IIT-JEE Mathematics

Binomial Theorem BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX Synopsis: 1. f x and a are real numbers, then for all n ∈ N ,

( x + a ) n = n Co x n a o + n C1 x n −1a1 + n C2 x n − 2 a 2 + .... +.... + n Cr x n − r a r + ... + n Cn −1 x1a n −1 + n Cn x o a n i.e n

Formula: ( x + a ) n = ∑ n Cr x n − r a r . r =0

2.

MIDDLE TERM IN A BINOMIAL EXPANSION

Synopsis: n  If n is an even natural number, then in the binomial expansion of ( x + a ) n ,  + 1 th 2  term is the middle term.  n +1   n+3 Formula: If is odd natural number , then   th and   th are the middle terms  2   2  In the binomial expansion of (x+a)n. 3.

GREATEST TERM

Synopsis: Let Tr+1 and Tr be(r+1)th and rth terms respectively in the expansion of (x+a)n. Then, Tr +1 = n Cr x n − r a r and Tr = nCr-1 xn-r+1 ar-1.

∴

n (r − 1)!(n − r + 1)! a n! n − r +1 a Tr +1 Cr x n − r a r . = × = . =n n − r +1 r −1 (n − r )!r ! n! x Tr Cr −1 x a r x

CASE1 When

Let

n +1 is an integer x 1+ a

n +1 = m. Then, from (i), we have x 1+ a Tr+1 > Tr for r=1,2,3,…..(m-1) ……(ii) Tr+1 = Tr for r = m

….(iii)

and, Tr+1 < Tr for r = m +1, ….n

…(iv)

∴ T2 > T1, T3 > T2, T4 > T3, ……, Tm>Tm-1 Tm+1 = Tm and, Tm+2 < Tm+1, Tm+3 < Tm+2, …Tn+1 < Tn

[From (ii)] [From (iii)] [From (iv)]

⇒

T1 < T2 < …. Tm+2 > …> Tn

This shows that mth and (m+1)th terms are greatest terms. Case II: When

n +1 is not an integer. x 1+ a

Let m be the integral part of

n +1 . Then, from (i), we have x 1+ a

Tr +1 > Tr for = 1, 2....., m

…(v)

and,

Tr +1 < Tr for r = m + 1, m + 2,....n

….(vi)

∴

T2 > T1 , T3 > T2 ,...., Tm +1 > Tm

[From (v)]

and, Tm + 2 < Tm +1 , Tm +3 < Tm + 2 ,..., Tn +1 < Tn ⇒ ⇒

4.

[From (vi)]

T1 < T2 < T3 < .... < Tm < Tm +1 > Tm + 2 > Tm +3... > Tn +1 (m+1)th term is the greatest term.

MULTINOMIAL THEOREM n

Using binomial theorem, we have x + a ) n = ∑ n Cr x n − r a r , n ∈ N r =0

n

=

n!

∑ (n − r )!r ! x

n−r

ar

r =0

=

∑

r + s =n

n! s r x a , where s=n-r. r !s !

This result can be generalized in the following form : ( x1 + x2 + .... + xk ) n =

∑

r1 + r2 +....+ rk = n

n! x1r1 x2r2 ....xkrk r1 !r2 !....rk !

The general term in the above expansion is n! x1r1 x2r2 x3r3 ....xkrk r1 !r2 !r3 !.....rk ! The number of terms in the above expansion is equal to the number of non-negative integral solution of the equation. r1+r2+…..+rk = n, because each solution of this equation gives a term in the above expansion. The number of such solutions is

n + k −1

Ck −1 .

5.

PARTICULAR CASES

(i) ( x + y + z ) n =

n! r s t x y z r + s + t = n r ! s !t !

∑

The above expansion has (ii)

( x + y + z + u )n =

∑

p+q+r + s=n

There are

n + 4 −1

n + 3−1

C3−1 = n + 2 C2 terms.

n! x p yq zr us p! q! r ! s!

C4−1 = n +3 C3 terms in the above expansion.

REMARK The greatest coefficient in the expansion of ( x1 + x2 + .... + xm ) n is

(q !)

m−r

n! , where q and [(q + 1)!]r

r are the quotient and remainder respectively when n is divided by m. PROPERTIES OF THE BINOMIAL COEFFICEINT PROPERTY I In the expansion of (1+x)n the coefficients of terms equidistant from the beginning and the end are equal. PROPERTY II The sum of the binomial coefficients in the expansion of (1+x)n is 2n. i.e, Co + C1 + C2 + ... + Cn = 2n or,

n

∑ r =0

n

Cr = 2 n .

PROPERTY III The sum of the coefficients of the odd terms in the expansion of (1+x)n is equal to the sum of the coefficients of the even terms and each is equal to 2n-1. i.e, Co+C2+C4+….=C1+C3+C5+……=2n-1. PROPERTY IV Prove that: n

n n n − 1 n−2 . Cr − 2 and so on. Cr = . n −1Cr −1 = . r r r −1

PROPERTY V Co-C1+C2-C3+C4-…+(-1)n Cn=0

1.

Find the coefficient of xm in the expression (1+x)n+2(1+x)n-1+3(1+x)n-2+….+(n-m+1) (1+x)m, where 0≤n.

2.

Find the sum of the series :

3.

Find the sum of the series

4.

If k and n be positive integers and sk =1k + …+ nk, then show that

 1 3r  7 r 15r r n − C + + + 4 r + ....upto m terms  ( 1) ∑ r  r 2r 3r 2 2 r =0 2 2 \ n

 1 3r  7 r 15r r n − C + + + 4 r + .... + to ∞  ( 1) ∑ r  r 2r 3r 2 2 r =0 2 2  n

m

∑ r =1

m +1

Cr sr = (n + 1) m +1 − (n + 1)

5.

 n n  n   n − 1  k − 2  n  n − 2  Prove that : 2k     − 2k −1     + 2    0 k   1   k − 1  2  k − 2   n n − k   n +…..+ +(-1)k-1 +.... + (−1)     =   , where  k  0   k 

n n   = Ck k 

6

In the expansion of the binomial expression (x+a)15, if the eleventh term is the geometric mean of the eighth and twelfth terms, which term in the expansion is the greatest.

7.

Prove that the greatest term in the expansion of (1+x)2n has also the greatest coefficient, then  n n +1 x ∈ , .  n +1 n 

8.

 1+ 2x   1 + 3x   1 + nx   1+ x  C − C + .... +(−1) n  C =0 1−   C1 +  2  2 3  3 n  n  1 + nx   (1 + nx)   (1 + nx)   (1 + nx) 

9.

If nCo, nC1, nC1, nC2, …..,nCn denote the binomial coefficients in the expansion of (1+x)n and p+q=1, then prove that n

n

(i)

∑ r nCr p r q n−r = np

(ii)

r =0

∑r

2 n

r =0

3r + 2 n Cr , Where 2 r r + + + 6 11 6 r =0 expansion of (1+x)n.

Cr p r q n − r = n 2 p 2 + npq

n

10.

11. 12.

Evaluate

∑r

(32) When 32

3

n

C0 , nC1 ,.....n Cn are the binomial coefficients in the

(32) is divided by 7, prove that the remainder is 4.

A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of P. A subset Q is again chosen at random. Find the number of ways selecting P and Q so that. (i) P and Q are disjoint sets.

(ii) P ∩ Q contains just one element

(iii) P ∪ Q contains just one element (iv) Q is a subset of P (v) P ∩ Q = φ (vi) P and Q have equal number of elements (vii)

Q contains just one element more than P

(viii) P ∪ Q = A

(ix) P = Q PASSAGE – 1

Numerically greatest Term in the Expansion of (x + a)n

Let Tr and Tr+1 be rth and (r+1)th terms respectively in the expansion of bionomial

(x+a)n. Then Tr = nCr-1 xn-r+1 ar-1 and Tr+1 = nCr xn-r ar

Tr +1 n − r + 1 a = Tr r x

∴

n +1 x  n − r +1 a − 1 >, =, < , i.e. Now, Tr+1 > , = , < Tr According as   >, =, < 1, i.e. according as r r a  x n +1 according as r . x +1 a So, if

n +1 is an integer, say p, then Tr+1 > Tr if r < p otherwise Tr+1 ≤ Tr x +1 a

So, Tp = Tp+1 (numerically) and these are greater than any other term in the expansion. Next, if

n +1 is a non-integer, suppose m be its integral part then Tr+1 < Tr if r ≤ m and x +1 a

Tr+1 < Tr if r > m. So, Tm+1 is the numerically greatest term among the terms of the expansion. Again we can also write that kth term is numerically greatest if Tk > Tk+1 and Tk-1.

1.

The numerically greatest term in the expansion of (1-2x)8, when x=2 is a) 8C6 46

2.

b) 8C4 44

c) 21 7

d) None of these

Magnitude wise the greatest term in the expansion of (3, -2x)9 when x=1 is a) 9C2 37 22

b) 9C3 36 23

c) 9C4 35 24

d) both (b) and (c) 10

3.

3   If x > 0 and the 4 term in the expansion of  2 + x  8   th

a) 2< x < 3 4.

5.

b) 3 < x <

10 3

has maximum value, then

c) 4 < x < 5

d) None of these

If n is even positive integer, then the condition that the numerically greatest term in the expansion of (1+x)n may have the greatest coefficient also is a)

n n+2 1 1 Prove that I = ∫ 2 dx =  a − 2a cos x + 1 0  π , if 0 < a < 1 1 − a 2  π π 2  2a 2 , if a > b > 0 sin x Prove that ∫ 2 dx =  a − 2ab cos x + b 2 0  π , if b > a > 0  2b 2 π 1 π dx = , a > 1. Also, deduce that Show that ∫ a x cos − a2 −1 0 π

(2)

(3)

(4) (5)

(6)

(7)

π

1

∫ (2 − cos x) 0

2

dx =

2π 3 3

1 1 2 2 If β + 2 ∫ x 2e− x dx = ∫ e− x dx , then find the value of β . 0 0 x log e t 1 For x > 0, let f(x) = ∫ dt. Find the function f(x) + f   and show that f (e) + 1+ t x 1 If in is positive integer, prove that 2 n a − x x n dx = n ! 1 − e − a 1 + a + a + .... + a  Also, deduce the value e ∫    2! n !    0 ∞ of ∫ e− x x n dx 0

1 1 f  = e 2

PROPERTIES OF DEFINITE INTEGRALS b

(12)

∫ ∫

f ( x) dx = ∫ f ( x) dx + ∫ f ( x) dx , where a < c < b.

a b

(13)

a

(14)

b

f ( x) dx = ∫ f (t ) dt i.e., integration is independent of the change of variable. a c

b

a

Property

c

b

b

a

a

∫ f ( x)dx =∫ f (a + b − x) dx.

 a Property ∫ f ( x) dx = 2∫ f ( x) dx, if f ( x) is an even function −a  0 a  a 2 ∫ f ( x) dx, if f ( x) is an even function f x dx ( ) =  0 ∫ −a 0 , if f ( x) is an odd function  π  cos x  π x sin 2 x sin  2   dx = 8 Prove that ∫ 2x − π π2 0 a

(15)

(16)

π

(17)

Evaluate

∫

−π

2a

(18)

Property

∫ 0

2a

(19)

Property Property

(19)

∫

−a

a

f ( x) dx = ∫ { f ( x) + f (− x)} dt 0

 a 2 f ( x) dx, if f (2a − x) = f ( x) f ( x) dx =  ∫0 0 , if f (2a − x) = − f ( x)  a

∫

∫

f ( x) dx = (b − a) ∫ f {(b − a ) x + a} dx

a

(21)

a

f ( x) dx = ∫ { f ( x) = f (2a − x)} dx

0 b

(20)

2 x (1 + sin x) dx 1 + cos 2 x

0

1

a

Property f(x) is a periodic function with period T, then nT

(i)

∫ 0

(ii)

(iii)

(iv)

(v)

T

f ( x) dx = n ∫ f ( x) dx, n ∈ Z 0

a + nT T ∫ f ( x) dx = n ∫ f ( x) dx, n ∈ Z , a ∈ R a 0 T nT ( ) ( ) f x dx = n − m ∫ ∫0 f ( x)dx, m, n ∈ Z mT a a + nT f x dx = ( ) ∫ ∫0 f ( x) dx, n ∈ Z , a ∈ R nT b b + nT ( ) f x dx = ∫ ∫a f ( x) dx, n ∈ Z , a ∈ R a + nT

INDEFINITE INTEGRALS (7)

Evaluate the following integral ∫

sin 2 x dx sin( x − π / 3) sin( x + π / 3)

(11) (12) (13) (14)

Evaluate the following integral

∫

1

dx sin x sin( x + a ) 2a sin x + b sin 2 x Evaluate the following integral ∫ dx (b + a cos x)3 Evaluate the following integral

∫

3

sin 3/ 2 x + cos3/ 2 x

sin 3 x cos3 x sin( x + θ ) Evaluate the following integral ∫ x13 / 2 (1 + x5 / 2 )1/ 2 dx

dx

(18)

 x −1  Evaluate ∫   dx  x +1  1 dx Evaluate ∫ 2 ( x + 2 x + 10)3 For any natural number m, evaluate ∫ ( x3m + x 2 m + x m )(2 x 2 m + 3 x + 6)1/ m dx, x > 0 ∫ sin 3 x cos 2 2 x dx Evaluate ∫ sec 25 /13 x cos ec 27 /13 x dx (19)

(20)

∫

4

(15) (16) (17)

(22) (24) (28)

x7 dx (1 − x 2 )5

1 dx cos x + sin 6 x cos3 x + cos5 x dx Evaluate ∫ 2 Prove that Evaluate ∫

6

1 dx sin xcs 5 x

(21)

∫

(23)

Evaluate ∫ ( tan x + cot x) dx

(25)

Evaluate ∫

3

x −1 ( x + 1) x3 + x 2 + x

 1  x 2 + b 2   tan −1  2 , if a 2 > b 2 2  − a b x  a 2 − b 2   ∫ dx  2 2 2 2 (x + a ) x + b  1 x2 + b2 − b2 − a 2 log , if a 2 < b 2  2 2 2 2 2 2 x +b + b −a  2 b − a

(29)

Evaluate ∫ cos ecx − 1 dx

(31)

Evaluate ∫

(33)

∫

(35)

∫ (sin −1 x) 2 dx

(36)

(37)

∫ cot −1 (1 − x + x 2 ) dx

(38)

(39)

∫ sin −1

x dx a+x

(40)

(41)

(43)

cos 2 x sin x dx sin x − cos x

sin −1 x dx (1 − x 2 )3/ 2

1+ x dx x

(30)

∫

(32)

Evaluate ∫

(34)

∫ sin x dx

cos 2 x + sin 2 x dx (2 cos x − sin x) 2

sin −1 x − cos −1 sin −1 x + cos −1  2 cos 2 θ ∫ tan −1   2 − sin 2θ

∫

∫

x x

 2  sec θ dθ 

cos 2 x + sin 2 x dx (2 cos x − sin x) 2

 x cos3 x − sin x  ∫ cos 2 x log(1 + tan x) dx ∫ esin x  (42  dx cos 2 x   If f(x) is a polynomial function of the degree prove that ∫ e x f ( x)dx = e x { f ( x) − f '( x) + f ''( x) − f '''( x) + .... + (−1) n f n ( x)} , where fn(x) =

d n f ( x) dx n

dx

(44)

(45) (47) (49)

 x 4 cos3 x − x sin x + cos x  Evaluate ∫ e x sin x + cos x .   dx x 2 cos 2 x   tan −1 x ∫ sec 2 x log(1 + sin 2 x) dx (46) ∫ 2 dx 2 x (1 + x ) ∫

( x + 1){ x 2 + 1 + 1}

e x dx

x +1 1 ∫ 6 dx 6 2 2 x − a + a x ( x2 − a2 ) 2

(48) (50)

1 dx sin x(2 cos 2 x − 1) x +1 ∫ dx x(1 + xe x ) 2 ∫

Limits FORMAL APPROACH TO LIMIT Before we proceed to formulate a definition of the limit of a function at a point on the basis of the discussion made in the previous article, we intend to discuss some basic concepts which will be used in further discussions. NEIGHBOURHOOD (NBD) OF A POINT Let a be a real number and let δ be a positive real number. Then the set of all real numbers lying between aδ and a+ δ is called the neighbourhood of a of radius ‘ δ ’ and is denoted by N δ (a). Thus N δ (a) = (a- δ , a+ δ ) = {x. R|a- δ f(a) for all x ∈ (a - δ , a + δ ), x ≠ a (or) f(x) – f(a0 > 0 for all x ∈ (a - δ , a + δ ), x ≠ a The value of the function at x=a i.e., f(a0 is called the local minimum value of f(x) at x = a. NECESSARY CONDITION FOR EXTREME VALUES: We have the following theorem which we state without proof. THEOREM A necessary condition for f (a) to be an extreme value of a function f(x) is that f’(a) = 0, in case it exists.  x 3 + x 2 + 10 x, x < 0 ILLUSTRATION Let f(x) =  , x≥0 −3sin x Investigate x = 0 for local maximum/minimum. PROPERTIES OF MAXIMA AND MINIMA (I) If f(x) is continuous function in its domain, then at least one maxima and one minima must lie between two equal values of x. (II) Maxima and Minima occur alternately, that is, between two maxima there is one minimum and viceversa. (III) If f(x) → ∞ as x → a or b and f’(x) = 0 only for one value of x (say c) between a and b, then f© is necessarily the minimum and the least value. If f(x) → ∞ as x → a or b, f(c) is necessarily the maximum the greatest value.

(1)

(2) (3)

The circle x2+y2 =1 cuts the x-axis at P and Q. Another circle with center at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of ∆ QSR. x2 y 2 P is a point on the ellipse 2 + 2 = 1 whose center is O and N is the foot of the perpendicular from a b O upon the tangent at P. Find the maximum area of ∆ OPN and the coordinates of P. Let A(p2, -p), B(q2, q) and C(r2, -r) be the vertices of the triangle ABC. A parallelogram AFDE is drawn with D, E and F on the lines segments BC, CA and AB respectively. Using calculus show that 1 the maximum area of such a parallelogram is (p+q) (q+r) (p-r) 4

(4)

(8)

(at12, 2ati); i=1, 2, 3 are the vertices of a triangle inscribed in the parabola y2 = 4ax. A parallelogram AFDE is drawn with D, E, F on the line segments BC, CA and AB respectively. Show that the a2 (t1 − t2 ) (t2 − t3 )(t1 − t3 ). maximum area of such a parallelogram is 2

From a fixed point P on the circumference of a circle of radius a, the perpendicular PM is let fall on 3 3 a2 the tangent at point Q. Prove that the maximum area of ∆PQM is . 8 (9) Find the values of p for which f(x) = x3 + 6(p-3)x2+3(p2-4)x+10 has positive point of maximum. (10) Find the condition that f(x) = x3 + ax2 + bx + c has (i) a local minimum at a certain x ∈ R+ (ii) a local maximum at a certain x ∈ R(iii) a local maximum at certain x ∈ R- and minimum at certain x ∈ R+. (11) If f(x) = cos3 x + λ cos2x, x ∈ (0, π ). Find the range of λ so that f(x) has exactly one maximum and exactly one minimum.

Monotonic Functions STRICTLY INCREASING FUNCTION A function f(x) is said to be a strictly increasing function on (a, b) if x1 < x2 ⇒ f ( x1 ) < f ( x2 ) all x1 , x2 ∈ (a, b) STRICTLY INCREASING FUNCTION A function f(x) is said to be a strictly decreasing function on (a, b) if x1 < x2 ⇒ f ( x1 ) > f ( x2 ) for all x1, x2 ∈ (a, b) NECESSARY CONDITION Let f(x) be a differentiable function defined on (a, b). Then f’(x) > 0 or < 0 according as f(x) is increasing or decreasing on (a, b). SUFFICIENT CONDITION THEOREM Let f be a differentiable real function defined on an open interval (a, b). COROLLARY Let f(x) be a function defined on (a, b) (a) If f’(x) > 0 for all x ∈ (a, b) except for a finite number of points, where f’(x)=0, then f(x) is increasing on (a, b). (b) If f’(x) < 0 for all x ∈ (a, b0 except for a finite number of points, where f’(x)=0, then f(x) id decreasing on (a, b). SOME USEFUL PROPERTIES OF MONOTONIC FUNCTIONS (1) If f (x) is strictly increasing function on an interval [a, b], then f-1 exists and it is also positive. (2) If f(x) is strictly increasing function on an interval [a, b] such that it is continuous, then f-1 I continuous on [f(a), f(b)]. (3) If f(x) is continuous on [a, b] such that f’(c ) ≥ 0 (f’©>0) for each c ∈ (a, b), then f(x) is monotonically (strictly) increasing function on [a, b]. (4) If f(x) is continuous on [a, b] such that f’ (c ) ≤ 0 (f’© < 0) for each ∈ (a, b), then f(x) is monotonically (strictly) decreasing function on [a, b] (5) If f(x) and g(x) are monotonically (or strictly) increasing (or decreasing) functions on [a, b], then gof(x) is a monotonically (or strictly) increasing function on [a, b] (6) If one of the two functions f(x) and g(x) is strictly (or monotonically) increasing and other a strictly (monotonically) decreasing, then g of (x) is strictly (monotonically) decreasing on [a, b].

(1)

(2) (3)

ax ,x ≤0  xe , where a is a positive constant. Let f(x) =  2 3  x + ax − x , x > 0 Find the intervals in which f’ (x) is increasing. If φ (x) =f(x) + f(1-x) and f’’ (x) < 0 for all x ∈ [0, 1]. Prove that φ (x) is increasing in [0, ½] and decreasing in (1/2, 1]. Let g(x) = 2f(x/2) + f(2-x) and f’’(x) < 0 for all x ∈ (0, 2). Find the intervals of increases and decrease of g(x).

Mean Values Theorems and Some Other Applications of Derivatives ROLE’S THEOREM STATEMENT Let f be a real valued function defined on the closed interval [a, b] such that (i) It is continuous on the closed interval [a, b], (ii) it is differentiable on the open interval (a, b), and (iii) f(a) = f(b). LAGRANGE’S MEAN VALUE THEOREM STATEMENT Let f(x) be a function defined on [a, b] such that it is continuous on [a, b] (ii) it is differentiable on (a, b). Then there exists a real number c ∈ (a, b) such that f '(c) =

(1) (2)

(3)

f (b) − f (a ) b−a

Prove that (b-a)sec2a G. PROPERTY II If A and G are respectively arithmetic and geometric means between two positive quantities a and b, then the quadratic equation having a, b as its roots is x2 – 2Ax + G2 = 0 PROPERTY III If A and G be the A.M. and G.M. between two positive numbers,

then the numbers are A ± A2 − G 2 ILLUSTRATIVE EXAMPLES

(1) If one geometric mean G and two arithmetic means A1 and A2 be inserted between two given quantities, prove that G2 = (2A1 – A2) (2A2-A1). ma + nb . Find m and n (2) The A.M. between m and n and the G.M. between a and b are each equal to m+n in terms of a and b. ARITHMETICO-GEOMETRIC SERIES Let a, (a + d)r, (a+2d)r2, (a+3d)r3, … be an arithmetico –geometric sequence. Then, a+(a+d)r+(a+2d)r2+(a+3d)r3+… is an arithmetico geometric series. 2 3 4 ILLUSTRATION Find the nth term of the series 1 + + 2 + 3 + ... 3 3 3 SUM OF n TERMS OF AN ARITHMETICO-GEOMETRIC SEQUENCE THEOREM The sum of n terms of an arthmetico-geometric sequence a, (a + d) r, (a+2d)r2, (a+3d)r3, … is given by  a (1 − r n −1 ) {a + (n − 1)d }r n + dr − , when r ≠ 1  1− r 1− r S n = 1 − r  n [2a + (n − 1)d ], when r = 1  2

 2n + 1   2n + 1  Show that the sum of the series 1 +   + 5  + ....to n terms is an even or an odd  2n − 1   2n − 1  number according as n is even or odd. 2

EXAMPLE

HARMONIC PROGRESSION DEFINITION A sequence a1, a2,…., an, …. Of non-zero number is called a Harmonic 1 1 1 1 progression, if the sequence , , ,..., ,... is an Arithmetic progression . a1 a2 a3 an nth TERM OF A HP The nth term of a H.P is the reciprocal of the nth term of the corresponding A.P. Thus, if a1, a2, a3, …, an, … is a HP and the common difference of the corresponding AP is d 1 1 1 − , then an = i.e. d = 1 an +1 an + (n − 1)d a1 1 1 1 2 1 1 2ac . If a, b, c are in HP, then , , are in AP. Therefore = + ⇒ b = b a c a b c a+c (1) If S1, S2 and S3 denote the sum up to n(>1) terms of three non-constant sequence in A.P., whose first 2 S S − S S − S 2 S3 terms are unity and common differences are in H.P., prove that n = 1 3 1 2 S1 − 2 S2 + S3 1 1 4 (2) If a, b, c are in HP and a > c, show that + > b−c a −b a −c (3) Let a, b, c be positive real numbers. If a, A1, A2, b are in A.P., a, G1, G2, b are in G.P. and a, H1, H2, b GG A + A2 (2a + b) (a + 2b) are in H.P., show that 1 2 = 1 = H1 H 2 H1 + H 2 9ab PROPERTIES OF ARITHMETIC, GEOMETRIC AND HARMONIC MEANS BETWEEN TWO GIVEN NUMBERS Let A, G and H be arithmetic, geometric and harmonic means of two positive numbers a and b. Then. 2ab a+b A= , G ab and H = . These three means possess the following properties : a+b 2

2ab A ≥ G ≥ H. a+b PROPERTY2 A, G, H form a GP i.e, G2 – AH. PROPERTY1 H =

PROPERTY3 The equation having a and b as its roots x2 – 2Ax + G2 = 0 PROPERTY4 If A, G, H are arithmetic, geometric and harmonic means between three given numbers a, b and c, then the equation having a, b, c as its roots is

3G 3 x − G3 = 0 H a n +1 + b n +1 For what value of n, is the harmonic mean of a and b? a n + bn If A1, A2; G1, G2; H1, H2 be two A.M.’s, G.M’s and H.M.s between two number a and b, then prove x 3 − 3 Ax 2 +

(1)

(2) that : (3) If H1, H2,….,Hn be n harmonic means between a and b and n is a root of the equation x2(1-ab)H n+r x(a2+b2)-(1+ab)=0, then prove that H1 = ab(a-b) 1 = H n nr + 1 METHOD OF DIFFERENCES: Sometimes the nth term of a sequence or a series cannot be determined by the methods discussed in the earlier sections. In such cases, we use the following steps to find the nth term Tn of the given sequence.

STEP 1 Obtain the terms of the sequence and compute the differences between the successive terms of the given sequence. If these differences are in A.P, then take Tn = an2 + bc +c, where a, b, c are constants. Determine a, b, c by putting n=1, 2, 3 and putting the values of T1, T2, T3. STEP 2 If the successive differences computed in step 1 are in G.P. with common ratio r, then take Tn = arn-1 + bn+c. STEP 3 If the differences of the differences computed in step 1 are in A.P., then take Tn = an3 + bn2 + cn + d and find the values of constants a, b, c, d. STEP 4 If the differences of the differences computed in step 1 are in G.P. with common ratio r, then take Tn = arn-1 + bn2 + cn + d The following examples will illustrate the above procedure.

(1)

Sum the following series to n terms : 5 + 7+ 13 + 31 + 85 +…. 1 1 1 (2) If H n = 1 + + + .... + and, 2 3 n n +1  1 2 3 n − 2 Hn ' = + + + ... + -  2 2.3   n(n − 1) (n − 1)(n − 2) (n − 2)(n − 3 show that Hn = Hn1

(3)

(4) (5)

1 (n 4 + 2n3 + 2n 2 + n − 1) n(n + 1) n n 1 n(n + 1)(n + 2)(n + 3) If ∑ Tr = . , where Tr denotes the rth term of the series. Find, lim ∑ n →∞ 12 r =1 r =1 Tr 1 1 1 Sum the series to n terms: + + + .... (1 + x)(1 + 2 x) (1 + 2 x)(1 + 3x) (1 + 3x)(1 + 4 x)

Find the sum of first n terms of the series whose nth term is

Scalar or Dot Product of Vectors DEFINITION





Let a and b be two non-zero vectors inclined at an angle θ . Then, the scalar product of a with b is



denoted by a .b and is defined as the scalar | a | | b |cos θ 1 Show that the angle between two diagonals of a cube is cos −1   . 3 (2) The length of the sides a, b, c of a triangle ABC are related as a2 +b2= 5c2. Prove, using vector methods, that the medians drawn to the sides a and b are perpendicular.
and (3) Determine the lengths of the diagonals of a parallelogram constructed on the vectors a
= 2α
− β
, where α
and β
are unit vectors forming an angle of 60o. b
= α
− 2 β





 (4) Two points A and B are given on the curve y=x2 such that OA .
i = 1 and OB .
i =-2. Find





 |2 OA − 3OB | (1)

VECTOR (CROSS) PRODUCT OF VECTORS DEFINITION

VECTOR (CROSS) PRODUCT Let a, b be two non –zero non-parallel vectors. Then



the vector product a × b, in that order, is defined as a vector whose magnitude is | a || b |



sin θ where θ is the angle between a and b and whose direction is perpendicular to the plane of a and b

in such a way that a, b and this direction constitute a right handed system. PROPERTIES OF VECTOR PRODUCT

The vector product has the following properties:

(1) Vector product I not commutative i.e. if a and b are any two vectors, then





a × b ≠ b × a , however - b × a .

(2) If a and b are two vectors and m is a scalar, then





ma × b = m(a × b) = a × mb









(3) If a, b are two vectors and m, n are scalars, then ma × nb = mn(a × b) = m(a × nb) = n(ma × b) (4) Vector over

product

is
distributive



vector
addition


i.e.
a × (b + c) = a × b + a × c and, (b + c) × a = b × a + c × a









(5) For any three vectors a, b, c , we have a × ( b − c) = a × b − a × c . (6) The vectors is she the null vector if they are collinear or parallel i.e.


vector
product

of two
non-zero a × b = 0 ⇔ a || b , where a, b are non-null vectors
i ×
i =
j ×
j = k
× k
= 0,

i ×
j = k
,
j ×
i = −k
,
j × k
=
i, k ×
j = i
, k
×
i =
j ,
i × k
= − j (7) (8)


i

 If a = a1 i + a2 k
, and b = b1 i + b2
j + b3 k
, then a × b
= a1 b1


j

k


a2 b2

a3 b3

SOME USEFUL RESULTS




RESULT I If a and b are two non-zero, non-parallel vectors, then unit vectors normal to the plane of a


a×b and b are ±

| a×b |





 a×b  RESULT II Vectors of magnitude λ normal to the plane of a and b are ± λ 

  | a×b | 



RESULT III The area of the parallelogram with adjacent sides a and b is | a × b |

1

RESULT IV The area of a triangle with adjacent sides a and b is | a × b | 2 1





 RESULT V Area of ∆ABC = | AB × AC | 2 1





 = | BC × BA | 2 1





 = | CA × CB | 2 1





 RESULT VI The area of a plane quadrilateral ABCD is | AC × BD | , where AC and BD are its diagonals. 2


If a, b, c are the position vectors of the vertices A, B, C of a triangle ABC, show that the area of 1




triangle ABC is | a × b + c + c × a | . 2
(2) Show that the perpendicular distance of the point c from the line joining







| b×c + c× a + a×b |

a and b is |b−a|

















 (3) If A, B, C, D be any four points in space, prove that | AB × CD + BC × AD + CA × BD | = 4 (Area of triangle ABC).


 


  


  (4) Let OA = a, OB = 10a + 2b, and OC = b where O is origin. Let p denote the area of the quadrilateral OABC and q denote the area of the parallelogram sides. Prove that p = 6q.


  with


 OA
 and


 OC as  adjacent
 (5) ABCD is a quadrilateral such that AB = b, AD = d , AC = mb + pd . Show that the area of the

1 quadrilateral ABCD is | m + p || b × d | 2

(1)

PRODUCT OF THREE VECTORS SCALAR TRIPLE
PRODUCT






DEFINITION Let a, b, c be three vectors. Then the scalar (a × b). c is called the scalar product of a , b



and c and is denoted by [a b c] .





Thus, [a b c] = (a × b) . c . PROPERTIES OF
SCALAR TRIPEL PRODUCT

PROPERTY-I If a, b, c are cyclically permuted the value of scalar triple product remains same,














i.e (b × c). a = (c × a ).b (or), [a b c] = [b c a ] = [c a b] PROPERTY-II The change of cyclic order of vectors in scalar triple product changes the sign of the scalar triple
product but



not
the
magnitude.




i.e.[a b c] = −[b a c] = −[c b a] = −[a c b] PROPERTY-III In scalar triple product the positions of dot and cross can be interchanged provided that the cyclic
order
of
the
vectors


remains same i.e. (a × b). c = a. a . (b × c)

PROPERTY-IV
The

scalar triple


product of three vectors is zero if any two of them are equal. [λ a b c ] = λ[ a b c ] PROPERTY-VI The scalar triple product of three vectors is zero if any two of them are parallel or collinear.













PROPERTY-VII If a b c d , are four vectors, then [a + b c d ] = [a c d ] + [b c d ]




PROPERTY-VIII The necessary and sufficient condition for three non-zero, non-collinear vectors a, b, c








to be coplanar is that [a b c] = 0. i.e. a, b, c are coplanar ⇔ [a b c] = 0



 PROPERTY-IX For points with position vectors a, b, c and d will be coplanar if
  
  
      [ d , b, c ] + [ d , c , a ] + [ d , a , b ] = [ a , b , c ]


PROPERTY-X If a = a, i + a2
j + a3 k
, b = b1 i + b2
j + b3 k
and c = c1 i + c2
j + c3 k
are three vectors, then

a1


[a b c] = b1

a2 b2

a3 b3

c1

c2

c3




(1) If the vectors α = ai = a
j + ck
, β = i + k
and γ = ci + c
j + bk
are coplanar, then prove that c is the geometric mean




of
a and b. (2) Let a, b, c three non-zero vectors such that c is a unit vector perpendicular to both a and b . If the




1
2
2 |a| |b| . angle between a and b is π / 6 , prove that [a b c]2 = 4
 (3) Consider the vectors: A = i + cos( β − α )
j + cos(γ − α )k

 B = cos(α − β )i +
j + cos(γ − β )k

 and, C = cos(α − γ ) i + cos( β − γ )
j + ak



 where α , β are γ are different angles in (0, π / 2) . If A, B, C are coplanar vectors, show that a is independent of α , β and γ .


(4) If a, b, c be three on –coplanar unit vectors equally inclined to one another at an angle θ such that






q2 a × b + b × c = pa + qb + rc, find p, q, r in terms of θ . Also, prove that p 2 + + r2 = 2 . cos θ



r a , b , c (5) If are three non-coplanar vectors, prove that any vector is expressible as









 [r b c 
 [r c a 
 [r b c 
r = 


 a + 


 b + 


 c. [a b c]  [a b c]  [a b c]  RECIPROCAL SYSTEM OF VECTORS





Let a, b, c be three non-coplanar vectors, vectors, so that [a b c] ≠ 0. We define another set of three










b×c
c×a a×b vectors a, b, c as given below a =


. b =


, c =


[a b c] [a b c] [a b c]

(1) that







If a, b, c are three non-coplanar vectors and a, b, c form a reciprocal system of vectors, then prove

(i)







a. a = b. b = c. c = 1













a. b = a. c = 0; b. c = b. a = 0; c. a = c. b = 0


1 (iii) [a b c] =


[a b c]





If a b c and a ' , b' , c ' be the reciprocal system of vectors, prove that












a. a + b. b + c.c = 3 a× a + b ×b + c × c = 0 (ii) (i)



If a and b are two vectors such that a. b ≠ 0, then solve the vector equations






r. a = 0, r. b = 1, [r a b] = 1.











 a × (d × c  If r × a + (r. b)c = d , then prove that r = λ a + a × 


2  , where λ is a scalar  (a. c)| a | 

(ii)

(2) (3)

(4)

VOLUME OF A TETRAHEDRON THEOREM (i) If two pairs of opposite edges of a tetrahedron are perpendicular, then the opposite edges of the third pair are also perpendicular to each other. (ii) The sum of the squares of two opposite edges is the same for each pair of opposite edges (iii) Any two opposite edges in a regular tetrahedron CENTROID OF A TETRAHEDRON THEOREM The volume V of a tetrahedron whose three coterminous edges in the right-handed system are


1


a, b c is given by V = [a b c] 6

(1)

A tetrahedron has three of its vertices of its vertices at A, B and C whre








 OA = 3i + 2
j , OB = i + 3
j − k
; OC = 2
j . Find the unit vector perpendicular to the face ABC. The











 fourth vertex D is such that DA. AB = 0 = DA. AC. Find the vector equation of AD. If the volume of the tetrahedron is 3 2 cubic units and D is on the same side as the origin, find the coordinates of D.

(2)

(3)

(4)

The position vectors of the vertices A, B and C of a tetrahedron ABCD are
i +
j + k
,
i and 3i
respectively. The altitude from vertex D to the opposite face ABC meets the median line through A of the triangle ABC at a point E. If the length of the side AD is 4 and the volume of the 2 2 cubic units, find the position vector of the point E for all its possible tetrahedron is 3 positions.

OABC is a regular tetrahedron. D is the circumcentre of ∆OAB and E is the middle point of the edge AC. Use vector method to find distance DE. . A pyramid with vertex at the point P whose position vector is 4
i + 2
j + 2 3 k
has a regular hexagonal base ABCDEF. The points A and B have position vectors
i and
i + 2
j respectively. The centre of hexagon has position vector
i +
j + 3 k
. Given that the volume of the pyramid is 6 3 and the perpendicular from the vertex meets the diagonal AD, locate the position vectors of the foot of this perpendicular.

(5)




Let a = a1 i + a2
j + a3 k
, b = b1 i + b2
j + b3 k
and c = c1 i + c2
j + c3 k
be three non-zero vectors such




that c is a unit vector perpendicular to both the vectors a and b . If the angle between a and b is

π

6

, prove that

a1 a2 a3 1 b1 b2 b3 = (a12 + a22 + a32) (b12 + b22 + b32) 4 c1 c2 c3



(6) If a, b, c are three on-coplanar vectors and r is any vector in space, then prove that






r. a

r. a

r. c

r =


(b × c) +


(c × a) +


(a × b) [a b c] [a b c] [a b c]



 (7) If a, b, c and d are four vectors, then prove that











(i) (a × b).(c × d ) + (b × c).(a × d ) + (c × a ).(b × d ) = 0
   
 
  
 d .[a × {b × c × d )}] = [b. d ][a c d ] (ii)

Tangents and Normals ILLUSTRATIVE EXAMPLES: (i) (ii) (iii)

(iv) (v) (vi) (vii)

The curve y=ax3+bx2+cx+5 touches the x-axis at P(-2, 0) and cuts the y-axis at the point Q where its gradient is 3. Find the equation of the curve completely. Find the equation of the normal to the curve y= (1=x)y + sin-1(sin2x)atx=0 Determine the constant c such that the straight line joining the points (0, 3) and 95, -2) is tangent to c the curve y = x +1 Prove that all normal to the curve x=a cost + at sint, y = a sint - at cost Find the points at which the tangents to the curves y=x3 – x – 1 and y=3x2 – 4x + 1 are parallel. Also, find the equations of tangents. Find the equation of the tangent to x3 = ay2 at the point A (at2, at3). Find also the point where this tangent meets the curve again. Tangent at point P1 (other than (0, 0) on the curve y=x3 meets the curve again at P2. The tangent at P2 meets the curve at P3 and so on. Show that the abscissae of P1, P2, P3. ….., Pn form a GP. Also, find area ∆P1 P2 P3 the ratio area ∆P2 P3 P4 x

(viii) For the function F(x) = ∫ 2 | t | dt, find the tangent lines which are parallel to the bisector of the angle 0

(ix)

in the first quadrant. If α , β are the intercepts made on the axes by the tangent at any point of the curve x=a cos3 θ , y=bsin3 θ , prove that

(x)

α2

+

β2

=1. a 2 b2 If x1 and y1 be the intercepts on the axes of X an Y cut off by the tangent to the curve a x  y   +   = 1, then prove that   a b  x1  n

(xi)

n

n / n −1

b +   y1 

n / n −1

= 1.

 c Show that the normal to the rectangular hyperbola xy=c2 at the point P  ct1 ,  meets the curve t1    c again at the point Q  ct2 ,  , if t13 t2 = -1 t2  

ANGLE OF INTERSECTION OF TWO CURVES The angle of intersection of two curves is defined to be the angle between the tangents to the two curves at their point of intersection. ORTHOGONAL CURVES If the angle of intersection of two curves is a right angle, the two curves are said to intersect orthogonally and the curves are called orthogonal curves. If the curves C1 and C2 are orthogonal, then φ = π / 2  dy   dy  m1 m2 = -1 ⇒     = −1 ∴  dx C1  dx C2 EXAMPLES: (i) Find the acute angle between the curves y = |x2 –1| and y=|x2 – 3| at their points of intersection (ii) Show that the curves x3 – 3xy2 = -2 and 3x2y-y3=2 cut orthogonally.

(iii)

Find the acute angles between the curves y = |2x2 – 4| and y=|x2 – 5|.

(iv)

Show that the curves y2=4ax and ay2=4x3 intersect each other at an angle of tan-1

1 and also if PG1 2 and PG2 be the normals to two curves at common point of intersection (other than the origin) meeting the axis of X in G1 and G2, then G1 G2 = 4a.

LENGTHS OF TANGENT, NORMAL, SUBTANGENT AND SUBNORMAL Let the tangent and normal at a point P(x, y) on the curve y=f(x), meet the x-axis at T and N respectively. If G is the foot of the ordinate at P, then TG and GN are called the Cartesian subtangent and subnormal, while the lengths PT and PN are called the lengths of the tangent and normal respectively. dy If PT makes angle Ψ with x-axis, then tan Ψ = . From Fig we find that dx y Subtangent = TG = y cot Ψ =  dy     dx  dy Subnormal = GN = y tan Ψ = y dx Length of the tangent = PT = y cosec Ψ

= y 1 + cot 2 Ψ

Theory of Equations SOME DEFINITIONS REAL POLYNOMIAL Let ao, a1, …., an be real numbers and x is a real variable. Then, f(x)=a0 + a1x+a2x2 +….+anxn is called a real polynomial of real variable x with real coefficients. For example, 2x3 – 6x2 + 11x-6, x2-4x+3 etc. are real polynomials. COMPLEX POLYNOMIAL If a0, a1, a2…..an be complex numbers and x is a varying complex number, then f(x) = a0 + a1x+a2x2 +….+an-1xn-1+anxn is called a complex polynomial or apolynomial of complex variable with complex coefficients. POLYNOMIAL EQUATION If f(x) is a polynomial, real or complex, then f(x) = 0 is called a polynomial equation. If f(x) is a polynomial of second degree, then f(x) =0 is called a quadratic equation. The general form of a quadratic equation is ax2+bx+c=0, where a, b, c ∈ C, set of all complex numbers, and a ≠ 0. ROOTS OF AN EQUATION The values of the variable satisfying the given equation are called its roots. In other words, x = α is a root of the equation f(x)=0, if f( α )=0. The real roots of an equation f(x)=0 are the x-coordinates of the points where the curve y=f(x) crosses xaxis. SOME RESULTS ON ROOTS OF AN EQUATION

The following are some results on the roots of a polynomial equation with rational coefficients: I II

III IV

An equation of degree n has n roots, real or imaginary Surd and imaginary roots always occur in pairs i.e. if 2-3i is a root of an equation, then 2+3i is also its root. Similarly, if 2 + 3 is a root of a given equation, then 2 − 3 is also its roots. An odd degree equation has at least one real root, whose sign is opposite to that of its last term provided that the coefficient of highest degree term is positive. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two reals, one positive and one negative.

POSITION OF ROOTS OF A POLYNOMIAL EQUATION If f(x) = 0 is an equation and a, b are two real numbers such that f(a) f(b) < 0, then the equation f(x) = 0 has at least one real root or an odd number of real roots between a and b. In case f(a) and f(b) are of the same sign, then either no real root or an even number of real roots f(x)=0 lie between a and b. DEDUCTIONS

1. 2. 3. 4.

Every equation of an odd degree has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of first term is positive. Every equation of an even degree whose last term is negative and the coefficient of first term positive, has at least two real roots, one positive and one negative. If an equation has only one change of sign, it has one positive root and no more. If all the terms of an equation are positive and the equation involves no odd powers of x, then all its roots are complex.

ILLUSTRATION 1 If a, b, c, d ∈ R such that a < b < c < d, then show that the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are real and distinct. ROOTS OF A QUADRATIC EQUTION WITH REALCOEFFICIENTS An equation of the form ax2 + bx + c= 0 where a ≠ 0, a, b, c ∈ R is a called a quadratic equation with real coefficients.

The quantity D=b2 – 4ac is known as the discriminant of the quadratic equation in (i) whose roots are given −b + b 2 − 4ac −b + b 2 − 4ac and β = 2a 2a The nature of the roots is as given below: 1. The roots are real and distinct if D > 0. 2. The roots are real and equal if D = 0. 3. The roots are complex with non-zero imaginary part if D < 0. 4. The roots are rational if a, b, c are rational and D is a perfect square. by α =

q ( p, q ∈ Q) if a, b, c are rational and D is not a perfect square.

5.

The roots are of the form P +

6.

If a = 1, b, c ∈ 1 and the roots are rational numbers, then these roots must be integers.

7.

If a quadratic equation in x has more than two roots, then it is an identity in x that is a = b = c = 0.

COMMON ROOTS

Let a1x2 + b1x+c1=0 and a2x2 + b2x+c2=0 be two quadratic equation such that a1, a2 ≠ 0 and a1b2 ≠ a2b1. Let α be the common root of these two equations. Then, a1α 2 + b1α + c1 = 0 a2α 2 + b2α + c2 = 0 Eliminating α , we get b1c2 − b2 c1  c1a2 − c2 a1  =  a1b2 − a2b1  a1b2 − a2b1 

2

SIGN OF A QUADRTIC EXPRESSION Let f(x) = ax2 + bx+c be a quadratic expression, where a, b, c ∈ R and a ≠ 0. In this section, we shall determine the sign of f(x) = ax2 + bx + c for real values of x. As the discriminate of f(x) = ax2 + bx + c can be positive, zero or negative. So, we shall discuss the following three cases. CASE I : When D = b2 – 4ac < 0 If D < 0, then it is evident from Figs 20.12 and 20.13 that f(x) > 0 iff a > 0 and f(x) < 0 iff a < 0. CASE II When D = b2 – 4ac = 0 From Figs, 20.10 and 20.11, we abserve that: When D = 0, we have f(x) ≥ 0 iff a > 0 and f(x) ≤ 0 iff a < 0. CASE III When D = b2 – 4ac > 0 From Fig. 20.8 and 20.9, we observe the following if D = b2 – 4ac > 0 and a > 0, then > 0 for x < α or x > β  f ( x) < 0 for α < x < β = 0 for x = α , β  < 0 for x < α or x > β  2 If D = b – 4ac > 0 and a < 0, then f ( x) > 0 for α < x < β  0 for x = α , β  CONDITION FOR RESOLUTION INTO LINEAR FACTORS

THEOREM: The quadratic function ax2 + 2hxy + by2 + 2gx + 2fy+c is resolvable into a h g 2 2 2 linear rational factors if abc+2fgh-af -bg -ch =0 i.e. h b f = 0

g

f

c

Probability ELEMENTARY EVENT If a random experiment is performed, then each of its outcomes is known as an elementary event. SAMPLE SPACE The set all possible outcomes of a random experiment is called the sample space associated with it and it is generally denoted by S. ILLUSTRATION Consider the experiment of tossing two coins together or a coin twice. In this experiment the possible outcomes are. Head on first and Head on second Head on first and Tail on second, Tail on first and Head on second, Tail on first and Tail on second. If we define HH = Getting head on both coins, HT = Getting head on first and tail on second TH = Getting tail on first and head on second, TT = Getting tail on both coins. COMPOUND EVENT A subset of the sample space associated to a random experiment is said to define a compound event if it is disjoint union of single element subsets of the sample space. NEGATION OF AN EVENT Corresponding to every event A associated with a random experiment we define an event “not A” which occurs when and only when A does not occur. PROBABILITY DEFINITION If there are n elementary events associated with a random experiment and m of them are favourable to an event A, then the probability of happing or occurrence of A is denoted by P(A) and is m defined as ratio . n m Thus, P(A) = n Clearly, 0 ≤ m ≤ n . Therefore, m 0 ≤ ≤1 n 0 ≤ p(a) ≤ 1 ⇒ If p(A) =1, then A is called certain event and A is called an impossible event, if P(A) = 0. The number of elementary events which will ensure the non-occurrence A i.e. Which ensure the occurrence of A is (n-m). Therefore. n−m P ( A) = n m = 1− n = 1 – P(A) P(A) + P ( A) = 1 ⇒ The odds in favour of occurrence of the event A are defined by m : (n-m) i.e, P(A) : P ( A) and the odds against the occurrence of A are defined n-m : m i.e, P ( A) : P(A)

1.

2. 3.

An unbiased die, with face numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n number showing up is noted. What is the probability that, among numbers, 1, 2, 3, 4, 5, 6, only three numbers appear in this list? Three six faced die are thrown together. Find the probability that the sum of the numbers appearing on them is k (9 ≤ k ≤ 14). In a bag there are three tickets numbered 1, 2, 3. A ticket is drawn at random and put back, and this is done four times. Find the probability of getting an even number as the sum.

GEOMETRICAL PROBABILITY 1. Two points are taken on a straight line AB of length unity. Prove that the probability that the distance between them exceeds (0 6, find the probability that the first four of those asked do not know the answer. (ii) Show that the probability that the rth person asked is the fist to know the answer is 3(n − r ) (n − r − 1) where 1 ≤ r ≤ n − 2 n(n − 1)(n − 2) 2. A die loaded so that the probability of throwing the number is proportional to i. Find the probability that the number 2 has occurred, given that when the die is recalled an even number has turned up. MORE ON INDEPENDENT EVENTS In section 40.6, we have defined independent events and we have seen that two events and B are independent if P(B/A)=P(B) and P(A/B)=P(A).

Also, P ( A ∩ B ) = P ( A) P ( B ) if A and B are independent events. In this section, we shall discuss about pair wise independence and mutual independence

of events. PAIRWISE INDEPENDENT EVENTS: Let A1, A2, …., An be n events associated to a random experiment. These events are said to be pair wise independent if P( A1 ∩ Aj ) = P ( Ai ) P ( Aj ) for i ≠ j; i, j = 1, 2..., n P ( Ai ∩ Aj ∩ Ak ) = P( Ai ) P( Aj ) P( Ak ), for i ≠ j ≠ k ; j, k = 1,2,….,n


P( A1 ∩ A2 ..... ∩ An ) = P( A1 ) P ( A2 )....P( An ) ILLUSTRATION 1 A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events A, B, C are defined as A : “the first bulb is defective”, B : “the second bulb is non-defective”, C : “the second bulb is non-defective”, Determine whether (i) A, B, C are pair wise independent, (ii) A, B, C are mutually independent. THEOREM If A and B are independent events associated with a random experiment, then prove that (ii) A and B are independent events (i) A and B are independent events (iii) A and B are also independent events

1.

2.

For three independent events A, B and C, the probability to A to occur is a, the probability that A, B and C will not occur is b, and the probability that at last one of thee three events will not occur is c. If p denotes the probability that c occurs but neither A nor B occurs, prove that p is a root of the equation (1 − a) 2 + ab ap2 + {ab +(1-a)(1-a-c)} p+b(1-a)(1-c)=0 and deduces that c > 1− a An urn contains five balls alike in every respect except colour. If three of these balls are white and two are black and we draw two balls at random from this urn without replacing them. If A is the event that the first ball drawn is white and B the event that the second ball drawn is black, are A and B independent?

THE LAW OF TOTAL PROBABILITY THEOREM (Law of total probability) Let S be the sample space and let E1, E2,….,En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or ….or En, then P(A) = P(E1) P(A/E1) + P(E2) P(A/E2) +…..P(En) P(A/En) n

= ∑ P( Er ) P ( A / Er ). r =1

1.

2.

Urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn is now white? An employer sends a letter to his employee but he does not receive the reply (It is certain that the employee would have replied if he did receive the letter). It is known that one out n letters does not reach its destination. Find the probability that the employee does not receive the letter.

BAYE’S RULE THEOREM (Baye’s Theorem) Let S be the sample space and let E1, E2…, En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or….or P(A / E k ).P(E k ) En, then P(E k / A) = n ∑ P(A / Ei ).P(Ei ) i =1

1.

A company has two plants to manufacture scooters. Plant 1 manufactures 70% of the scooters and Plant II manufactures 30%. At Plant I, 80% of the scooters are rated as of standard quality and at Plant II, 90% of the scooters are rated as of standard quality A scooter is chosen at random and is found to be of standard quality. What is the probability that it has come from Plant II?

RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION DEFINITION Let S be the sample space associated with a given random experiment. Then a real valued function X which assigns to each even w ∈ S to a unique real number X (w) is called a random variable. . In other words, a random variable is a real valued function having domain as the sample space associated with a random experiment. MEAN AND VARIANCE OF A RANDOM VARIABLE MAEN If X is discrete random variable which assumes values x1, x2, x3,….,xn with respective probabilities p1, p2, p3,….,pn, then the mean X of X is defined as n

X = p1 x1 + p2x2 +…..+pnxn or, X = ∑ pi xi i =1

VARIANCE If X is a discrete random variable which assumes values x1, x2, x3,…, xn With the respective probabilities p1, p2, ….,pn, then variance of X is defined as Var (X) = p1 ( x1 − X ) 2 + p2 ( x2 − X ) 2 + .... + pn ( xn − X ) 2 …… + pn ( xn − X ) 2 n

n

i =1

i =1

= ∑ pi ( xi − X ) 2 , Where X = ∑ pi xi is the mean of X.

Now, n

= ∑ pi xi − X

2

2

i =1

BINOMIAL DISTRIBUTION A random variable X which takes values 0. 1, 2,.., n is said to follow binomial distribution if its probability distribution function is given by P(X = r) = n Cr p r q n − r , r = 0,1, 2,....n, where p, q > 0 such that p + q = 1 1. An urn contains 25 balls of which 10 balls bear a mark ‘A’ and the remaining 15 balls bear a mark ‘B’. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that (i) all will bear ‘A’ marked (ii) not more than 2 will bear ‘B’ mark (iii) then number of balls with ‘A’ mark and ‘B’ mark will be equal (iv) at least one ball will bear ‘B’ mark 2. Numbers are selected at random one at a time, from the numbers 00, 01, 02,…, 99 with replacement. An event A occurs if the product of the two digits of the selected number is 18. If four numbers are selected, find the probability that A occurs at lest 3 time.

Areas of Bounded Regions THEOREM Let f(x) be a continuous function defined on [a, b]. Then, the area bounded by the curve y=f(x), the x-axis and the ordinates x = a and x = b is given by b b ∫ f ( x) dx or , ∫ y dx a a The area bounded by the curve x = f(y), the y axis and the abscissae y = c and y = d is given by d d ∫ f ( y ) dy or , ∫ x dy c c (1) Let f(x) = maximum {x2, (1-x)2, 2x(1-x)}. Determine the area of the region bounded by the curve y=f(x), x-axis, x = 0 and x = 1. (2) Let f(x) be defined by f(x) = max (4sinx, 4-2 sinx), 0 ≤ x ≤ 2 π . Draw a sketch of y=f(x) and compute the area bounded by the curves y=f(x), the y-axis, the x-axis and the ordinate at x = 2 π . (3) Find the area lying on the same side of the axis of x, as the positive part of the axis of y and which is contained by y2 = 4ax, x2 + y2 = 2ax and x=y + 2a..

Complex Numbers COMPLEX NUMBER: If a, b are two real numbers, then a number of the form a+ib is called a complex number. A complex number z is purely real if its imaginary part is zero i.e. Im(z)=0 and purely imaginary if its real part is zero i.e. Re (z) = 0. SET OF COMPLEX NUMBERS: The set of all complex numbers is denoted by C I.e. C = {a + ib | a, b ∈ R}. Since a real number ‘a’ can be written as a+0 i, therefore every real number is a complex number. Hence, » ⊂ » , where » is the set of all real numbers. DEFINITION: Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if a1 = a2 and b1 = b2 i.e. Re (z1) = Re (z2) and Im(z1) = Im(z2). POLAR FORM OF z = x + iy FOR DIFFERENT SIGNS OF x and y.  y Let | z | = r and be the acute angle given by tan −1   . Let θ be the argument of z. x CASE I Polar form of z = x + iy when x > 0 and y > 0 : In this case, we have θ = α . So, the polar form of z = x + iy is r(cos α + isin α ). CASE II Polar form of z = x + iy when x < 0 and y > 0 : In this case, we have θ = π − α . So the polar form of z=x+iy is r[cos(π − α ) + i sin(π − α ) or, r (− cos α + i sin α ) CASE III Polar form of z = x +iy when x < 0 and y < 0 : In this case, we have θ = −(π − α ). So, the polar form of z is r[cos(π − α ) + i sin(−(π − α ))] or r[− cos α − i sin α ) CASE IV Polar form of z=x+iy when x > 0 and y v(x1) for some x1 and f(x) > g(x) for all x > x1, prove that any point (x, y), where x > x1, does not satisfy the equation y=u(x) and y = v(x). Let u(x) and v(x) satisfy the differential equation

EQUATIONS REDUCIBLE TO LINEA FORM BERNOULLI’S DIFFERENTIAL EQUATIONS

dy + Py = Qy n Where P and Q are constants or functions of x alone and n dx is a non-zero constant other than unity, are known as Bernoulli’s equations. dy dy y y + log y = 2 (log y ) 2 +xsin2y = x3 cos2 y (1) Solve (2) Solve dx dx x x Solve each of the following differential equations: dy y 3 dy (3) (4) + y 2 − y secx = y3 tanx dx x dx 3 dy y (5) (6) + = xe x y 2 ( xy 2 − e1/ x )dx − x 2 y dy = 0 dx x The equations of the form

EQUATIONS SOLVABLE FOR Y If the given differential equation is expressible in the form y = f ( x, p ) then we say that it is solvable for y. dy dp  dp    Differentiating (i) with respect to x, we get = f  x, p  or p = f  x, p,  dx dx  dx    This equation contain two variables x and p. Solving this equation, we obtain φ ( x, p , c ) = 0 The solution of differential equation (i) is obtained by eliminating p between (i) and (iii). Following examples will illustrate the above procedure.

(1)

Solve the differential equation y=(1+p)x+ap2, where P=

dy . dx

(2) Solve the differential equation x2p2 + xyp – 6y2 = 0 (3) A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionally constant = k > 0).

Functions INTERVALS: Let a and b two given real numbers such that a < b. Then the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b]. i.e. [a, b] = {x ∈ r |a ≤ x ≤ b} For example, [1, 2] = {x ∈ R| 1 ≤ x ≤ 2} i.e., the set of all real numbers lying between 1 and 2, including the end points. OPEN INTERVAL: Let a and b be two given real numbers such that a < b. Then the set of all real numbers x such that a ≤ x b is called a closed interval and is denoted by (a, b). i.e (a, b) = {x ≤ R | a ≤ x ≤ b} REAL FUNCTION: If the domain and co-domain of a function are subsets R9st of all real numbers). It is called a real valued function or in short a real function. EXAMPLES:

1 1 (i) If for non-zero x, a f(x) +b f   = -5, where a ≠ b then find f(x). x  x (ii) Let g : R → R be given by g(x) = 4x + 3. If g n ( x) = gogog....og ( x), show that n −times

n

n

n

g (x) = 4 x + (4 -1). If g-n(x) denotes the inverse of gn (x), prove that g − n ( x) = 4− n x + (4− n − 1) for all x ∈ R. DOMAIN: Generally real functions in calculus are described by some formula and their domains are not explicitly stated. In such cases to find the domain of a function f (say) we use the fact that domain is the set of all real numbers x for which f (x) is a real number. In other words, determining the domain of a function f means finding all real numbers x for which f(x) is real. For example, if f(x) = 2 − x , then f(x) is real for all x ≤ 2. For x > 2, f(x) is not real. So, domain of f(x) is the set of all real numbers less than or equal to 2 i.e. (∞ , 2] EXAMPLE: Find the domain and range of function f ( x) =

1 . 2 − cos 3x

SOME STANDARD REAL FUNCTIONS CONSTANT FUNCTION: Let k be a fixed real number. Then a function f(x) given by f(x) =k for all x ∈ R is called a constant function. GREATEST INTEGER FUNCTION: For any real number x, we denote [x], the greatest integer less than or equal to x. PROPERTIES OF GREATEST INTEGER FUNCTION: If n is an integer and x is any real number between n and n+1, then the greatest integer function has the following properties : (i) [-n] = -[n] (ii) [x + n] = [x] + n (iii) [-x] =-[x]-1 − 1, if x ∉ Z 2[   x] , if x ∈ Z (v)  (iv) [x]=[-x]=  0 , if x ∈ Z 2[ x] + 1, if x ∉ Z (vi) [x] ≥ n ⇒ x ≥ n, where n ∈ Z (vii) [x] ≤ n ⇒ x < n + 1, n∈Z (viii) [x] > n ⇒ x ≥ n + 1, n∈Z (ix) [x] < n ⇒ x < n, n∈Z

(x) [x+y] = [x] + [y=X-[X]] for all x, y ∈ R. n − 1 1  2   (xi) [x] +  x +  +  x +  + .... +  x + =[nx], n ∈ N n  n n    (1) If [x] and [x] denote respectively the fractional and integral parts of a real number x. Solve the equation 4[x] = x+[x] (2) I f[x] and [x] denote the fractional and integral parts of x and (x) is defined as 2[ x] − [ x], x < 0 follows ( x)  [ x] + 3[ x], x ≥ 0 then solve the equation : (x) = x + {x} SIGNUM FUNCTION: | x |  , x≠0 The function defined by f(x) =  x 0 , x = 0

(Or)

1, x > 0  f(x) = 0 , x = 0 0 , x = 0 

is called signum function. RECIPROCAL FUNCTION: The function that associates each nonzero real number x to its reciprocal 1/x is called the reciprocal function. LOGARITHMIC FUNCTION: If ‘a’ is a positive real number, then the function that associates every positive real number to loga x i.e. f(x) = loga x is called the logarithmic function. EXPONENTIAL FUNCTION: If a is positive real number, then the function which associates every real number x to ax i.e. f(x) = ax is called the exponential function. SQUARE ROOT FUNCTION: The function that associates every positive real number x to + x is called the square root function, i.e., f(x) = + x . POLYNOMIAL FUNCTION: A function of the form f(x) = aoxn +a1 xn-1 +…+an-1 x+an, where ao, a1, a2, …..an are real numbers, ao ≠ 0 and n ∈ N , is called polynomial function of degree n.

The domain of a polynomial function is always R. RATIONAL FUNCTION: A function of the form f(x) =

P( x) , where p(x) and q(x) are polynomials and q( x)

q(x) ≠ 0, is called a rational function. SUM Let f and g be two real functions with domain D1 and D2 respectively. Then, we define their sum f + g as that function from D1 ∩ D2 to R which associates each x ∈ D1 ∩ D2 to the number f(x) + g(x). Thus, f+g: D1 ∩ D2 → R such that (f+g) (x) =f(x) + g(x) for all x ∈ D1 ∩ D2 . Similarly, we define the difference, product and quotient as follows: DIFFERENCE f-g : D1 ∩ D2 → R such that (f-g) 9x) = f(x) –g(x) for all x ∈ D1 ∩ D2 PRODUCT fg : D1 ∩ D2 → R such that (fg) (x)=f9x0 g(x) for all x ∈ D1 ∩ D2 QUOTIENT

 f  f ( x) f : D1 ∩ D2 − {x | g ( x) = 0} → R such that   ( x) = for all x ∈ D1 ∩ D2 − {x | g ( x) = 0}. g ( x) g g

SCALAR MULTIPLE for any real number c, the function cf is defined by

(cf) (x) = c.f(x) for all x ∈ D1. REMARK Note that the above operations are defined here are true only for real functions. For general functions from one set to another, these do not make sense. COMPOSITION OF FUNCTIONS: Let f and g be two functions with domain D1 and D2 respectively. If range (f) ⊂ domain g (g), we define gof by the rule (gof) (x) = g(f(x)) for all x ∈ D1. Also, if range (g) ⊂ domain (f), we define fog by the rule (fog) (x) = f(g(x)) for all x ∈ D2 It follows from the above discussion that if f(x) and g(x) are two real functions with domains D1 and D2 respectively. Then (ii) Domain of ( fg ) = D1 ∩ D2 (i) Domain of ( f ± g ) = D1 ∩ D2

 f  (iii) Domain of   = D1 ∩ D2 − {x | g 9 x) = 0} g

(1) For what real values of ‘a’ does the range of the function f ( x) = belonging to the interval [-1, -1/3] ? (2) For what real values of ‘a’ does the range of the function f(x) =

x −1 not contain any values a − x2 + 1

x −1 not attain any value from the 1 − x2 − a

interval [-1, 1]? Fin the domains of definition of the following functions: PERIODIC FUNCTIONS: PERIOD If f(x) is a periodic function, then the smallest positive real number T is called the period or fundamental period of function f(x) if. F(x+T) = f(x) for all x ∈ R. (1) Prove that the function (x) = x-[x] is a periodic function. Also find its period. (2) Let f(x) be a real valued function with domain R such that f(x + p) = 1+[2-3 f(x) + 3 (f(x))2 – (f(x))3]1/3 hold good for all x ∈ R. and some positive constant p, then prove that f(x) is a periodic function. SOME USEFUL RESULTS ON PERIODIC FUNCTIONS RESULT 1 If f(x) is a periodic function with periodic. T and a, b, ∈ R such that a ≠ 0 , then af(x) + b is periodic with period T. RESULT 2 If f(x) is a periodic function with period T and a, b ∈ R such that a ≠ 0, then f(ax+b0 is periodic with period T |a|. RESULT 3 Let f(x) and g(x) be two periodic functions such that : m Period of f(x) = , where m, n ∈ N and m, n are co-prime. n and, r Period of g(x) = , where r ∈ N and s ∈ N are coprime 3 s LCM of (m, r ) , provided that there does not Then, (f+g) (x) is periodic with period T given by T = HCF of (n, s ) exist a positive number k < T for which f(k+x) = g (x) and g(k=x)=f(x), else k will be the period of (f+g) (x).

EXAMPLE Prove that f(x) = sin-1 (sinx) is a periodic function EVEN FUNCTIONS A function f(x) is said to be an even function if f(-x) = f(x) for all x. ODD FUNCTION A function f(x) is said to be an odd function if (-x) = -f(x) for all x. (1) If f is an even function defined on the interval [-5, 5], then find the real values of  x +1  x satisfying the equation f(x) = f  .  x+2 (2) Extend f(x) = x2 + x defined in [0, 3] onto the interval [-3, 3] so that f(x) (i) even (ii) odd.