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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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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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IIT- MATHEMATICS-SET-IV
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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IIT- MATHEMATICS-SET-IV
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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IIT-MATHEMATICS-SET IV
1-A PERMUTATIONS ANDCOMBINATIONS
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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
a) 19 b) 185 c) 201 d) none of these
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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IIT-MATHEMATICS-SET IV
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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PERMUTATIONS AND COMBINATIONS
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
a) 1296 b) 625 c) 671 d) none of these
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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IIT-MATHEMATICS-SET IV
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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PERMUTATIONS AND COMBINATIONS
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 :
a) 17 b) 18 c) 9 d) None of these
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
a) 240 b) 320 c) 360 d) None of these
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
a) 105 b) 45 c) 51 d) None of these
23. If
n
r r
nn
C
a
0
1then
n
r r
n
C
r
0
equals :
a) na1n b) nna c) nna21
d) None of the above
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
a) 3 b) 4 c) 5 d) None of these
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
d) None of the above
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
a) 756 b) 1512 c) 3024 d) none of the above
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
a) 535 b) 601 c) 728 d) none of these
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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PERMUTATIONS AND COMBINATIONS
SECTION B -MULTIPLE
ANSWER TYPE QUESTIONS
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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IIT-MATHEMATICS-SET-IV
2 BINOMIAL THEOREM
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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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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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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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2-A BINOMIAL THEOREM
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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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SECTION A -SINGLE
ANSWER TYPE QUESTIONS
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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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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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
a) 10 b) 1 c) 100 d) none of the above
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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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=
a) 6 b) 5 c) 4 d) none of these
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 :
a) 9 b) 6 c) 12 d) None of these
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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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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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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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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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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IIT-MATHEMATICS-SET-IV
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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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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IIT-MATHEMATICS SET IV
3-A CIRCLES
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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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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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IIT-MATHEMATICS SET IV
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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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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IIT-MATHEMATICS SET IV
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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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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IIT-MATHEMATICS SET IV
SECTION A -SINGLE
ANSWER TYPE QUESTIONS
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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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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IIT-MATHEMATICS SET IV
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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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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IIT-MATHEMATICS SET IV
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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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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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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IIT-MATHEMATICS SET IV
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