BANSAL CLASSESMATHEMATICSTARGET IIT JEE 2007XI (P, Q, R, S)

COMPOUND ANGLESTrigonometry Phase - /i : - .

I

CONTENTSKEY-CONCEPTS EXERCISE-I EXERCISE -II EXERCISE-III ANSWER KEY

KEYBASIC TRIGONOMETRIC IDENTITIES : 2 2

CONCEPTS-l 1) has exactly n roots & if the equation has more than n roots, it is an identity. If the coefficients of the equation f(x) = 0 are all real and a + ip is its root, then a - ip is also a root. i.e. imaginary roots occur in conjugate pairs. If the coefficients in the equation are all rational & a + ^P is one of its roots, then a - ^P is also a root where a, p Q & p is not a perfect square. 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'. Every equation f(x) = 0 of degree odd has atleast one real root of a sign opposite to that of its last term.

(vi)

fa Bansa! Classes

Quadratic Equations

12.

LOCATION OF ROOTS : Let f(x) = ax2 + bx + c , where a > 0 & a, b, c e R . (i) Conditions for both the roots of f(x) = 0 to be greater than a specified number'd' are b2 - 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 b2 - 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) . b2 - 4ac > 0 ; f ( p ) > 0 ; f ( q ) > 0 & p < ( - b / 2 a ) < q . LOGARITHMIC INEQUALITIES (i) For a> 1 the inequality 0 < x < y & log a x< logay are equivalent. (ii) For 0 < a < l the inequality 0 < x < y & log a x>log a y are equivalent. (iii) If a> 1 then log a x 0 p => x > ap (v) If 0 < a < 1 then log a x x>ap (vi) If 0 < a < 1 then log a x>p => 0 0), prove that b lies between (1/4) (a2 - c2) & (l/4)a 2 . At what values of'a' do all the zeroes of the function, f (x) = (a - 2) x2 + 2 a x + a + 3 lie on the interval ( - 2,1)? If one root of the quadratic equation ax2+ bx + c = 0 is equal to the nth power of the other, then show that (acn)1/(n+l) + (a 1 ^) 1 ^ 1 ) + b = 0 .

r3 - 5 ^ ' s4 q4 ? q J - 5 p - 2 ' p-2 q - 2 ' q-2 r-2 r - 2 / and \ s - 2 pqrs = 5(p + q + r + s) + 2 (pqr + qrs + rsp + spq).

V

P

s3-5A s - 2 y are collinear if

Q.21 Q.22 Q.23 Q.24 Q.25

The quadratic equation x2 + px + q = 0 where p and q are integers has rational roots. Prove that the roots are all integral. If the quadratic equations x 2 +bx+ca = 0 & x 2 +cx+ab = 0 have a common root, prove that the equation containing their other root is x2 + ax + be = 0 . If a , p are the roots of x 2 +px+q = 0 & x 2n +p n x n + qn = 0 where n is an even integer, show that a/p, p/a are the roots of xn +1 + (x + l) n = 0 . If a , p are the roots of the equation x2 - 2x + 3 = 0 obtain the equation whose roots are a 3 - 3 a 2 + 5a - 2 , p 3 - p 2 + p + 5. If each pair of the following three equations x2 + p 1 x+q 1 = 0 ,x 2 +p 2 x + q 2 =0 & x 2 +p 3 x + q3 = 0 has exactly one root common, prove that ; (PI + P2 + P3)2 = 4 [P1P2 + P2P3 + P3P1 ~ ~ % ~ Show that the function z = 2x 2 +2xy+y 2 -2x + 2y+2 is not smaller than - 3 . If (1/a) + (1/b) + (1/c) = l/(a+b + c) & n is an odd integer, show that ; (l/an) + (l/b n ) + (l/c n ) = l/(an + bn + cn) .

Q.26 Q.27

ii Bansal Classes

Quadratic Equations

[9]

Q.28

Find the values o f ' a ' f o r which - 3 < [ ( x 2 + a x - 2 ) / ( x 2 + x + l ) ] < 2 is valid for all real x. b ( 1 6 1 > -2 X + ~6 1 xJ \ x; Find the minimum value of --r ^ forx>0 / 1\ ' 1 3 1 + X +rXH xJ V

(X H

0

Q.29

0 30

Let f (x) = ax2 + bx + c = 0 has an irrational root r. If u = be any rational number, where a, b, c, p and 1 q are integer. Prove that < | f (u) |. q

EXERCISE-IIQ.l (a) (c) (d) (f) (h) Q.2 Solve the following where x e R . ( x - l ) | x 2 - 4 x + 3| + 2 x 2 + 3 x - 5 = 0 (b) 3 I x2 - 4x + 2 | = 5x - 4 For a < 0, determine all real roots of the equation x 2 - 2 a | x - a j - 3 a 2 = 0. |x 2 + 4 x + 3 | + 2 x + 5 = 0 (e) fx + 3). |x + 2 | + |2x+31 + 1 = 0 | (x + 3) |. (x +1)+12x + 5 | = 0 (g) | x 3 + 1 1 + x2 - x - 2 = 0 2 |x+21 - |2X+1 - 1 | = 2 X + 1 +1 Let a, b, c, d be distinct real numbers and a and b are the roots of quadratic equation x2 - 2cx 5d=0. If c and d are the roots of the quadratic equation x2 - 2ax - 5 6 = 0 then find the numerical value of a + b + c + d. Find the true set of values of p for which the equation p 2cosZx + p 2'cos2x - 2 = 0 Q.4 Q.5 Q.6 Q.7 has real roots.

Q.3

Prove that the minimum value of [(a+x)(b+x)]/(c+x),x>-c is U a - c + 7b-cj . If Xj, x2 be the roots of the equation x2 - 3x + A = 0 & x 3 , x 4 be those of the equation x2 - 12x + B = 0 & X j , x 2 , x 3 , x 4 are in GP. Find A & B . If ax 2 +2bx + c = 0 & a,x2 + 2b 1 x + c1 = 0 have a common root & a/a 1 ,b/bj,c/c, are inAP, show that a t , bj & c t are in GP . If by eleminating x between the equation x 2 +ax+b = 0& xy+/(x+y) + m = 0 , a quadratic in y is formed whose roots are the same as those of the original quadratic in x . Then prove either a = 21 & b = m or b + m = a / . . , 2a cos x -2xcosa + l 7 ~ lies between and n r x 2x cos (3 + 1 , (3 23 sin cos 2 sin

Q.8

If x be real, prove that

2

2

Q.9 Q.10 Q.ll

Solve the equations, ax +bxy + cy = bx + cxy+ ay = d . Find the values of K so that the quadratic equation x 2 + 2 ( K - l ) x + K + 5 = 0 has atleast one positive root. Findtiievalues of *b'for which the equation 2 log , (bx + 28) = -log 5 (l2-4x-x 2 jhasonlyonesolution.25

2

2

2

2

ii Bansal Classes

Quadratic Equations

[9]

Q.12 Q.13

Find all th e v aiues of the parameter 'a' for which both roots of the quadratic equation x2 - ax + 2 = 0 belong to the interval ( 0 , 3 ) . Find all the values ofthe parameters c for which the inequality has at least one solution. 1 + log2 2x2 + 2x + - > log2 (cx2 + c) . \ 2) " Find the values of K for which the equation x4 + (1 - 2 K) x2 + K2 - 1 = 0 ; (a) has no real solution (b) has one real solution Find the values of p for which the equation 1 + p sin x = p2 - sin2 x has a solution. Solve the equation -4.3" | x ~ 2 ' - a = 0 for every real numbera.

Q.14 Q.15 Q.16

Q.17 Find the integral values of x & y satisfying the system of inequalities; y - 1 x 2 -2x | + (1/2)> 0 & y+1 x - 1 1 5 & r ^ 0 has roots a } , a 2 , a 3 ,n

an.

Denoting (i) (ii)

il Calculate S7 & deduce that the roots cannot all be real. Prove that Sn + pS2 + qS, +nr = 0 & hence find the value of S n .

a f k by Sk.

EXERCISE-IIISolve the inequality. Where ever base is not given take it as 10. Q.l Q.3 Q.5 Q.7 ' x5'2 (log2x)4- l o g l T 201og 2 x+148 < 0. 2 y 2 (log 100 x) + (log 10 x) 2 +log x < 14 logx2 . log2x2 . log2 4x > 1. log1/2x + log 3 x>l.4 x 4- S

Q.2 Q.4 Q.6 Q.8

x 1 / l o 8 x .logxlog 2 (2-x). log1/5 (2x 2 + 5 x + 1) 1 Q12. log [(x+6)/3] [log 2 {(x-l)/(2+x)}]>0 ~ < 1 is a l s o a

X tX

Find out the values of 'a' for which any solution of the inequality, solution of the inequality, x2 + (5 - 2 a) x < 10a.2

Q.14 Q.15

Solve the inequality log

N (x

-10x + 22) > 0 .

Find the set of values of'y' for which the inequality, 2 log0 5 y2 - 3 + 2 x log0 5 y2 - x2 > 0 is valid for atleast one real value of'x'.

ii Bansal Classes

Quadratic Equations

[9]

EXERQ. 1 Prove that the values of the function s i n xcos

CISE-IV3 do not lie from - & 3 for any real x. 13

sin3x cosx

Q.2 Q.3

The sum of all the real roots of the equation |x - 2| + |x - 2| - 2 = 0 is

2

.

[JEE '97,5] [JEE '97,2]

Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c & d denote the lengths of the sides of the quadrilateral, prove that : 2 < a2 + b2 + c2 + d2 < 4. [JEE '97,5] In a college of 300 students, every student reads 5 news papers & every news paper is read by 60 students. The number of news papers is: (A) atleast 30 (B) atmost 20 (C) exactly 25 (D) none ofthe above [JEE'98,2] If a, p are the roots of the equation x2 - bx + c = 0, then find the equation whose roots are, (a 2 + p2) (a 3 + p3) & a 5 p3 + a 3 p5 - 2a 4 p4. [REE'98,6]

Q.4

Q.5

Q.6(i) Let a + ip ; a, p e R, be a root of the equation x3 + qx + r = 0; q, r e R. Find a real cubic equation, independent of a & p, whose one root is 2 a . (ii) Find the values o f a & p , 0 < a , P< n/2, satisfying the following equation, cos a cos p cos (a + p) = - 1 / 8 . Q.7(i) In a triangle PQR, ZR = ~ . If tan ( ^ j & tan ax2 + bx + c = 0 (a*0) then: (A) a + b = c (B) b + c = a

[REE '99,3 + 6]

are the roots of the equation (C)a + c = b (D)b = c

(ii) If the roots of the equation x2 - 2ax + a2 + a - 3 = 0 are real & less than 3 then (A) a < 2 (B) 2 < a < 3 (C)3 2 b = a + c. [2 a+ (n - 1 )d] = ^ [a+/].

(vii)

GEOMETRIC PROGRESSION (GP): GP is a sequence of numbers whosefirstterm 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, ar2,ar3, ar4, is a GP with a as thefirstterm & r as common ratio. (i) nth term = ar n _ 1 (ii) (iii) (iv) (v) a(r n -l) Sum of the I n terms i.e. Sn = , if r * 1 . r-1 Sum of an infinite GP when | r | < 1 when n oo rn 0 if | r | < 1 therefore, >st

S ^ f l r K l ) . If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP. 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/r3, a/r, ar, ar3 & so on. If a, b, c are in GP => b2 = ac.

(vi)

Bansal Classes

Sequence & Progression

[2]

HARMONIC PROGRESSION (HP): A sequence is said to HP if the reciprocals of its terms are in AP. If the sequence a,, a2, a 3 ,...., an is an HP then l/a l5 l/a 2 ,...., l/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 isb, then111 term is tn =b + (n-l)(a-b) 2ac a + c

If a, b, c are in HP => b =

or = Tc

a

a - b d- c

.

MEANSARITHMETIC 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,, a 2 ,..., an is ; A = a ' + a 2 + a ^ ++a

" .

n-ARITHMETIC MEANS BETWEEN TWO NUMBERS : Ifa,b are any two given numbers & a,A15A2,.... ,An, b are inAP thenA,, A2, ...Anare then AM's between a & b . A1 = a + ^ - , A2 = a + ^ i l 2n + 1 ' n+ 1 '

,

'

,An= a +

n (b - a) n +1 b-a

= a + d,NOTE

=a+2d ,

, A = a + nd, where d = n

n+1

: Sum of n AM's inserted between a & b is equal to n times the single AM between a & bn i.e. X Ar = nA where A is the single AM between a & b. r= l

GEOMETRIC

MEANS:.

If a, b, c are in GP, b is the GM between a & c. b2 = ac, therefore b = Ja c ; a > 0, c > 0. n-GEOMETRIC MEANS BETWEEN a, b : If a, b are two given numbers & a, G}, G2, , Gn, b are in GP. Then Gj, G2, G 3 ,...., Gn are n GMs between a & b . G, = a(b/a)1/n+1, G2 = a(b/a)2/n+1, = ar , = ar 2 ,NOTEn

, Gn = a(b/a)n/n+1 = arn, where r = (b/a)1/n+1

: The product of n GMs between a & b is equal to the nth power of the single GM between a & b i.e. ^ G r =(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.

fa B ansa/ Classes

Sequence & Progression

[3]

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 + 5x2 + 7x3 + Here 1,3,5,.... are in AP& l,x,x 2 ,x 3 areinGP. Standart appearance of an Arithmetico-Geometric Series is Let Sn = a + (a + d)r + (a + 2 d) r2 + + [a + ( n - l ) d ] r""1 SUM TO INFINITY :If

|r|

1, and the runs scored in [JEE 2005,2]

,

fa Ban sal Classes

Sequence & Progression

[11]

ANSWER KEY

EXERCISE-IQl. 1 Q 2. x = 105, y = 10 Q3. ji= 14 Q 4. S = (7/81){10n+1 - 9n - 10} Q 5. 35/222 Q6. n(n+l)/2(n 2 + n + 1 ) Q 7. 27 Q 10. (14 n - 6)/(8 n + 23) Q 11. 1 Q 14. 9 Q 16. a = 5 , b = 8 , c = 12 Q 18. ( 8 , - 4 , 2, 8) Q 19. n2 Q20. (i) 2n+1 - 3 ; 2 n + 2 - 4 - 3 n (ii) n 2 + 4 n + l ; ( l / 6 ) n ( n + l ) ( 2 n + 1 3 ) + n Q 21. 120,30 Q 22. 6 , 3 Q 23. (i) sn = (1/24) - [l/{6(3n+ 1) (3n + 4) >] j s ^ l / 2 4 (ii) (l/5)n(n+ l)(n + 2)(n+ 3)(n + 4) (iii) n/(2n+1) (iv) Sn =2

1 2

1.3.5 (2n-l)(2n + l) 2.4.6 (2n)(2n + 2)

; Son = 1 "

Q 24. (a) (6/5) (6n - 1) (b) [n (n + 1) (n + 2)]/6

EXERCISE-IIQ6. 8problems, 127.5 minutes Q.8 C = 9 ; (3,-3/2 ,-3/5) Q 12. (iii) b = 4 , c = 6 , d = 9 OR b = - 2 , c = - 6 , D = - 18 Q 15. (a) a = 1, b = 9 O b = 1, a = 9 ; (b) a = 1 ; b = 3 or vice versa R

Q 23. (a) 1 Q 24. n = 38

(x + 1) (x + 2)

(x + n)

(b) 1 - (l + a i ) ( l + a 2 ) EXERCISE-III

(l + a n )

Q 25. 931

Ql.

|(2n-l)(n+l)2 Where a = 1 - x"1/3 & b = 1 - y"1/4 Q3. p < (1/3) ; y > -(1/27)

Q 2. S =

Q 4. - 3 , 7 7 Q 6. (a) C (b) B

Q 5. 8,24,72,216,648 Q 7. (a) B (b) D

Q 8. r = 1/9 ; n = 2 ; a = 144/180 OR r = 1/3 ; n = 4 ; a = 108 OR r = 1/81 ; n = 1 ; a = 160 Q9. (a) D Q l l . A.P. Q13. (a) A, (b)C, (c) D, (d)[(A,, A 2 , Q14. (a) D Q.16 B (b) A Q 10. A = 3 ; B = 8 Q 12. x = 2V2 andy = 3 A n ) (Hl5 H2, Q.18 n=7 H n )]

fa Ban sal Classes

Sequence & Progression

[11]

BANSAL CLASSESMATHEMATICS ITARGETIITJEE 2007

XI ( > Q, R, S) P

r

*

-yfi

i I

i n ca csi I Le n

CONTENTSKEY- CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY

KEYSTANDARD RESULTS:

CONCEPTS

1.

EQUATION OF A CIRCLE IN VARIOUS FORM : (a) The circle with centre (h, k) & radius 'r' has the equation; (x-h) 2 + ( y - k ) 2 =r 2 . (b) The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 with centre as : (-g, -f) & radius = ^ g 2 + f 2 - c . Remember that every second degree equation in x & y in which coefficient of x2 = coefficient of y2 & there is no xy term always represents a circle. If g2 + f 2 - c > 0 => real circle. 2 2 g + f - c = 0 => point circle. 2 2 g + f - c < 0 => imaginary circle. Note that the general equation of a circle contains three arbitrary constant s, g, f & c which corresponds to the fact that a unique circle passes through three non collinear points. (c) The equation of circle with (x,, y}) & , y2) as its diameter is :( X - X l ) ( x - x ^ + ( y - y i ) ( y - y 2 ) = 0.

Note that this will be the circle of least radius passing through 2.

, yj) & (xj, y2).

INTERCEPTS MADE BY A CIRCLE ON THE AXES : The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the co-ordinate axes are 2 vg2 - c & 2 ^ / f ^ c respectively. NOTE : circle cuts the x axis at two distinct points. => If g2 - c > 0 2 circle touches the x-axis. If g =c 2 => circle lies completely above or below the x-axis. If g r < > the line does not meet the circle i. e. passes out side the circle. = (ii) p = r o the line touches the circle. (iii) p < r o the line is a secant of the circle. (iv) p = 0 => the line is a diameter of the circle. PARAME TRIC EQUATIONS OF A CIRCLE: The parametric equations of (x - h)2 + (y - k)2 = r2 are: x = h + rcos9 ; y = k + rsin9 ; -TC < 0 < T where (h, k) is the centre, C r is the radius & 9 is a parameter. Note that equation of a straight line joining two point a & (3 on the circle x2 + y2 = a2 is a+B , . a+B a-B x cos + y sin = a cos - .

5.

(!%Bansal

Classes

Circles

[12]

6. (a)

TANGENT & NORMAL: The equation of the tangent to the circle x 2 +y 2 = a2 at its point (x t , y t ) is, xxj + y y, = a2. Hence equation of a tangent at (a cos a, a sina) is; x cos a + y sin a = a. The point of intersection of the tangents at the points P(a) and Q(f3) isacos 2a-fl

a+ff

asrn^-11a-p

.

a+6

(b) (c)

C S x - vcos - 2 O uo The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at its point (x t , yj) is XX, + yyj + g (x + X j ) + f (y + yj) + c = 0. y = mx + c is always a tangent to the circle x 2 +y 2 =a 2 if c2 = a2 (1 + m2) and the point of contact ( a 2 a 2\ m isc cJ

(d)

If a line is normal/orthogonal to a circle then it must pass through the centre of the circle. Using this fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at(x ; , y,) is

y_y = Z (X-Xl). ix,+g 7. (a) (b) (c) A FAMILY OF CIRCLES : The equation of the family of circles passing through the points of intersection of two circles St = 0 & S2 = 0is : S ! + K S 2 = 0 (K*-l). The equation of the family of circles passing through the point of intersection of a circle S = 0 & a line L = 0 is given by S+KL = 0. The equation of a family of circles passing through two given points (x}, y t ) & in the form: , y2) can be written

y i (x-x 1 )(x-x 2 ) + ( y - y 1 ) ( y - y 2 ) + K Y 1 = 0 where K is a parameter. i x2 y2 i(d) The equation of a family of circles touching afixedline y - yL = m (x - x}) at thefixedpoint (xL, yj) is (x - x t ) 2 + (y - yx)2 + K [y - yj - m (x - Xj)] = 0, where K is a parameter. In case the line through (xj, yj) is parallel to y - axis the equation of the family of circles touching it at (Xj, y t ) becomes (x - x,)2 + (y - y^ 2 + K (x - Xj) = 0. Also if line is parallel to x - axis the equation of the family of circles touching it at (xiYi) becomes ( x - X j ) 2 + ( y - y ^ 2 + K ( y - y i ) = 0. Equation of circle circumscribing a triangle whose sides are given by Lj = 0 ; L2 = 0 & L3 = 0 is given by; LjL2 + A. L2L3 + \x L3Lj = 0 provided co-efficient of xy = 0 & co-efficient of x2 = co-efficient of y2. Equation of circle circumscribing a quadrilateral whose side in order are represented by the lines Lj = 0, L2 = 0, L 3 = 0 & L4 = 0 is L,L 3 + A L 2 L 4 = 0 provided co-efficient of x2 = co-efficient of y2 and co-efficient of xy=0. LENGTH OF A TANGENT AND POWER OF A POINT : The length of a tangent from an external point (x t , y^ to the circle S = x2 + y2 + 2gx + 2fy + c = 0 is given by L= Jx 1 2 +y 1 2 +2gx,+2f 1 y+c = Js^. Square of length of the tangent from the point P is also called THE POWER OF POINT w.r.t. a circle. Power of a point remains constant w.r.t. a circle. Note that : power of a point Pis positive, negative or zero according as the point 'P'is outside, inside or on the circle respectively.

(e)

(f)

8.

(!%Bansal

Classes

Circles

[12]

9.

DIRECTOR CIRCLE

:

The locus ofthe point of intersection of two perpendicular tangents is called the DIRECTOR CIRCLE ofthe given circle. The director circle of a circle is the concentric circle having radius equal to V2 times the original circle.10. EQUATION OF THE CHORD WITH A GIVEN MIDDLE POINT:

The equation of the chord of the circle S = x2 + y2 + 2gx + 2fy + c = 0 in terms of its mid point M(xj, yj) is y - yj = - - 1 - (x - Xj). This on simplication can be put in the form Yj+r xxj + yyj + g (x + Xj) + f (y + yj) + c = Xj2 + Y!2 + 2gx} + 2fyj + c which is designated by T = S,. Note that : the shortest chord of a circle passing through a point 'M' inside the circle, is one chord whose middle point is M. 11. CHORD O F C O N T A C T : If two tangents PTj & PT2 are drawn from the point P (x ]; y t ) to the circle S = x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact T t T 2 is: xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.Xi 2

REMEMBER : (a) Chord of contact exists only if the point 'P' is not inside. 2LR (b) Length of chord of contact T, T2 =

(c)

RL3 Area ofthe triangle formed by the pair of the tangents & its chord of contact = |> 2+T2 Where R is the radius of the circle & L is the length of the tangent from (x1; y}) on S = 0.

(d) (e) (f)

Angle between the pair of tangentsfrom(xt, yj) = tan 1

' 2RL ^

vL2-*2/

where R=radius ; L = length of tangent. Equation of the circle circumscribing the triangle PTj T2 is: (x-Xj) (x + g) + ( y - y i ) (y + f) = 0. The joint equation of a pair of tangents drawn from the point A(xj,yj)to the circle x2 + y2 + 2gx + 2fy + c = 0 is : S S ^ T 2 . Where S s x2 + y2 +2gx + 2 f y + c ; Sj =Xj2 + y2 + 2gXj + 2fyj + c T= xxj + yyl + g(x + XJ) + f(y + y^ + c.POLE & POLAR:

12.

(i) (ii)

If through a point P in the plane of the circle, there be drawn any straight line to meet the circle in Q and R, the locus of the point of intersection of the tangents at Q & R is called the POLAR O F THE POINT P ; also P is called the POLE O F THE POLAR. The equation to the polar of a point P (xj, y,) w.r.t. the circle x2 + y2 = a2 is given by xxj + yy t =s a 2 , & if the circle is general then the equation of the polar becomes xx1 + yy,+g(x + Xj) + f (y + y^ + c = 0. Note that if the point (x t , yj) be on the circle then the chord of contact, tangent & polar will be represented by the same equation. Pole of a given line Ax + By + C = 0 w.r.t. any circle x + y = a is2 2 2

(iii)

Aa2

Ba 2 ^

(!%Bansal

Classes

Circles

[12]

(iv) (v) 13. (i) (ii) (iii)

If the polar of a point P pass through a point Q, then the polar of Q passes through P. Two lines L, & L2 are conjugate of each other if Pole of Lj lies on L2 & vice versa Similarly two points P & Q are said to be conjugate of each other if the polar of P passes through Q & vice-versa. COMMON TANGENTS TO TWO CIRCLES: Where the two circles neither intersect nor touch each other, there are FOUR common tangents, two of them are transverse & the others are direct common tangents. When they intersect there are two common tangents, both of them being direct. When they touch each other: (a) EXTERNALLY : there are three common tangents, two direct and one is the tangent at the point of contact. (b) INTERNALLY: only one common tangent possible at their point of contact. Length of an external common tangent & internal common tangent to the two circles is given by:L e Wd 2

(iv)

- (

r

. -

r

2 )

2

&

L

int=

A

/d2-(r

1 +

r2)

2

.

(v)

Where d = distance between the centres of the two circles. ^ & r2 are the radii of the two circles. The direct common tangents meet at a point which divides the line joining centre of circles externally in the ratio of their radii. Transverse common tangents meet at a point which divides the line joining centre of circles internally in the ratio of their radii. RADICAL AXIS & RADICAL CENTRE : The radical axis of two circles is the locus of points whose powers w.r.t. the two circles are equal. The equation of radical axis of the two circles S j = 0 & S2 = 0 is given; S 1 - S 2 = 0 i.e. 2 ( g 1 - g 2 ) x + 2 ( f 1 - f 2 ) y + (c 1 -c 2 ) = 0. NOTE THAT: If two circles intersect, then the radical axis is the common chord of the two circles. If two circles touch each other then the radical axis is the common tangent of the two circles at the common point of contact. Radical axis is always perpendicular to the line joining the centres of the two circles. Radical axis need not always pass through the mid point of the line joining the centres of the two circles. Radical axis bisects a common tangent between the two circles. The common point of intersection of the radical axes of three circles taken two at a time is called the radical centre of three circles. A system of circles, every two which have the same radical axis, is called a coaxal system. Pairs of circles which do not have radical axis are concentric. ORTHOGONALITY OFTWO CIRCLES: Two circles St = 0 & S 2 =0 are said to be orthogonal or said to intersect orthogonally if the tangents at their point of intersection include a right angle. The condition for two circles to be orthogonal is : 2 g[ g2 + 2 f j f 2 = Ci + C2 .

14.

(a) (b) (c) (d) (e) (f) (g) (h) 15.

Note : (a) Locus of the centre of a variable circle orthogonal to twofixedcircles is the radical axis between the twofixedcircles. (b) . If two circles are orthogonal, then the polar of a point 'P' onfirstcircle w.r.t. the second circle passes through the point Q which is the other end of the diameter through P . Hence locus of a point which moves such that its polars w.r.t. the circles S j = 0, S2 = 0 & S3 = 0 are concurrent in a circle which is orthogonal to all the three circles.

(!%Bansal

Classes

Circles

[12]

EXERCISE-IQ 1. Q 2. Q 3. Q 4. Find the equation of the circle circumscribing the triangle formed by the lines ; x + y = 6,2x + y = 4& x + 2y = 5, without finding the vertices of the triangle. If the lines a, x + b, y + Cj = 0 & a2X + b2y + c 2 =0 cut the coordinate axes in concyclic points. Prove that aj bj b2. One of the diameters of the circle circumscribing the rectangle ABCD is4y=x + 7. If A& B are the points (-3,4) & (5,4) respectively. Then find the area of the rectangle. Lines 5x + 12y - 10 = 0 & 5 x - 12y-40 = 0 touch a circle Cj of diameter 6. If the centre of Cj lies in the first quadrant, find the equation of the circle C2 which is concentric with C} & cuts interceptes of length 8 on these lines. Find the equation of the circle inscribed in a triangle formed by the lines 3x + 4y = 12; 5x + 12y = 4 & 8y = 15x + 10 withoutfindingthe vertices of the triangle. Find the equation of the circles passing through the point (2,8), touching the lines 4x - 3y - 24 = 0 & 4x + 3y - 42 = 0 & having x coordinate of the centre of the circle less than or equal to 8. Find the equation of a circle which is co-axial with circles 2x2 + 2y2 - 2x + 6y - 3 = 0 & x2 + y2 + 4x + 2y + 1 = 0. It is given that the centre of the circle to be determined lies on the radical axis of these two circles. Let A be the centre of the circle x2 + y2 - 2x - 4y - 20 = 0. Suppose that the tangents at the points B(1,7) & D(4,-2) on the circle meet at the point C. Find the area of the quadrilateral ABCD. The radical axis of the circles x2 + y2 + 2gx + 2fy + c = 0 and 2x2 + 2y2 + 3x + 8y + 2c = 0 touches the circle x2 + y2 + 2x - 2y + 1 = 0. Show that either g = 3/4 or f = 2 .

Q 5. Q 6. Q 7.

Q 8. Q 9.

Q. 10 Find the equation of the circle through the points of intersection of circles x 2 +y 2 -4x-6y-12=0 and x2 + y2 + 6x + 4y - 12 = 0 & cutting the circle x2 + y2 - 2x - 4 = 0 orthogonally. Q 11. Consider a curve ax 2 +2 hxy + by2 = 1 and a point P not on the curve. Aline is drawnfromthe point P intersects the curve at points Q & R. If the product PQ. PR is independent of the slope of the line, then show that the curve is a circle. Q 12, Find the equations of the circles which have the radius Vl3 & which touch the line 2x-3y+1 = 0at(l, 1). Q 13. A circle is described to pass through the origin and to touch the lines x = 1, x + y = 2. Prove that the radius of the circle is a root of the equation ^3 - 2-j2j t 2 - 2 j 2 t + 2 = 0. Q 14. The centre of the circle S = 0 lie on the line 2x-2y + 9 = 0 & S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through twofixedpoints &findtheir coordinates. Q 15. Show that the equation x2 + y2 - 2x - 2 Ay - 8 = 0 represents, for different values of A, a system of circles passing through twofixedpoints A B on the x - axis, andfindthe equation of that circle of the system the tangents to which at A & B meet on the line x + 2y + 5 = 0.

(!%Bansal

Classes

Circles

[12]

Q 16. Find the equation ofthe circle which passes through the point (1, 1) & which touches the circle x2 + y2 + 4x - 6y - 3 = 0 at the point (2, 3) on it. Q 17. Find the equation of a circle which touches the lines 7x2 - 18xy + 7y2 = 0 and the circle x2 + y2 - 8x- 8y = 0 and is contained in the given circle. Q 18. Find the equation of the circle which cuts the circle x2 + y2 -14x - 8y + 64 = 0 and the co-ordinate axes orthogonally. Q 19. Obtain the equations ofthe straight lines passing through the point A(2,0)&making 45 angle with the tangent at A to the circle (x + 2)2 + (y - 3)2 = 25. Find the equations of the circles each of radius 3 whose centres are on these straight lines at a distance of 5 V2 from A. Q 20. Find the equations of the circles whose centre lie on the line 4x + 3y - 2 = 0 & to which the lines x + y + 4 = 0 & 7 x - y + 4 = 0 are tangents. Q 21. Find the equations to the four common tangents to the circles x2 + y2 = 25 and (x-12) 2 + y2 = 9. Q 22. If 4/2 - 5m2 + 6/ + 1 = 0. Prove that /x + my + 1 = 0 touches a definite circle. Find the centre & radius of the circle. Q 23. Find the condition such that the four points in which the circle x2 + y2 + ax + by + c = 0 and x2 + y2 + a'x + b'y + c' = 0 are intercepted by the straight lines Ax + By + C = 0 & A'x + B'y + C' = 0 respectively, lie on another circle. Q 24. Show that the equation of a straight line meeting the circle x2 + y2 = a2 in two points at equal distances d2 'd' from a point (x}, yj) on its circumference is xxt + yy} - a + = 0.2

Q 25. If the equations of the two circles whose radii are a & a' be respectively S = 0 & S - 0, then prove that S S' the circles + = 0 will cut each other orthogonally, a a Q 26. Let a circle be given by 2x (x - a) + y (2y - b) = 0, (a * 0, b ^ 0). Find the condition on a & b if two chords, each bisected by the x-axis, can be drawn to the circle from

v ^j

Q 27. Prove that the length of the common chord of the two circles x2 + y2 = a2 and (x - c)2 + y2 = b2 is -7(a+b+c)(a-b+c)(a+b-c)(-a+b+c) . c Q 28. Find the equation of the circle passing through the points A (4,3) & B (2, 5) & touching the axis of y. Also find the point P on the y-axis such that the angle APB has largest magnitude. Q 29. Find the equations of straight lines which pass through the intersection of the lines x - 2y - 5 = 0, 7x + y = 50 & divide the circumference of the circle x2 + y2 = 100 into two arcs whose lengths are in the ratio 2:1. Q 30. Find the equation of the circle which cuts each of the circles x2 + y2 = 4, x2 + y 2 - 6 x - 8 y + 10=0 & x2 + y2 + 2 x - 4 y - 2 = 0 at the extremities of a diameter.

(!%Bansal Classes

Circles

[12]

EXER CISE-IIQ 1. A point moves such that the sum of the squares of its distancefromthe sides of a square of side unity is equal to 9. Show that the locus is a circle whose centre coincides with centre of the square. Find also its radius. A triangle has two of its sides along the coordinate axes, its third side touches the circle x2 + y2 - 2ax - 2ay + a2 = 0. Prove that the locus of the circumcentre of the triangle is : a2 - 2a (x + y) + 2xy = 0. A variable circle passes through the point A (a, b) & touches the x-axis; show that the locus ofthe other end of the diameter through A is (x - a)2 = 4by. (a) Find the locus of the middle point of the chord of the circle, x2 + y2 + 2gx + 2fy + c = 0 which subtends a right angle at the point (a, b). Show that locus is a circle. Let S= x 2 +y 2 + 2gx+2fy+c=0 be a given circle. Find the locus of the foot of the perpendicular drawnfromthe origin upon any chord of S which sustends a right angle at the origin.

Q 2.

Q 3. Q 4.

(b) Q 5.

A variable straight line moves so that the product ofthe perpendiculars on itfromthe twofixedpoints (a, 0) & (- a, 0) is a constant equal to c2 . Prove that the locus ofthe feet of the perpendiculars from each of these points upon the straight line is a circle, the same for each. Showthat the locus of the centres of a circle which cuts two given circles orthogonally is a straight line & hence deduce the locus of the centers of the circles which cut the circles x2 + y2 + 4x - 6y + 9 = 0 & x2 + y2 - 5x + 4y + 2 = 0 orthogonally . Afixed circle is cut by afamilyofcirclespassingthroughtwofixedpointsA(x1,y1)andB(x2,y2). Show that the chord ofintersection ofthefixedcircle with any one ofthe circles offamily passes through afixedpoint. The sides of a variable triangle touch the circle x2 + y2 = a2 and two of the vertices are on the line y2 - b2 = 0 (b > a > 0) . Show that the locus of the third vertex is; (a2 - b2) x2 + (a2 + b2) y2 = (a(a2 + b2))2. Show that the locus of the point the tangentsfromwhich to the circle x2 + y2 - a2 = 0 include a constant angle a is (x2 + y2 - 2a2)2 tan2 a = 4a2 (x2 + y2 - a 2 ).

Q 6.

Q 7.

Q 8.

Q 9.

Q 10. ' O' is afixedpoint & P a point which moves along afixedstraight line not passing through O; Q is taken on OP such that OP. OQ=K(constant) . Prove that the locus of Q is a circle. Explain how the locus of Q can still be regarded as a circle even if thefixedstraight line passes through 'O'. Q 11. P is a variable point on the circle with centre at C. CA & CB are perpendiculars from C on x-axis & y-axis respectively. Show that the locus of the centroid of the triangle PAB is a circle with centre at the centroid of the triangle CAB & radius equal to one third of the radius ofthe given circle. Q 12. A(-a, 0) ; B(a,0) arefixedpoints. C is a point which divides AB in a constant ratio tana. If AC & CB subtend equal angles at P, prove that the equation ofthe locus of P is x2 + y2 + 2ax sec2a + a2 = 0.

Bansal Classes

Circles

Q 13. The circle x2 + y2 +2ax- c2 = 0 and x2 + y2 + 2bx- c2 = 0 intersect at Aand B. Aline through Ameets one circle at P and a parallel line through B meets the other circle at Q. Show that the locus of the mid point of PQ is a circle. Q 14. Find the locus of a point which is at a least distance from x2 + y2 = b2 & this least distance is equal to its distance from the straight line x = a. Q 15. The base of a triangle is fixed. Find the locus of the vertex when one base angle is double the other. Assume the base of the triangle as x-axis with mid point as origin & the length ofthe base as 2a. Q 16. An isosceles right angled triangle whose sides are 1, 1, V2 lies entirely in thefirstquadrant with the ends of the hypotenuse on the coordinate axes. If it slides prove that the locus of its centroid is (3x-y) 2 + ( x - 3 y ) 2 = - 3 ^ . Q 17. The circle x2 + y2 - 4x - 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the coordinate axes. The locus of the circumcentre ofthe triangle is: x + y - xy + K ^ / P + y 1 = 0. Find K.*

Q 18. Find the locus of the point ofintersection of two perpendicular straight lines each of which touches one of the two circles (x - a)2 + y2 = b 2 , (x + a)2 + y2 = c2 and prove that the bisectors of the angles between the straight lines always touch one or the other of two otherfixedcircles. Q.19 Find the locus ofthe mid point of the chord of a circle x2 + y2 = 4 such that the segment intercepted by the chord on the curve x2 - 2x - 2y = 0 subtends a right angle at the origin. Q 20. TheendsAB ofafixedstraightlineoflength'a'&endsA'&B'ofanotherfixedstraightlineoflength 'b' slide upon the axis ofx&the axis ofy (one end on axis of x& the other on axis of y). Find the locus of the centre of the circle passing through A, B, A' & B'. Q 21. The foot of the perpendicularfromthe origin to a variable tangent of the circle x 2 +y 2 - 2x = 0 is N. Find the equation ofthe locus ofN. Q 22, Find the locus of the mid point of all chords of the circle x 2 +y 2 - 2x - 2y = 0 such that the pair of lines joining (0,0) & the point of intersection of the chords with the circles make equal angle with axis of x. Q 23. P (a) & Q (p) are the two points on the circle having origin as its centre & radius 'a' & AB is the diameter along the axis of x. If a - p = 2 y, then prove that the locus of intersection of AP & BQ is x2 + y2 - 2 ay tany = a2. Q 24. Show that the locus of the harmonic conjugate of a given point P (xl5 y t ) w.r.t. the two points in which any line through P cuts the circle x2 + y2 = a2 is xxj + yy} = a2. Q 25. Find the equation of the circle which passes through the origin, meets the x-axis orthogonally & cuts the circle x2 + y2 = a2 at an angle of 45.

(!%Bansal

Classes

Circles

[12]

EXERCISE-IIIQ.l (a) (b) The intercept on the line y=x bythe circle x 2 + y 2 - 2 x = 0 isAB . Equation ofthe circle with AB as a diameter is . The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x - 6y + 9 sin2a + 13 cos2a = 0 is 2 a. The equation of the locus of the point P is : (A) x2 + y2 + 4 x - 6 y + 4 = 0 (B) x2 + y2 + 4x - 6y - 9 = 0 2 2 (C) x + y + 4 x - 6 y - 4 = 0 (D) x2 + y2 + 4 x - 6y + 9 = 0 Find the intervals of values of a for which the line y + x = 0 bisects two chords drawn from a ' l + V2a 1-V2a 1 rpoint , - to the circle; 2x 2 +2y 2 - (1+V2 a) x - (1 - V2 a) y = 0.V2

(c)

J

[JEE'96, 1+1+5] Q.2 Q.3 Atangent drawnfromthe point (4,0)tothecircle x2+y2 = 8 touches it at apointAin the first quadrant. Find the coordinates of the another point B on the circle such that AB = 4. [ REE '96, 6 ] (a) (b) The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to the circle x2 + y2 = 1 pass through the point . Let C be any circle with centre (o, 42 ) Prove that at the most two rational point can be there on C. (A rational point is a point both of whose co-ordinate are rational numbers). [JEE '97, 2+5] 2 2 2 2 The number of common tangents to the circle x + y = 4 & x + y - 6x - 8y = 24 is : (A) 0 (B) 1 (C) 3 (D) 4 Cj & C2 are two concentric circles, the radius of C2 being twice that of Cj. From a point P on C2, tangents PA & PB are drawn to C}. Prove that the centroid of the triangle PAB lies on C j. [ JEE '98, 2 + 8 ]

Q.4

(a) (b)

Q. 5 Q.6

Find the equation of a circle which touches the line x + y = 5 at the point (-2, 7) and cuts the circle x2 + y2 + 4x - 6y + 9 = 0 orthogonally. [ REE '98, 6 ] (a) If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq q) are bisected by the x-axis, then: (A) p2 = q2 (B) p2 = 8q2 (C) p2 < 8q2 (D)p2>8q2 Let Lj be a straight line through the origin and L2 be the straight line x+y = 1. If the intercepts made by the circle x2 + y2 - x + 3y = 0 on L} & L2 are equal, then which of the following equations can represent Lj? (A) x + y = 0 (B)x-y = 0 ( C ) x + 7y = 0 (D)x-7y = 0 Let Tj, T2 be two tangents drawnfrom( - 2,0) onto the circle C: x2 + y2 = 1. Determine the circles touching C and having T t , T2 as their pair of tangents. Further,findthe equations of all possible common tangents to these circles, when taken two at a time. [ JEE '99, 2 + 3 + 10 (out of200) ] 2 2 The triangle PQR is inscribed in the circle, x + y = 25. IfQ and Rhave co-ordinates (3,4) & ( - 4,3) respectively, then Z QPR is equal to : (A) (b) (B) f (C) f (D) f If the circles, x2 + y2 + 2x + 2ky + 6 = 0 & x2 + y2 + 2 ky + k = 0 intersect orthogonally, then 'k' is: (A) 2 or - | (B) - 2 or - | (C) 2 or | (D) - 2 or \ [ JEE '2000 (Screening) 1 + 1 ]

(b)

(c)

Q.7

(a)

^Bansal Classes

Circles

[10]

Q.8

(a) (b)

Extremities of a diagonal of a rectangle are (0,0) & (4,3). Find the equation ofthe tangents to the circumcircle of a rectangle which are parallel to this diagonal. A circle of radius 2 units rolls on the outerside of the circle, x2 + y2 + 4 x = 0, touching it externally. Find the locus ofthe centre ofthis outer circle. Alsofindthe equations of the common tangents of the two circles when the line joining the centres ofthe two circles makes on angle of 60 with x-axis. [REE '2000 (Mains) 3 + 5] Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. IfPS and RQ intersect at a point X on the circumference of the circle then 2r equals

Q.9

(a)

[JEE'2001 (Screening) 1 out of 35] (b) Let 2x + y - 3xy = 0 be the equation of a pair of tangents drawn from the origin 'O' to a circle of radius 3 with centre in thefirstquadrant. IfAis one ofthe points of contact,findthe length of OA [JEE '2001 (Mains) 5 out of 100] Find the equation of the circle which passes through the points of intersection of circles x2 + y2 - 2x - 6y + 6 = 0 and x2 + y2 + 2x - 6y + 6 = 0 and intersects the circle x2 + y2 + 4x + 6y + 4 0 orthogonally. [ REE '2001 (Mains) 3 out of 100 ] Tangents TP and TQ are drawnfroma point T to the circle x2 + y2 = a2. If the point T lies on the line px + qy = r,findthe locus of centre of the circumcircle of triangle TPQ. [ REE '2001 (Mains) 5 out of 100 ] If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the length of PQ is (A) 4 (B)2V5 (C)5 (D)3V5 If a > 2b > 0 then the positive value of m for which y = mx-b-Jl + m is a common tangent to x2 + y2 = b2 and (x - a)2 + y2 = b2 is 2b v'a2 - 4 b 2 2b(C)2 2

Q. 10 (a)

(b)

Q. 11 (a)

(b)

b

W - z T -

T-Tib [ JEE '2002 (Scr)3 + 3 out of270]

Q .12 The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle x2 + y2 2x 6y + 6 = 0 (A)l (B)2 (C)3

(D)V3

[JEE'2004 (Scr)] Q.13 Line2x + 3y+ 1 = 0 is a tangent to a circle at (1,-1). This circle is orthogonal to a circle which is drawn having diameter as a line segment with end points (0,-1) and (- 2,3). Find equation of circle. [JEE'2004, 4 out of 60] 2 2 Q.14 A circle is given by x + (y -1) = 1, another circle C touches it externally and also the x-axis, then the locus of its centre is (A){(x,y):x 2 = 4y}u{(x,y):y = 0} (B) {(x, y) : x2 + (y - 1)2 = 4} u {x, y): y = 0} (C) {(x, y): x2 = y} VJ {(0, y): y = 0} (D*) {(x, y): x2 = 4y} u {(0, y): y = 0} [JEE '2005 (Scr)]

(!%Bansal

Classes

Circles [12]

ANSWERQ 1. x + y - 17x - 19y + 50 = 0 Q 5. x2 + y2 - 2x - 2y + 1 - 0 Q 7. 4x2 + 4y2 + 6x + lOy- 1 = 02 2

KEYQ 4. x2 + y2 - lOx-4y + 4 = 0f

EXERCISE-IQ 3. 32 sq. unit Q 6. centre (2 ,3), r = 5 ; centre Q 8. 75 sq.units182 "2 0 5

Q 10. x2 + y2 + 16x+ 14y- 12 = 0 Q 14. ( - 4, 4) ; f _ 1 V

Q 12. x2 + y2 - 6x + 4y = 0 OR x2 + y2 + 2 x - 8 y + 4 = 0

Q 15. x2 + y2 - 2x - 6y - 8 = 0 Q 16. x2 + y2 + x - 6y + 3 = 0 2 2 Q 17. x + y - 12x -12y + 64 = 0 Q18. x2 + y2 = 64 Q 19. x - 7y = 2, 7x + y = 14 ; (x - l) 2 + (y - 7)2 = 32 ; ( x - 3 ) 2 + (y + 7)2 = 32 ; (x - 9)2 + (y - l) 2 = 32 ; (x + 5)2 + (y + I)2 = 32 Q 20. x2 + y2 - 4x + 4y = 0 ; x2 + y2 + 8x - 12y + 34 = 0 Q 21. 2 x - V 5 y - 1 5 = 0, 2 x + V 5 y - 1 5 = 0 , x - ^ I J y - 3 0 = 0, x + ^/35 y - 3 0 = 0 Q 22. Centre = (3, 0), 1 (radius) = S =2 2 2 2

I

2' 2 J

a-a' b-b' Q 23. A B C A' B'

c-c' C

Q 26. (a2 > 2b2)

Q 28. x + y - 4x - 6y + 9 = 0 OR x + y - 2 0 x - 2 2 y + 1 2 1 =0, P(O,3),0 = 45 Q 29. 4x - 3y - 25 = 0 OR 3x + 4 y - 2 5 = 0 Q 30. x2 + y 2 - 4 x - 6 y - 4 = 0

EXER CISE-IIQl. r = 2 Q 4. (b) 2(x + y ) + 2gx + 2fy + c = 0 Q 6. 9 x - 10y + 7 = 0 Q 10. aline 2 2 2 2 Q 14. y = (b + a) (b + a - 2x) OR y = (b - a) (b - a + 2x) Q 15. 3 x - y 2 a x - a 2 = 0 Q 17. K = 1 Q 18. { cy + b(x + a)}2 + { - by c(x - a)}2 = (a2 - x2 - y2)2 Q 19. x2 + y2 - 2x - 2y = 0 Q 20. (2ax - 2by)2 + (2bx - 2ay)2 = (a2 - b2)2 Q 21. (x2 + y2 - x ) 2 = x2 + y2 Q 22. x + y = 2 Q 25. x2 + y2 aV2 x = 02 2

EXER CISE-II I( n2 f if 1 Q.l (a) x + yI = - , (b) D, (c) ( - 00, -2) u (2,00) Q.2 (2, -2) or (-2, 2) Q.3 (a) (1/2, 1/4) 2J 2 V 2 y V 2 4- 2 2 + y2 . 7 x - l l y +38 = 0 Q.4 (a) B Q.5 x + Q.6 (a) D (b) B, C (c)Cl

: ( x - 4 ) 2 + y2 = 9 ; c2 :

+

+y2= I

common tangent between c & Cj : Tj = 0; T2 = 0 and x - 1 = 0 ; common tangent between c & c2 : T t = 0; T2 = 0 and x + 1 = 0 ; common tangent between c, & c2 : T, = 0 ; T, = 0 and y = -jL= ^x + where ^ : x - V 3 y + 2 = 0 and T2 : x + v / 3 y + 2 = 0 Q.7 Q.8 (a) C (b) A (a) 6 x - 8 y + 25 = 0 & 6 x - 8 y - 2 5 = 0 (b) x2 + y2 + 4 x - 12= 0, T. : V3x-y + 2^3 +4 = 0, T2 : V 3 x - y + 2V3-4 = 0(D.C.T.) T 3 : x + V 3 y - 2 = 0, T4 : x + V3y + 3 = 0 (T.C.T.) Q.9 (a) A ; (b).OA=3(3 +V10) Q.ll (a) C ; (b). A Q.12 C Q.IO (a) x2 + y2 + 14x-6y + 6 = 0 ; (b) 2px + 2qy = r Q.13 2x2 + 2 y 2 - 1 0 x - 5 y + 1 = 0

(!%Bansal Classes

Circles

[12]

BANSAL CLASSESMATHEMATICSTARGET IIT JEE 2007 XI(PQRS)

PERMUTATION AND COMBINATION

CONTENTSKEY- CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY

KEYDEFINITIONS: 1. PERMUTATION 2.

CONCEPTS

: Each ofthe arrangements in a definite order which can be made by taking some or all ofa number of things is called a P E R M U T A T I O N .COMBINATION:

Each of the groups or selections which can be made by taking some or all of a number ofthings without reference to the order ofthe things in each group is called a C O M B I N A T I O N .FUNDAMENTAL PRINCIPLE OF COUNTING:

If an event can occur in'm' different ways, following which another event can occur in'/?' different ways, then the total number of different ways of simultaneous occurrence of both events in a definite order is m x n. This can be extended to any number of events.RESULTS:

(i)

A Useful Notation :n! = n ( n - l ) ( n - 2 ) 3. 2. 1 ; n! =n. ( n - 1) ! n 0! = 1! = 1 ; (2n)! = 2 . n ! [1. 3. 5. 7...(2n- 1)] Note that factorials of negative integers are not defined. If nPr denotes the number of permutations of n different things, taking r at a time, then n! n Pr = n (n - 1) (n - 2) ( n - r + 1)= ( n _ r ) j Note that, nPn = n !. If nCr denotes the number of combinations of n different things taken r at a time, then n n! p n Cr = . = L where r < n ; n e N and r e W . r!(n-r)! rj The number ofways in which (m+n) different things can be divided into two groups containing m & n things respectively is : ( m + n ) - if m=n, the groups are equal & in this case the number of subdivision m!n! ; for in any one way it is possible to interchange the two groups without obtaining a new n!n!2! distribution. However, if 2n things are to be divided equally between two persons then the number of (2n)! ways = n!n! Number ofways in which (m + n + p) different things can be divided into three groups containing m, n & p things respectively is m! n!p!5

(ii)

(iii)

(iv)

is

(v)

m ^ n ^ p./A \ I

If m = n = p then the number of groups^

(vi)

(3n)! However, if 3 n things are to be divided equally among three people then the number of ways = . (n!) The number ofpermutations ofn things taken all at a time whenp of them are similar & of one type, q of them are similar & of another type, r of them are similar & of a third type & the remaining I n - (p + q + r) are all different is: -. p!q!r! The number of circular permutations ofn different things taken all at a time is; (n-1)!. Ifclockwise& anti-clockwise circular permutations are considered to be same, then it is Note : Number of circular permutations ofn things when p alike and the rest different taken all at a time distinguishing clockwise and anticlockwise arrangement is^. p!

n!n!n!3!'

(vii)

(!iBansalClasses

Permutation and Combination

[7]

(viii)

n

Given n different objects, the number of ways of selecting atleast one of them is , C[ + nC2 + nC3 + + nCn = 2n - 1. This can also be stated as the total number of combinations of n distinct things.

(ix)

Total number of ways in which it is possible to make a selection by taking some or all out of p+q+r+ things, where p are alike of one kind, q alike of a second kind, r alike of third kind & so on is given by: (p+ l ) ( q + l ) ( r + 1) -1. Number of ways in which it is possible to make a selection ofm + n + p = N things, where p are alike of one kind, m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr in the expansion of(1 + X + X 2 + + X ? ) ( 1 + X + X2+ + X m ) (1 + X + X2 + +xn).

(x)

Note : Remember that coefficient ofx r in (1 -x) _n = n+r_1 C r (n e N). For example the number ofways in which a selection of four letters can be madefromthe letters of the word PROPORTION is given by coefficient of x4 in (1 + x + x2 + x3) (1 + x + x2) (1 + x + x2) (1 + x) (1 + x) (1 + x). (xi) (xii) (xiii) Number ofways in which n distinct things can be distributed to p persons if there is no restriction to the number of things received by men = pn. Number of ways in which n identical things may be distributed among p persons if each person may receive none, one or more things is; n+p_1 Cn. a. c. (xiv) (xv)n n n

C r = n C n _ r ; n C 0 = nCn = 1 c r + nCr_! = n+1Cr

;

b.

n

Cx = nCy =>x = y orx + y = n

Cr is maximum if: (a) r = y if n is even, (b) r = o r

- y - if n is odd.

Let N = p8- qb- r- where p, q, r. are distinct primes & a, b, c are natural numbers then: (a) The total numbers of divisors ofN including 1 & N is = (a + 1 )(b + 1 )(c + 1) (b) (c) The sum ofthese divisors is - (p + p1 + p2+.... +p a )(q+q 1 + q2+.... + qb) (r +r 1 + r 2 +....+r).... Number of ways in which N can be resolved as a product of two . factors is (d) 4(a + l)(b + l)(c +1).... if N is not a perfect square j [(a + l)(b + l)(c +1).... +1] if N is a perfect square

(xvi)

Number of ways in which a composite number N can be resolved into two factors which are relatively prime (or coprime) to each other is equal to 2 n_I where n is the number of different prime factors inN. [ Refer Q.No.28 of Ex-I ] Grid Problems and tree diagrams.

DEARRANGEMENT: Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own envelope is = n!1 1

2!

+ + 3! 4!

1

/ IV +(-1) . v ' n!

1

(xvii) S ome times studentsfindit difficult to decide whether a problem is on permutation or combination or both. Based on certain words / phrases occuring in the problem we can fairly decide its nature as per the following table:PROBLEMS OF COMBINATIONS PROBLEMS OF PERMUTATIONS

Selections, choose Distributed group is formed Committee Geometrical problems

Arrangements Standing in a line seated in a row IB problems on digits Problems on letters from a word

(!i Bansal Classes

Permutation and Combination

[7]

EXERCISE-IQ.l (a) (b) (c) In how many ways 8 persons can be seated on a round table If two of them (say Aand B) must not sit in adjacent seats. If 4 of the persons are men and 4 ladies and if no two men are to be in adjacent seats. If 8 persons constitute 4 married couples and if no husband and wife, as well as no two men, are to be in adjacent seats? A box contains 2 white balls, 3 black balls & 4 red balls. In how many ways can 3 balls be selected from the box if atleast 1 black is to be included in the draw ? How manyfivedigits numbers divisible by 3 can be formed using the digits 0, l,2,3,4,7and8 ifeach digit is to be used atmost once. During a draw of lottery, tickets bearing numbers 1, 2, 3, , 40, 6 tickets are drawn out & then arranged in the descending order of their numbers. In how many ways, it is possible to have 4th ticket bearing number 25. In how many ways can a team of 6 horses be selected out of a stud of 16, so that there shall always be 3 out of AB C A' B' C ' , but never A A ' , B B' or C C' together. 5 boys & 4 girls sit in a straight line. Find the number ofways in which they can be seated if 2 girls are together & the other 2 are also together but separatefromthefirst2. In how many ways can you divide a pack of 52 cards equally among 4 players. In how many ways the cards can be divided in 4 sets, 3 of them having 17 cards each & the 4th with 1 card. Find the number ofways in which 2 identical kings can be placed on an 8 x 8 board so that the kings are not in adjacent squares. How many on n x m chessboard? The Indian cricket team with eleven players, the team manager, the physiotherapist and two umpires are to travelfromthe hotel where they are staying to the stadium where the test match is to be played. Four of them residing in the same town own cars, each a four seater which they will drive themselves. The bus which was to pick them up failed to arrive in time after leaving the opposite team at the stadium. In howmany ways can they be seated in the cars ? In how many ways can they travel by these cars so as to reach in time, if the seating arrangement in each car is immaterial and all the cars reach the stadium by the same route.

Q.2

Q.3

Q.4

Q.5

Q.6 Q.7

Q. 8 Q.9

Q.IO How many 4 digit numbers are there which contains not more than 2 different digits? Q.ll An examination paper consists of 12 questions divided into parts A & B. Part-A contains 7 questions & Part - B contains 5 questions. A candidate is required to attempt 8 questions selecting atleast 3fromeach part. In how many maximum ways can the candidate select the questions ? A crew of an eight oar boat has to be chosen out of 11 menfiveof whom can row on stroke side only, four on the bow side only, and the remaining two on either side. How many different selections can be made?

Q.12

faBansal Classes

Permutation and Combination

[4]

Q.13

There are p intermediate stations on a railway line from one terminus to another. In how many ways can a train stop at 3 of these intermediate stations if no 2 of these stopping stations are to be consecutive ? The straight lines l x , l2 & /3 are parallel & lie in the same plane. A total of m points are taken on the line /j, n points on l2 & k points on /3. How many maximum number oftriangles are there whose vertices are at these points?

Q.14

Q. 15 Prove that if each of m points in one straight line be joined to each of n in another by straight lines terminated by the points, then excluding the given points, the lines will intersect mn(m - l)(n -1) times. 4 Q.16 Afirmof Chartered Accountants in Bombay has to send 10 clerks to 5 different companies, two clerks in each. Two of the companies are in Bombay and the others are outside. Two of the clerks prefer to work in Bombay while three others prefer to work outside. In how many ways can the assignment be made if the preferences are to be satisfied. Q.17 Find the number of words each consisting of 3 consonants & 3 vowels that can be formed from the letters of the word "Circumference". In how many of these c's will be together. Q.18 There are n straight lines in a plane, no 2 of which parallel, & no 3 pass through the same point. Their point of intersection are joined. Show that the number of fresh lines thus introduced is

n(n-l)(n-2)(n-3)8 Q. 19 Find the number of distinct throws which can be thrown with 'n' six faced normal dice which are indistinguishable among themselves. Q . 20 There are 2 women participating in a chess tournament. Every participant played 2 games with the other participants. The number of games that the men played between themselves exceeded by 66 as compared to the number of games that the men played with the women. Find the number of participants & the total numbers of games played in the tournament. Q.21 Find the number of ways 10 apples, 5 oranges & 5 mangoes can be distributed among 3 persons, each receiving none, one or more. Assume that the fruits ofthe same species are ail alike. Q.22 All the 7 digit numbers containing each of the digits 1,2,3,4, 5, 6,7 exactly once, and not divisible by 5 are arranged in the increasing order. Find the (2004)th number in this list. Q. 23 (a) (b) (c) How many divisors are there of the number x = 21600. Find also the sum of these divisors. In how many ways the number 7056 can be resolved as a product of 2 factors. Find the number of ways in which the number 300300 can be split into 2 factors which are relatively prime.

Q. 24 There are 5 white, 4 yellow, 3 green, 2 blue & 1 red ball. The balls are all identical except for colour. These are to be arranged in a line in 5 places. Find the number of distinct arrangements.

(!i Bansal Classes

Permutation and Combination

[7]

Q.25

(1) 00 (iii) (iv) (v)

Prove that: nPr = n"1Pr + r. n"1Pr_1 If 20 C r+2 = 20 C 2r _ 3 find 12 C r Find the ratio 20Cr to 25Cr when each of them has the greatest value possible. Prove that n_1 C3 + C4 > nC3 if n > 7. Find r if 15C3r = 15Cr+3

Q. 26 In a certain town the streets are arranged like the lines of a chess board. There are 6 streets running north & south and 10 running east & west. Find the number ofways in which a man can gofromthe north-west corner to the south-east corner covering the shortest possible distance in each case. Q.27 A train goingfromCambridge to London stops at nine intermediate stations. 6 persons enter the train during the journey with 6 different tickets of the same class. How many different sets ofticket may they have had? Q.28 How many arrangements each consisting of 2 vowels & 2 consonants can be made out of the letters of the word4 DEVASTATION' ? Q. 29 0 If'ri things are arranged in circular order, then show that the number ofways of selecting four of the things no two ofwhich are consecutive is n(n - 5) (n - 6) (n - 7) 4! If the 'ri things are arranged in a row, then show that the number of such sets of four is (n-3)(n-4)(n-5)(n-6) 4!

(ii)

Q. 3 0 There are 20 books on Algebra & Calculus in our library. Prove that the greatest number of selections each ofwhich consists of 5 books on each topic is possible only when there are 10 books on each topic in the library.

EXERCISE-IIQ. 1 There are 5 balls of different colours & 5 boxes of colours same as those of the balls. The number of ways in which the balls, 1 in each box could be placed such that a ball does not go to the box ofits o^/n colour. How many integral solutions are there for the equation ;x + y + z + w = 29 when x > 0, y > 1, z > 2 & w>0. There are counters available in 7 different colours. Counters are all alike except for the colour and they are atleast ten of each colour. Find the number ofways in which an arrangement of 10 counters can be made. How many of these will have counters of each colour. A man has 7 relatives, 4 of them are ladies & 3 gentlemen; his wife has also 7 relatives, 3 of them are ladies & 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies & 3 gentlemen so that there are 3 of the man's relative & 3 of the wife's relatives? Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed using the letters ofthe word "DIFFERENTIATION".

Q.2

Q. 3

Q.4

Q. 5

(!iBansalClasses

Permutation and Combination

[7]

Q.6

A shop sells 6 different flavours of ice-cream. In how many ways can a customer choose 4 ice-cream cones if (1) they are all of different flavours (ii) they are non necessarily of different flavours (iii) they contain only 3 different flavours (iv) they contain only 2 or 3 different flavours? 6 white & 6 black balls of the same size are distributed among 10 different urns. Balls are alike except for the colour & each urn can hold any number of balls. Find the number of different distribution ofthe balls so that there is atleast 1 ball in each urn. There are 2n guests at a dinner party. Supposing that the master an d mistress of the house have fixed seats opposite one another, and that there are two specified guests who must not be placed next to one another. Show that the number of ways in which the company can be placed is (2n - 2)! ,(4n2 - 6n+4). Eachof3 committees has 1 vacancy which is to befilledfroma group of 6 people. Find the number of ways the 3 vacancies can befilledi f ; (l) Each person can serve on atmost 1 committee. (ii) There is no restriction on the number of committees on which a person can serve. (iii) Each person can serve on atmost 2 committees.

Q.7

Q. 8

Q.9

Q . 10 A party of 10 consists of 2 Americans, 2 Britishmen, 2 Chinese & 4 men of other nationalities (all different). Find the number of ways in which they can stand in a row so that no two men ofthe same nationality are next to one another. Find also the number of ways in which they can sit at a round table, Q.ll 5 balls are to be placed in 3 boxes. Each box can hold all 5 balls. In how many different ways can we place the balls so that no box remains empty if, (i) balls & boxes are different (ii) balls are identical but boxes are different (iii) balls are different but boxes are identical (iv) balls as well as boxes are identical (v) balls as well as boxes are identical but boxes are kept in a row. In how many other ways can the letters of the word MULTIPLE be arranged; without changing the order of the vowels keeping the position of each vowelfixed& without changing the relative order/position ofvowels & consonants.

Q.12 (i) (ii) (iii)

Q.13 Find the number of ways in which the number 3 0 can be partitioned into three unequal parts, each part being a natural number. What this number would be if equal parts are also included. Q. 14 In an election for the managing committee of a reputed club, the number of candidates contesting elections exceeds the number of members to be elected by r (r > 0). If a voter can vote in 967 different ways to elect the managing committee by voting atleast 1 ofthem & can vote in 55 different ways to elect (r - 1 ) candidates by voting in the same manner. Find the number of candidates contesting the elections & the number of candidates losing the elections. Q.15 Find the number of three digits numbersfrom100 to 999 inclusive which have any one digit that is the average ofthe other two.

(!i Bansal Classes

Permutation and Combination

[7]

Q.16 Prove by combinatorial argument that: n+I (a) C r = Cr + C r _ 1 /TA n + nif> Jif . mr 4. n/ . m^ c c W 0 r 1 r- 1 r Q.17

4. 2

. C

r-2

>

C

r

. C

0'

A man has 3 friends. In how many ways he can invite one friend everyday for dinner on 6 successive nights so that nofriendis invited more than 3 times. 12 persons are to be seated at a square table, three on each side. 2 persons wish to sit on the north side and two wish to sit on the east side. One other person insists on occupying the middle seat (which may be on any side). Find the number of ways they can be seated. There are 15 rowing clubs; two of the clubs have each 3 boats on the river;fiveothers have each 2 and the remaining eight have each 1;findthe number ofways in which a list can be formed ofthe order ofthe 24 boats, observing that the second boat of a club cannot be above the first and the third above the second. How many ways are there in which a boat of the club having single boat on the river is at the third place in the list formed above? 25 passengers arrive at a railway station & proceed to the neighbouring village. At the station there are 2 coaches accommodating 4 each & 3 carts accommodating 3 each, Find the number ofways in which they can proceed to the village assuming that the conveyances are always fully occupied & that the conveyances are all distinguishablefromeach other. An 8 oared boat is to be manned by a crew chosen from 14 men ofwhich 4 can only steer but can not row & the rest can row but cannot steer. Of those who can row, 2 can row on the bow side. In how many ways can the crew be arranged.

Q.18

Q.19

Q.20

Q.21

Q. 22 How many 6 digits odd numbers greater than 60,0000 can be formed from the digits 5, 6, 7, 8,9,0 if (i) repetitions are not allowed (ii) repetitions are allowed. Q. 23 Find the sum of all numbers greater than 10000 formed by using the digits 0 1 , 2 , 4 , 5 no digit being repeated in any number. Q. 24 The members of a chess club took part in a round robin competition in which each plays every one else once. All members scored the same number of points, except four juniors whose total score were 17. J. How many members were there in the club? Assume that for each win a player scores 1 point, for di uw 1/2 point and zero for losing. Q.25 In Indo-Pak one day International cricket match at Shaijah, India needs 14 runs to win just before the start ofthefinalover. Find the number ofways in which India just manages to win the match (i.e. scores exactly 14 runs), assuming that all the runs are made off the bat & the batsman can not score more than 4 runs off any ball.

Q.26 A man goes in for an examination in which there are 4 papers with a maximum of m marks for each paper; show that the number of ways of getting 2m marks on the whole is I (m+ l)(2m 2 + 4m + 3). Q.27 The number of ways in which 2n things of 1 sort, 2n of another sort & 2n of a 3rd sort can be divided between 2 persons so that each may have 3 n things is 3 n 2 +3 n + I.

(!iBansalClasses

Permutation and Combination

[7]

Q. 28

Six faces of an ordinary cubical die marked with alphabets A, B, C, D, E and F is thrownntimes and the list of n alphabets showing up are noted. Find the total number ofways in which among the alphabets A, B, C, D, E and F only three of them appear in the list.

Q.29 Find the number of integer betwen 1 and 10000 with at least one 8 and atleast one 9 as digits. Q.30 The number of combinations n together of 3n letters of which n are 'a' and n are 'b' and the rest unlike is (n + 2). 2"- 1 .

EXERCISE-IIIQ.l Let n & k be positive integers such that n > kfr+1). The number of solutions (xj.xj,.... , x k ) , x j > 1, X j > 2,... , x k > k , all integers, satisfying Xj + X2+.... +x k =n, is . [ JEE '96,2 ] Q. 2 Q.3 (l) (ii) Find the total numb er of ways of selectingfivelettersfromthe letters of the word INDEPENDENT. [REE'97, 6] Select the correct alternative(s). Number of divisors of the form 4n + 2 ( n > 0) of the integer 240 is (A) 4 (B) 8 (C)10 [ JEE '98, 2 + 2 ] (D)3

An n-digit number is a positive number with exactly 'n' digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5&7. The smallest value ofn for which this is possible is : (A) 6 (B)7 (C)8 (D)9 How many different nine digit numbers can be formedfromthe number 2233 55888 by rearranging its digits so that the odd digits occupy even positions ? [JEE '2000, (Scr)] (A) 16 (B) 36 (C) 60 (D) 180 Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of ' n' sides. If T n + 1 - Tn = 21, then V equals: [ JEE '2001, (Scr) ] (A) 5 (B)7 (C)6 (D)4 The number of arrangements of the letters of the word BANANAin which the two N's do not appear adj acently is [JEE 2002 (Screening), 3 ] (A) 40 (B) 60 (C) 80 (D) 100 Number of points with integral co-ordinates that lie inside a triangle whose co-ordinates are (0, 0), (0, 21) and (21,0) [JEE 2003 (Screening), 3] (A) 210 (B) 190 (C) 220 (D)None (n 2 ) ! Using permutation or otherwise, prove that . .. is an integer, where n is a positive integer. (n!) [JEE 2004, 2 out of 60] A rectangle with sides 2m - 1 and 2n - 1 is divided into squares ofunit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is (A) (m + n+ l) 2 (B) 4m + n ~ 1 (C) m2n2 (D) mn(m + l)(n + 1) [JEE 2005 (Screening), 3]

Q.4

Q. 5

Q.6

Q.7

Q. 8 Q.9

(!iBansalClasses

Permutation and Combination

[7]

ANSWER KEYEXERCISE-IQ.l Q.2 Q.3 (a) 5-(6!), (b) 3! 4!, (c) 12 6 if the balls of the same colour are alike & 64 if the balls of the same colour are different 744 Q.424

C2 . 15C3

Q.5

960

Q.6

43200 111.4!

Q J

Q 8

'

n[*C2-(m-l) + m[C2-(n-l)] 420 Q.12 145

Q.9

12!;(3!)42|

Q.IO Q.14

576m+n+k

Q.ll C 3 - (mC3 + nC3 + k C 3 ) Q.20

Q.13 P~2C.

Q.16 13 , 156

5400

Q,17

22100,52

Q.19

n+5

C,

Q.21

29106

Q.22

4316527

Q.23 (a) 72 ; 78120 ; (b) 23 ; (c) 32

Q.24

2111

Q.25 Q.28

(ii) 792 ; (iii) ^ 1638

; (v) r = 3

Q.26

(14)! 5!9!

Q.27

45

Cfi

EXERCISE-IIQ.l Q.5 44 532770 Q.2 Q.6 Q. 10 2600 Q.3 49 710 ; | | 10 Q.4 Q.7 485 26250

(i) 15, (ii) 126, (iii) 60, (iv) 105 (i) linear: (47) 8! ; (ii) circular: (244). 6!

Q.9

120, 216, 210

Q.ll Q.12

(i) 150 ; (ii) 6 ; (iii) 25; (iv)2; (v) 6 (i) 3359 ; (ii) 59; (iii) 359 Q.13 61,75 Q.14 10,3

(!iBansalClasses

Permutation and Combination

[7]

Q.15

121

Q.17

510

Q.18

2! 3! 8!

Q.22 240,15552

Q.23 31199766

Q.24 - 2) - 3C2]

27

Q.25

1506

Q.28

C3[3n - ^ ^

Q.29 974 Q.l Q.2 Q.5 Q.9m

EXERCISE-III

Ck_, where m = (1/2) (2n - k2 + k - 2) Q.3 Q.6 (i) A; (ii) B A Q.4 Q.7 C B

72 B C

(!i Bansal Classes

Permutation and Combination

[7]

ft

BANSAL CLASSESMATHEMATICSTARGET IIT JEE 2007XI (P. Q, R, S)

BINOMIAL

CONTENTSKEY- CONCEPTS EXERCISE - 1(A) EXERCISE - 1(B) EXERCISE-II EXERCISE - 111(A) EXERCISE - 111(B) EXERCISE-IV ANSWER-KEY

KEY1.

CONCEPTS

BINOMIAL EXPONENTIAL & LOGARITHMIC SERIES BINOMIAL THEOREM : The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as BINOMIAL THEOREM . If x,y e R and n e N , then ;n

(x + y) = C0 x + Cj x"- y + C2 x " y + This theorem can be proved by Induction .

n

n

n

n

1

n

n 2 2

+ Crx " y +

n

n r r

+ Cny = X n C r x n " r y r .

n

n

r0 =

OBSERVATIONS : (i) The number of terms in the expansion is (n + 1) i.e. one or more than the index. (ii) The sum of the indices of x & y in each term is n . (iii) The binomial coefficients of the terms n C 0 , nC j.... equid istant from the beginning and the end are equal. 2. (i) (iii) (i) (ii) IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE : General term (ii) Middle term Term independent of x & (iv) Numerically greatest term The general term or the(r+ l) th term in the expansion of (x + y)n is given by; Tr+i = nCr x n - r . yr The middle term(s) is the expansion of (x + y)n is (are) : (a) If n is even, there is only one middle term which is given by ;T

(b) (iii) (iv)

If n is odd, there are two middle terms which are :T(n+l)/2 & T[(n+l)/2]+l

(n+2)/2

= nf1

n/2

x

Yn/2

'l

n/2

Term independent of x contains no x; Hencefindthe value of r for which the exponent of x is zero. To find the Numerically greatest term is the expansion of (1 + x) n , n e N findn T . C1x r n-r+ 1 ^ = r = x . Put the absolute value of x &findthe value ofr Consistent with the T Cf_jX

T.j inequality - y > 1. Note that the Numerically greatest term in the expansion of (1 - x) n , x > 0, n e N is the same as the greatest term in (1 +x) n . 3. If ( A + b)"= I + f where I & n are positive integers, n being odd and 0 < f < l , then V > (I + f). f = Kn where A - B 2 = K > 0 & V A - B < 1 . If n is an even integer, then (I + f) (1 - f) = Kn. 4. (i) (ii)(iii)

BINOMIAL COEFFICIENTS :C 0 + CJ + C 2 + C0 + C2 + C4 + + CN = 2* = CJ + C 3 + C 5 +2N

= 211-1 ^( ! (n+ r^ _r).

C 0 2 + C J 2 + C 2 2 + .... + C N 2 =

CN =

(iv)

C0.Cr + Cj.C^j + C2.Cr+2 + ... + C n _ r .C n -

?fe\Bansal

Classes

Binomial

[6]

REMEMBER : (i) (2n)!=2 n .n! [1.3.5 5.

(2n-l)]

BINOMIAL THEOREM FOR NEGATIVE OR FRACTIONAL INDICES : . , n ( n - l ) 2 n ( n - l ) ( n - 2 )J 3s I f n e Q , then (1 +x)n= l + nx + ^-x ^ -x + .. ,, G Provided | x | < 1. .O

Note : (i) When the index n is a positive integer the number of terms in the expansion of (1 +x) n is finite i.e. (n+ 1) & the coefficient of successive terms are : np np np np np M> 2' 3 n (ii) When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 +x) n is infinite and the symbol nCr cannot be used to denote the Coefficient of the general term. (iii) Following expansion should be remembered (| x | < 1). (a) (1 +X)"1 = 1 - x + x 2 - x 3 + x 4 -.... oo (b) (1 - x ) - 1 ^ 1 + x + x2 + x3 + x 4 +.... oo (c) (1 + x)~2 = 1 - 2 X + 3 X 2 - 4 X 3 + . . . . oo (d) (1 -x)~ 2 = 1 + 2 X + 3X 2 +4X 3 + oo The expansions in ascending powers of x are only valid if x is 'small'. If x is large i.e. | x | > 1 then we may find it convinient to expand in powers of , which then will be small. X APPROXIMATIONS : (1 +x) n = 1 +nx+ - 1 ) ( n - 2 ) x3 1.2 1.2.3 If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx, approximately. This is an approximate value of (1 +x)n. x* + 7.(i)n(n

(iv)

6.

EXPONENTIAL SERIES: x x2 x3 ex=l + + + +2

( lV oo ; where x may be any real or complex & e = ^u^t \ \ + J3

(ii) Note:/ X (a)

ax = 1 + In a + / n 2 a + / n 3 a + 1! 2! 3!1 1 1

oo where a > 0

(b) (c) (d) (e)

4 e = l1 + . + + + oo 1! 2! 3! e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is 2.7182818284.

e + e" = 2

1

' 1 1 1 ,+1 + + +V 2! 4! 6!

0 .0

N

y

-1=0 r i i i 00 e - e"1 = 2 l + _ + _ + _ + ^ 3! 5! 7! J Logarithms to the base 'e' are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm.

N

?fe\Bansal

Classes

Binomial

[6]

8.(i) (ii)

LOGARITHMIC

SERIES:

/n(l+x) = x - ~ In v(1- x) = - x (1 + x) f

2

2^

12

3

4

33

4 3

b4

oo w h e r e - l < x < l oo where - 1 < x < 1 X n 7 , n e N , n>2 n~i 1 Binomial

te Bonsai Classes

[4]

vn Q.12 In the expansion of \1 + x + - J find the term not containing x.

Q.13

Show that coefficient of x5 in the expansion of (1 +x 2 ) 5 . (1 + x) 4 is 60.

Q.14 Find the coefficient of x4 in the expansion o f :(i) ( l + x + x 2 + x 3 )n

(ii) ( 2 - X + 3 X 2 ) 6

Q.15 Find numerically the greatest term in the expansion of : (i) (2 + 3x)9 when x = Q.16 Given s n = l + q + q2 + prove thata+1

(ii) (3 - 5x)15 when x = j + qn & Sn = 1 + ^ + + .... + ( ^ j , q * 1,

C t + n+1C2.S! + n+1C3.s2 +....+ n+1Cn+1.sn = 2 n . S n .

Q.17 Prove that the ratio of the coefficient of x10 in (1 -x 2 ) 1 0 & the term independent of x in x-|j is 1 : 32 .3x 2 2 3 xj

Q.18 Find the term independent of x in the expansion of (1 + x + 2x3) " (l + rlog 10) Q. 19 Prove that for n e N X (~l) r nCr 7 = 0. r=0 (l+log e 10 n ) Q.20 Prove the identity +

=

. Use it to prove

=

Q. 21 If the coefficient of a r_1 , a r , afTl in the expansion of (1 + a)n are in arithmetic progression, prove that n2 - n (4r + 1) + 4r2 - 2 = 0. (1 - x n )(1 - x n_1 )(1 - x n " 2 )Q22 If nJ = r

(1 - x n_r+1 )

"iMO^Xi-x3)K=0

...,.(i-xr)~~'provethatnJ- = nJ-

n Q.23 Prove that ^ n C K sinKx. cos(n - K)x = 2 n_1 sin nx. Q.24 The expressions 1 + x, 1+x + x2, 1 + x + x2 + x3, 1 + x + x2 + + xn are multiplied together and the terms ofthe product thus obtained are arranged in increasing powers ofx in the form of a0 + ajX + a^x2 + , then, (a) how many terms are there in the product. (b) show that the coefficients of the terms in the product, equidistantfromthe beginning and end are equal. (c) (n + 1)! show that the sum of the odd coefficients = the sum of the even coefficients =2

Q.25 Find the coeff. of

(a) (b) (c)x3r

x6 in the expansion of (ax2 + bx+c) 9 . x2 y3 z4 in the expansion of (ax - by + cz)9 . a2 b3 c4 d in the expansion of (a - b - c + d)10.

2n

Q.26 If 2>r( - ) S r( - ) & a k = 1 for all k>n, then show that bn = 2n+1Cn+1. r=0 r0 =

x 2 r= b

2n

?fe\Bansal

Classes

Binomial [6]

/=k-l /. ^ n Q.27 If P k (x)= 2 x1' then prove that, n C k P k (x) = 2""1-Pn' /o = \ ^j k1 = Q.28 Find the coefficient of xr in the expression of : (x + 3)n_1 + (x + 3)n"2 (x + 2) + (x + 3)n"3 (x + 2)2 + + (x + 2)n~l

fx 2V Q.29(a) Find the index n of the binomial I - + -1 if the 9th term of the expansion has numerically the greatest coefficient (n e N). (b) For which positive values of x is the fourth term in the expansion of (5 + 3x)10 is the greatest. Q.30 Prove that (72)1 (36!

f

- 1 is divisible by 73.

Q. 31 If the 3rd, 4th, 5th & 6th terms in the expansion of (x+y) n be respectively a, b, c & d then prove that b 2 -ac 5a c 2 ~bd 3c' Q.32 Find x for which the (k+ l) th term of the expansion of (x + y)n is the greatest if x + y = 1 andx>0, y>0. *Q,33 If x is so small that its square and higher powers may be neglected, prove that:

(l - 3 ) 2 ( ~ x)5/a xy+i*Q.34 w (a) (b)

=

j

Mr+Mr5

= 1 + (Ji)x+,)x or

If x = - + + HJL + _L11L +3 3.6 3 . 6 . 9 3.6.9.12

o then F o prove that x2 + 2 x - 2 = 0. then find the value of y2 + 2y.

tfy=!

+~if)

+

*Q.35 If p = q nearly and n >1, show that (n-l)p + (n + l)q (*) Q.l Only for CBSE. Not in the syllabus of HT JEE.

P U

EXERCISE-I (B)Show that the integral part in each of the following is odd. n e N( A ) (5 + 2 Ve)" ( B ) (s + 3 V7) n ( C ) (e + V3?) n

Q.2

Show that the integral part in each of the following is even, n e N (A) (3V3 + 5 p ' (B) (5V5 + ll)2n+1 p+P where n & p are positive integers and P is a proper fraction show that

Q.3 Q.4

If (7 + 4-^3

=

(1-P)(p + P)=l. If x denotes (2 + V3) , n e N & [x] the integral part of x then find the value of : x - x 2 + x[x].

?fe\Bansal Classes

Binomial

[6]

Q.5

If P = (s + 3V7) and f = P - [P], where [ ] denotes greatest integer function. Prove that: P (1 - f) = 1 (n e N) If (6V6 + 14)2n+1 = N & F be the fractional part of N, prove that NF = 202n+1 (n e N) Prove that if p is a prime number greater than 2, then the difference (2 + V5)P - 2p+1 is divisible by p, where [ ] denotes greatest integer.

Q.6 Q. 7

Q.8 Q.9

Prove that the integer next above (v'3 + lj contains 2n+1 as factor (n e N) Let I denotes the integral part & F the proper fractional part of (3 + denotes the rational part and 0 the irrational part of the same, show that p=|(I+l)and2 n

where n e N and if p

a =

(I + 2 F - 1).

Q.IO Prove that

C

is an integer, V n e N .

n+1

EXERCISE-II(NOT IN THE SYLLABUS OF I IT-JEE) PROBLEMS ON EXPONENTIAL & LOGARITHMIC SERIES For Q.l TO Q.15, Prove That: Q.l Q2 Q3~A Q.4 v 2! 4! 6!

> 2 , 1 1 1 , + + + - . 11+1,1 1 + +3! [ 1 1 1 + + +, V2! 4! 6!

+ +. 5! 7!

1

e-1 e + 1

+ + +, .1! 3! 5!1 1 1+ + + +,11

1

1 1

e2 1 - 1 = ( _ + _1 + _1 + , e + 1 U 3! 5!

2!

4!

6!

1.1+2 1+2+3 1+2+3+4 ! + _ _ + + + 2! 3! 4!

= \2J

6

Q5 Q.6 07 V Q.9 Q.llQ.12

_L

1+

11.2.3.4.5.7 4!

1.3 + 1 . 2 . 3 . 5

, 1 + 2 1 + 2 + 22 1 + + 3! + 2!1 + 2! + 3! +4! + 23

1 + 2 + 22 + 23 + . = e2- e

3

3

4

3

5e&e

Q8.^

+ + + + +1! 2! 3! 4! 5!

2

3

6

11

18

=3 ( e - 1 ) v'

,\

+ + +2.3 4.5 6.7 1.2 21

0 = 1 - log 2 02 1.2.3 3.4.5

Q 10.

1 +2 14t +3.2 5.2

+

r + 7.2 6

1

loge3

1 1 1 + + + ... = 1 +1 1 +1 3.4 5.61

+

5.6.7

_ + .... = .In 21 1 1

14.2

2.2

3.2

+

- ln3 - ln2

nn13. - + - + ^ + T1+ 1 Q3 3.3 5.3 7.3

=

In 2[6]

?fe\Bansal

Classes

Binomial

Q1 .4

2 \2

V

4 V2

3 /

6V2 3

3

Q.15 If y = x - +

+

where|x|< 1, then prove that x = y +

+ 2L +

+

EXERCISE-HI (A)If C 0 , CJ , C 2 , , Cn are the combinatorial coefficients in the expansion of (1 +x) n , n s N , then prove the following : Q.l Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 v C - + CV + C - + + cn 2

=-^

( n! 2)

C0 q + Cj C2 + C2 C3 +....+Cn_j Cn = Cj + 2C2 + 3C3 + C0 + 2Cj + 3C2 + C0 + 3Cj + 5C2 + + n . Cn = n . 2n_1 + (n+l)Cn = (n+2)2n~1 + (2n+l)Cn = (n+1) 2n (Cn_j+Cn) C a _, 2

(C0+C1)(C1+C2)(C2+C3)C0+

C,

+

C2

+

+

Q8 v

0

2

+

^

3

+

+

n+1

n+1

V

'

0

2

3

4

n+1( n

n+1

Q. 10 C0Cr + CjCr+1 + C2Cr+2 + .... + C n . r Cn = Q.ll2

_r^

+ r) ,

+

3

+ (-Dn

C

n+1

" "

1

n+1r|

Q.12 C 0 - C j + C 2 - C 3 + .... + ( - l ) r . Cr< Q.13 Q. 14 C 0 2 -C, 2 + C 2 2 -C 3 2 +

(1 _( _ 1! -X n - ) (n r 1)s

C0 - 2Cj + 3C2 - 4C3 + .... + (-l) n (n+l) Cn = 0 + (-l) n C n 2 = 0 or (-l) a / 2 Ca/2 according as n is odd or even.

Q.15 If n is an integer greater than 1, show that ; a - n Cj(a-l) + nC2(a-2) + (~l) n (a - n) = 0 Q. 16 (n-1) 2 . Cj + (n-3) 2 . C3 + (n-5) 2 . C5 + Q. 17 V 1 . C 2 + 3 . C,2 + 5 . C 2 + 0 1 2'

= n (n + l)2n~3l)

+ V(2n+l) Cn2 = -(n + n

n!n!

Q.18 If a 0 , a;, a,, be the coefficients in the expansion of ( l + x + x2)n in ascending powers of x, then prove that : (1) a0 a, - aj % + % % - .... = 0 . + (ii) a^-aja3 + a ^ ^n - 2 *2n = an +1 o r an-in-! (iii) Ej = E2 = E 3 = 3 ; where E t = a0 + % + a6 + ; E2 = a1 + a 4 +a 7 +

E a2 + a + a + ^ ^ gr=o '

&

1-2 Q.19 Prove that : Z(" c r ^+2) = xn 2

( n! 2)( n - 2 ) ! (n+2)!

Q.20 If (l+x) = C0 + CjX + C2x + .... + C n x n , then show that the sum of the products of the C .' s taken two at a time, represented by 2XC C . ' .j0 < i < j < n

is equal to 22n_1 n

2n! 2(n!) 2

ItBansat Classes

Binomial

[8]

Q.21

J c ^ + yfc7 +

+

+ Jc~n2.

EXERCISE-III (B)Q.l If ( l + x y ^ C o + Cj. x + C2. x 2 +.... + C15. x15, then find the value o f : C Q.22

+ 2C3 + 3 C

4

+ ....

+14Cn

1 5

If(1 + x + x 2 + . . . + x P ) a,+2a2

= a0 + a1x + a2x2+..,+anp.xnP

,

then

find

the value of :

t- 3 a 3 + . . . . + n p . a n pn

Q.3 Q.4

l 2 . C 0 + 2 2 . C t + 3 2 . C 2 + 4 2 . C 3 + .... + ( n + 1 ) 2 C ^ r 2 . C r = n ( n + l)2n-2

= 2n~2(n+1) (n+4) .

t=0

Q.5

Given

p + q = l

, showthat

j>>2.nCr.p1 .q'w = n p [ ( n i=0

l)p+l]

IIQ.6 S h o w that (1 +X)n.

2r

10 =

^C

(2r-n)

= n.2n

w h e r e Cf denotes the combinatorial coeff. in the expansion

of

Q.7 v

r

0

2

+

3

4

+

n + 1

=

(1

+

x n+ ) ' (n + l ) x

Q.8

v

P r o v e t h a t , 2 . C n + . c 1 + . C2, + ,

0

2

3

+

11

. C100 = ,

11

Q.9

If(l+x)

n

=

r0 =|

Z c

r

. x '

t h e n p r o v e that ;

22.C0 1.2

23.Ct 2.3

;

24,C2 3.4

|

|

2n+2.Cn

=

31""2 - 2 n - 5

(n+1) (n+2) ~ (n+1) ( n + 2 )

QO I .

+

+

+

2n n+1 13

Q 1 Si _Sl + Si _ Si + 11 5 9 a i ^ -Ca + 9+ + Q.12 2 -+ Ci O Ci 2 3 4 5

( i)"Cn n+2

4n+l

1.5.9.13

(4n-3)(4n+l)

4n.n!

l+n.2 (n+1) (n+2)1

Q.13 Si_SL+Sl_Si +2 3 4 5

+ (_!)" . . C _ nn+2v y

(n+1) (n+2)

Q.14 Si _ Si + Si _ Si +vQ. 15

+ ( !i .Si = I + I + i + A + _) n+ Cnxn

+I

1q o - x ) -

2

3

42

2 3 4, then showx )

nthat : _x ) +

I f ( l + x ) n = C 0 + CjX + C 2 x 2 + % ( l - x ) + %

(1-x)3 -....+ (-l)n-l I ( l _

n =

( 1

I

( 1

_

X

2

) +

I (1x3) +

+ ^(1-X)

?fe\Bansal

Classes

Binomial [6]

Q. 16 Prove that , \ - C r f C2+ j C3n

+

+^+ (-l)n

n+1n

^

. Cn= - 1=

n+1 n!

Q.17

If n e N ; show that

x+1

C.L -i C +

n

C

x+2

x+n

x ( x + l) (x + 2) .... ( x + n )

Q.18 Prove that, ( ^ C ^ 2 . (2nC2)2 + 3 . (2nC3)2 + ... + 2n. (2nC2n) Q.19 2n If(l+x+x ) = X a r x r , n s N , then prove that2 nr=0

(4n - 1! ) [(2n-l)!]2

(r+l)ar+1 =(n-r)ar + (2n-r+l)ar_j. Q.20 Prove that the sum to (n + 1) terms of n(n+1)

(0 0

Domain (i. e. values taken by x)

Range (i. e. values taken by f (x) )

(v i)

a1/x, a > 0

R R- { 0 } R

R -{ 0}

R+ R+- { 1 } R+

R+-{1)

Logarithmic Functions

() i(ii)

logx,(a>0)(a * 1 )logxa=]^ (a > 0 ) (a * 1)

+ R

RR-{0}

R+-{1}

F.

Integral Part Functions Functions (i) [x] R I

^

R-[0,1)

g,nSI-{0}j

Fractional Part Functions

0 )

(*) 1to

RR-I

[,) 01(, O 1 Q )

H.

Modulus Functions () i (ii) |x| j^j R R-{ 0 } R+w{0} R+

1

I.

Signum Function sgn(x)=^,x*0A.

R

{-1,0,1}

=0,x=0 J. Constant Function say f (x) = c R {c}

^Bansal Classes

Functions &

Trig.-IV

[5]

5. (i) (ii) (iii)

6.

EQUAL OR IDENTICAL FUNCTION : Two functions f & g are said to be equal if: The domain of f = the domain of g. The range of f = the range of g and f(x) = g(x) , for every x belonging to their common domain, eg. 1 x f(x) = & g(x) = are identical functions . x x2 CLASSIFICATION OF FUNCTIONS : One-One Function (Injective mapping): A function f: A-^Bis said to be a one-one function or injective mapping if different elements of A have different f images in B . Thus for x p x2 e A& f(Xj), f(x2) e B, f(x,) = f(x2) < > x, = x, or x, * x2 o f(x,) * f(x2). = Diagramatically an injective mapping can be shown as A B A B

Note : (i)

Any function which is entirely increasing or decreasing in whole domain, then f(x) is one-one. (ii) If any line parallel to x-axis cuts the graph of the function atmost at one point, then the function is one-one. Many-one function: Afunction f: A>B is said to be a many one function iftwo or more elements ofA have the same f image in B . Thus f: A- B is many one if for ; x p x 2 e A, f(Xj) = f(x2) but x] ^ x2 . Diagramatically a many one mapping can be shown as A B A B

Note : (i) (ii)

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 ofthe function atleast at two points, then f is many-one. 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 inA then we say that fis a function ofA'onto'B . Thus f : A ^ B i s suijective iff V b e B, 3 some a e A such that f (a) = b . Diagramatically surjective mapping can be shown as A B A B

B is said to be a constant function if every element ofA has the same f image inB . Thus f: A B ; f(x) = c, V x e A, c e 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 functio