) + P( A .B) + P(A.B) = 1 − P( A. B )
(ii) Opposite of′ " atleast A or B " is NIETHER
A NOR B .e. (^) A + B= 1-(A or B) = (^) A ∩B Note that P(A+B) + P( (^) A ∩ B) = 1.
(iii) If A & B are mutually exclusive then P(A∪B) = P(A) + P(B). (iv) For any two events A & B, P(exactly one of A , B occurs)
= P A( ∩^ B^ ) +^ P B( ∩^ A^ ) =^ P A(^ )^ +^ P B(^ )^ −^2 P A(^ ∩B) = P A( ∪ B) − P A( ∩ B) = P A( c^ ∪ B c^ ) − P A( c^ ∩Bc)
(v) If A & B are any two events P(A∩B) = P(A).P(B/A) = P(B).P(A/B), Where P(B/A) means conditional probability of B given A & P(A/B) means conditional probability of A given B. (This can be easily seen from the figure) (vi) DE MORGAN'S LAW : − If A & B are two subsets of a universal set U , then (a) (A∪B)c^ = Ac∩Bc^ & (b) (A∩B)c^ = Ac∪Bc (vii) A ∪ (B∩C) = (A∪B) ∩ (A∪C) & A ∩ (B∪C) = (A∩B) ∪ (A∩C)
P B A
P B P A B i P B P A B i i i i
/
( ). / ( ). /
=
RESULT −−−− 6 If p 1 and p 2 are the probabilities of speaking the truth of two indenpendent witnesses A
and B thenP (their combined statement is true) = p p p p p p
1 2 1 2 +^ (^1 −^1 )(^1 − 2 )^
. In this case it has been
assumed that we have no knowledge of the event except the statement made by A and B. However if p is the probability of the happening of the event before their statement then P (their combined statement is true) =
p p p p p p p p p
1 2 1 2 +^ (^1 −^ )(^1 −^1 )(^1 − 2 )^
Here it has been assumed that the statement given by all the independent witnesses can be given in two ways only, so that if all the witnesses tell falsehoods they agree in telling the same falsehood. If this is not the case and c is the chance of their coincidence testimony then the Pr. that the statement is true = P p 1 p 2 Pr. that the statement is false = (1−p).c (1−p 1 )(1−p 2 ) However chance of coincidence testimony is taken only if the joint statement is not contradicted by any witness. RESULT −−−− 7 (i) A PROBABILITY DISTRIBUTION spells out how a total probability of 1 is distributed over several values of a random variable .www.MathsBySuhag.com , www.TekoClasses.com
(ii) Mean of any probability distribution of a random variable is given by : μ =^ =
p x p
i i p x i
i i
( Since Σ pi = 1 ) (iii) Variance of a random variable is given by, σ² = ( xi − μ)². pi
σ² = pi x²i − μ² ( Note that SD = (^) + σ^2 ) (iv) The probability distribution for a binomial variate
‘‘ ‘‘ XXXX ’’’’ i si si si s gggg i vi vi vi v eeee nnnn bbbb yyyy ; P; P; P; P (((( XXXX ==== rrrr )))) ====^ nCr pr^ qn−r^ where all symbols have the same meaning as given in result 4. The
recurrence formula P r P r
n r r
p q
( ) ( )
1 1
, is very helpful for quickly computing
P(1) , P(2). P(3) etc. if P(0) is known. (v) Mean of BPD = np ; variance of BPD = npq. (vi) If p represents a persons chance of success in any venture and ‘M’ the sum of money which he will receive in case of success, then his expectations or probable value = pM expectations = pM
RESULT −−−− 8 : GEOMETRICAL APPLICATIONS : The following statements are axiomatic :
(i) If a point is taken at random on a given staright line AB, the chance that it falls on a particular segment PQ of the line is PQ/AB. (ii) If a point is taken at random on the area S which includes an area σ , the chance that the point falls on σ is σ/S.
9.FUNCTIONS
THINGS TO REMEMBER : 1. GENERAL DEFINITION : If to every value (Considered as real unless other−wise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only one finite value of the quantity y, then y is said to be a function (Single valued) of x or a dependent variable defined on the set E ; x is the argument or independent variable. If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word "FUNCTION” is used only as the meaning of a single valued function, if not otherwise stated. Pictorially : x input
f x y output
( ) = → , y^ is called the image of x & x is the pre-image of y under f. Every function from^ A^ →^ B
satisfies the following conditions. (i) f ⊂ A x B (ii) ∀ a ∈ A (a, f(a)) ∈ f and (iii) (a, b) ∈ f & (a, c) ∈ f b = c
2. DOMAIN, CO −−−− DOMAIN & RANGE OF A FUNCTION : Let f : A → B, then the set A is known as the domain of f & the set B is known as co-domain of f. The set of all f images o f elements of A is known as t he range of f. Thus
Domain of f = {a a ∈ A, (a, f(a)) ∈ f} Range of f = {f(a) a ∈ A, f(a) ∈ B} It should be noted that range is a subset of co−domain. If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.
3. IMPORTANT TYPES OF FUNCTIONS : (i) POLYNOMIAL FUNCTION : If a function f is defined by f (x) = a 0 xn^ + a 1 xn−^1 + a 2 xn−^2 + ... + an− 1 x + an where n is a non negative integer and a 0 , a 1 , a 2 , ..., an are real numbers and a 0 ≠ 0, then f is called a polynomial function of degree n NOTE : (a) A polynomial of degree one with no constant term is called an odd linear function. i.e. f(x) = ax , a ≠ 0 (b) There are two polynomial functions , satisfying the relation ; f(x).f(1/x) = f(x) + f(1/x). They are : (i) f(x) = xn^ + 1 & (ii) f(x) = 1 − xn^ , where n is a positive integer. (ii) ALGEBRAIC FUNCTION : y is an algebraic function of x, if it is a function that satisfies an algebraicequation of the formP 0 (x) yn^ + P 1 (x) yn−^1 + ....... + Pn− 1 (x) y + Pn (x) = 0 Where n is a positive integer and P 0 (x), P 1 (x) ........... are Polynomials in x. e.g. y = x is an algebraic function, since it satisfies the equation y² − x² = 0. Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called TRANSCEDENTAL FUNCTION .www.MathsBySuhag.com , www.TekoClasses.com
(iii) FRACTIONAL RATIONAL FUNCTION : A rational function is a function of the form. y = f (x) = g x h x
( ) ( )
where g (x) & h (x) are polynomials & h (x) ≠ 0. (iv) ABSOLUTE VALUE FUNCTION : A function y = f (x) = x is called the absolute value function or
Modulus function. It is defined as : y = x=
x if x x if x
≥ − <
0 0 (V) EXPONENTIAL FUNCTION : A function f(x) = ax^ = ex^ l n a^ (a > 0 , a ≠ 1, x ∈ R) is called anexponential function. The inverse of the exponential function is called the logarithmic function. i.e. g(x) = loga x. Note that f(x) & g(x) are inverse of each other & their graphs are as shown.
(vi) SIGNUM FUNCTION : A function y= f (x) = Sgn (x) is defined as follows :
y = f (x) =
1 0 0 0 1 0
for x for x for x
>
− <
It is also written as Sgn x = |x|/ x ; x ≠ 0 ; f (0) = 0 (vii) GREATEST INTEGER OR STEP UP FUNCTION : The function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x. Note that for : − 1 ≤ x < 0 ; [x] = − 1 0 ≤ x < 1 ; [x] = 0 1 ≤ x < 2 ; [x] = 1 2 ≤ x < 3 ; [x] = 2 and so on. Properties of greatest integer function : (a) [x] ≤ x < [x] + 1 and x − 1 < [x] ≤ x , 0 ≤ x − [x] < 1
45º (1, 0)
(0, 1)
45º
(0, 1)
(1, 0)
g(x) = loga x
f(x) = ax^ , 0 < a < 1
y = 1 if x > 0
y = −1 if x < 0
y = Sgn x
> x
− 3 − 2 − 1 1 2 3
x
y
º º
º
º
3 2 1
− 1 − 2
º
− 3
graph of y = [x]
f image in B. Thus f : A → B is many one if for ; x 1 , x 2 ∈ A , f(x 1 ) = f(x 2 ) but x 1 ≠ x 2 Diagramatically a many one mapping can be shown as
OR
Note : (i) Any continuous function which has atleast one local maximum or local minimum, then f(x) is many−one. In other words, if a line parallel to x−axis cuts the graph of the function atleast at two points, then f is many−one .www.MathsBySuhag.com , www.TekoClasses.com (ii) If a function is one−one, it cannot be many−one and vice versa. Onto function (Surjective mapping) : If the function f : A → B is such that each element in B (co−domain) is the f image of atleast one element in A, then we say that f is a function of A 'onto' B. Thus f : A → B is surjective iff ∀ b ∈ B, ∃ some a ∈ A such that f (a) = b. Diagramatically surjective mapping can be shown as
OR
Note that : if range = co−domain, then f(x) is onto. Into function : If f : A → B is such that there exists atleast one element in co−domain which is not the image of any element in domain, then f(x) is into. Diagramatically into function can be shown as
OR
Note that : If a function is onto, it cannot be into and vice versa. A polynomial of degree even will always be into. Thus a function can be one of these four types :
(a) one−one onto (injective & surjective)
(b) one−one into (injective but not surjective)
(c) many−one onto (surjective but not injective)
(d) many−one into (neither surjective nor injective)
Note : (i) If f is both injective & surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions. (ii) If a set A contains n distinct elements then the number of different functions defined from A (^) → A is nn^ & out of it n! are one one. Identity function : The function f : A → A defined by f(x) = x ∀ x ∈ A is called the identity of A and is denoted by IA. It is easy to observe that identity function is a bijection. Constant function : A function f : A → B is said to be a constant function if every element of A has the same f image in B. Thus f : A → B ; f(x) = c , ∀ x ∈ A , c ∈ B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into.
7. ALGEBRAIC OPERATIONS ON FUNCTIONS : If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in A ∩ B. Now we define f + g , f − g , (f. g) & (f/g) as follows :
(i) (f ± g) (x) = f(x) ± g(x) (ii) (f. g) (x) = f(x). g(x)
(iii)
f g
(x) =^
f x g x
domain is {x x ∈ A ∩ B s. t g(x) ≠ 0}.
8. COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS : Let f : A → B & g : B → C be two functions. Then the function gof : A → C defined by (gof) (x) = g (f(x)) ∀ x ∈ A is called the composite of the two functions f & g.
Diagramatically (^) →x f x ( )→ → g (f(x)) .Thus the image of every x ∈ A under the function gof is the g−image of the f−image of x. Note that gof is defined only if ∀ x ∈ A, f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f & g, the range of f must be a subset of the domain of g. PROPERTIES OF COMPOSITE FUNCTIONS : (i) The composite of functions is not commutative i.e. gof ≠ fog. (ii) The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog) oh are defined, then fo (goh) = (fog) oh. (iii) The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection.www.MathsBySuhag.com , www.TekoClasses.com
9. HOMOGENEOUS FUNCTIONS : A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables. For example 5 x^2 + 3 y^2 − xy is homogeneous in x & y. Symbolically if , f (tx , ty) = tn^. f (x , y) then f (x , y) is homogeneous function of degree n. 10. BOUNDED FUNCTION : A function is said to be bounded if f(x) ≤ M , where M is a finite quantity. 11. IMPLICIT & EXPLICIT FUNCTION : A function defined by an equation not solved for the dependent variable is called an IMPLICIT FUNCTION. For eg. the equation x^3 + y^3 = 1 defines y as an implicit function. If y has been expressed in terms of x alone then it is called an EXPLICIT FUNCTION. 12. INVERSE OF A FUNCTION : Let f : A → B be a one−one & onto function, then their exists a unique function g : B → A such that f(x) = y ⇔ g(y) = x, ∀ x ∈ A & y ∈ B. Then g is said to be inverse of f. Thus g = f−^1 : B → A = {(f(x), x) (x, f(x)) ∈ f}. PROPERTIES OF INVERSE FUNCTION : (i) The inverse of a bijection is unique. (ii) If f : A → B is a bijection & g : B → A is the inverse of f, then fog = IB and gof = IA , where IA & IB are identity functions on the sets A & B respectively. Note that the graphs of f & g are the mirror images of each other in the line y = x. As shown in the figure given below a point (x ',y ' ) corresponding to y = x^2 (x >0) changes to (y ',x ' ) corresponding to (^) y = + x, the changed form of x = y.
(iii) The inverse of a bijection is also a bijection. (iv) If f & g two bijections f : A → B , g : B → C then the inverse of gof exists and (gof)−^1 = f−^1 o g−^1
13. ODD & EVEN FUNCTIONS : If f (−x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x ; g (x) = x² + 3. If f (−x) = −f (x) for all x in the domain of ‘f’ then f is said to be an odd function. e.g. f (x) = sin x ; g (x) = x^3 + x. NOTE : (a) f (x) − f (−x) = 0 => f (x) is even & f (x) + f (−x) = 0 => f (x) is odd. (b) A function may neither be odd nor even. (c) Inverse of an even function is not defined (d) Every even function is symmetric about the y−axis & every odd function is symmetric about the origin.
(e) Every function can be expressed as the sum of an even & an odd function.
e.g. f x
f x f x f x f x ( )
(f) only function which is defined on the entire number line & is even and odd at the same time is f(x)= 0. (g) If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd.
14. PERIODIC FUNCTION : A function f(x) is called periodic if there exists a positive number T (T > 0) called the period of the function such that f (x + T) = f(x), for all values of x within the domain of x e.g. The function sin x & cos x both are periodic over 2π & tan x is periodic over π NOTE : (a) f (T) = f (0) = f (−T) , where ‘T’ is the period. (b) Inverse of a periodic function does not exist .www.MathsBySuhag.com , www.TekoClasses.com (c) Every constant function is always periodic, with no fundamental period.
(d) If f (x) has a period T & g (x) also has a period T then it does not mean that f (x) + g (x) must have a period T. e.g. f (x) = sinx + cosx.
(e) If f(x) has a period p, then 1 f x( )
and (^) f x( ) also has a period p.
(f) if f(x) has a period T then f(ax + b) has a period T/a (a > 0).
15. GENERAL : If x, y are independent variables, then : (i) f(xy) = f(x) + f(y) f(x) = k l n x or f(x) = 0. (ii) f(xy) = f(x). f(y) f(x) = xn^ , n ∈ R (iii) f(x + y) = f(x). f(y) f(x) = akx^. (iv) f(x + y) = f(x) + f(y) f(x) = kx, where k is a constant.
10.INVERSE TRIGONOMETRY FUNCTION
GENERAL DEFINITION(S):1. sin−^1 x , cos−^1 x , tan−^1 x etc. denote angles or real numbers whose sine is x , whose cosine is x and whose tangent is x, provided that the answers given are numerically smallest available. These are also written as arc sinx , arc cosx etc. If there are two angles one positive & the other negative having same numerical value, then positive angle should be taken.
2. PRINCIPAL VALUES AND DOMAINS OF INVERSE CIRCULAR FUNCTIONS :(i) y = sin−^1 x
where − 1 ≤ x ≤ 1 ; (^) − π^ ≤ ≤π 2 2 y and^ sin y = x.
(ii) y = cos−^1 x where − 1 ≤ x ≤ 1 ; 0 ≤ y ≤ π and cos y = x.
(iii) y = tan−^1 x where x ∈ R ; − π^ < <π 2 2
x and tan y = x.
(iv) y = cosec−^1 x where x ≤ − 1 or x ≥ 1 ; (^) − π^ ≤ ≤π 2 2
y , y^ ≠^ 0 and cosec y = x
(v) y = sec−^1 x where x ≤ −1 or x ≥ 1 ; 0 ≤ y ≤ π ; y ≠ π 2
and sec y = x.
(vi) y = cot−^1 x where x ∈ R , 0 < y < π and cot y = x. NOTE THAT : (a) 1st quadrant is common to all the inverse functions. (b) 3rd quadrant is not used in inverse functions.
(c) 4th quadrant is used in the CLOCKWISE DIRECTION i.e. (^) − π≤ ≤ 2
y 0.
3. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS : P −−−− 1 (i) sin (sin−^1 x) = x , − 1 ≤ x ≤ 1 (ii) cos (cos−^1 x) = x , − 1 ≤ x ≤ 1
(iii) tan (tan−^1 x) = x , x ∈ R (iv) sin−^1 (sin x) = x , (^) − π^ ≤ ≤π 2 2
x
(v) cos−^1 (cos x) = x ; 0 ≤ x ≤ π (vi) tan−^1 (tan x) = x ; − π^ < <π 2 2
x
P −−−− 2 (i) cosec−^1 x = sin−^1 1 x ; x ≤ −1 , x ≥ 1 (ii) sec−^1 x = cos−^1 1 x ; x ≤ −1 , x ≥ 1
(iii) cot−^1 x = tan−^1
1 x ;^ x > 0^ =^ π^ + tan
− 1 1 x ;^ x < 0 P −−−− 3 (i) sin−^1 (−x) = − sin−^1 x , − 1 ≤ x ≤ 1 (ii) tan−^1 (−x) = − tan−^1 x , x ∈ R (iii) cos−^1 (−x) = π − cos−^1 x , − 1 ≤ x ≤ 1 (iv) cot−^1 (−x) = π − cot−^1 x , x ∈ R P −−−− 4 (i) sin−^1 x + cos−^1 x = π 2 − 1 ≤ x ≤ 1 (ii) tan−^1 x + cot−^1 x = π 2 x ∈ R
(iii) cosec−^1 x + sec−^1 x = π 2
x ≥ 1www.MathsBySuhag.com , www.TekoClasses.com
P −−−− 5 tan−^1 x + tan−^1 y = tan−^1 x y x y
1 − where x > 0 , y > 0 & xy < 1
= π + tan−^1
x y x y
1 − where x > 0 , y > 0 & xy > 1
tan−^1 x − tan−^1 y = tan−^1
x y x y
− 1 + where x > 0 , y > 0
P −−−− 6 (i) sin−^1 x + sin−^1 y = sin−^1 x 1 −^ y^2 +^ y^1 −x^2 where x ≥ 0 ,y≥0 & (x^2 + y^2 ) ≤ 1
Note that : x^2 + y^2 ≤ 1 0 ≤ sin−^1 x + sin−^1 y ≤
π 2
(ii) sin−^1 x + sin−^1 y = π − sin−^1 x 1 −^ y^2 +^ y^1 −x^2 where x≥0,y ≥ 0 & x^2 + y^2 > 1
Note that : x^2 + y^2 >
π 2
< sin−^1 x + sin−^1 y < π
(iii) sin–1x – sin–1y = (^) sin −^1 [ x 1 −y^2 −y 1 −x^2 ]where x > 0 , y > 0
(iv) cos−^1 x + cos−^1 y = cos−^1 [x y 1 −x^2 1 −y^2 ] where x ≥ 0 , y ≥ 0
P −−−− 7 If tan−^1 x + tan−^1 y + tan−^1 z = tan−^1
x y z x y z x y y z z x
1 if, x >0,y>0,z>0 & xy+yz+zx<
Note : (i) If tan−^1 x + tan−^1 y + tan−^1 z = π then x + y + z = xyz
(ii) If tan−^1 x + tan−^1 y + tan−^1 z = π 2 then xy + yz + zx = 1
P −−−− 8 2 tan−^1 x = sin−^1 2 1 2
x
= cos−^1 1
2 2
−
x x
= tan−^1 2 1 2
x − x
Note very carefully that :
sin−^1 2 1 2
x
( )
2 1 2 1 2 1
1 1 1
tan tan tan
− − −
≤ − > − + < −
x if x x if x x if x
π π
cos−^1 1 1
2 2
−
x x
2 0 2 0
1 1
tan tan
− −
≥ − <
x if x x if x
tan−^1 2 1 2
x − x
( )
−π− >
π+ <−
−
−
−
2 tan x if x 1
2 tan x if x 1
2 tan x if x 1
1
1
1
REMEMBER THAT : (i) sin−^1 x + sin−^1 y + sin−^1 z = 3 2
π x = y = z = 1 (ii) cos−^1 x + cos−^1 y + cos−^1 z = 3π x = y = z = − 1
(iii) tan−^1 1 + tan−^1 2 + tan−^1 3 = π and tan−^1 1 + tan−^1 12 + tan−^1 13 = π 2
11. Limit and Continuity
& Differentiability of Function
THINGS TO REMEMBER :
1. Limit of a function f(x) is said to exist as, x→a when Limit x → a− f(x) = Limit x → a+ f(x) = finite quantity.. 2. FUNDAMENTAL THEOREMS ON LIMITS :
Let Limit x → a f (x) = l & Limit x →a g (x) = m. If l & m exists then :
(i) Limit x → a f (x) ± g (x) = l ± m (ii) Limit x → a f(x).^ g(x) = l. m
(iii) Limit x →a
f x g g m
( ) ( )
= ^ , provided m ≠ 0www.MathsBySuhag.com , www.TekoClasses.com
(iv) Limit x → a k f(x) = k Limit x → a f(x) ; where k is a constant.
(v) Limit x → a f [g(x)] = f (^) ^ Limit g x x →a
( ) (^) = f (m) ; provided f is continuous at g (x) = m.
For example Limit x → a l n (f(x) = l n Limit f x x →a ^
( ) (^) l n l ( l > 0).
3. STANDARD LIMITS :
(a) Limit x→ 0
sinx x
= 1 = Limit x→ 0
tan x x
= Limit x→ 0 tan−^1 x x
= Limit x→ 0 sin−^1 x x
[ Where x is measured in radians ]
(b) Limit x→ 0 (1 + x)1/x^ = e= Limit x→∞ 1 +^1 ^ ^
x ^
x note however the re
Limit hn →→ ∞ (^0) (1 - h )n^ = 0and
Limit h n → → ∞ (^0) (1 + h )n^ → ∞
(c) If Limit x → a f(x) = 1 and Limit x → a φ (x) = ∞ , then ; x a
Limit
→ [^ ]^
f (x)φ^ (x^ )=eLimitx→aφ(x)[f(x)−^1 ]
(d) If Limit x → a f(x) = A > 0 & Limit x → a φ (x) = B (a finite quantity) then ;
Limit x → a [f(x)]^ φ(x)^ = ez^ where z =^ Limit x → a φ^ (x). ln[f(x)] = eBlnA^ = AAB
(e)^ Limitx → 0 a x
x (^) − 1 = 1n a (a > 0). In particular Limitx → 0 e x
x (^) − 1 = 1 (f)^ Limitx →a x a x a
n a
n (^) − n (^) n −
= −^1
4. SQUEEZE PLAY THEOREM :
If f(x) ≤ g(x) ≤ h(x) ∀ x & Limit x → a f(x) = l = Limit x → a h(x) then Limit x → a g(x) = l.
5. INDETERMINANT FORMS : 0 0
, ∞ , 0 , 0 , , 1 ∞
x ∞ ° ∞° ∞ − ∞ and ∞ Note : (i) We cannot plot ∞ on the paper. Infinity (∞) is a symbol & not a number. It does not obey the laws of elementry algebra. (ii) ∞ + ∞ = ∞ (iii) ∞ × ∞ = ∞ (iv) (a/∞) = 0 if a is finite (v) a 0 is not defined , if a ≠ 0. (vi) a b = 0 , if & only if a = 0 or b = 0 and a & b are finite.
6. The following strategies should be born in mind for evaluating the limits: (a) Factorisation (b) Rationalisation or double rationalisation (c) Use of trigonometric transformation ; appropriate substitution and using standard limits (d) Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below :
(i) a x n a x n a x n a x (^) = 1 + + + + a> 1 1
1 2!
1 3
0
2 2 3 3 !!
......... (ii) e x x^ x^ x = 1 + + + + 1 2! 3
2 3 !!
............
(iii) ln (1+x) = x − x^ + x^ − x^ + for − < x≤
2 3 4 2 3 4
......... 1 1 (iv) sin !
x = x − x^ + x^ − x +.......
3 5 7 3 5! 7!
(v) cos^ !!
x = 1 − x^ + x^ − x +...... 2! 4 6
2 4 6 (vi) tan x = x x x
3 5 3
2 15 ........ (vii) tan-1x = x x x x − + − +
3 5 7 3 5 7
.......
(viii) sin-1x = x + x + x + x + 1 3
1 3 5!
1 3 5 7!
(^2 3 2 2 5 2 2 2 ) !
... ....... (ix) sec-1x = 1 2!
5 4
61 6
2 4 6
......
(CONTINUITY) THINGS TO REMEMBER :
1. A funct ion f(x) is said to be co ntinuous at x = c, if x c
Limit →
f(x) = f(c). Symbolically
f is continuous at x = c if h 0
Limit →
f(c - h) = h 0
Limit →
f(c+h) = f(c). i.e. LHL at x = c = RHL at x = c equals Value of ‘f’ at x = c. It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighbourhood of x = a, not necessarily at x = a.
2. Reasons of discontinuity: www.MathsBySuhag.com , www.TekoClasses.com (i) x c
Limit →
f(x) does not exist
i.e. (^) − x →c
Limit f(x) ≠ (^) + x →c
Limit f (x)
(ii) f(x) is not defined at x= c (iii) x c
Limit →
f(x) ≠ f (c) Geometrically, the graph of the function will exhibit a break at x= c. The graph as shown is discontinuous at x = 1 , 2 and 3.
3. Types of Discontinuities : Type - 1: ( Removable type of discontinuities) In case x c
Limit →
f(x) exists but is not equal to f(c) then the function is said to have a removable discontinuity
or discontinuity of the first kind. In this case we can redefine the function such that x c
Limit →
f(x) = f(c) &
make it continuous at x= c. Removable type of discontinuity can be further classified as : (a) MISSING POI NT DISCO NTIN UITY : Where x a
Limit →
f(x) exists finitely but f(a) is no t defined.
e.g. f(x) =
( 1 x)
( 1 x)( 9 x^2 ) −
− − has a missing point discontinuity at x = 1 , and f(x) = sin^ x x
has a missing point
discontinuity at x = 0
(b) ISOLATED POINT DISCONTINUITY : Where x a
Limit →
f(x) exists & f(a) also exists but ; x a
Limit →
≠ f(a). e.g. f(x)
x 4
x^216 −
, x ≠ 4 & f (4) = 9 has an isolated point discontinuity at x = 4.
Similarly f(x) = [x] + [ –x] =
0
1
if x I
if x I
∈
− ∉
has an isolated point discontinuity at all x^ ∈^ I.
Type-2: ( Non - Removable type of discontinuities) In case x c
Limit →
f(x) does not exist then it is not possible to make the function continuous by redefining it.
Such discontinuities are known as non - removable discontinuity oR discontinuity of the 2nd kind. Non-removable type of discontinuity can be further classified as :
(a) Finite discontinuity e.g. f(x) = x − [x] at all integral x ; f(x) = tan−^1
1 x
at x = 0 and f(x) = x
1 12
1
at x = 0 (
note that f(0+) = 0 ; f(0–) = 1 )
(b) Infinite discontinuity e.g. f(x) =
x − 4
or g(x) =
( x − 4 )^2
at x = 4 ; f(x) = 2tanx^ at x =
π 2
and f(x) =
cosx x
at
x = 0.
(c) Oscillatory discontinuity e.g. f(x) = sin x
(^1) at x = 0.
In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but
x a
Limit →
does not exist.
Note: From the adjacent graph note that
- f is continuous at x = – 1
- f has isolated discontinuity at x = 1
- f has missing point discontinuity at x = 2
- f has non removable (finite type) discontinuity at the origin. 4. In case of dis-continuity of the second kind the non-negative difference between the value of the RHL at x = c & LHL at x = c is called THE JUMP OF DISCONTINUITY. A function having a finite number of jumps in a given interval I is called a PIECE WISE CONTINUOUS or SECTIONALLY CONTINUOUS function in this interval. 5. All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains. 6. If f & g are two functions that are continuous at x= c then the functions defined by : F 1 (x) = f(x) ± g(x) ; F 2 (x) = K f(x) , K any real number ; F 3 (x) = f(x).g(x) are also continuous at x= c.
Further, if g (c) is not zero, thenF 4 (x) =
f x g x
( ) ( ) is also continuous atx= c.
7. The intermediate value theorem: Suppose f(x) is continuous on an interval I , and a and b are any two points of I. Then if y 0 is a number between f(a) and f(b) , their exists a number c between a and b such that f(c) = y 0. NOT E VE RY CAREFULLY THAT : www.MathsBySuhag.com , www.TekoClasses.com (a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function φ(x) = f(x).^ g(x)
is not necessarily be discontinuous at x = a. e.g. f(x) = x & g(x) = sin π x x x
≠
0 0 0
(b) If f(x) and g(x) both are discontinuous at x = a then the product function φ(x) = f(x).^ g(x) is not necessarily
be discontinuous at x = a. e.g.f(x) = − g(x) =
1 0 1 0
x x
≥ − <
(c) Point functions are to be treated as discontinuous. eg. f(x) = 1 − x + x− 1 is not continuous at x = 1.
(d) A Continuous function whose domain is closed must have a range also in closed interval. (e) If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is continuous at x = c. eg.
f(x) =
x x x
sin (^2) + 2 & g(x) =^ x^ are continuous at x = 0 , hence the composite (gof) (x) =^
x x x
sin (^2) + 2 will also be
continuous at x = 0 .www.MathsBySuhag.com , www.TekoClasses.com
7. CONTINUITY IN AN INTERVAL : (a) A function f is said to be continuous in (a , b) if f is continuous at each & every point ∈(a,b).
(b) A function f is said to be continuous in a closed interval[ a b, ]if :
(i) f is continuous in the open interval (a , b) &
(ii) f is right continuous at ‘a’ i.e. Limitx →a+^ f(x) = f(a) = a finite quantity..
(iii) f is left continuous at ‘b’ i.e. Limitx →b−^ f(x) = f(b) = a finite quantity..
Note that a function f which is continuous in[ a b, ]possesses the following properties :
(i) If f(a) & f(b) possess opposite signs, then there exists at least one solution of the equation f(x) = 0 in the
open interval (a , b). (ii) If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open inetrval (a , b).
8. SINGLE POINT CONTINUITY: Functions which are continuous only at one point are said to exhibit single point continuity
eeee. g. g. g. g .... ffff (((( xxxx )))) ==== x if x Q x if x Q
∈ − ∉
and g(x) =
x if x Q if x Q
∈ 0 ∉ are both continuous only at x = 0.
DIFFERENTIABILITY THINGS TO REMEMBER :
1. Right hand & Left hand Derivatives ; By definition:
f ′(a) = Limith → 0 h
f (a+ h)−f(a ) if it exist
(i) The right hand derivative of f ′ at x = a denoted by f ′(a+) is defined by :
f ' (a+) = Limith → 0 + h
f (a+ h)−f(a ) ,
provided the limit exists & is finite.www.MathsBySuhag.com , www.TekoClasses.com (ii) The left hand derivative : of f at x = a denoted by
f ′(a+) is defined by : f ' (a–) = Limith → 0 + h
f(a h) f(a ) −
, Provided the limit exists & is finite.
We also write f ′(a+) = f ′ + (a) & f ′(a–) = f ′_(a).
- This geomtrically means that a unique tangent with finite slope can be drawn at x = a as shown in the figure.www.MathsBySuhag.com , www.TekoClasses.com
(iii) Derivability & Continuity : (a) If f ′(a) exists then f(x) is derivable at x= a f(x) is continuous at x = a. (b) If a function f is derivable at x then f is continuous at x.
For : f ′(x) = Limith → 0 h
f (x+ h)−f(x ) exists.
Also .h[h 0 ] h
f(x h) f(x ) f (x h) f(x) ≠
Therefore : [f (x+ h)−f(x)]= Limith → (^0) h .h f'(x).^00
f (x h) f(x ) = =
Therefore Limith → 0 [f (x+ h)−f(x)]= 0 Limith → 0 f (x+h) = f(x) f is continuous at x. Note : If f(x) is derivable for every point of its domain of definition, then it is continuous in that domain. The Converse of the above result is not true : “ IF f IS CONTINUOUS AT x , THEN f IS DERIVABLE AT x ” IS NOT TRUE.
e.g. t he functions f(x) = x & g(x) = x sin x
(^1) ; x ≠ 0 & g(0) = 0 are continuous at
x = 0 but not derivable at x = 0. NOTE CAREFULLY : (a) Let f ′+(a) = p & f ′_(a) = q where p & q are finite then : (i) p = q f is derivable at x = a f is continuous at x = a. (ii) p ≠ q f is not derivable at x = a. It is very important to note that f may be still continuous at x = a. In short, for a function f : Differentiability Continuity ; Continuity (^) / derivability ;
derivative of y w. r. t. x , if it exists, is defined by
d y d x
d dx
d y d x
3 3
2 = (^) 2
It is also denoted by f ′′′(x) or y ′′′.
12. If F(x)=
f x g x h x l x m x n x u x v x w x
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
,where f ,g,h,l,m,n,u,v,w are differentiable functions of x then
F ′(x) =
f x g x h x l x m x n x u x v x w x
'( ) '( ) ' ( ) ( ) ( ) ( ) ( ) ( ) ( )
f x g x h x l x m x n x u x v x w x
( ) ( ) ( ) ' ( ) '( ) '( ) ( ) ( ) ( )
f x g x h x l x m x n x u x v x w x
( ) ( ) ( ) ( ) ( ) ( ) '( ) '( ) ' ( )
13. L’ HOSPITAL’S RULE : www.MathsBySuhag.com , www.TekoClasses.com If f(x) & g(x) are functions of x such that : (i) Limit x → a f(x) = 0 = Limit x → a g(x) OR Limit x → a f(x) = ∞ = Limit x → a g(x) and (ii) Both f(x) & g(x) are continuous at x = a &
(iii) Both f(x) & g(x) are differentiable at x = a & (iv) Both f ′(x) & g ′(x) are continuous at x = a , Then Limit x →a
f x g x
( ) ( ) = Limit x →a f x g x
'( ) '( ) = Limit x →a f x g x
"( ) "( ) & soon till indeterminant form vanishes.
14. ANALYSIS AND GRAPHS OF SOME USEFUL FUNCTIONS :
(i) y = f(x) = sin−^1 1 2
x + x
( )
2 1 2 1 2 1
1 1 1
tan tan tan
− − −
≤ − > − + < −
x x x x x x
π π
HIGHLIGHTS : (a) Domain is x ∈ R &
range is −
^
π π 2 2
,
(b) f is continuous for all x but not diff. at x = 1 , - 1
(c) d y d x
2 1
2 1
2
2
1 1 1
<
− >
x
x
for x non existent for x for x
(d) I in (- 1 , 1) & D in (- ∞ , - 1) ∪ (1 , ∞)
ii) Consider y = f (x) = cos-^ 1 1
2 2
− ^ +
x x
2 0 2 0
1 1
tan tan
− −
≥ − <
x if x x if x HIGHLIGHTS : (a) Domain is x ∈ R & range is [0, π) (b) Continuous for all x but not diff. at x = 0
(c) d y d x
2 1
2 1
2
2
0 0 0
>
− <
x
x
for x non existent for x for x
(d) I in (0 , ∞) & D in (- ∞ , 0)
(iii) y = f (x) = tan-^ 2 1 2
x − x
( )
2 1 2 1 2 1
1 1 1
tan tan tan
− − −
<
x x x x x x
π π
HIGHLIGHTS : (a) Domain is R - {1 , -1} &
range is − ^ ^
^
π π 2 2
, (b) f is neither continuous nor diff. at x = 1 , - 1
(c) d^ y d x
2 1 2 1 1
x x non existent x (d) I ∀ x in its domain (e) It is bound for all x
(IV) y = f (x) = sin−^1 (3 x − 4 x^3 ) =
− (^) ( + (^) ) − ≤ ≤ − − ≤ ≤ − ≤ ≤
− − −
π
π
3 1 3 3 1
(^1 ) 2 (^1 ) 2
1 2 (^1 ) 2
sin sin sin
x if x x if x x if x
HIGHLIGHTS : www.MathsBySuhag.com , www.TekoClasses.com (a) Domain is x ∈ [− 1 , 1] &
range is −
π π 2 2
,
(b) Not derivable at x^ =^
1 2
(c) d y d x
( ) ( ) ( )
3 1
1 2
1 2 3 1
1 2 1 2
2 2 1 1
−
−
∈ −
− ∈ − −
x
x
if x
if x
,
, ,
(d) Continuous everywhere in its domain
(v) y = f (x) = cos-1^ (4 x^3 - 3 x) =
3 2 1 2 3 3 1
(^1 ) 2 (^1 ) 2 1 2 (^1 ) 2
cos cos cos
− − −
− − ≤ ≤ − − − ≤ ≤ ≤ ≤
x if x x if x x if x
π π
HIGHLIGHTS :
(a) Domain is x ∈ [- 1 , 1] & range is [0 , π]
(b) Continuous everywhere in its domain
but not derivable at x = 1 2
1 2
, −
(c) I in − ^
1 2
1 2 , & D in
1 2 1 1 1 2 , , ^
− − ^
(d) d y d x
( ) ( ) ( )
3 1
1 2
1 2 3 1
1 2 1 2
2 2 1 1
− −
∈ − − ∈ − −
x x
if x if x
, , ,
GENERAL NOTE :
Concavity in each case is decided by the sign of 2nd^ derivative as :
d y d x
2 2 > 0^ Concave upwards^ ;^
d y d x
2 2 < 0^ Concave downwards
D = DECREASING ; I = INCREASING
13. APPLICATION OF DERIVATIVE (AOD). TANGENT & NORMAL
THINGS TO REMEMBER : I The value of the derivative at P (x 1 , y 1 ) gives the slope of the tangent to the curve at P. Symbolically
f′ (x 1 ) = d x x 1 y 1
dy = Slope of tangent at P (x 1 y 1 ) = m (say).
II Equation of tangent at (x 1 , y 1 ) is ; y − y 1 = dy dx (^) x 1 y 1 (x^ −^ x^1 ).
III Equation of normal at (x 1 , y 1 ) is ; y − y 1 = − 1
1 1
d y d x (^) x y
(x − x 1 ).
NOTE : www.MathsBySuhag.com , www.TekoClasses.com
1. The point P (x 1 , y 1 ) will satisfy the equation of the curve & the eqation of tangent & normal line. 2. If the tangent at any point P on the curve is // to the axis of x then dy/dx = 0 at the point P. 3. If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = ∞ or dx/dy = 0. 4. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx = ± 1. 5. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = – 1. 6. Tangent to a curve at the point P (x 1 , y 1 ) can be drawn even through dy/dx at P does not exist. e.g. x = 0 is a tangent to y = x2/3^ at (0, 0). 7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation. e.g. If the equation of a curve be x^2 – y^2 + x^3 + 3 x^2 y − y^3 = 0, the tangents at the origin are given by x^2 – y^2 = 0 i.e. x + y = 0 and x − y = 0. IV Angle of intersection between two curves is defined as the angle between the 2 tangents drawn to the 2 curves at their point of intersection. If the angle between two curves is 90° every where then they are called ORTHOGONAL curves.
V (a) Length of the tangent (PT) =
[ ] f(x )
y 1 f(x)
1
2 1 1 ′
(b) Length of Subtangent (MT) = f(x)
y
1
1 ′
(c) Length of Normal (PN) = y 1 1 +[f ′(x 1 )]^2 (d) Length of Subnormal (MN) = y 1 f ' (x 1 )
VI DIFFERENTIALS :
The differential of a function is equal to its derivative multiplied by the differential of the independent variable. Thus if, y = tan x then dy = sec^2 x dx. In general dy = f ′ (x) d x. Note that : d (c) = 0 where 'c' is a constant. d (u + v − w) = du + dv − dw d (u v) = u d v + v d u Note :
1. For the independent variable 'x' , increment ∆ x and differential d x are equal but this is not the case with the dependent variable 'y' i.e. ∆ y ≠ d y. 2. The relation d y = f ′ (x) d x can be written as dx
dy = f ′ (x) ; thus the quotient of the differentials of 'y' and
'x' is equal to the derivative of 'y' w.r.t. 'x'.
MONOTONOCITY (Significance of the sign of the first order derivative) DEFINITIONS :
1. A function f (x) is called an Increasing Function at a point x = a if in a sufficiently small neighbourhood
around x = a we have
f a h f a and f a h f a
( ) ( ) ( ) ( )
increasing;
Similarly decreasing if
f a h f a and f a h f a
( ) ( ) ( ) ( )
decreasing.
2. A differentiable function is called increasing in an interval (a, b) if it is increasing at every point within the interval (but not necessarily at the end points). A function decreasing in an interval (a, b) is similarly defined.www.MathsBySuhag.com , www.TekoClasses.com 3. A function which in a given interval is increasing or decreasing is called “Monotonic” in that interval. 4. Tests for increasing and decreasing of a function at a point : If the derivative f ′(x) is positive at a point x = a, then the function f (x) at this point is increasing. If it is negative, then the function is decreasing. Even if f ' (a) is not defined, f can still be increasing or decreasing.
Note : If f ′(a) = 0, then for x = a the function may be still increasing or it may be decreasing as shown. It has to be identified by a seperate rule. e.g. f (x) = x^3 is increasing at every point. Note that, dy/dx = 3 x².
5. Tests for Increasing & Decreasing of a function in an interval : SUFFICIENCY TEST : If the derivative function f ′(x) in an interval (a , b) is every where positive, then the function f (x) in this interval is Increasing ;www.MathsBySuhag.com , www.TekoClasses.com If f ′(x) is every where negative, then f (x) is Decreasing.
General Note : (1) If a function is invertible it has to be either increasing or decreasing. (2) If a function is continuous the intervals in which it rises and falls may be separated by points at which its derivative fails to exist. (3) If f is increasing in [a, b] and is continuous then f (b) is the greatest and f (c) is the least value of f in [a, b]. Similarly if f is decreasing in [a, b] then f (a) is the greatest value and f (b) is the least value.
6. (a) ROLLE'S THEOREM : Let f(x) be a function of x subject to the following conditions : (i) f(x) is a continuous function of x in the closed interval of a ≤ x ≤ b. (ii) f ′ (x) exists for every point in the open interval a < x < b. (iii) f (a) = f (b). Then there exists at least one point x = c such that a
