Iit 4 Unlocked

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Document Number : SRMGW - IIT-JEE - MATH-04

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Preface

About the book


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IIT - MATHS

SET - 4


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INDEX

1. PERMUTATION COMBINATION......................................................................

2. BINOMIAL THEOREM......................................................................................

3. CIRCLE ..............................................................................................................

4. PARABOLA ........................................................................................................

5. ELLIPSE .................................................................................................... .........

6. HYPERBOLA.................................................................................................... .

7. PROBABILITY...................................................................................................

2

20

34

72

112

142

174


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2

IIT- MATHEMATICS-SET-IV

1 Permutation Combination


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3

PERMUTATION COMBINATION

FUNDAMENTAL PRINCIPLES OF COUNTING

(i) Multiplication Principle: If an operation can be performed in m different ways following

which a second operation can be performed in n different ways, then the two operations in

succession can be performed in m n ways. This can be extended to any finite number of

operations.

Permutation:

Definition:

Each of the different arrangement which can be made by taking some (or all) number of

given things is called a permutation of things.

Note :

Factorial Notation :

The product of first n natural numbers is generally written as n! and is read as factorial n.

i.e., n! = n (n 1)(n 2) . . . 3. 2. 1

some important properties

(i)n! = n (n 1) ! (ii) (2n)! = 2n. n! [1. 3. 5. 7. . . . (2n 1)]

(iii)0! = 1! = 1 (iv) factorials of negative integers are not defined

Important Results

(i) The number of permutations of n different things, taking r at a time is denoted bynPr

or P (n, r)

nPr= nn r

!!

0 r n

= n (n 1) (n 2) . . . (n r + 1).

Note:

(i) The number of permutations of n different things taken all at a time = nPn= n!

(ii) nP0= 1, nP

1= n and nP

n 1= nP

n= n!.

(iii) nPr= n (n 1P

r 1) = n (n 1) (n 2)(n 2P

r 2) = n (n 1) (n 2) (n 3P

r 3) = . . .

Alternative :

(Combinational proof) nPrdentoes the number of ways of arranging r-objects out of

n-objects, in a line. This work can be done in the following way also.

Suppose the objects are a1, a

2, . . ., a

n.First we find the number of permutations, in which a

1

does not appear. Number of such permutations isn 1Pr 1

.Further we consider those arrange-

ments, in which a1necessarily appears. Number of such permutation is r. n 1P

r 1, as we can

arrange (r 1) objects out of (n 1) objects in n 1Pr 1

ways, and then in any such permutation

we can fix the position of a1in r - ways.

Now using the principle of addition, the required number is n 1Pr+ r. n 1Pr 1.

(ii) The number of permutations of n things taken all at a time, p are alike of one kind, q are alike


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of second kind and r are alike of a third kind and the rest n (p + q + r) are all different is

!

! ! !

n

p q r

(iii) The number of permutations of n different things taken r at a time when each things may be

repeated any number of times is nr.

COMBINATION

Defination :

Each of the different group or selection which can be made by taking some (or all) number

of given things, without reference to the order of the things in each group, is called a combina-

tion.

Important Results

(i) The number of combinations of n different things taken r at a time is denoted by

nCror C (n, r) or

n

r

nCr=

n!

r! n r ! 0 r n

=n

rP

r!

=

n n 1 n 2 . . . n r 1

r r 1 r 2 . . . 2.1

If r > n, then nCr= 0.

Note:

(i) nCris a natural number.

(ii) nC0= nC

n= 1, nC

1= n

(iii) nCr= nC

n r

(iv) nCr+ nC

r 1= n + 1C

r

(v) nCx= nC

y x = y or x + y = n

(vi) n. n 1Cr 1

= (n r + 1). nCr 1

(vii) If n is even then the greatest value of nCris

nn

2

C

(viii) If n is odd then the greatest vlaue of nCris n 1 n 1

n n

2 2

C or C .


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6


4 3

36!9!

2! .4! 5! .3!.6!.7!

.

PRINCIPLE OF INCLUSION AND EXCLUSION

If A1, A

2, . . . , A

mare m sets and n(S) denotes the number of elements in the set S, then

k

1 2 s

m sms 1

k k i j i

k 1 1 i j m i i i ... i mk 1 k 1

n A n A n A A . . . 1 n A . . .

m

m 1k

k 1

1 n A

Note that if

m

k

k 1x A

, then x belongs to at least one of AAk, 1 k m.

As another application of the principle of inclusion and exclusion, number of derangement of

n objects (number of ways in which n numbered balls (from 1 to n) can be placed in n

numbered boxes (from 1 to n), one in each box, so that no ball goes to its corresponding

numbered box), denoted by n can be obtained. Infact we have

n = n n1 1 1 1

1 . . . 11 2 3 n

.

MULTINOMIAL THEOREM

Consider the equation x1+ x

2+ . . . +x

r= n, where i i i i, i ia x b ; a b ,x I;i 1, 2, . . ., r. In

order to find the number of solutions of the given equation satisfying the given conditions we

observe that the number of solutions is the same as the coefficient of xnin the product

1 1 1 1a a 1 a 2 bx x x . . .x 2 2 2 2a a 1 a 2 bx x x . . .x

3 3 3 3a a 1 a 2 bx x x . . .x . . . . . . . r r r r a a 1 a 2 bx x x . . .x .

For example, if we have to find the number of non-negative integral solutions of

x1+ x

2+ . . . + x

r= n, then as above the required number is the coefficient of xn in

0 1 n 0 1 n 0 1 nx x . . . x x x . . . x . . . x x . . . x r brackets

= Coefficient of xnin (1 + x + x2+ . . . + xn)r

= Coefficient of xnin (1 + x + x2+ . . . )r

= Coefficient of xnin (1 x) r

=Coefficient of xnin


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PERMUTATION COMBINATION

2 3r r 1 r r 1 r 2

1 r x x x . . .2! 3!

= Coefficient of xninr r 1 2 r 2 3

1 2 31 C x C x C x . . .

n r 1 n r 1n r 1C C .

Note :

If there are objects of one kind, m objects of second kind, n objects of third kind and so on;

then the number of ways of choosing r objects out of these objects is the coefficient of xrin the

expansion of

(1 + x + x2+ x3+ . . . + x ) (1 + x + x2+ . . . + xm) (1 + x + x2+ . . . + xn)

Further if one object of each kind is to be included, then the number of ways of choosing r

objects out of these objects is the coefficient of xrin the expansion of

(x + x2+ x3+ . . . + x ) (x + x2+ x3+ . . . + xm) (x + x2+ x3+ . . . + xn) . . .

EXPONENT OF PRIME IN n!

Let p be a given prime and n any positive integer, then maximum power of p present in n! is

n

p

+ 2 3n n

. .. ,p p

where [ ] denotes the greatest integer function.

The proof of the above formula can be obtained using the fact that nm

gives the number of

integral multiples of m in 1, 2, . . . , n; for any positive integers n and m.

The above formula does not work for composite numbers. For example if we have to find the

maximum power of 6 present in 32!, then the answer is not 232 32

. . . 56 6

, as 5 is the

number of integral multiples of 6 in 1, 2, . . . , 32; and 6 can be obtained on multiplying 2 by

3 also. Hence for the required number, we find the maximum powers of 2 and 3 (say r and s)

present in 32!. Using the above formula r = 31 and s = 14. Hence 2 and 3 will be combined

(to form 6) 14 times. Thus maximum power of 6 present in 32! is 14.


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8

IIT-MATHEMATICS-SET IV

1-A PERMUTATIONS ANDCOMBINATIONS


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9

PERMUTATIONS AND COMBINATIONS

WORKEDOUT ILLUATRATION

ILLUSTRATION : 01

A polygon has 44 diagonals, then the number of its sides are

a) 11 b) 7 c) 8 d) none of these

Solution :

We have

44 = n)1n(n2

1nC2

n

or 088n3n 2 or 0)8n)(!1n(

n = 11, since n 8

ILLUSTRATION : 02

If 7 points out of 12 are in the same straight line, then the number of triangles formed is


Solution :

Numbers of s=3.2.1

5.6.7

3.2.1

10.11.12CC 3

73

12 = 220 35 = 185

ILLUSTRATION : 03

All the letters of the word EAMCET are arranged in all possible ways. The number of such arrange-

ments in which no two vowels are adjacent to each other is

a) 360 b) 144 c) 72 d) 54

Solution :

Gap Method. Consonants M, C, T in 3! = 6 ways and 4 gaps and 3 vowels (2 alike) in

12!2

1.P3

4 ways = 12 x 6 = 72

ILLUSTRATION : 04

Out of 10 red and 8 white balls, 5 red and 4 white balls can be drawn in number of ways

a) 410

58 CxC b) 4

85

10 CxC c) 918 C d) None

Solution :

Ans : 48

510 CxC

ILLUSTRATION : 05

7 men and 7 women are to sit round a table so that there is a man on either side of a women. The

number of seating arrangement isa) (7 !)2 b) (6!)2 c) 6!. 7! d) 7!


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Solution :

Ans : 6! 7!

ILLUSTRATION : 06

The number of ways in which we can post 5 letters in 10 letter boxes is

a) 50 b) 510

c) 105

d) none of these

Solution :

We can post the first letter in 10 ways, the second letter in 10 ways and so on. Thus, the number of

ways of posting 5 letters in 10 letter boxes is 10 x 10 x 10 x 10 x 10 = 105

ILLUSTRATION : 07

A class has 30 students. The following prizes are to be awarded to the students of this class. First and

second in Mathematics; first and second in Physics first in Chemistry and first in Biology. If N denote

the number of ways in which this can be done, then

a) 400/N b) 600/N

c) 8100/N d) N is divisible by four distinct prime numbers

Solution :

First and second prizes in Mathematics (Physics) can be awarded in 2230230 PP ways. First prize inChemistry (Biology) can be awarded in 30 (30) ways.

Therefore, 2442422

2

30

5.3.22930)30(PN . 29

2

Since 400 = 24. 52. 600 = 23. 3. 52and 8100 = 22. 34. 52we get N is divisible by each of 400, 600 and

8100. Also N is divisible by four distinct primes, viz., 2,3,5 and 29.

ILLUSTRATION : 08

A letter lock consists of three rings marked with 15 different letters. If N denotes the number of ways

in which it is possible to make unsuccessful attempts to open the lock, then

a) 482/N b) N is product of 3 distinct prime numbers

c) N is product of 4 distinct prime numbers d) none of these

Solution :

Since each ring has 15 positions, the total number of attempts that can be made to open the lock is 153.

Out of these, there is just one attempt in which the lock will open. Therefore, N = 1531 = (15 1)

(152+ 15 + 1) = 2.7.241 clearly, 482|N and N is product of three distinct prime numbers.

ILLUSTRATION : 09

The tens digit of 1! + 2! + 3! + +49! Is

a) 1 b) 2 c) 3 d) 4


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Solution :

We have 1! + 2! + 3! + 4! = 33. Also 5! = 120, 6! = 720, 7! = 5040, 8! = 40320 and 9! = 326880.

Thus, the tens digit of 1! + 2! + + 9! Is 1.

Also, note that n! is divisible by 100 for all n 10, so that tens digits of 10! + 11! + + 49! is zero.

Therefore, tens digit of 1! + 2! ++49! Is 1

ILLUSTRATION : 10

Four dice are rolled. The number of possible outcomes in which at least one die shows 2 is


Solution :

The total number of possible outcomes is 64. The number of possible outcomes in which 2 does not

appear on an die is 54, so that tens digit of 10! + 11! + +49! is zero. Therefore, the number of

possible outcomes in which at least one die shows a 2 is 64

54

= 1296 625 = 671.


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SECTION A -SINGLE

ANSWER TYPE QUESTIONS

1. How many numbers greater than 1000 but not greater than 4000 can be formed from with the digits0,1,2,3, 4; repetition of digits being allowed :

a) 374 b) 375 c) 370 d) 376

2. A candidate is required to answer 6 out of 10 questions which are divided into 2 groups each containing5 questions and he is not permitted to attempt more than 4 from each group. In how many ways can hemake up his choice?

a) 200 b) 225 c) 250 d) 275

3. The number of ways in which 5 male and 2 female members of a committee can be seated around around table so that two females are not seated together is :

a) 480 b) 600 c) 720 d) 840

4. Let nT denote the number of triangles, which can be formed by using the vertices of a regular polygon ofn sides. If 211 nn TT then nequals.

a) 5 b) 7 c) 6 d) 4

5. The numbers are picked at random from the numbers 1,2,3,.,100. The number of ways of select-ing the two numbers whose product is a multiple of 3 is

a) 528 b) 2211 c) 2739 d) none of the above

6. The number of different signals which can be given from 6 flags of different colours taking one or moreat a time is

a) 1958 b) 1956 c) 16 d) 64

7. If yx, and rare positive integers, then ryy

r

xy

r

x

r

x CCCCCC

.....2211 =

a)!

!!

r

yxb)

!

!

r

yx c)

r

yx C d)r

xyC

8. There are 16 points in a plane, no three of which are in a straight line except 8 which are all in a straightline. The number of triangles that can be formed by joining them equals:

a) 504 b) 552 c) 560 d) 1120

9. The number of integral solutions of 0 zyx with 5,5,5 zyx is

a) 135 b) 136 c) 455 d) 105

10. The sides AB, BC, CA of a triangle ABC have 3,4 and 5 interior points respectively on them. The totalnumber of triangles that can be constructed by using these points as vertices is

a) 220 b) 204 c) 195 d) 205

11. There were two women participating in a chess tournament. Every participant played two games with

the other participants. The number of games that the men played between themselves proved to exceedby 66 the number of games that the men played with the women. The number of participants is


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a) 6 b) 11 c) 13 d) None of these

12. Two women and three men are to be seated on chairs numbered 1 to 8. First the women choose thechairs from amongst the chairs maked 1 to 4 and then men select the chairs from the remaining. Thenumber of possible arrangements are :

a) 24

34 x CC b) 3

42

4 x PC c) 34

24 x PP d) 3

62

4 PxP

13. The number of ways in which a team of 11 players can be selected from 22 players including 2 of themand excluding 4 of them is :

a)211

22

C b) 516C c) 9

16C d) 820C

14. In a foot ball championship, 153 matches were played. Every team played one match each with eachother. The number of teams participating in the championship is :


15. In an examination, there are three multiple choice questions and each question has 4 choices. Number of

ways in which a student can fail to get all answers correct is :

a) 11 b) 12 c) 27 d) 63

16. The number of integers greater than a million can be formed by using the digits 2,3,0,3,4,2,3 is


17. If a denotes the number of permutations ofx+2 things taken all at a time, b the number of permutationsofxthings taken 11 at a time andc, the number of permutations ofx11 things taken all at a time suchthat a = 182bc then the value ofxis

a) 15 b) 12 c) 10 d) 18

18. A man has 8 children to take them to a zoo. He takes three of them at a time to the zoo as he can withouttaking the same 3 children together more than once. How many times will he have to go to the zoo?How many times a particular child will go?

a) 56,21 b) 56,36 c) 31,41 d) none of these

19. Among 2nobjects, nare indentical. The number of ways to select nobjects out of these 2nobjects is

a) n2 b) n1n211n201n2 C.......CC

c) the number of proper subsets of n

aaa .......,21

d) None of the above

20. The value of the expression

5

13

524

47

J

JCC is equal to

a) 547C b) 3

52C c)4

52C d) 552C

21. If one quarter of all three element subsets of the set A= naaaa ......., 321 contains the element 3a , then

n =a) 10 b) 12 c) 14 d) 16


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22. Six points in a plane be joined in all possible ways by indefinite straight lines and if no two of them becoincident or parallel and no three pass through, the same point (with the exception of the original 6).The number of distinct points of nis equal to


23. If

n

r r

nn

C

a

0

1then

n

r r

n

C

r

0

equals :

a) na1n b) nna c) nna21


24. The number of triangles whose vertices are at the vertices of an octagon but none of whose sideshappen to come from the sides of the octagon is :

a) 24 b) 52 c) 48 d) 16

25. Between two junction stations A and B, there are 12 intermediate stations. The number of ways in which

a train can be made to stop at 4 of these station so that no two of these halting stations are consecutiveis :

a)4

8C b) 49C c) 44

12C d) None of these

26. There are nstraight lines in a plane, no two of which are parallel and no three pass through the samepoints. Their points of intersection are joined. Then the number of fresh lines thus obtained is

a)

8

21 nnnb)

6

321 nnnn c)

8

321 nnnnd) None of the above

27. There are three piles of identical red, blue and green balls and each pile contains at least 10 balls. Thenumber of way of selecting 10 balls if twice as many red balls as green balls are to be selected is :

a) 3 b) 4 c) 6 d) 8

28. The number of non-negative integral solutions of the equation 333 zyx is

a) 120 b) 135 c) 210 d) 520

29. If N is the number of positive integer of solution of 7704321 xxxx Then

a) N is divisible by 4 distinct primes b) N is a perfect cube

c) N is a perfect 4thpower d) N is a perfect 6thPower

30. The total number of seven digit numbers the sum of whose digits is even is:

a) 610x9 b) 510x45 c) 510x81 d) 510x9

31. The number of integral co-ordinates (integral point means both the co-ordinates must be integers) thatthe exactly in the interior of the triangle with vertices (0,0) (0,2a) and (21,0) is

a) 133 b) 190 c) 233 d) 105

32.The number of ways in which candidates A1. A2,.A10can be ranked if A1is always above A2isa) 45 .8! b) 45. 2! c) 45 d) 45 + 8!


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33. In a class of 10 students, there are 3 girls A, B,C the number of different ways that they can be arrangedin a row such that no two of the three girls are consecutive is :

a) 10! b) 7 ! 8! c)!5

!8!7d)

!5

!3!8!3

34. In an election, three wards of a town are convassed by 4, 5 and 3 men respectively. If 20 men volunteer,

then the number of ways they can be allotted to different wards is :

a) 513

417

320 .. CCC b) 5

134

203

17 .. CCC c) 513

417

320 CCC d) 5

134

203

17 CCC

35. There are four oranges, five apples and six mangoes in a fruit basket. The number of ways can a personmake a selection of fruits among the fruits in the basket ?

a) 210 b) 290 c) 209 d) 920

36. The number of positive integers satisfying the inequality 10011

21

n

n

n

nCC is

a) 9 b) 5 c) 8 d) none of these37. A regular polygon has 23 vertices and consequently 23 sides. The number of additional lines need be

drawn so that every pair of vertices will be connected is

a) 253 b) 230 c) 256 d) 276

38. The number of permutations that can be formed by arranging all the letters of the word NINETEEN inwhich no two Es occur together is:

a)!3!.3

!8b) 3

8x!3

!5C c) 3

6x!3

!5C d) 3

6x!5

!8C

39. The greatest number of points of intersection of 8 straight lines and 4 circles is

a) 32 b) 64 c) 128 d) 104

40. If nis a natural number, then the number of non negative integral solutions of nzyx is

a)

2

1nnb)

2

21 nnc)

2

1nnd) None of the above

41. A student takes tests in three different subjects. Maximum marks of a test in any one subject is 15. The

number of ways in which the student can score 15 marks in all the subjects is :a) P(45,15) b) C(45,15)

c) Coefficient of 15x in 3152 .....1 xxx d) None of the above

42. There are nwhite and nblack balls marked 1,2,3,n. The number of ways in which we canarranged these balls in a row so that the neighbouring balls are of different colours is

a) n! b) 2n! c) 2(n!)2 d) (2n!)(n!)2

43. A double decked bus can accommodate 75 passengers, 35 in the upper deck and 40 in the lower deck.

The number of ways can the passengers be accommodated if 5 refuse to sit in the upper deck and 8refuse to sit in the lower deck are


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a) 2562C b) 27

62C c) 240

375 . CC d) None of these

44. A word is formed by using four letters from the letters of the word MATHEMATICS. In how mayways can it be done if exactly two letters are identical and other two are different :

a)2!

4!xx3 2

5C b)2!

4!xx3 2

7C c)2!

4!x2

7C d) None of the above

45. If rn

r

n

r

n

r

n

r

n

r

nCCCCCC

132

121

21

21

11 2..... then the value of rand the minimum

value of nare

a) 11,10 b) 11,12 c) 12,12 d) 12, 13

46. Ifrrr

CCC465

111 then requals


47. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy on e chair each. First thewomen choose the chairs from amongst the chairs marked 1 to 4 and then the men select from theremaining. The number of possible arrangements is

a) 24

36 x CC b) 3

42

4 x PC c) 36

24 x PP d) None of the above

48. A library has a copies of one book, b copies of each of two books, c copies of each of three books andsingle copies of d books. The total number of ways in which these books can be distributed is

a)

!!!

!

cba

dcba

b)

32 !!!.

!32

cba

dcba

c)

!!!

!32

cba

dcba


49. The number of ways in which an examiner can assign 30 marks out of 8 questions, giving not more than2 marks to any question is

a) 721C b) 7

20C c) 722C d) None of these

50. If ........,, zyx are (m+1) distinct prime numbers, the number of factors of ......... zyx n is :

a) 1nm b) mn2 c) mn 21 d) mn 2.

51. The number of ways in which n distinct objects can be put into two different boxes so that no boxremains empty is

a) 12 n b) 12n c) 22 n d) 2

2n

52. The adjacent figure is to be coloured using three different colours. The number of ways in which this canbe done if no two adjacent triangles have the same colours is

a) 81 b) 24c) 6 d) none of these


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53. A student is allowed to select at most n books from a collection of 2n +1 books. If the total number ofways in which he can select at least one book is 255, then the value ofnis

a) 3 b) 4 c) 5 d)6

54. The number of ways in which a mixed double game can be arranged from amongst 9 married couples,if no husband and wife play in the same game is


55. Note the arrangement (1,1) (1,2) (1,3) (2,3) (3,3),(3,4)(4,4). Here we start from (1,1) then increase

one of the co-ordinates by 1 and repeat the same until we reach (4,4). For ILLUSTRATION(1,1)(2,1)(2,2)(2,3)(3,3)(3,4)(4,4) is another such arrangement. The number of such arrangements is

a) 66 P b) 6

6C c)!3!3

!6d) 6

56. In a certain test, there are nquestions. In this test kn2 students gave wrong answers to at least k

questions, where k = 1,2,3,..n If the total number of wrong answers given is 2047, then n is equalto :

a) 10 b) 11 c)12 d)13

57. A person goes in for an examination in which there are four papers with a maximum ofmmarks fromeach paper. The number of ways in which one can get 2mmarks is

a) 332 Cm b) 1421

3

1 2 mmm

c) 34213

1 2 mmm d) 3421

3

1 2 mmm

58. In a plane, there are 37 straight lines of which 13 pass through the point A and 11 pass through the point

B. Besides no three lines pass through one point, no line pass through both points A and B and no twoare parallel. Then the number of intersection points the lines have is equal to


59. The sum 10ji0

ij

j10 C.C is equal to

a) 1210 b) 102 c) 1310 d) 103

60. The number of ordered pairs (m,n), m, n{1,2100} such that m7 + 7nis divisible by 5 is

a) 1250 b) 2000 c) 2500 d) 5000


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61. The number of distinct terms in the expansion of 3n21 x........xx is

a)3

1n C b) 32n C c) 3

3n C d) none of these


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19


SECTION B -MULTIPLE


1. The number of ways of painting the faces of a cube with six different colours is

a) 1 b) 30 c) 6! d) 68 C

2. The product of r consecutive integers is divisible by

a) r b)

1r

1k

k c) r ! d) none of these

3. Sanjay has 10 friends among whom two are married to each other. She wishes to invite 5 of the them fora party. If the married couple refuse to attend separately then the number of different ways in which shecan invite five friends is

a) 58 C b) 2 x 38 C c) 48510 Cx2C d) none of these

4. There are n seats round a table marked 1, 2, 3 .... n. The number of ways in which m ( n) persons cantake seats is

a)n

n P b) )!1m(xCnn

c) !mxCnn d)

1n1n P

5. The number of ways in which 10 candidates A1, A

2.... A

1-0can be ranked so that A

1is always above A

2

is

a) 2

!10b) 8 ! x 210 C c) 210 P d) 210 C

6. In a class tournament when the participants were to play one game with another, two class players fell ill,having played 3 games each. If the total number of games played is 84, the number of participants at the

beginning was

a) 15 b) 30 c)2

6 C d) 48

7. The number of ways of distributing 10 different books among 4 students (S1 - S

4) such that S

1and S

2

get 2 books each and S3and S

4get 3 books each is

a) 12600 b) 25200 c)4

10 C d) !3!3!2!2

!10

8. Ten persons, amongst whom are A, B & C are speak at a functional. The number of ways in which it canbe done of A wants to speak before B, and B wants to speak before C is

a)6

!10b) 21870 c) !3

!10d) 7

10 P


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20

IIT-MATHEMATICS-SET-IV

2 BINOMIAL THEOREM


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21

BINOMIAL THEOREM

DEFINITION OF BINOMIAL EXPRESSION AND BINOMIAL EXPANSION

An expression containing two terms, is called a binomial expression. For examplea + b/x, x + 1/y, a y2etc. are binomial expressions. Expansion of (x + a)nis called BinomialExpansion.

Expression containing three terms are called trinomials. For example x + y + z is a trinomialexpression. In general an expression containing more than two terms is called a multinomial.

Definition of binomial theorem

If n is a positive integer and x, y are two complex numbers, then

n

n n n r r r

r 0

x y C x y

= nC0xn+ nC

1xn 1 y + nC

2xn 2y2+ . . . + nC

nyn . . . (i)

The coefficients nC0, nC

1, . . . , nC

nare called binomial coefficients, while (i) is called the

binomial expansion.

Some important facts regarding Binomial Expansion

1. There are (n + 1) terms in the expansion.

2. The sum of the exponents of x and y in any term of the expansion is equal to n.

3. The binomial coefficients of terms equidistant from the beginning and the end are equal,since nC

r= nC

n r.

4. The term nCrxn ryris the (r + 1)th term from the beginning of the expansion. It is usually

denoted by Tr + 1and is called the general term of the expansion.5. The rth term from the end is equal to the (n r + 2)th term from the beginning, i.e.,

nCn r

+ 1

xr 1 yn r + 1 .

6. If n is even, then the expansion has only one middle term, then

12

th term i.e.,

n n / 2 n / 2n / 2C x y .

If n is odd, then the expansion has two middle terms, the

n 1

2

th term and the

n 3

2

th term

i.e., n 1 /2 n 1 /2n

n 1 / 2C x y

and n 1 /2 n 1 / 2n

n 1 / 2C x y

.

SOME STANDARD EXPANSIONS

1. Consider the expansion

nn n n r r

rr 0

x y C x y

. . . (i)


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22


If we replace y by y in equation (i), we have

n

n rn n r r r

r 0

x y C 1 x y

. . . (ii)

= nn n n n 1 n n 2 2 n n

0 1 2 nC x C x y C x y C 1 y . . . (ii)

2. Adding equations (i) and (ii), we have

n nn n n n 2 2 n n 4 4

0 2 41

C x C x y C x y x y x y2

. . . (iii)

and substracting equations (ii) from (i) we have,

n nn n 1 n n 3 3 n n 5 5

1 3 51

C x y C x y C x y x y x y2

. . . (iv)

3. Putting x = 1 and y = 1 in equation (i), we have

n n n n n n0 1 2 n 1 nC C C C C 2 . . . (v)

Thus, we see that the sum of the binomial coefficients of (x + y)nis 2n.

4. Putting x = 1 and y = 1 in equation (iii) and (iv), we have

n n n n 1 n n n0 2 4 1 3 5C C C 2 C C C

. . . (vi)

5. Putting x = 1and replacing y by x in equation (i), we have

(1 + x)n= nC0+ nC

1x + nC

2x2+ . . .+ nC

nxn . . . (vii)

Replacing x by x in equation (vii), we have

(1 + x)n= nC0 nC

1x + nC

2x2 . . . + nC

n( 1)nxn . . . (viii)

GREATEST BINOMIAL COEFFICIENT

The greatest coefficient depends upon the value of n.

n no. of greatest coefficient (s) Greatest coefficient

Even 1 nCn/2

Odd 2 n 1n

2

C and n 1

n

2

C

(Values of both these coefficeitns are equal)

Clearly greatest binomial coefficient corresponds to the coefficient of middle term.

NUMERICALLY GREATEST TERM OF BINOMIAL EXPANSION

(a + x)n= C0an+ C

1an 1x + . . . C

n 1

a xn 1+ Cnxn


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23

BINOMIAL THEOREM

nr 1 r

nr r 1

T C x n r 1 x

T a r aC

Ifn r 1 x

1r a

, for given a, x and n, then r n 1

a1

x

So numerically greatest term will be Tr + 1

, where r =n 1

a1

x

[ ] denotes the greatest integer function.

Note :

Ifn 1

a1

x

itsel

Thus, we get | x | 2 and | x | 64 1

321 21

. So x 64 64

, 2 2,21 21

SERIES OF BINOMIAL CO-EFFICIENT

Sum of the series by the use of differentiation

Gernally we use the method of differentiation when the coefficient of binomial expansion Ckis

a polynomial in k

Sum of the series by the use of integration

Generally we use integration for the series having terms of the form m kC

rm 1

or of the form

m kCr

m 1 m 2 . . . m j

.

Sum of the series by comparing the co-efficients of some power of x in an expansion.

In this method we use the fact that coefficient of same power of x in an appropriate identity isthe given series.

Sum of the series by equating the real and imaginary parts


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24


2-A BINOMIAL THEOREM


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25

BINOMIAL THEOREM

WORKEDOUT ILLUATRATIONS

In the expansion of (1+x)43, the coefficients of the (2r+1)th and the (r+2)th terms are equal, then the value of r,

is

a) 14 b) 15 c) 16 d) 17

1. If (1-x+x2)n = a0+a

1x+a

2x2+.+a2

nx2nthen a

0+a

2+a

4++a

2n equals

a)2

1(3n+1) b)

2

1(3n 1) c)

2

1(1-3n) d)

2

1+3n

2. The no. of terms in the expansion of (a+b+c)n. nN, is

a) 2n b) 2n c) n d)( n 1)( n 2 )

2

3. The first integral term in the expansion of (is its

a) 2ndterm b) 3rdterm c) 4thterm d) 5thterm

4. The coefficient of a3b4c in the expansion of (1+a+b-c)9is

a) 2.9C7. 7C

4b) 2.9C

2. 7C

3c) 9C

7. 7C

4d) None of these

5. The coefficient of xmin : (1+x)m+ (1+x)m+1+(1+x)n, mn is

a) n+1Cm+1

b) n-1Cm-1

c) nCm

d) nCm+1

6. The value of C02+ 3C12+ 5C22+ to (n+1) terms, is (given that CrnCr

a) 2n-1Cn-1

b) (2n+1). 2n-1Cn

c) 2(n+1). 2n-1Cn

d) 2n-1Cn+(2n+1). 2n-1C

n-1

7. For 1 r n, the value of nCr+ n-1C

r+ n 2C

r+ ..+ rC

ris

a) nCr+1

b) n+1Cr

c) n+1Cr+1

d) None of these

8. The remainder of 7103when divided by 25 is

a) 7 b) 25 c) 18 d) 9


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26


SECTION A -SINGLE


1. The sum of the coefficients in the expansion of 171231 xx is

a) 0 b) 1 c) -1 d) 2

2. The term independent ofxin the expansion of n

n

xxx

1

1 is

a) 22221

20 1....32 nCnCCC b) nCCCC .....321

c) 22221

20 ..... nCCCC d) nnCCCC .....32 321

3. The value of .....642

531 CCC

is equal to

a)1

12

n

n

b)1

2

n

n

c)1

12

n

n

d)1

12

n

n

4. If coefficients of 12 r th and 2r th terms are equal in the expansion of 431 x then the value ofrwill be

a) 14 b) 15 c) 13 d)16

5. IfnCCC ,....., 21 are coefficients in the expansion of

n

n

nxCxCxCx ........11 221 then the

value of 2223

22

21 2.....32 nnCCCC :

a) 2n b) nn11 c) n

nnCn

121 ..1 d) 2n

6. If the second, third and fourth terms in the expansion of nax are 240, 720 and 1080 respectively,,then the value of n is :

a)15 b)20 c)10 d)5

7. The value of .....5.3 22

2

1

2

0

CCC to 1n terms is

a) 112

n

n C b) nn

Cn1212

c) 112.12

n

n Cn D) 11212 12

nn

n

nCnC

8. If the second, third and fourth terms in the expansion of nba are 135, 30 and 1310

respectively, then

a) n = 4 b) n = 7 c) n = 6 d) n = 5

9. Let k and n be positive integers and put kkkkk n.....321S then the value of

mm1m

221m

111m SC.......SCSC is


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27

BINOMIAL THEOREM

a) n1n1 m b) n1n1 m1 c) 1mm n1n1 d) 1mn1

10. If n, r N and 1rn3

r1n C3kC

, then k lies in the interval

a) 3,3 b) ,2 c) ),3[ d) 2,3

11. The number of integral terms in the expansion of 1024

8/12/1 75 isa) 128 b)129 c)130 d)131

12. If the fourth term in the expansion of

6

2/11log

1

xx x is equal to 200 and x >1, thenxis equal to

a) 210 b) 10 c) 410 d) None of the above

13. a, b, c, d are any four consecutive coefficients of any binomial expansion thenc

dc

b

cb

a

ba ,, are in

a) A.P. c) H.P.b) G.P. d) Arithmetic geometric progression

14. If the coefficients of 2nd, 3rd, 4thterms in the expansion of nx 21 are in A.P. then :

a) 0792 2 nn b) 0792 2 nn c) 0792 2 nn d) 0792 2 nn

15. The term independent ofxin the expansion of 81111 11 xtxt is

a)

3

1

156

t

tb)

3

1

156

t

td)

4

1

170

t

td)

4

1

170

t

t

16. If the nineth term in the expansion of 1015log8/1725log 331

31

3

xx

is equal to 180 andx>1 thenx

equals

a) log-10

15 b) log515 c) 15log

ed) None of the above

a) 1 b) 1 c) n d) None of theabove

18. The coefficient of the term independent ofx in the expansion of

10

2/13/13/2

1

1

1

xx

x

xx

xis

a) 210 b) 105 c) 70 d)112

19. The total number of dissimilar terms in the expansion of 321 ..... nxxx is


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28


a) 3n b) 4

3 23 nn c)

6

21 nnnd)

4

1 22 nn

20. The coefficient of nmxm 0 in the expansion of nxxx 1....111 2 is

a) 11

m

n C b) m

n C c)m

1n C d) mnn C

21. Coefficient of 4x in

10

2

3

2

3

x

is :

a)226

405b)

259

504c)

263

450d) None of these

22. If the coefficient of 2x and 3x in the expansion of 93 ax are the same, then the value of a is :

a) 9/7 b) 7/9 c) 9/7 d) 7/9

23. The number of terms in the expansion of 233 44 yxyx is :a) 6 b) 7 c) 8 d) 32

24. If the 4thterm in the expansion of

n

xax

1

is2

5then the values of a and n:

a) are2

1, 6 b) are 1,3 c) are

2

1, 3 d) can not be found

25. If the sixth term in the expansion of

8

102

3/8log

1

xxx

is 5600, then the value ofxis


26. Let pS and qS be the coefficient of px and qx respectively in qpx 1 then :

a) qp SS b) qp Sp

qS

c) qp Sq

pS

d) qp SS

27. The coefficient of 7x in the expansion of 94 11 xx is:

a) 27 b) 24 c) 48 d)-48

28. If the sum of the coefficients in the expansion of 5122 12 xx vanishes, then the values of is :

a) 2 b)-1 c) 1 d) -2

29. If the coefficients of three consecutive terms in the expansion of n

x1 are in the ratio1 : 7 : 42, then the value of nis


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29

BINOMIAL THEOREM

a) 60 b)70 c)55 d)None of these

30. The value ofn

nn

n

n

n

n

nn CCC

81

10.......

81

10.

81

10.

81

10

81

1 23

23

22

2

12 is :

a) 2 b) 0 c) 1/2 d) 1

31. If ,.......1 2210n

n

nxCxCxCCx then

1......

3221

0

n

CCCC n is equal to

a) 12 n b) 1

12 1

n

n

c)1

12

n

n

d) 12 1 n

32. If

n

r

r

r

nxCx

0

1 then

11

2

0

1 1......11n

n

C

C

C

C

C

C

a) !1

1

nn

n

b) !111

nn

n

c) !1

nn

n

d) !1

1

nn

n

33. The sum of the all the coefficients of the expansion of nyx is 1024. Then the greatest coefficient inthe rth term where r=


34 . If the ratio of the 7thterm from the beginning to the seventh term from the end in the expansion of

x

3

3

3

12

is 6

1

then x is :


35. If a be the sum of odd terms and b be the sum of even terms in the expansion of even terms in the

expansion of ,nxa then nx 21 is equal to

a) 22 ba b) 22 ba c) 2

22ba

d) 22 ab

36. The value of6432.3.616.9.158.27.204.181.152.243.63

25.7.18.37186

33

is

a) 10 b) 1 c) 2 d) 20

37. If 4321 ,,, aaaa are the coefficients of any four consecutive terms in the expansion of n

x1 then

43

3

21

1

aa

a

aa

a

is equal to

a)32

2

aaa b) 32

2.21

aaa c) 32

22aa

a d) 32

32aa

a


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30


38. The coefficient of 99x in 199........531 xxxx is

a) 1+2+3+..+99 b)1+3+5+..+99 c)1.3.599 d) None of the above

39. If nxxf then the value of

!

1......

!2

1

11

21

n

ffxff

n

where xf r denotes the rth order

derivative of xf with respect ofxisa) n b) n2 c) 12 n d) None of these

40. If mn 121466 andf is the fractional part of m,thenfmis equal to

a) 115 n b) 1220 n c) n25 d) None of these

41. The coefficient of 24t in 2412122 t1t1t1 is

a) 3C612

b) 1C612

c) 612

C d) 2C612

42. If 210 C,C,C nC are co-efficients in the expansion of (1+x)nthen the value of

n21

n1n2110

C.........CC

CC......CCCC is equal to

a)

!n

1n nb) !1n

nn

c)

!1n

1n n

d) none of these

43. If n is an even positive integer and x > 0, then the condition that the greatest term in the expansion of(1+x)nmay have the greatest co-efficient also, is

a)2n

n

< x c, the circle will meet the x-axis at two distinct

points, say (x1, 0) and (x

2, 0) where x

1+ x

2= 2g and x

1x

2= c.

The intercept made on x-axis by the circle = |x1 x

2| = cg2 2 .

In the similar manner if f 2>c, intercept made on y-axis = cf2 2 .

TANGENTS AND NORMALS

Definition :

A tangent to a curve at a point is defined as the limiting positions of a secant obtained by

joining the given point to another point in the vicinity on the curve as the second point tends to

the first point along the curve or as the limiting position of a secant obtained by joining two

points on the curve in the vicinity of the given point as both the points tend to the given point.

Two tangents, real or imaginary, can be drawn to a circle from a point in the plane. Thetangents are real and distinct if the point is outside the circle, real and coincident if the point is

on the circle, and imaginary if the point is inside the circle.

The normal to a curve at a point is defined as the straight line passing through the point and

perpendicular to the tangent at that point. In case of a circle, every normal passes through the

centre of the circle.

Chord of Contact :

From a point P(x1, y1) two tangents PA and PB can be drawn to the circle. The chord ABjoining the points of contact A and B of the tangents from P is called the chord of contact of

P(x1, y

1) with respect to the circle.

Equations of Tangents and Normals :

If S = 0 be a curve than S1= 0 indicate the equation which is obtained by substituting

x=x1and y = y

1in the equation of the given curve, and T = 0 is the equation which is obtained

by substituting x2= xx1,

y2= yy1, 2xy = xy

1+ yx

1, 2x = x + x

1, 2y = y + y

1in the equation

S = 0.

If S x2+ y2+ 2gx + 2fy + c = 0 then S1 x

12+ y

12+ 2gx

1+ 2fy

1+ c, and

T xx1+ yy

1+ g (x + x

1) + f (y + y

1) + c

* Equation of the tangent to x2+ y2+ 2gx + 2fy + c = 0 at A(x

1, y

1) is xx

1+ yy

1+ g(x + x

1)

+ f (y + y1) + c = 0.

* The condition that the straight line y = mx + c is a tangent to the circle x2+ y2= a2is

c2= a2 (1 + m2) and the point of contact is (a2m/c, a2/c) i.e. y = mx a 2m1 is always a

tangent to the circle x2+ y2= a2whatever be the value of m.

* The joint equation of a pair of tangents drawn from the point A(x1, y

1) to the circle


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37

CIRCLE

x2+ y2 + 2gx + 2fy + c = 0 is T2= SS1.

* The equation of the normal to the circle x2+ y2+ 2gx + 2fy + c = 0 at any point (x1, y

1) lying

on the circle is gx

xx

1

1

= fy

yy

1

1

.

In particular, equations of the tangent and the normal to the circle x2+ y2= a2at (x1, y

1)

are xx1

+ yy1

= a2; and1x

x=

1y

yrespectively..

* The equation of the chord of the circle S 0, whose mid point (x1, y

1) is T = S

1.

* The length of the tangent drawn from a point (x1,y

1) outside the circle S = 0, to the circle is

1S .

Director Circle:

The locus of the point of intersection of perpendicular tangents is called director circle.

If 2)x( + 2)y( = r2 is the equation of a circle then its director circle is

2)x( + 2)y( = 2r2

The position of a point with respect to a circle :

The point P(x1, y1) lies outside, on, or inside a circle S x2+ y2+ 2gx + 2fy + c = 0, according

as S1 x

12+ y

12+ 2gx

1+ 2fy

1+ c > = or < 0.

p = )m1(

b1m22

.......(2)

or, by means of (1) p = )m1(

m792

.

Hence, m is given by (9 7m)2= 13(1 + m2) or 18m2 63m + 34 = 0

or (3m 2)(6m 17) = 0

The gradients are 2/3 and 17/6; the tangents are

y 10 =3

2(x 5) and y 10 =

6

17(x 5)

or 2x 3y + 20 = 0 and 17x 6y 25 = 0. ........(3)

The corresponding normals are

y 1 = 2

3(x + 2) and y 1 =

17

6(x + 2)


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38


or 3x + 2y + 4 = 0 and 6x + 17y 5 = 0.

The coordinates of the point of contact of the first tangent in (3) are obtained by solving

2x 3y + 20 = 0 and 3x + 2y + 4 = 0;

the coordinates are (4, 4).

Similarly, the point of contact of the second tangent is (7/5, 1/5)

RADICAL AXIS

The radical axis of two circles is the locus of a point from which the tangent segments to the

two circles are of equal length.

Equation to the Radical Axis

In general S S = 0 represents the equaion of the radical Axis to the two circles i.e.

2x(g g ) + 2y (f f) + c c = 0 where S= x2+ y2+ 2gx + 2fy + c = 0 and S= x2+

y 2 + 2 g x

+2fy + c = 0

* If S = 0 and S = 0 intersect in real and distinct point then S S = 0 is the equation of the

common chord of the two circles.

* If S = 0 and S = 0 touch each other, then S S = 0 is the equation of the common tangent

to the two circles at the point of contact.

(-g,-f) (-g -f ),

Common tangent

(-g,-f) (-g -f ),

Common chord

The radical axes of three circles, taken in pairs, are concurrent

Let the equations of the three circles S1, S

2and S

3be

S1 x2+ y2+ 2g

1x + 2f

1y + c

1= 0,

S2 x

2

+ y

2

+ 2g2x + 2f2y + c2= 0and S

3 x2+ y2+ 2g

3x + 2f

3y + c

3= 0. ........(1)

Now, by the previous section, the radical axis of S1and S

2is obtained by subtracting the

equations of these circles ; hence it is

S1 S

2= 0 .......(2)

Similarly, the radical axis of S2and S

3is

S2 S3= 0 .......(3)

The lines (2) and (3) meet at a point whose coordinates say, (X, Y) satisfy


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39

CIRCLE

S1 S

2= 0 and S

2 S

3= 0;

hence the coordinates (X, Y) satisfy

(S1 S

2) + (S

2 S

3) = 0;

that is , (X, Y) satisfy S1 S

3= 0 ........(4)

But (4) is the radical axis of the circles S1and S

3and hence the three radical axes are concur-

rent. The point of concurrency of the three radical axes is called the radical centre.

FAMILY OF CIRCLES :

* If S x2+ y2+ 2gx + 2fy + c = 0 and S x2+ y2+ 2gx + 2fy + c = 0 are two intersecting

circles , then S + lS = 0, l 1, is the equation of the family of circles passing through the

points of intersection S = 0 and S = 0.

* If S x2+ y2+ 2gx + 2fy + c = 0 is a circle which is intersected by the straight line

ax + by + c = 0 at two real and distinct points, then S + l = 0 is the equation of the family

of circles passing through the points of intersection of S = 0 and = 0. If = 0 touches

S = 0 at P, then S = l = 0 is the equation of the family of circles, each touching = 0 at P.

* The equation of a family of circles passing through two given points (x1, y

1) and (x

2, y

2) can be

written in the form.

(xx1) (x x

2) + (y y

1)(y y

2) + l

1yx

1yx

1yx

22

11 = 0 where l is a parameter..

* The equation of the family of circles which touch the line y - y1

= m(x x1

) at (x1

, y1

) for any

value of m is (x x1)2+ (y y

1)2+ l(y - y

1 m (x x

1)) = 0.

THE CONDITION THAT TWO CIRCLES SHOULD INTERSECT

A necessary and sufficient condition for the two circles to intersect at two distinct points isr

1+ r

2> C

1C

2> |r

1 r

2|, where C

1, C

2 be the centres and r

1, r

2 be the radii of the two circles.

External and Internal Contacts of Circles :

If two circles with centres C1(x

1, y

1) and C

2(x

2, y

2) and radii r

1and r

2respectively, touch each

other externally, C1C

2= r

1+ r

2. Coordinates of the point of contact are

A

21

1221

21

1221

rr

yryr,

rr

xrxr.


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40


The circles touch each other internally if C1C

2= r

1 r

2.

Coordinates of the point of contact are

T

21

1221

21

1221

rr

yryr,

rr

xrxr.

Common Tangents to Two Circles :

(a) The direct common tangents to two circles meet on the line of centres and divide itexternally in the ratio of the radii.

(b) The transverse common tangents also meet on the line of centres and divide it internallyin the ratio of the radii.

Note :

(i) When two circles are real and non-intersecting, 4 common tangents can be drawn.

(ii) When two circles touch each other externally, 3 common tangents can be drawn to the circle.

(iii) When two circles intersect each other, two common tangents can be drawn to the circles.

(iv) When two circles touch each other internally 1 common tangent can be drawn to the circles.

ORTHOGONAL CIRCLES

Two circles are said to be orthogonal if the tangents to the circles at either point of intersectionare at right angles.

Q2Q1

r2r1

(-g,-f) (-G,-F)

A

B

In fig. Q1and Q

2are the centres of the circles

S1 x2+ y2+ 2gx + 2fy + c = 0 .......(1)

S2 x2+ y2+ 2Gx + 2Fy + C = 0 ........(2)

the circles, S1and S

2, intersect at A and B.

The tangent at A to the circles S1is perpendicular to the radius Q1A, and the tangent at A to S2is perpendicular to the radius Q

2A. Hence, if the two tangents are at right angles, then the radii


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41

CIRCLE

Q1A and Q

2A must also be at right angles. Accordingly, the condition that S

1and S

2should be

orthogonal is that Q1AQ

2should be 90; by Pythagoras theorem this condition is equivalent

to

Q1Q

22= Q

1A2+ Q

2A2= r

12+ r

22 .......(3)

(g G)2+ (f F)2= g2+ f2 c + G2+ F2 C

or, on simplification, 2(gG + fF) = c + C. .......(4)

Since, AQ2is perpendicular to the radius Q

1A, the tangent at A to the circle S

1passes through

the centre of the circle S2; similarly, the tangent at A to S

2passes through the centre of S

1.

In numerical examples the procedure of solution should be based on the condition expressed

by (3).


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42

IIT-MATHEMATICS SET IV

3-A CIRCLES


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43

CIRCLES

WORKEDOUT ILLUATRATION

ILLUSTRATION : 01

If 62, is an interior point of the circle 012822 pyxyx and the circle neither cuts nortouches any one of the axes of co-ordinates then

A) p (36,47) B) p (16, 47) C) p (16, 36) D) none of theseSolution :

We have 012822 pyxyx then centre and radius of the circle are (4,6) and p52

respectively.

Circle neither cuts nor touches any one of the axes of coordinates then

x coordinates of centre > radius i.e., p 524

36p ..(1)

& y-coordinate of centre > radius

p 526 ,

16p .(2)

D is interior point of the circle then

CD < radius p 525

47p (3)from (1), (2) & (3) we obtain

4736 p

p (36, 47).

ILLUSTRATION : 02

The abscissas of two points A and B are the roots of the equations 02 22 baxx and theirordinates are the roots of the circle with AB as diameter is

A) 2222 qpba B) 22 pa C) 22 qb D) None of these


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45

CIRCLES

parametric angles differ by3

is

A) 222 324 ryx B) 13 22 yx C) 222 32 ryx D) 2 23 x y 42

Solution

Circle is 222222 sinrcosryx

222 ryx

Equation of tangent at is

rsinycosx .(1)

and at

3is

rsinycosx

33

rsincosysincosx

2

3

2

1

2

3

2

1

rcosysinxsinycosx 233 rcosysinxr 23

or 3

rcosysinx .(2)

squaring and adding (1) & (2) then we get

3

4 2

22 ryx 222 43 ryx

ILLUSTRATION : 05

If one circle of a co-axial system is 02222 cfygxyx and one limiting point is (a, b) then

equation of the radical axis will be

A) 022 bacybfxag B) 0202 222 bacygfxbagC) 022 22 bacfygx D) None of these

Solution :

Given circle 1

S 02222 cfygxyx .(1)

and let a circle whose limiting point is (a, b)

02222

'cy'fx'gyxCentre of circle (a, b) and radius = 0


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022 'cf'g '

22 'f'g'c 22a'c

Equation of the second circle is

022 2222

2 babyaxyxS (2)

From (1) and (2), Equation of radical axis is

021

SS

022 22 bacybfxag

ILLUSTRATION : 06

If the circle 02222 cfygxyx cuts each of the circles ,yx 0422 and

024222 yxyx at the extremities of a diameter, then

A) c = -4 B) 1 cfg C) 1722 cfg D) 6gf

Solution :

Let S 02222 cfygxyx

0422

1 yxS

0108622

2 yxyxS

024222

3 yxyxS

Common chords are

04221

cfygxSS ..(1)

01082622

cyfxgSS ..(2)

0242223

cyfxgSS ..(3)

For cutting at the extremities of diameter, chords (1), (2) & (3) pass through the centers of321

S&S,S

respectively, then

36204 g,c + 010482 cf

& 02242122 cfg

after solving 324 f,g,c

ILLUSTRATION : 07

The locus of the point of intersection of the lines

2

2

1

1

t

t

ax and t

at

y 1

2

represents t being a

parameter.


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47

CIRCLES

A) Circle B) Parabola C) Ellipse D) Hyperbola

Solution :

2222

22

2

2241

1

attt

ayx

222 ayx

Radical axis or common chord

021

CC

0884 yx or 022 yx or 2 = 0

Centres of1

C and2

C are (1,2) and (-1, -2) respectively..

Slope of 21CC =2

11

22

Slope of ( 0L ) is2

1

slope of 21CC slope of 10 L

Hence L is perpendicular to the line joining centers of and .

ILLUSTRATION : 08

The equation of a circle is .yx 422 The centre of the smallest circle touching this circle and the line

25yx has the coordinates

(a)

22

7

22

7, (b)

2

3

2

3, (c)

22

7

22

7, (d) None of these

Solution :

Here, OB = radius = 2.

The distance of (0,0) from 25yx is 5.

the radius of the smallest circle =2

3

2

25

and2

7

2

32 OC

The slope of OA = 1= tan

1 1

cos ,sin

2 2


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C 0 OC .cos ,0 OC .sin =7 7

,2 2 2 2

00, O2

C

B

A 25yx

ILLUSTRATION : 09

The equation of the locus of the middle point of a chord of the circle yxyx 222 such that thepair of lines joining the origin to the point of intersection of the chord and the circle are equally inclinedto the x-axis is

A) 2yx B) 2yx C) 12 yx D) none of these

Solution :

Solving mxy and 02222 yxyx , we get

022222 mxxxmx ,x 0

2

1

12

m

m

02222

yxyx

O

B mxy

mxy A

x

Similarly, solving mxy and the equation of the circle, we get

21

120

m

m,x

221

12

1

12

m

mm,

m

mA and

221

12

1

12

m

mm,

m

mB

Let the middle point of AB be , . Then


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49

CIRCLES

221

12

1

12

2

1

m

m

m

m and

221

12

1

12

2

1

m

mm

m

m

2

2

21

2

1

2

m

m,

m

; Eliminating mfrom these, 2 .

ILLUSTRATION : 10

Let1

L be a straight line passing through the origin and2

L be the straight lien 1yx . If the inter-

cepts made by the circle 0322 yxyx on and are equal then which of the following equations

can represent ?

A) x y 0 B)x y 0 C) x 7 y 0 D)x 7 y 0

Solution :

Let the line be 221 xmy .

Its distance from the centre 00, =2

1

2

12

m

m

So, the length of the intercept =

2

2

2

1

2

12

12

m

m

..(1)

The distance of the lien from the other centre (4, 0) =2

1

2

124

m

mm

the length of the intercept in this case

2

2

2

1

2

1

652

m

m..(2)

(1) and (2) are equal. Hence,

22

2

2

14

1125

14

141

m

m

m

m

4

14

14112

2

22

m

mm

017 2 mm

* * *


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50


SECTION A -SINGLE


1. The two conics

ax2+ 2hxy + by2= c and px2+ 2kxy + qy2= r intersect in four concyclic points, then

A) k

h

qp

ba

B) h

k

qp

ba

C) k

h

qp

ba

D) h

k

qp

ba

2. The values of for which the circle

x2+ y2+ 6x + 5 + (x2+ y2- 8x + 7) = 0 dwindles into a point are

A)3

243 B)

3

246 C)

3

234 D) none

3. If 2x2+ pxy + 2y2+ (p - 4) x + 6y - 5 = 0 is the equation of a circle, then its radius is

A) 2 3 B) 2 2 C) 232

1D) none

4. Two circles x2+ y2- 4x - 6y - 8 = 0 and x2+ y2- 2x - 3 = 0 are such that they

A) touch B) cut C) one lies inside the other D) do not intersect

5. Radius of a circle is 5. It cuts x-axis at two points at a distance 3 from the origin. Its centre is

A) (0, 4) B) (0, 3) C) (0, 5) D) none

6. 3x + 4y - 7 = 0 is common tangent at (1, 1) to two equal circles of radius 5. Their centres are the points

A) (4, 5), (-2, -3) B) (4, -3), (-2, 5) C) (4, -5, (-2, 3) D) none

7. The equation of the circumcircle of the triangle formed by the lines y+x=6, y-x=6 and y = 0 is

A) x2+ y2- 4y = 0 B) x2+ y2+ 4x = 0 C) x2+ y2- 4y - 12 = 0 D) x2+ y2+ 4x = 12

8. The equation of the image of the circle x2+ y2+ 16x - 24y + 183 = 0 by the line mirror 4x+7y+13 = 0

is

A) x2+ y2+ 32x - 4y + 235 = 0 B) x2+ y2+ 32x + 4y - 235 = 0

C) x2+ y2+ 32x - 4y - 235 = 0 D) x2+ y2+ 32x + 4y + 235 = 0

9. The circle passing through three distinct points (1, k), (k, 1) and (k, k) passes through the points

A) (1, 1) B) (-1, -1) C) (-1, 1) D) (1, -1)

10. If the lines1 1 1

a x b y c =0 and2 2 2

a x b y c 0 cut the coordinate axes in concyclic points, then

A)1 2 1 2

a a b b B)1 1 2 2

a b a b C)1 2 1 2

a / a b / b D) none

11. If a circle passes through the points where the lines 3lx - 2y - 1 = 0 and 4x - 3y + 2 = 0 meet the


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CIRCLES

coordinate axes then =

A) -1 B) -1/2 C) 1/2 D) 1

12. Four distinct points (2, 3), (1, 0), (0, 1) and (0, 0) lie on a circle for

A) all integral values of l B) 0 < < 1 C) < 0 D) only one value of

13. If a chord of the circle x2+ y2= 8 makes equal intercepts of length a on the coordinate axes then

A) |a| < 8 B) |a| < 4 C) |a| < 4 D) |a| > 4

14. The end A of diameter AB of a circle is (1, 1) and B lies on the line x + y - 3 = 0. The locus of the centre

of the circle is

A) x - y = 1 B) x + y = 1 C) 2x + 2y - 5 = 0 D) 2x - 2y - 5 = 0

15. If one end of a diameter of a circle x2+ y2- 4x - 6y + 11 = 0 is (3, 4), then the other end is

A) (-1, -1) B) (1, 2) C) (4, 3) D) none

16. The co-ordinates of A and B are (1 1

x , y ) and (2 2

x , y ) and O is the origin. If circles e described on OA,

OB as diameters, then length of common chord is

A)1 2 2 1

( x y x y ) /AB B)1 1 2 2

( x y x y ) /AB

C)1 2 2 1

( x y x y ) /AB D)1 1 2 2

( x y x y ) /AB

17. If the abscissas and ordinates of two points P and Q are the roots of the equations 2 2x 2ax b =0 and2 2x 2 px q =0 respectively, then equation of the circle with PQ as diameter is

A) 2 2 2 2x y 2ax 2 py b q =0 B) 2 2 2 2x y 2ax 2 py b q =0

C) 2 2 2 2x y 2ax 2 py b q 0 D) 2 2 2 2x y 2ax 2 py b q 0

18. Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ

intersect at a point X on the circumference of the circle, then 2r equals

A) PQ.RS B) PQ RS 2 C)

2PQ.RS

PQ RS d)2 2

PQ RS 2

19. If a straight line through C (- 8 , 8) making an angle of 1350with the x-axis cuts the circle x = 5 cos

, y = 5 sin in points A and B, then the length of AB is

A) 6 B) 8 C) 10 D) none

20. A square is inscribed in the circle x2+ y2- 2x + 4y + 3 = 0.

Its sides are parallel to the co-ordinates axes. Then one vertex of the square is

A) (1 + 2 , -2) B) (1 - 2 , - 2) C) (1, -2 + 2 ) D) none of these


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21. A square is formed by following two pairs of straight lines y2- 14y + 45 = 0 and x2-8x+12=0. A circle

inscribed in it. The centre of circle is

A) (7, 4) B) (4, 7) C) (3, 7) D) (3/8, 4)

22. A, B, C are three points on the unit circle x2+ y2= 1 whose parametric angles, and respectively.

Through a point P (-1, 0) on the circle chords PA, PB, PC are drawn whose lengths are in G.P. then cos

2

, cos

2

, cos

2

are in

A) A.P. B) G.P. C) H.P. D) None of these

23. Two vertices of an equilateral triangle are (-1, 0) and (1, 0) and its third vertex lies above the x-axis. The

equation of its circumcircle is

A)2 2 2xx y 1

3 =0 B)

2 2 2xx y 13

=0

C) 2 2 2 yx y 1

3 =0 D)

2 2 2 yx y 13

=0

24. (-1, 2) is the vertex of an equilateral triangle whose centroid is (1, 1), then the equation of its circum-

circle is

A) x2+ y2+ 2x + 2y - 3 = 0 B) x2+ y2+ 2x - 2y - 3 = 0

C) x2+ y2- 2x - 2y - 3 = 0 D) none

25. If the equation of incircle of an equilateral triangle is x2+ y2+ 4x - 6y + 4 = 0, then the equation of

circumcircle of the triangle is

A) (-2, -3), 6 B) (-2, 3), 6 C) (2, 3), 6 D) none

26. The triangle PQR is inscribed in the circle x2+ y2= 25. If Q and R have co-ordinates (3, 4) and (-4, 3)

respectively, then QPR is equal to

A) /2 B) /3 C) /4 D) /6

27. The vertices of a triangle ABC are the points (6, 0), (0, 6) and (7, 7). The equation of the circle inscribed

in the triangle is

A) x2+ y2- 9x - 9y + 36 = 0 B) x2+ y2- 9x - 9y - 36 = 0

C) x2+ y2- 9x + 9y + 36 = 0 D) x2+ y2+ 9x - 9y + 36 = 0

28. ABCD is a square of side a. The centre of the circle which circumscribes the square on taking AB and

AD as axes is

A) (a, -a) B) (-a, a) C) (a/2, a/2) D) none

29. The equation of a circle with centre at origin and passing through the vertices of an equilateral triangle

whose median is of length 3a is


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CIRCLES

A) x2+ y2= a2 B) x2+ y2= 4a2 C) x2+ y2= 9a2 D) none

30. A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is

A) a2/ 3 B) a2/4 C) a2/6 D) none

31. The lines 3x - 4y + 4 = 0 and 3x - 4y - 5 = 0 are tangents to the same circle. The radius of this circle is

A) 9/5 B) 9/10 C) 1/5 D) 1/10

32. Equation of the circle whose radius is 5 and which touches externally the circle x2+ y2- 2x

- 4y - 20 = 0 at the point (5, 5) is

A) (x-9)2+ (y - 6)2= 52 B) (x-9)2+ (y-8)2= 52C) (x - 7)2+ (y - 3)2= 52 D) none

33. Two circles each of radius 5 units touch each other at (1, 2). If the equation of their common tangents is

4x + 3y = 10, then the centres of the two circles are

A) (3, 4), (-1, 0) B) (5, 7), (-3, -3) C) (5, 5), (-3, 1) D) none of these

34. The centre of the two circles each of radius 13 units and having a common tangent

5x + 12y - 17 = 0 at (1, 1) are

A) (-6, -13) and (8, 15) B) (6, 13) and (-4, -11) C) (5, 12) and (-3, -10) D) none of these

35. A variable circle passes through a fixed point A (a, b) and touches the axis of x. Locus of the other end

of the diameter through A is

A) circle B) parabola C) ellipse D) none of these

36. If the two circles x2+ y2= 9 and x2+ y2- 8x - 6y +2= 0 have exactly two common tangents, then the

number of integral values of is

A) 2 B) 8 C) 9 D) none

37. Locus of a point which moves such that sum of the squares of its distances from the sides of a square of

side unity is 9, is

A) straight line B) circle C) parabola D) none

38. The locus of a point which moves such that sum of the squares of its distances from the three vertices of

a triangle is constant is

A) circle B) straight line C) ellipse D) none

39. The equation x = a cos + b sin and y = a sin - b cos , 0 2together represent

A) parabola B) straight line C) ellipse D) circle

40. Locus of the centre of the circle which always passes through the fixed points (a, 0) and (-a,0) is


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54


A) x =1 B) x + y = 6 C) x + y = 2a D) x = 0

41. Circle are drawn through the point (-5, 0) to cut the x-axis on +ive side and making an intercept of 10

units on x-axis. Locus of the centre of such circles is

A) x = 0 B) y = 0 C) x + y = 0 D) x - y = 0

42. The equation x2+ y2+ 4x + 6y + 13 = 0 represent is

A) a circle B) a pair of two distinct straight lines

C) a pair of coincident straight lines D) a point

43. The line joining (5, 0) to (10 cos q, 10 sinq) is divided internally in the ratio 2 : 3 at P. If q varies, then the

locus of P is

A) a pair of straight lines B) a circle C) a straight line D) none of these

44. Let AB be a chord of the circle x2+ y2= r2subtending a right angle at the centre. Then the locus of thecentroid of the triangle PAB as P moves on the circle is

A) a parabola B) a circle C) an ellipse D) a pair of straight lines

45. If (2, 5) is an interior point of the circle x2+ y2- 8x - 12y + k = 0 and the circle neither cuts nor touches

any one of the axes of co-ordinates, then

A) k (36, 47) B) k (16, 47) C) k (16, 36) D) none of these

46. The point (, 4) lies outside the circles S1

= x2+ y2+ 10x = 0 and S2

= x2+ y2- 12x + 20 = 0 then

belong to

A) (-, -8) (-2, ) B) (-8, -2) C) (-, -8) (-2, 6) (6, ) D) none

47. The point on the circle

x2+ y2- 2x - 4y - 11 = 0 which is farthest from the origin is

A)8 4

1 ,25 5

B)

4 81 ,2

5 5

C)

8 42 ,1

5 5

D) none

48. If the equation x cos+ y sin= p represents the equation of common chord APQB of the circles x2+

y2= a2and x2+ y2= b2(a > b) then AP =

A) 2 2 2 2a p b p B) 2 2 2 2a p b p

C) 2 2 2 2a p b p D) 2 2 2 2a p b p

49. The common chord of the circles x2+ y2- 4x - 4y = 0 and x2+ y2- 16 = 0 subtends at the origin an angle

equal to

A) 300 B) 450 C) 600 D) 900


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CIRCLES

50. A diameter of x2+y2-2x-6y+6=0 is a chord to circle centre (2, 1), then radius of the circle is

A) 1 B) 2 C) 3 D) 4

51. If the line lx + my = 1 be a tangent to the circle x2+ y2= a2, then the point (l, m) lies on

A) ellipse B) parabola C) circle D) none

52. The angle between two tangents from the origin to the circle (x - 7)2+ (y + 1)2= 25 is

A) /3 B) /6 C) /2 D) 0

53. The condition so that the line (x + g) cos + (y + f) sinq = k is a tangent to x2+y2+2gx+2fy+c=0 is

A) g2+ f2= c + k2 B) g2 + f2= c2+ k C) g2+ f2= c2+ k2 D) g2+ f2= c + k

54. The angle between a pair of tangents drawn from a point T to the circle

x2+ y2+ 4x - 6y + 9 sin2a + 13 cos2a = 0 is 2a. The equation of the locus of the point T is

A) x2+ y2 +4x - 6y + 4 = 0 B) x2+ y2+ 4x - 6y - 9 = 0

C) x2+ y2+ 4x - 6y - 4 = 0 D) x2 + y2+ 4x - 6y + 9 = 0

55. From any point on the circle x2+ y2= a2tangents are drawn to the circle x2+ y2= a2sin2. The angle

between them is

A) /2 B) C) 2 D) none of these

56. If 3x + y = 0 is a tangent to a circle which has its centre at the point (2, -1), then the equation of the other

tangent to the circle from the origin is

A) x - 3y = 0 B) x + 3y = 0 C) 3x - y = 0 D) x + 2y = 0

57. The length of the chord of the circle x2+ y2= 25 joining the points, tangents at which intersect at an angle

of 1200is

A) 5/2 B) 5 C) 10 D) none of these

58. If the two circles x2+ y2+ 2gx + 2fy = 0 and x2+ y2+ 2x + 2y = 0 touch each other, then

A) f2+ g2= 21f +

2

1g B) ff1= 1gg C) 1 1f / f g / g D) none of these

59. A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. The

distances from the end points of the side AB to the line touching the circle at the origin O are equal to p

and q respectively. The diameter of the circle is

A) p (p + q) B) q (p + q) C) p + q D) 1/2 (p + q)

60. P and Q are two symmetrical points about the tangent at origin to the circle x2+y2-x+y=0.

If P be (-5, 6), then Q is

A) (6, 5) B) (5, 6) C) (6, -5) D) (-6, 5)


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56


61. Tangents drawn from the point (4, 3) to the circle x2+ y2- 2x - 4y = 0 are inclined at an angle.

A) /6 B) /4 C) /3 D)/2

62. Tangents are drawn to the circle x2+ y2- 2x - 4y - 4 = 0 from the point (1, 7), then slopes are

A) 4/3 B) 3/4 C) 1, 2 D) 3, 0

63. If a > 2b > 0 then the positive value of m for which y = mx - b 21 m is a common tangent to x2+ y2

= b2and (x - a)2+ y2= b2is

A) 2 22b

a 4bB)

2 2a 4b

2b

C)

2b

a 2bD)

b

a 2b

64. The number of tangents that can be drawn from the point (0,1) to the circle x2+y2-2x-4y=0 is

A) 0 B) 1 C) 2 D) none

65. The equation of the circle which has a tangent 2x - y - 1 = 0 at (3, 5) on it and with the centre on x + y

= 5, is

A) x2+ y2+ 6x - 16y + 28 = 0 B) x2+ y2- 6x + 16y - 28 = 0

C) x2+ y2+ 6x + 6y - 28 = 0 D) x2+ y2- 6x - 6y - 28 = 0

66. Equation of a circle touching the line |x - 2| + |y - 3| = 4 is (x -2)2+ (y-3)2=R2where R2=

A) 4 B) 8 C) 10 D) 12

67. A variable circle always touches the line y = x and passes through the point (0, 0). The common chords

of above circle and x2+ y2+ 6x + 8y - 7 = 0 will pass through a fixed point whose co-ordinates are

A) (1, 1) B) (2, 2) C) (1/2, 1/2) D) none

68. The equation of a circle which has its centre on the positive side of x-axis and cuts off a chord of length

2 along the line - x = 0 and also touches the line y = x is

A) x2+ y2- 4x + 1 = 0 B) x2+ y2- 4x +2 = 0 C) x2+ y2- 8x + 8 = 0 D) x2+ y2- 8x + 4 = 0

69. The locus of the point of intersection of tangents to the circle x = a cosq, y = a sinq at the points, whoseparametric angles differ by p/2, is

A) straight line B) circle C) ellipse D) none

70. If the tangent from a point P to the circle x2+ y2= 1 is perpendicular to the tangent from P to x2+ y2=

3 then the locus of P is a circle of radius

A) 4 B) 3 C) 2 D) none

71. The locus of the point of intersection of the perpendicular tangents of x2+ y2= 4 is

A) x2+ y2= 8 B) x2+ y2= 12 C) x2+ y2= 16 D) x2+ y2= 4


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72. If1 2, be the inclination of tangents with x-axis drawn from the point P to the circle x2+ y2= a2,

then the locus of P, if given that cot1 2

cot = c is

A) c (x2- a2) = 2xy B) c (x2- a2) = y2- a2 C) c (y2- a2) = 2xy D) none

73. The length of the chord joining the points (4 cos, 4 sin) and [4 cos (+600), 4 sin(+ 600)] of the circle

x2

+ y2

= 16 is

A) 4 B) 6 C) 2 D) 8

74. If the circle x2+ y2+ 2gx + 2fy + c = 0 is touched by y = x at P such that OP = 6, then the value of c is

A) 36 B) 144 C) 72 D) none of these

75. Two tangents OA and OB are drawn to the circle x2+ y2+ 4x + 6y + 12 = 0 from origin O. The circum-

radius of triangle OAB is

A) 1/2 B) 1 C) 2 D) 1/2

76. Circles are drawn through the point (2, 0) to cut intercept of length 5 units on the x-axis. If their centres

lie in the first quadrant, then their equation is

A) x2+ y2- 9x + 2fy + 14 = 0 B) 3x2+ 3y2+ 27x - 2fy + 42 = 0

C) x2+ y2- 9x - 2fy + 14 = 0 D) x2+ y2- 2fx - 9y + 14 = 0

77. Tangent to the parabola y=x2+6 at (1,7) touches the circle x2+y2+16x+12y+c=0 at the point

A) (-6, -9) B) (-13, -9) C) (-6, -7) D) (13, 7)

78. The equation of the circle which touches both the axes and the straight line 4x + 3y = 6 in the first

quadrant and lies below it, is

A) 4x2+ 4y2- 4x - 4y + 1 = 0 B) x2+ y2- 6x - 6y + 9 = 0

C) x2+ y2- 6x - y + 9 = 0 D) 4 (x2+ y2- x - 6y) + 1 = 0

79. The equation of the circle passing through (2, 1) and touching co-ordinate axes is

A) x2+ y2- 2x - 2y + 1 = 0 B) x2+ y2+ 2x + 2y + 1 = 0

C) x2+ y2- 2x - 2y - 1 = 0 D) x2+ y2+2x + 2y - 1 = 0

80. The equation of common tangent to the circles

x2+ y2+ 14x - 4y + 28 = 0 and x2+ y2- 14x + 4y - 28 = 0 is

A) x = 7 B) y = 7 C) 7x - 2y + 14 = 0 D) 2x - 7y + 14 = 0

81. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes one

complete roll on x-axis along the positive direction of x-axis, then its equation in the new position is

A) x2 +y2+20px - 10y + 100p2= 0 B) x2+ y2+ 20px + 10y + 100p2= 0

C) x2+ y2- 20px -10y + 100p2= 0 D) none of these


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82. The equation of the circle passing through the intersection of the circles

x2+ y2- 6x + 2y + 4 = 0, x2+ y2+ 2x - 4y - 6 = 0 and having its centre on the line y = x is

A) 3 (x2+ y2) - 5x - 5y + 2 = 0 B) 7(x2+ y2) - 10x - 10y - 12 = 0

C) x2+ y2- 2x - 2y + 1 = 0 D) x2+ y2- 6x - 6y + 12 = 0

83. The equation of the circle which passes through the origin and the points of intersection of the circle x2

+ y2= 4 and the line x + y = 2 is

A) x2+ y2= 4 (x + y) B) x2+ y2= 2 (x + y) C) x2+ y2= 3 (x + y) D) x2+ y2= (x + y)

84. If the two curves ax2+ 2hxy + by2+ 2gx + 2fy + c = 0 and ' 2 ' ' 2 ' ' ' a x 2h xy b y 2g x 2 f y c 0

intersect in four concylic points, then

A)' '

'

a b a b

h h

B)

' '

'

a b a b

h h

C) h (a - b) = h(a+b) D) h (a+b) = h(a - b)

85. One of the limit point of the coaxial system of circles containing x2+ y2- 6x - 6y + 4 = 0, x2+ y2- 2x

- 4y + 3 = 0 is

A) (-1, 1) B) (-1, 2) C) (-2, 1) D) (-2, 2)

86. The two lines through (2, 3) from which the circle x2+ y2= 25 intercepts chords of length 8 units have

equations

A) 2x + 3y = 13, x + 5y = 17 B) y = 3, 12x + 5y = 39

C) x = 2, 9x - 11y = 51 D) none of these

87. The length of the common chord of the circles (x-a)2+(y-b)2= c2and (x- b)2+ (y - a)2= c2is

A) 2 2c ( a b ) B) 2 24c 2( a b ) C) 2 22c ( a b ) D) 2 24c ( a b )

88. The intercept on the line y = x by the circle x2+ y2- 2x = 0 is AB. Equation of the circle with AB as a

diameter is

A) x2+ y2- x - y = 0 B) x2+ y2- x + y = 0 C) x2+ y2+ x - y = 0 D) x2+ y2+ x + y = 0

89. The centre of a circle passing through the point (0, 1) and touching the curve y=x2at (2, 4) is

A)16 27

,5 10

B)16 5

,7 10

C)16 53

,5 10

D) none of these

90. A variable circle is described to pass through the point (a, 0) and touch the line x+y = 0. Locus of the

centre of the above circle is

A) parabola B) ellipse C) hyperbola D) none of these

91. The equation of tangent drawn from origin to the circle x2+ y2- 2ax - 2by + b2= 0 are perpendicular if


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A) a2= b2 B) a2+ b2= 1 C) 2a = b D) 2b = a

92. The equation of the circle passing through (2, 0) and (0, 4) and having the minimum radius is

A) x2+ y2+ 2x + 4y = 0 B) x2+ y2- 2x + 4y = 0

C) x2+ y2- 2x - 4y = 0 D) x2+ y2+ 2x - 4y = 0

93. A circle cuts the circles x2+ y2= 4, x2+ y2- 6x - 8y + 10 = 0 and x2+ y2+ 2x - 4y - 2 = 0 at the endsof diameter. The co-ordinates of its centre are

A) (2, 3) B) (-2, -3) C) (4, 6) D) (-4, -6)

94. The locus of the feet of the perpendiculars from (1, 2) to the family of lines (a+3b)x-(2a -b) y - (a - 4b)

= 0 where a, b R is

a) a straight line B) a circle C) a parabola D) a hyperbola

95. There exist two circles passing through (1, 2) and touching both the axes. The length of the commonchord is

A) 2 B) 1/ 2 C) 1 D) 2 2

96. If the lines ax + by + c = 0 and cx + by + a = 0 meet the coordinate axes in four concyclic points then

A) a, b, c are in GP B) b, c, a are in GP C) c, a, b are in GP D) None

97. The values of a for which the point (2, +1) is in the interior of the larger segment of the circle - 2x -

2y - 8 = 0 made by the chord x - y + 1 = 0 is

A)9

,05

B)9

0,5

C)9

1,5

D)9

,15

98. The point on the y axis, where the line segment joining A (1, 0) and B(3, 0) subtends greatest angle is

A) (0, 2) B) (0, ) C) (0, ) D) (0, 3/2)

99. The equation of the circle having the pair of lines as its normals and having the size just sufficient to

contain the circle x (x - 4) + y (y - 3) = 0 is2 2

x y 6 x 3 y k 0 the k equalsA) -35 B) -45 C) -15 D) -25

100. The equation of the smallest circle which passes through (2, 1) and touches the x axis is

A) x2+ y2- 2x - 2y - 1 = 0 B) x2+ y2- 2x - 2y + 1 = 0

C) x2+ y2- x - y - 1 = 0 D) x2+ y2- x - y + 1 = 0

101. The radius of the largest circle which passes through (1, 2) and (3, 4) and lies completely in the first

quadrant is

A) 3 B) 2 C) D) none


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102. The circle 2 2x y 4x + 4y + 4 = 0 is inscribed in a triangle, whose two sides are along the axes and

the locus of the circumcentre of the triangle is xy + x + y + k 2 2x y 0 then k equals

A) 1 B) -1 C) 1 D) 0

103. The abscissae and ordinates of two points A and B are respectively the roots of the quadratic equations

f(x) = 0 and g(x) = 0 andx

f ( x )Lt

g( x ) =

2

3 then the equation of the circle on AB as diameter is

A) 2 f(x) + 3 g(y) = 0 B) 3 f(x) + 2 g(y) = 0 C) 2 f(x) - 3 g(y)=0 D) 3 f(x) - 2 g(y) = 0

104. A ray of light incident at the point (3, 1) on the tangent at (0, 1) to the circle gets reflected and the

reflected ray touches the circle. The equation of the reflected ray is

A) 3x - 4y + 13 = 0 B) 3x + 4y - 13 = 0 C) 4x - 3y - 5 = 0 D) 4x - 3y - 13 = 0

105. Tangents AP and AQ are drawn from A (1, 2) to the circle x2+ y2+ 4x + 2y + 1 = 0 then the equationof the line joining the mid points of AP and AQ is

A) x + y = 1 B) x + y + 1 = 0 C) 3x + 3y = 2 D) 3x + 3y + 2 = 0

106. The number of distinct chords of the circle x2+ y2- 4x - 3y = 0 passing through (4, 3) which are

bisected by the x axis is

A) 0 B) 1 C) 2 D) 3

107. A set of circles, each of radius 2 have their centres on the circle x

2

+ y

2

= 36 then the points in the setsatisfy

A) 16 x2+ y264 B) 4 x2+ y281 C) 0 x2+ y264 D) 36 x2+ y2144

108. The number of rational point(s) (a point (a,b) is ratio

Iit 4 Unlocked

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