Bansal Classes Mathematics Study Material for IIT JEE

BANSAL CLASSES

MATHEMATICS TARGET IIT JEE 2007

XI (P, Q, R, S)

COMPOUND ANGLES Trigonometry Phase - /

i : - .

I CONTENTS

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

KEY CONCEPTS BASIC TRIGONOMETRIC IDENTITIES :

(a) sin29 + cos 2 9=1 ; - l < s i n 9 < l ; - l < c o s 9 < l V 9 e R (b) sec2 9 - tan2 9 = 1 ; | sec 0 | > 1 V 9 e R (c) cosec2 9 - cot2 9=1 ; | cosec9 |> l V 9 e R

IMPORTANT T' RATIOS :

(a) sinn7t = 0 ; cosn7T = (-l)n ; tann7t = 0 where n e I

. (2n + l)7i (b)

(c)

sin- ;(- l)n & cos(2n + 1)Tt = Q where n e I

sin 15 or sin 12

cos 15 or cos 12

V3-1 __0 5n r=- = cos 75 or cos 2V2 12

V 3 + 1 . _ . 0 . 5 7X r=~ = sm 75 or sin 2V2 12

(d)

(e)

tan 15 = - = 2 - V 3 = cot 75 ; tan75= = 2 + V 3 =cot 15 V3 + 1 V3 - 1

. 11 J2 - ^ 71 1 / 2 + V 2 71 r- 3t t rz , = v = J L - - ; tan - = V2 - 1 ; tan = V2 + 1 sm cos 8 8

71 1 0 0 _ V5 - 1 sm or sin 18 = 10 4

P ICO 71 V J + l & cos 36 or cos = 5 4

TRIGONOMETRIC FUNCTIONS O F ALLIED ANGLES :

If 9 is any angle, then - 9, 90 9 , 180 9 , 270 9 , 360 9 etc. are called ALLIED ANGLES

4 .

5 .

(a) sin ( - 9) = - sin 9 ; cos ( - 9) = cos 9 (b) sin (90- 9) = cos 9 ; cos (90 - 9) = sin 9 (c) sin (90+ 9) = cos 9 ; cos (90+ 9) = - sin 9 (d) sin(180-9) = sin9 ; cos (180-9) =-cos 9 (e) sin (180+ 9) = - sin9 ;cos(180+9) = -cos9 (f) sin (270- 9) = - cos 9 ; cos (270- 9) = - sin 9 (g) sin (270+ 9) = - cos 9 ; cos (270+9) = sin9

TRIGONOMETRIC FUNCTIONS O F SUM O R DIFFERENCE O F T w o ANGLES

(a) sin (A B) = sinA cosB cosA sinB (b) cos (A B) = cosA cosB + sinA sinB

sin2A - sin2B = cos2B - cos2A = sin (A+B). sin (A- B)

sine & cosec only + ve All +ve

tan & cot cos & sec only + v e only + v e

(c) (d) cos2A - sin2B = cos2B - sin2A = cos (A+B). cos (A - B)

(e) tan (A B): t a n A t a n B (f) cot (A B) = c o t A c o t B + 1

1 + t a n A t a n B " ~ ' c o t B + c o t A

FACTORISATION O F T H E SUM O R DIFFERENCE O F T w o SINES O R COSINES :

(a) sinC + sinD = 2sinC + P cos

(c) cosC + cosD = 2 cos C + D cos

C + D C (b) sinC - sinD = 2 cos sin D

D C + D (d) cosC-cosD = -2s in sm

2

C - D

^Bansal Classes Trig.--1 or , ^ ^ Sin [2]

TRANSFORMATION O F PRODUCTS INTO SUM O R DIFFERENCE O F SINES & COSINES:

(a) 2 sinA cosB = sin(A+B) + sin(A-B) (b) 2 cosA sinB = sin(A+B) - sin(A-B) (c) 2 cosA cosB = cos(A+B) + cos(A-B) (d) 2 sinA sinB = cos(A-B) - cos(A+B) MULTIPLE ANGLES A N D HALF ANGLES :

9 9 (a) sin 2A = 2 sinA cosA ; sin9 = 2s in -cos -

(b) cos 2 A = cos2A - sin2 A = 2cos2A - 1 = 1 - 2 sin2A;

cos 6 = cos2 - sin2-^ = 2cos2^- - 1 = 1 - 2sin2^-. 2 2 2 2

. l -cos2A 2 cos2A = 1 + cos 2A, 2sin2A = 1 - cos 2A ; tan2A = l + cos2A

9 9 2 cos2 = 1 + cos 9 , 2 sin2 = 1 - cos9. 2 2 ? T A ^ t a n f

(c) tan2A= 2 t a n ? ; tan9= , ~ r i w 1 - t an 2 A 1 - tan T

(d) sin2A = 2 t a n A , cos2A= 1 - t a a [ A (e) sin3A = 3 sinA-4 sin3A W 1 + tan A 1 + tan A 3 tan A - tan3 A (f) cos 3A = 4 cos3A - 3 cosA (g) tan 3A , , w 1 - 3 t a n ' A

8 . THREE ANGLES :

t a n A + t a n B + t a n C - t a n A t a n B t a n C (a) tan (A+B+C) = 1 - t a n A t a n B - t a n B t a n C - t a n C t a n A

NOTE IF : (i) A+B+C = n then tanA+tanB + tanC = tanA tanB tanC (ii) A+B+C = ~ then tanA tanB + tanB tanC + tanC tanA= 1

(b) If A+B + C = 7t then: (i) sin2A + sin2B + sin2C = 4 sinA sinB sinC A B C (ii) sinA + sinB + sinC = 4 cos cos cos v ' 2 2 2

9 . MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS:

(a) Min. value of a2tan29 + b2cot29 = 2ab where 9 R (b) Max. and Min. value of acos9 + bsin9 are ,ja 2 52 and^Ja 2 + ^2

(c) If f(9) = acos(a + 9) + bcos(B + 9) where a, b, a and P are known quantities then -\Ja2 + b2 + 2abcos(ct-p) -I [3]

10. Sum of sines or cosines of n angles, nB , / \ sin^ f- ( n-1 sin a + sin (a + P) + sin (a + 2p ) + + sin ^a + n - i p j = sin I a +--p sin ^

n

P / / s i n ( n-1 cos a + cos (a + P) + cos(a + 2p ) + + cos la + n-1 pi = 1- cos a + sin j V 2 EXERCISE-I

Q.l Prove that cos2a + cos2 (a + P) - 2cos a cos p cos (a + P) = sin2P

Q.2 Prove that cos 2a = 2 sin2P + 4cos (a + P) sin a sin p + cos 2(a + P)

Q.3 Prove that, tan a + 2 tan 2a + 4 tan 4a + 8 cot 8 a - cot a .

Q.4 Prove that : (a) tan 20 . tan 40 . tan 60 . tan 80 = 3

(b) tan9 -tan27 -tan63 + tan81 = 4 . (c) sin4 ~ + sin4 ^ + sin4 ~ + sin4 ~ = | 16 16 16 16 2

Q.5 Calculate without using trigonometric tables :

(a) cosec 10 - S sec 10 (b) 4 cos 20 - V3 cot 20 (c) 2 cos40 - cos20 sin20

(d) 2V2 sin 10 sec 5 cos 40 . 2sin35 27t 4TI 6 7 1 7

(e) c o s + c o s + cos 2 sin 5

(f) tan 10 - tan 50 + tan 70 Q.6 (a) Showthat 4 sin 17 . sin 43 . sin 77 = sin 51

(b) Prove that sin2120 + sin221 + sin2390 + sin248= l+sin29 + sin218 .

Q.7 Showthat: (a) c o t 7 y or tan 82 y = (V3 + V2) (V2 + 1) or V2+V3+V4+V6

(b) tanl42y =2 +V2-V3-V6 .

Q.8 If mtan(9-30) = ntan(6+ 120), show that cos20 m + n 2(m - n)

Q.9 If cos9= cosu , prove that, tan = tan . 1 - e c o s u 2 V1 _ e 2

4 5 71 Q.10 If cos (a + P) = j ; sin (a - P) = & a , P lie between 0 & , then find the value of tan 2a .

Q.ll Prove that if the angles a & P satisfy the relation p) = ~ (M > lnl) then

. tang

1 + tana _ 1 - tana tanP m+n m-n

Q.12 (a) If y = 10 cos2x - 6 sin x cos x + 2 sin2x, then find the greatest & least value of y. (b) If y = 1 + 2 sin x + 3 cos2 x, find the maximum & minimum values of y V x e R. (c) If y = 9 sec2x +16 cosec2x, find the minimum value of y V x 6 R.

y'vfl/ < a ses Trig.-(f>-1 [4]

Q.13 (a) Prove that 3 cos (e + y j + 5 cos 0 + 3 lies from - 4 & 10 .

(b) Prove that (2V3 + 4) sin 0 + 4 cos 0 lies between - 2(2 + Vs ) & 2(2 + VJ).

Q.14 If a + p = c where a, p > 0 each lying between 0 and 7i/2 and c is a constant, find the maximum or minimum value of (a) sin a + sin p (b) sin a sin p (c) tan a + tan p (d) cosec a + cosec p

Q.15 Let A,., A2, , An be the vertices of an n-sided regular polygon such that ; 1 + - . Find the value of n.

A.j A2 ^i .A] A.^

Q. 16 Prove that: cosec 0 + cosec 20 + cosec 22 0 + + cosec 2 n " 1 0 = cot (0/2) - cot 2 n"!0

Q.17 For all values of a , p , y prove that ;

x , a+B P+Y Y+a cos a + cos p + cos y + cos (a + p + y) = 4 cos . cos . cos . 2 . : 2 .2

18 S h t h t * + + C 0 S ^ - 2 s i n A - 2 s i n B Q.18 Showthat C0Sj^ 1 - sinB sin(A-B) + cosA - cosB

A i n I ( , Q _ t ana + tan y . . s in2a + sin2y Q.19 If tan P = , prove that sin 2B = ; 7 - ^ 1 + t a n a . t a n y 1 + s in2a . sm2y

Q.20 If a + p = y , prove that cos2 a + cos2 p + cos2 y = 1 + 2 cos a cos P cos y .

Q.21 If a + p + y = | , show that

Q.22 If A + B + C = 7t and cot0 = cot A + cot B + cot C, showthat, sin (A - 0). sin (B - 0). sin (C - 0) = sin30 .

% 3n 571 17TC Q.23 IfP= cos + cos + cos + + cos and V 19 19 19 19

271 47t 6tz 20n Q= cos+ cos+ COS + + cos-^ -? then find P - Q.

- * Q.24 Without using the surd value for sin 18 or cos 36, prove that 4 sin 36 cos 18 V5

Q.25 For any three angles a , p and y prove that:

' cos(a + P) cos(a + y)V ( sin(a + P) sin(a + y)^2

cos(a - P) cos(a - y )J \ cos(a - P) cos(a - y)> = sec2(a - p) sin2(P - y) sec2(y - a)

( l - t a n f ) ( l - tanf ) | i l - t an | j ' _ s ina + sin(3 + siny 1 (l + tanf ) (l + tan f)l l 1 + t a n i cosa + cosp + cosy

Bamal Classes Trig.--! [5]

EXERCISE-II Q.l If tana = p/q where a = 6p , a being an acute angle, prove that ;

- (p cosec 2 P - q sec 2 P) = /^p2 + q2 .

Q.2 If + = + = 1, where 9 & 5 do not differ by an even multiple of n, then prove cosa s ina cosa s ina

cosB . cos5 sin9 . sin5 that i + + 1 = 0 . cos a sm a

cos 30 + cos3(j) Q.3 Prove that 2 c o s ( 0 - 1. [Hint: E t a n - . t a n | = 1]

feBansal Classes Trig.- [6]

Q.20

Q.21

If A+B+C = n (A, B , C > 0), prove that sin^ sinf . sin^ < ^ . 2 2 2 8

Show that eliminating x&y from the equations, sinx + siny = a ; 8a b

cosx + cosy = b & tanx + tany = c gives = c 4 a '

Q.22 Show that

Q.23 Evaluate :

2sec 6 + 3tan 6 + 5sin 9 - 7 c o s 0 + 5 l - c o s 0

2 t an 8 + 3sec 0 + 5cos 9 + 7sin 0 + 8 sin 0

. x , tan

2n

n=12n-1COS X 2 n- l

Q.24 If a + (3+y = 7t & tan f p + y - a ^ | tan ryn h a - tan

I 4 J V 4 I 4 j = 1, then prove that;

1 + cos a + cos P + cos y = 0 .

Q.25

Q.l

V x e R, find the range of the function, f (x) = cos x (sin x + ^sin2 x + sin2 a ) ; a e [0, n] EXERCISE-III

Q.2

(a)

(b)

(a)

(b)

sec29 = 4 x y , is true if and only if : (X + y ) 2

(A) x+y*0 (B) x = y , x * 0 (C) x = y

Find all values of 9 in the interval

(1 - tan 9) (1 + tan 9) sec2 9 + 2tan2 9 = 0 . ~ s a t i s f y i n g the equation ;

(D) x * 0 , y * 0

n ; [JEE'96,1 + 2]

Let n be an odd integer. If sin n9 = b sinr 9, for every value of 9, then: r = 0

(A) b0 = 1, b, = 3 (B) bQ = 0, bj = n (C) b0 = - 1, bj = n (D) b0 = 0, bj = n2 - 3n + 3 Let A0 Aj A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A0 Aj, A0 A2 & A0 A4 is :

(B)3V3 (C) 3 (D) ^

(c) Which of the following number(s) is/are rational ? (A) sin 15 (B) cos 15 (C) sin 15 cos 15 (D) sin 15 cos 75

(d) Prove that a triangle ABC is equilateral if & only if tan A+tan B + tan C = 3 V3 [JEE '98,2 + 2 + 2 + 8 = 14outof200]

Q.3 For a positive integer n, let.fn (9) =

(B ) / , (A)/2^]=l n

3 V32.

t 9 tan

= 1

(1+ sec9)(1+ sec29)(1+sec49)....(1 + sec2n9) Then

f n WfAm. 1 [JEE '99,3]

Q.4 (a) Let f(9) = sin9 (sin9 + sin 3 9). Then f(9)

(A) > 0 only when 9 > 0 (C) > 0 for all real 9

(B) < 0 for all real 9 (D) < 0 only when 9 < 0.

[ JEE 2000 Screening. 1 out of 35 ]

Bansal Classes Trig.-(j>-1 m

(b)

Q.5 (a)

A B C A B C Ei any triangle A B C , prove that, cot + cot + cot = cot cot cot .

Z* Z* Zi Zi Zi

[ JEE 2000 Mains, 3 out of 100 ]

Find real values of x for which, 27cos2x. 81sul2x is minimum. Also find this minimum value. (b) Find the smallest positive values of x & y satisfying, x - y = , cot x + cot y = 2 .

71 Q.6 If a + (3 = and p + y = a then tana equals

(A) 2(tanP + tany) (B) tanp + tany (C) tanp + 2tany

[ REE 2000, 3 + 3 ]

(D) 2tanP + tany [JEE 2001 (Screening), 1 out of 35 ]

1 1 Q.7 If 9 and (j) are acute angles satisfying sin9 = , cos = , then 9 + e

(A) 7C 71

3 ' 2 (B) [ f n 2%^ 2'T (C)

r2n 5nx

T'T (D) 571

v6 . [JEE 2004 (Screening)]

Q. 8(a) In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they A touch each other and also the sides of the triangle. Area of the triangle is

(A) 4 + 2 V3 (B) 6 + 4 V3

7V3 7V3 (C) 12 + (D)3 + 4 i r 4 (h) cos(a - P) = 1 and cos(a + P)= 1 /e, where a, P e [ - n, 7t], numbers of pairs of a, P which satisfy both

the equations is (A)0 (B)l (C) 2 (D)4

[JEE 2005 (Screening)] ANSWER SHEET

EXERCISE-I Q 5. (a) 4 (b) -1 (c) V3 (d) 4 (e) (f) fi Q10. | |

13 Q12- (a> ymax= 1 1 i ymm = 1 (b) ymax= y ; y min = - \ , (c)49 mm ' v ' Q14. (a) max = 2 sin (c/2), (b) max. = sin2 (c/2), (c) min = 2 tan (c/2), (d) min = 2 cosec (c/2) Q 15. n = 7 Q23. 1

EXERCISE -II

Q 23. 1 s i n 2 x 2n_I sin X

Q-25 - V1 + sin2 a ^ y ^ Vl + si sin a n-1

EXERCISE-III Q.l (a) B (b) Q.2 (a) B, (b) C, (c) C Q.3 A, B, C, D Q.4 (a) C

5 7Z 71 Q.5 (a) Minimum value = 3 ~5; maximum value = 35,(b) x= ; y = Q.6 C 12 6

Q.8 (a) B, (b)D

Q.7 B

^Bansal Classes Trig.-1 [8]

BANSAL CLASSES

MATHEMATICS TARGETIIT JEE 2007

XI (P, Q, R, S)

QUADRATIC EQUATIONS

CONTENTS EXERCISE-I EXERCISE-II EXERCISE-III EXERCISE-IV ANSWER KEY

KEY CONCEPTS The general form of a quadratic equation in x is, ax2 + bx + c = 0 , where a , b , c e R&a^O. RESULTS :

2.

4.

5.

6.

The solution of the quadratic equation, ax2 + bx + c = 0 is given by x = - b /b2 - 4ac

2a The expression b2-4ac=D is called the discriminant of the quadratic equation. If a & (3 are the roots of the quadratic equation ax2 + bx + c = 0 , then ; (i) a + p = - b/a ' (ii) a p = c/a (iii) a -B = Vd/a . N A T U R E O F R O O T S : (A) Consider the quadratic equation ax2 + bx + c = 0 where a, b, c e R & a ^ O then ;

(i) D > 0 roots are real & distinct (unequal) . (ii) D = 0 roots are real & coincident (equal) . (iii) D 0 & is a perfect square , then roots are rational & unequal. (ii) If a = p + ^ /q is one root in this case, (where pis rational & y'q is a surd)

then the other root must be the conjugate of it i.e. P = p - /^q & vice versa.

A quadratic equation whose roots are a & p is (x - a)(x - P) = 0 i.e. x 2 - ( a + p)x + a,p = 0i.e. x2-(sumof roots)x+ product of roots = 0 . Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity. Consider the quadratic expression, y = ax2 + bx + c , a ^ 0 & a, b, c e R then ; (i) The graph between x, y is always a parabola . If a > 0 then the shape of the

parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards. V x e R, y > 0 only if a > 0 & b 2 - 4 a c < 0 (figure 3). V x e R, y < 0 only if a < 0 & b2 - 4ac < 0 (figure 6). Carefully go through the 6 different shapes of the parabola given below .

(ii) (iii)

Fig. 1 Fig. 2 Fig. 3 y

\

\ a > 0 / \ D = 0 / V y ,

X o

Roots are coincident

Fig. 5

O x

Roots are complex conjugate

Fig. 6

y y

0 X 0 X

/ a < 0 \ / a < 0 \ X /I D = 0 / ' D < 0 \

Roots are real & distinct Roots are coincident Roots are complex conjugate

ii Bansal Classes Quadratic Equations [9]

7. S O L U T I O N O F Q U A D R A T I C I N E Q U A L I T I E S : ax2 + bx + c > 0 (a * 0). (i) If D > 0, then the equation ax2 + bx + c = 0 has two different roots x, < x2.

Then a > 0 => x e (-co, Xj) u (x2, oo) a < 0 => x e (Xj, x2)

(ii) If D = 0, then roots are equal, i.e. Xj = x2 . In that case a > 0 => x e (-co, x,) u (xp oo)

a < 0 => x e (j)

9.

10.

11.

(iii)

intervals.

P(x) < Inequalities of the form 0 >can be quickly solved using the method of Q(x)

8. MAXIMUM & MINIMUM VALUE of y = ax2+.bx + c occurs at x = - (b/2a) according as ;

a < 0 or a > 0 . y e 4ac - b 4a

00 if a > 0 & y - c o , 4 ac - b 4a

if a < 0

C O M M O N R O O T S O F 2 Q U A D R A T I C E Q U A T I O N S [ONLY O N E C O M M O N R O O T ] : Let a be the common root of ax2+bx + c = 0 & a'x2 + b'x + c'= 0 .Therefore

a a2 + ba + c = 0 ; a'a2 + b'a + c' = 0 . By Cramer's Rule 1 be' - b'c a ' c - a c ' a b ' - a ' b

ca ' - c'a be' - b'c Therefore, a = ab ' - a'b a'c - ac '

So the condition for a common root is (ca'-c'a)2 = (ab'-a'b)(bc'-b'c). The condition that a quadratic function f (x, y) - ax2+2 hxy + by2 + 2 gx + 2 fy + c may be resolved into two linear factors is that ;

abc + 2 fgh - af2 - bg2 - ch2 = 0 OR a h g h b f g f c

= 0 .

T H E O R Y O F E Q U A T I O N S : If a j , a 2 , a 3 , an are the roots of the equation ; f(x) = a0xn + ajX""1 + a,xn"2 + .... + an_,x + an = 0 where a0, a,,.... an are all real & a0 * 0 then,

X OCj

Note : (i)

(ii)

(iii)

(iv)

(V)

(vi)

, X a2 = + , X a, a2 a3 = - OC| (X^ .a (-If a a0 a0 a0 a0

If a is a root ofthe equation f(x) = 0, then the polynomial f(x) is exactly divisible by (x - a) or (x - a) is a factor of f(x) and conversely. Every equation of nth degree (n > 1) has exactly n roots & if the equation has more than n roots, it is an identity.

If the coefficients of the equation f(x) = 0 are all real and a + ip is its root, then a - ip is also a root. i.e. imaginary roots occur in conjugate pairs. If the coefficients in the equation are all rational & a + ^P is one of its roots, then a - ^P is also a root where a, p Q & p is not a perfect square. If there be any two real numbers 'a' & 'b' such that f(a) & f(b) are of opposite signs, then f(x) = 0 must have atleast one real root between 'a' and 'b'. Every equation f(x) = 0 of degree odd has atleast one real root of a sign opposite to that of its last term.

fa Bansa! Classes Quadratic Equations

12. LOCATION OF ROOTS : Let f(x) = ax2 + bx + c , where a > 0 & a, b, c e R . (i) Conditions for both the roots of f(x) = 0 to be greater than a specified number'd' are

b2 - 4ac > 0 ; f (d) > 0 & (-b/2a)>d. (ii) Conditions for both roots of f(x) = 0 to lie on either side of the number'd' (in other words

the number'd' lies between the roots of f (x) = 0) is f (d) < 0. (iii) Conditions for exactly one root of f(x) = 0 to lie in the interval (d,e) i.e. d < x < e are

b2 - 4ac > 0 & f (d). f (e) < 0 . (iv) Conditions that both roots of f (x) = 0 to be confined between the numbers p & q are

(p < q) . b2 - 4ac > 0 ; f (p )>0 ; f (q )>0 & p < ( - b / 2 a ) < q .

13. LOGARITHMIC INEQUALITIES (i) For a> 1 the inequality 0 < x < y & logax< logay are equivalent. (ii) For 0 < a < l the inequality 0 < x < y & logax>logay are equivalent. (iii) If a> 1 then logax 0 < x < a p

(iv) If a > 1 then logax > p => x > ap

(v) If 0 < a < 1 then logax x>a p

(vi) If 0 < a < 1 then logax>p => 0 < x < a p

EXERCISE-I

Q.l If the roots of the equation [ 1 /(x+p)] + [ 1 /(x+q)] = 1/r are equal in magnitude but opposite in sign, show that p + q = 2r&that the product of the roots is equal to (-l/2)(p2 + q2).

Q.2 If x2 - x cos (A + B) + 1 is a factor of the expression, 2x4 + 4x3 sin A sin B - x2 (cos 2A + cos 2B) + 4x cos A cos B - 2 . Then find the other factor.

Q.3 a , (3 are the roots of the equation K(x2-x)+x+5 = 0 . If K, & K2 are the two values of K for which the roots a , (3 are connected by the relation (a/p) + (p/a) = 4/5 . Find the value of (K(/K2) + (K2/Kj) .

Q.4 If the quadratic equations, x2 + bx + c = 0 and bx2 + cx + 1 = 0 have a common root then prove that either b + c+ l = 0orb 2 + c 2 + l = b c + b + c.

f P2^

Q.5 If the roots of the equation I l-q+

p2 = 4q .

P2 X2 +p(l+q)x + q(q-l) + = 0 are equal then show that

Q.6 If one root of the equation ax2 + bx + c = 0 be the square of the other, prove that b3 + a2c + ac2 = 3abc.

ax2 +2(a + l)x + 9a + 4 Q. 7 Find the range of values of a, such that f (x) = ^ - 8 x + 32 i s a l w a y s n eg a t i v e-

Q.8 Let a, b, c be real. If ax2 + bx + c = 0 has two real roots a & P, where a < - 1 & P > 1 then show that 1 + c/a + | b/a | < 0.

6x2 - 22x + 21 , , . Q.9 Find the least value of ; for all real values of x, using the theory of quadratic

5x -18x + 17 equations.


Q.IO

Q.ll

Q.12

Find the least value of (2p2 + 1 )x2 + 2(4p2 - 1 )x + 4(2p2 + 1) for real values of p and x.

If a be a root of the equation 4x2 + 2x - 1 = 0 then prove that 4a3 - 3a is the other root.

If the equations x2+px + q = 0 & x2 + p'x + q' = 0 have a common root, show that it must be equal to (pq'-p'q)/(q-q') or (q-q')/(p'-p)

Q.13 If a,(3 are the roots of ax2+bx + c = 0 &a ' ,~P are the roots of a'x2+b'x + c' = 0, showthat

a , a ' are the roots of "b b' -1 2 "b b'"

+ x + x + + a a' c c'

-l = 0

Q.14

Q.15

Q.16

If a , P are the roots of x2 - px + 1 = 0 & y, 5 are the roots of x2 + qx + 1 = 0 , show that ( a - y ) ( P - y ) ( a + 5)(P + 8) = q 2 -p 2 .

Show that if p , q , r & s are real numbers & pr = 2(q+s), then at least one of the equations x2 + px + q = 0, x 2 + r x + s = 0 has real roots .

If a & b are positive numbers, prove that the equation + ' + = 0 has two real roots, x x-a x+b one between a/3 & 2a/3 and the other between - 2b/3 & - b/3 .

If the roots of x2 - ax + b = 0 are real & differ by a quantity which is less than c (c > 0), prove that b lies between (1/4) (a2 - c2) & (l/4)a2.

At what values of'a' do all the zeroes of the function, f (x) = (a - 2) x2 + 2 a x + a + 3 lie on the interval ( - 2,1)?

If one root of the quadratic equation ax2+ bx + c = 0 is equal to the nth power of the other, then show that (acn)1/(n+l) + (a1^)1^1) + b = 0 .

V P p - 2 ' p - 2

q4 ? q J - 5 q - 2 ' q - 2

r3 - 5^ and

' s4 s 3 -5 A

s - 2 s - 2 are collinear if

Q.21

Q.22

Q.23

Q.24

Q.25

Q.26

Q.27

r - 2 r - 2 / \ y pqrs = 5(p + q + r + s) + 2 (pqr + qrs + rsp + spq).

The quadratic equation x2 + px + q = 0 where p and q are integers has rational roots. Prove that the roots are all integral. If the quadratic equations x2+bx+ca = 0 & x2+cx+ab = 0 have a common root, prove that the equation containing their other root is x2 + ax + be = 0 .

If a , p are the roots of x2+px+q = 0 & x2n+pnxn + qn = 0 where n is an even integer, show that a/p, p/a are the roots of xn +1 + (x + l)n = 0 .

If a , p are the roots of the equation x2 - 2x + 3 = 0 obtain the equation whose roots are a 3 - 3 a 2 + 5a - 2 , p 3 - p 2 + p + 5.

If each pair of the following three equations x2 + p1x+q1 = 0 ,x2+p2x + q2=0 & x2+p3x + q3 = 0 has exactly one root common, prove that ; (PI + P2 + P3)2 = 4 [P1P2 + P2P3 + P3P1 ~ ~ % ~ Show that the function z = 2x2+2xy+y2-2x + 2y+2 is not smaller than - 3 .

If (1/a) + (1/b) + (1/c) = l/(a+b + c) & n is an odd integer, show that ; (l/an) + (l/bn) + (l/cn) = l/(an + bn + cn) .


Q.28 Find the values of 'a ' for which -3-c is Ua-c + 7b-cj .

Q.5 If Xj, x2 be the roots of the equation x2 - 3x + A = 0 & x3 , x4 be those of the equation x2 - 12x + B = 0 & Xj ,x 2 , x 3 , x 4 are in GP. Find A & B .

Q.6 If ax2+2bx + c = 0 & a,x2 + 2b1x + c1 = 0 have a common root & a/a1,b/bj,c/c, are inAP, show that a t , bj & ct are in GP .

Q.7 If by eleminating x between the equation x2+ax+b = 0& xy+/(x+y) + m = 0 , a quadratic in y is formed whose roots are the same as those of the original quadratic in x . Then prove either a = 21 & b = m or b + m = a / .

. , 2 a x2 - 2 x c o s a + l s i n 7 c o s ~

Q.8 If x be real, prove that nr lies between and x 2x cos (3 + 1 , (3 2 3 sin cos -

2 2 Q.9 Solve the equations, ax2+bxy + cy2 = bx2+ cxy+ ay2 = d .

Q.10 Find the values of K so that the quadratic equation x 2 +2(K- l )x+K + 5 = 0 has atleast one positive root.

Q.ll Find tiie values of *b'for which the equation 2 log , (bx + 28) = -log5(l2-4x-x2jhasonlyonesolution. 25


Q.12 Find all th e v aiues of the parameter 'a' for which both roots of the quadratic equation x2 - ax + 2 = 0 belong to the interval (0 ,3 ) .

Q.13 Find all the values ofthe parameters c for which the inequality has at least one solution.

1 + log2 2x2 + 2x + - > log2 (cx2 + c) . \ 2) " Q.14 Find the values of K for which the equation x4 + (1 - 2 K) x2 + K2 - 1 = 0 ;

(a) has no real solution (b) has one real solution Q.15 Find the values of p for which the equation 1 + p sin x = p2 - sin2 x has a solution. Q.16 Solve the equation -4.3" | x~2 ' - a = 0 for every real numbera. Q.17 Find the integral values of x & y satisfying the system of inequalities;

y-1 x 2-2x | + (1/2)> 0 & y+1 x - 1 1 5 & r ^ 0 has roots a}, a2, a3, an.

n

Denoting a fk by Sk. i l

(i) Calculate S7 & deduce that the roots cannot all be real. (ii) Prove that Sn + pS2 + qS, +nr = 0 & hence find the value of Sn.

EXERCISE-III Solve the inequality. Where ever base is not given take it as 10.

Q.l ( log 2x) 4 -' x 5 ' 2 l o g l T

2 y 201og2x+148 < 0. Q.2 x1/ lo8x.logxlog2(2-x). Q.5 logx2 . log2x2 . log2 4x > 1. Q.6 log1/5 (2x2 + 5 x + 1)l. Q.8 logx2(2+x)< 1

4 x 4- S

Q-9 l o g x ^ < - l Q.10 (log|x+6|2). log2 ( x 2 -x -2 )> 1

Q 11- Q12. log[(x+6)/3][log2{(x-l)/(2+x)}]>0 X t X

Q.13 Find out the values of 'a' for which any solution of the inequality, ~ < 1 is a l s o a

solution of the inequality, x2 + (5 - 2 a) x < 10a. Q.14 Solve the inequality log N (x 2 -10x + 22) > 0 .

Q.15 Find the set of values of'y' for which the inequality, 2 log0 5 y2 - 3 + 2 x log0 5 y2 - x2 > 0 is valid for atleast one real value of'x'.


EXER CISE-IV

3 1 Q. 1 Prove that the values of the function s i n x c o s do not lie from - & 3 for any real x. sin3x cosx 3

[JEE '97,5] Q.2 The sum of all the real roots of the equation |x - 2|2 + |x - 2| - 2 = 0 is . [JEE '97,2]

Q.3 Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c & d denote the lengths of the sides of the quadrilateral, prove that : 2 < a2 + b2 + c2 + d2 < 4.

[JEE '97,5]

Q.4 In a college of 300 students, every student reads 5 news papers & every news paper is read by 60 students. The number of news papers is: (A) atleast 30 (B) atmost 20 (C) exactly 25 (D) none ofthe above

[JEE'98,2] Q.5 If a, p are the roots of the equation x2 - bx + c = 0, then find the equation whose roots are,

(a2 + p2) (a3 + p3) & a5 p3 + a 3 p5 - 2a4 p4. [REE'98,6]

Q.6(i) Let a + ip ; a, p e R, be a root of the equation x3 + qx + r = 0; q, r e R. Find a real cubic equation, independent of a & p, whose one root is 2 a .

(ii) Find the values o f a & p , 0 a, then the equation, (x - a) (x - b) - 1 = 0, has : (A) both roots in [a, b] (B) both roots in ( - oo, a) (C) both roots in [b, oo) (D) one root in ( - oo, a) & other in (b, + oo)

[JEE 2000 Screening, 1 + 1 + 1 out of 35]

faBansal Classes Quadratic Equations [8]

(d) If a, P are the roots of ax2 + bx + c - 0, (a * 0) and a + 5, P + 5, are the roots of, Ax2 + Bx + C = 0, (A * 0) for some constant 5, then prove that,

h2 4a r R2 4 A C = 2 [JEE 2000, Mains, 4 out of 100]

3. A

Q.10 The number of integer values of m, for which the x co-ordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is (A) 2 (B) 0 (C)4 (D)l

[JEE 2001, Screening, 1 out of 35] Q. 11 Let a, b, c be real numbers with a * 0 and let a, p be the roots of the equation

ax2 + bx + c = 0. Express the roots of a3x2 + abcx + c3 = 0 in terms of a, p. [JEE 2001, Mains, 5 out of 100]

Q.12 The set of all real numbers x for which x2 - |x + 2j + x > 0, is

(A) (-a>, -2) U (2, oo) (B) (-oo, -V2) U (V2 , 00)

(C) (-00,-1) 11 (1, 00) (D) (V2,00) [JEE 2002 (screening), 3]

Q. 13 If x2 + (a - b)x + (1 - a - b) = 0 where a, b e R then find the values of 'a' for which equation has unequal real roots for all values of 'b'. [JEE-03, Mains-4 out of 60]

[ Based on M. R. test] Q.14 (a) If one root of the equation x2 + px + q = 0 is the square of the other, then

(A) p3 + q2 - q(3p + 1) = 0 (B) p3 + q2 + q(l + 3p) = 0 (C) p3 + q2 + q(3p - 1) = 0 (D) p3 + q2 + q(l - 3p) = 0

(b) If x2 + 2ax + 10 - 3a> 0 for all x e R, then (A) - 5 < a < 2 (B) a < 5 (C)a>5 ( D ) 2 < a < 5

[JEE 2004 (Screening)]


ANSWER KEY EXERCISE-I

Q.2 2x2 + 2x cos (A - B) - 2 Q.3 254 * Q.7 a e - o o , - - Q.9 1

u { 2 } u ( 5 , 6 ] Q.IO minimum value 3 when x = 1 and p = 0 Q.18

Q.24 x 2 - 3 x + 2 = 0 Q.28 - 2 < a < 1 Q-29y m i n = 6

EXERCISE-II

Q.l (a) x = l (b) x = 2 or 5 (c) x = ( l - f i ) a o r (Ve - l )a (d) x = - 4 o r - ( 1 + V3)

(e) x = ( - 7 - V l 7 ) / 2 (f) x = - 2 or - 4 o r - ( l + V ^ ) (g) x = - l or 1 (h) x > - l or x = - 3

"4 Q.2 30 Q.3 :,1 Q.5 A = 2 or - 1 8 , B = 3 2 o r - 2 8 8

Q.9 x2 = y2 = d/(a+b+c) ; x/(c - a) = y/(a - b) = K where K2a (a2+b2+c2 - ab - be - ca) = d

Q 10. K < - 1 Q 11. ( - 0 0 , - 1 4 ) u {4} u 14

00 Q 12. 2V2 < a < 11

Q.13 (0,8] Q 14. (a) K < - 1 or K > 5/4 (b) K = - 1 Q 1 5 . - 2 < p < - 2 / V ? or 2/VJ

BANSAL CLASSES *

MATHEMATICS TARGETIIT JEE 2007

XI (P, Q,R, S)

SEQUENCE & PROGRESSION

CONTENTS KEY CONCEPTS

EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY

KEY CONCEPTS

DEFINITION: A sequence is a set of terms in a definite order with a rule for obtaining the terms, e.g. 1,1/2,1/3, ,1/n, is a sequence.

AN ARITHMETIC PROGRESSION (AP): AP is a sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference. If a is the first term & d the common difference, then AP can be written as a, a + d, a + 2d, a + ( n - l)d, nth term of this AP tn = a + (n - 1 )d, where d = an - a n l .

The sum ofthe first n terms of the AP is given by ; Sn = [2 a+ (n - 1 )d] = ^ [a+/].

where I is the last term.

NOTES: (i) If each term of an A.P. is increased, decreased, multiplied or divided by the same non zero number, then

the resulting sequence is also an AP. (ii) Three numbers in AP can be taken a s a - d , a, a+d; four numbers in AP can be taken as a - 3 d,

a -d , a+d, a+3d; five numbers in APare a - 2 d , a - d , a, a+d, a+2d & six terms in APare a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d etc.

(iii) The common difference can be zero, positive or negative.

(iv) The sum of the two terms of an AP equidistant from the beginning & end is constant and equal to the sum of first & last terms.

(v) Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it. (vi) t r = S r - S M

(vii) If a, b, c are in AP => 2 b = a + c.

GEOMETRIC PROGRESSION (GP): GP is a sequence of numbers whose first term is non zero & each of the succeeding terms is equal to the proceeding terms multiplied by a constant. Thus in a GP the ratio of successive terms is constant. This constant factor is called the COMMON RATIO of the series & is obtained by dividing any term by that which immediately proceeds it. Therefore a, ar, ar2,ar3, ar4, is a GP with a as the first term & r as common ratio.

(i) nth term = arn_1

a(rn-l) (ii) Sum of the Ist n terms i.e. Sn = , if r * 1 . r-1 (iii) Sum of an infinite GP when | r | < 1 when n oo rn > 0 if | r | < 1 therefore,

S ^ f l r K l ) .

(iv) If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP.

(v) Any 3 consecutive terms of a GP can be taken as a/r, a, ar ; any 4 consecutive terms of a GP can be taken as a/r3, a/r, ar, ar3 & so on.

(vi) If a, b, c are in GP => b2 = ac.

Bansal Classes Sequence & Progression [2]

HARMONIC PROGRESSION (HP): A sequence is said to HP if the reciprocals of its terms are in AP. If the sequence a,, a2, a3,...., an is an HP then l/al5 l/a2,...., l/an is an AP & converse. Here we do not have the formula for the sum of the n terms of an HP. For HP whose first term is a & second term

isb, then111 term is tn = b + ( n - l ) ( a - b )

2ac a a - b If a, b, c are in HP => b = or = T . a + c c d - c

MEANS ARITHMETIC MEAN:

If three terms are in AP then the middle term is called the AM between the other two, so if a, b, c are in AP, b is AM of a & c .

AM for any n positive number a,, a2,..., an is ; A = a ' + a 2 + a^+ + a" .

n-ARITHMETIC MEANS BETWEEN TWO NUMBERS : Ifa,b are any two given numbers & a,A15A2,.... ,An, b are inAP thenA,, A2, ...Anare then AM's between a & b .

A1 = a + ^ - , A2 = a + ^ i l , ,A = a + n (b - a)

n + 1 ' 2 n + 1 ' ' n n + 1

b - a = a + d , = a + 2 d , , A = a + nd, where d = -

n n + 1

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

r = l

G E O M E T R I C M E A N S : . If a, b, c are in GP, b is the GM between a & c. b2 = ac, therefore b = Ja c ; a > 0, c > 0.

n-GEOMETRIC MEANS BETWEEN a, b : If a, b are two given numbers & a, G}, G2, , Gn, b are in GP. Then Gj, G2, G3,...., Gn are n GMs between a & b . G, = a(b/a)1/n+1, G2 = a(b/a)2/n+1, , Gn = a(b/a)n/n+1

= ar , = ar2, = arn, where r = (b/a)1/n+1

NOTE : The product of n GMs between a & b is equal to the nth power of the single GM between a & b n

i.e. ^ Gr=(G)n where G is the single GM between a & b.

HARMONIC MEAN : If a, b, c are in HP, b is the HM between a & c, then b = 2ac/[a+c].

THEOREM: If A, G, H are respectively AM, GM, HM between a & b both being unequal & positive then, (i) G2 = AH (ii) A > G > H (G > 0). Note that A, G, H constitute a GP.

fa B ansa/ Classes Sequence & Progression [3]

ARITHMETICO-GEOMETRIC SERIES: A series each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series, e.g. 1 + 3x + 5x2 + 7x3 + Here 1,3,5,.... are in AP& l,x,x2 ,x3 areinGP. Standart appearance of an Arithmetico-Geometric Series is Let Sn = a + (a + d)r + (a + 2 d) r2 + + [a + (n- l )d] r""1

SUM TO INFINITY : a dr

If | r | < 1 & n oo then Limit rn = 0 . S . - + 2 . (1 - r )

SIGMA NOTATIONS THEOREMS :

(i) Z ( a r b r ) = a r b, r = l r = 1 r = 1

(ii) I ka r k ar. r = 1 r = 1

(iii) ]T k = nk ; where k is a constant. r = 1

RESULTS n n (n + 1)

(i) X r = 9 (sum of the first n natural nos.) r = l 2

(ii) Y = n (n+1) ^ s u m 0f the squares of the first n natural numbers)

(iii) r3 = (" + 1)2 T = \ 4 r = 1

(sum of the cubes of the first n natural numbers)

(iv> z r4 = - ( n + l)(2n+l)(3n2 + 3 n - l ) r = l '

METHOD OF DIFFERENCE : If T j , T 2 , T 3 , , T N are the terms of a sequence then some times the terms T 2 - T j , T 3 - T 2 , constitute an AP/GP. nth term of the series is determined & the sum to n terms of the sequence can easily be obtained. Remember that to find the sum of n terms of a series each term of which is composed of r factors in AP, the first factors of several terms being in the same AP, we "write down the nth term, affix the next factor at the end, divide by the number of factors thus increased and by the common difference and add a constant. Determine the value of the constant by applying the initial conditions".

r


EXERCISE-I

Q 1. If the 1 Oth term of an HP is 21 & 2 V* term ofthe same HP is 10, then find the 210th term.

Q 2. Solve the following equations for x & y, logio x + log10 x1/2 + log,6 X1/4 + = y

1+3+5+ +(2y-l ) 20 4+7+10+ +(3y+l) 71og]0 x

Q 3. There are n AM's between 1 & 31 such that 7th mean: (n - 1 )th mean =5:9 , then find the value of n. Q 4. Find the sum of the series, 7 + 77 + 777 + to n terms.

Q 5. Express the recurring decimal 0.1576 as a rational number using concept of infinite geometric series.

1 2 3 Q 6. Find the sum of the n terms of the sequence 1 + 12+14 1+22 +24 1 + 32 +34

Q 7. The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369. The first and the ninth term of a geometric progression coincide with the first and the ninth term ofthe arithmetic progression. Find the seventh term of the geometric progression.

Q 8. If the pth, qth & rth terms of an AP are in GP. Show that the common ratio of the GP is q~r

p - q

Q 9. If one AM 'a' & two GM's p & q be inserted between any two given numbers then show that p3+ q3 = 2 apq .

Q 10. Thesumofn terms oftwo arithmetic series are in the ratio of (7 n+1): (4 n+27). Find the ratio oftheir nth term.

Q l l . If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of p2 !) Q 12. The first and last terms of an A.P. are a and b. There are altogether (2n + 1) terms. A new series is

formed by multiplying each of the first 2n terms by the next term. Show that the sum ofthe new series is (4n2 - l)(a2 + b2) + (4n2 + 2)ab

6n Q 13. In an AP of which' a' is the 1st term, if the sum of the 1st p terms is equal to zero, show that the sum

of the next q terms is - a (p + q) q/(p -1) .

Q 14. The interior angles of a polygon are in AP. The smallest angle is 120 & the common difference is 5. Find the number of sides of the polygon,

Q 15. An AP & an HP have the same first term, the same last term & the same number of terms; prove that the product of the V th term from the beginning in one series & the rth term from the end in the other is independent of r.

Q 16. Find three numbers a, b, c between 2 & 18 such that ; (i) their sum is 25 (ii) the numbers 2, a, bare consecutive terms of an AP & (iii) the numbers b, c, 18 are consecutive terms of a GP .

Q 17. Given that ax = by = cz = du & a , b , c , d are in GP,show that x , y , z , u a r e i n H P . Q 18. In a set of four numbers, the first three are in GP & the last three are in AP, with common difference 6.

If the first number is the same as the fourth, find the four numbers.

Q 19. Find the sum of the first n terms of the sequence: l + 2(1+] +3[1+] +4(1+] +


Q 20. Find the nth term and the sum to n terms of the sequence ; (i) 1 + 5 + 13+29 + 61 + (ii) 6+13 + 22 + 33 +

Q 21. The AM of two numbers exceeds their GM by 15 & HM by 27 . Find the numbers.

Q 22. The harmonic mean of two numbers is 4. The airthmetic mean A & the geometric mean G satisfy the relation 2 A + G2 = 27. Find the two numbers.

Q 23. Sum the following series to n terms and to infinity:

(i) 5;+ + + (ii) I r (r+ l)(r+2) (r + 3) w 1.4.7 4 .7 .10 7.10.13 r _ i

n i 1 1.3 1.3.5 I 7 ^ 7 7 - + - +

r=1 4r - 1 4 4.6 4.6.8 Q 24. Find the value ofthe sum

n n

(a) Y y 02 r3s where 5 is zero if r ^ s&8 r is one if r = s. v iti""""' < 7t/2, if : oo co z = 1l cos2n (j) sin2n (j) then : Prove that n = 0 n = 0 n = 0

(i) xyz = xy + z (ii) xyz = x + y + z

EXERCISE-II

Q 1. The series of natural numbers is divided into groups (1), (2,3,4), (5,6,7,8,9), & so on. Show that the sum of the numbers in the nth group is (n - 1 )3 + n3.

Q 2. The sum of the squares of three distinct real numbers, which are in GP is S2. If their sum is a S, show that a2 e (1/3 ,1) u (1,3).

Q 3. If there be m AP's beginning with unity whose common difference is 1,2.3 .... m. Show that the sum of their nth terms is (m/2) (mn - m + n + 1).

Q 4, If Sn represents the sum to n terms of a GP whose first term & common ratio are a & r respectively, then

prove that S, + S, + S, + + S7 , = - a r . ^ 1 3 5 2 n- ! 1 - r (1-r) (l + r)

Q 5. A geometrical & harmonic progression have the same pth, qth & rlh terms a, b, c respectively. Show that a (b - c) log a + b (c - a) log b + c (a - b) log c = 0.

Q.6 A computer solved several problems in succession. The time it took the computer to solve each successive problem was the same number of times smaller than the time it took to solve the preceding problem. How many problems were suggested to the computer if it spent 63.5 min to solve all the problems except for the first, 127 min to solve all the problems except for the last one, and 31.5 min to solve all the problems except for the first two?

Q 7. If the sum of m terms of an AP is equal to the sum of either the next n terms or the next p terms ofthe same APprove that (m+n)[(l/m)-(l/p)] = (m + p)[(l/m)-(l/n)] (n*p)

Q 8. If the roots of 1 Ox3 - cx2 - 54x -21 = 0 are in harmonic progression, then find c & all the roots.


Q 9,(a) Let a,, a2, a3 an be an AP . Prove that :

1 1 1 1 + + + + al an a2 an-l a3 an-2 an al a l+ an

1 1 1 1 1 h + + al a2 a3 an

(b) Show that in any arithmetic progression al, a^ a3 a.2 - a22 + a32 - a42 + ...... + a22K _ , - a22K = [K/(2 K - 1)] (&12 - a22K).

Q10. Let a{, a2 , , an , an + i , be an A.P. Let S, = ax + a2 + a3 + + ap

52 = an+l + an+2 + + a2n 53 = a2n+I + a2n+2 + + S3n '

Prove that the sequence Sj, S2 , S3 , is an arithmetic progression whose common difference

is n2 times the common difference of the given progression.

Q l l . If a, b, c are in HP, b, c, d are in GP & c, d, e are in AP, Show that e = ab2/(2a - b)2.

Q 12. If a, b, c, d, e be 5 numbers such that a, b, c are in AP ; b, c, d are in GP & c, d, e are in HP then: (i) Prove that a, c, e are in GP . (ii) Prove that e = (2 b - a)2/a . (iii) If a = 2 & e = 18 , find all possible values of b, c, d .

1 1 1 1 Q13. If A = 1 + - + - + + - + and

2 3 n n + 1

b _ 2 2 + + < n z J ) l 2 [(n + l)n n (n - l ) ( n - i ) ( n - 2 ) 3-2 j '

then show that A = B. Q 14. If n is a root of the equation x2(l -ac)-x(a 2+c 2)-( l+ac) = 0&if n HM's are inserted between

a & c, show that the difference between the first & the last mean is equal to ac(a - c). Q15. (a) The value of x + y + z is 15 if a , x , y , z , b are in APwhile the value of;

(l/x)+(l/y)+(l/z) is 5/3 if a, x, y, z, b are in HP . Find a & b . (b) The values of xyz is 15/2 or 18/5 according as the series a, x, y, z, b is an AP or HP. Find

the values of a & b assuming them to be positive integer.

Q 16. An AP, aGP &aHP have' a' & 'b' for their first two terms. S how that their (n+2)th terms will be

h2 n + 2- a 2 n + 2 n+1 inGP if a n+1 ba(b2n-a2n) n

13 3.5 5.7 7.9 Q 17. Prove that the sum of the infinite series h ~ + T + o=23 .

2 2 2 2

Q 18. If there are n quantities in GP with common ratio r & Sm denotes the sum ofthe first m terms, show that the sum of the products of thesemterms taken two&two together is [r/(r+1)] [Sm] [Sm_,].

Q 19. Consider an A.P., with the first term 'a', the common difference'd' and a G.P. with the first term 'a', the common ratio 'r' such that a, d, r >0 and both these progressions have same number of terms as well as the equal extreme terms. Show that the sum of all the terms of A.P. > the sum of all the terms ofthe G.P.


Q 20. If n is even & a+(3, a - (3 are the middle pair of terms, show that the sum of the cubes of an arithmetical progression is n a {a2+ (n2-1) (32}.

Q 21. If a, b, c be in GP & logc a, logb c, loga b be in AP, then show that the common difference of the AP must be 3/2.

Q 22. If ax = 1 & for n> 1, an = an_, + 1 , then show that 12 < aJ5 < 15. a n - l

1 2x 3x2 Q 23. Sum tonterms: (i) - + - +

(ii) ^ +

x + 1 (x + 1) (x + 2) (x + 1) (x + 2) (x + 3) a,

l + i (1 + ai)(1 + a2) (l + ai)(l + a2)(l + a3) Q 24. In a GPthe ratio of the sum of the first eleven terms to the sum ofthe last eleven terms is 1/8 and the

ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is 2. Find the number of terms in the GP.

Q 25. Given a three digit number whose digits are three successive terms ofa G.P. If we subtract 792 from it, we get a number written by the same digits in the reverse order. Now if we subtract four from the hundred's digit of the initial number and leave the other digits unchanged, we get a number whose digits are successive terms of an A.P. Find the number.

EXERCISE-III

Q.l For any odd integer n> l , n 3 - ( n - 1)3 + + ( - l ) n ~ 1 l 3 = _ . [JEE '96,1]

Q.2 x= l+3a + 6a2+ 10a3+ | a | < l y = 1+ 4b + 10b2 + 20b3 + |b | 1, z > 1 are in GP, then - , - , - are in : 1 + f n x l + ny 1 + ftiz

(A) AP (B) HP (C) GP (D) none ofthe above

(c) Prove that a triangle ABC is equilateral if & only if tanA+tanB + tanC = 3


Q.7

(a) The harmonic mean of the roots of the equation (5+Jl j x2 - (4 + V5 j x + 8 + 2%/5 =0 is

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

(b) Letajjaj,...., a10, be in A.P. & h,,h2, ,h10 beinH.P. If a1 = h 1 =2&a ] 0 = h10 = 3 then a4h7 is: (A) 2 (B) 3 (C) 5 (D) 6

[ JEE '99,2 + 2 out of200 ]

Q.8 The sum of an infinite geometric series is 162 and the sum of its first n terms is 160. If the inverse of its common ratio is an integer, find all possible values ofthe common ratio, n and the first terms of the series.

[REE'99,6]

Q.9 (a) Consider an infinite geometric series with first term 'a' and common ratio r . If the sum is 4 and the

second term is 3/4, then:

(A) a = ~ , r = | (B) a = 2 , r = |

(C) ^ , x = \ (D) a = 3,

(b) If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation: (A) 0 < M < 1 (B) 1 < M < 2 (C) 2 < M < 3 (D) 3 < M < 4

[ JEE 2000, Screening, 1 + 1 out of 35 ]

(c) The fourth power of the common difference of an arithmetic progression with integer entries added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

[ JEE 2000, Mains, 4 out of 100 ]

Q.10 Given that a , y are roots of the equation, A x 2 - 4 x + l = 0 and fi, 5 the roots of the equation, B x2 - 6 x + 1 = 0, find values of A and B, such that a, p, y & 8 are in H.P.

[REE 2000, 5.outof 100]

Q.ll The sum of roots of the equation ax2+bx+c = 0 is equal to the sum of squares of their reciprocals. Find whether be2, ca2 and ab2 in A.P., G.P. or H.P.? [ REE 2001, 3 out of 100 ]

Q.12 Solve the following equations for x and y log2x + log4x + log16x + =y

5 + 9 + 13+ +(4y +1) 1 + 3 + 5 + +(2y-l) = 4 l 0 g4 x [ R E E 2 0 0 1 ' 5 o u t o f l 0 ]

Q.13 (a) Let a, p be the roots of x2 - x + p = 0 and y, 8 be the roots of x2 - 4x + q = 0. If a, P, y, 6 are in G.P.,

then the integral values of p and q respectively, are (A)-2,-32 (B) -2 ,3 (C)-6,3 (D)-6,-32

(b) If the sum ofthe first 2n terms of the A.P. 2,5,8, is equal to the sum ofthe first n terms of the A.P. 57,59,61, , then n equals

(A) 10 (B) 12 (C)l l (D) 13 fa Ban sal Classes Sequence & Progression [11]

(c) Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bed are (A) NOT in A.P./G.P./H.P. (B) in A.P. (C) in G.P. (D)H.P.

[JEE 2001, Screening, 1 + 1 + 1 out of 3 5 ]

(d) Let al5 a2 be positive real numbers in G.P. For each n, let An, Gn, Hn, be respectively, the arithmetic mean, geometric mean and harmonic mean of a v a,, a3, an. Find an expression for the G.M. of Gj, G2, Gn in terms of A p A2 An, Hj, H2, Hn.

[JEE 2001 (Mains);5] Q.14 3 (a) Suppose a, b, c are in A.P. and a2, b2, c2 are in G.P. If a < b < c and a + b + c = - , then the value of

a is

[JEE 2002 (Screening), 3]

(b) Let a, b be positive real numbers. If a , A, , A7 , b are in A.P. ; a , a! , a2 , b are in G.P. and a, Hj , H2 , b are in H.P., show that

G J G J A!+A2 (2a + b)(a+2b) H H - H + H - [ JEE 2002 , Mains , 5 out of 60 ]

c Q.15 If a, b, c are in A.P., a2, b2, c2 are in H.P., then prove that either a = b = c or a, b, - form a G.P.

[JEE-03, Mains-4 out of 60]

Q.16 The first term of an infinite geometric progression is x and its sum is 5. Then

(A)010

[JEE 2004 (Screening)] Q.17 If a, b, c are positive real numbers, then prove that [(1 + a) (1 + b) (1 + c)]7 > 77 a4 b4 c4.

[JEE 2004,4 out of 60]

(n + \) , Q.18 If total number of runs scored in n matches is (2n+1 - n - 2) where n > 1, and the runs scored in

V 4 J the kth match are given by k-2n+1"k, where 1 < k < n. Find n. [JEE 2005,2]

fa Ban sal Classes Sequence & Progression [11]

A N S W E R K E Y

EXERCISE-I

Q l . 1 Q 2. x = 105, y = 10 Q3. ji= 14 Q 4. S = (7/81){10n+1 - 9n - 10} Q 5. 35/222 Q6. n(n+l)/2(n2 + n+1) Q 7. 27 Q 10. (14 n - 6)/(8 n + 23) Q 11. 1 Q 14. 9 Q 16. a = 5 ,b = 8 , c = 12 Q 18. ( 8 , - 4 , 2, 8) Q 19. n2

Q20. (i) 2n+1 - 3 ; 2 n + 2 - 4 - 3 n (ii) n 2 + 4 n + l ; ( l /6)n(n+l ) (2n+13) + n Q 21. 120,30 Q 22. 6 ,3 Q 23. (i) sn = (1/24) - [l/{6(3n+ 1) (3n + 4) >] j s ^ l / 2 4 (ii) ( l /5)n(n+ l)(n + 2)(n+ 3)(n + 4)

(iii) n/(2n+1) (iv) Sn = 2

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

EXERCISE-II

1 1.3.5 (2n-l)(2n + l) 2 2.4.6 (2n)(2n + 2) ; S = 1 " on

Q6. 8problems, 127.5 minutes Q.8 C = 9 ; (3,-3/2 ,-3/5) Q 12. (iii) b = 4 , c = 6 , d = 9 OR b = - 2 , c = - 6 , D = - 18 Q 15. (a) a = 1, b = 9 OR b = 1, a = 9 ; (b) a = 1 ; b = 3 or vice versa

Q 23. (a) 1 - (b) 1 -(x + 1) (x + 2) (x + n) (l + a i ) ( l + a2) (l + a n ) Q 24. n = 38 Q 25. 931

EXERCISE-III

Q l . | ( 2 n - l ) ( n + l ) 2

Q 2. S = Where a = 1 - x"1/3 & b = 1 - y"1/4 Q3. p < (1/3) ; y > -(1/27)

Q 4. - 3 , 7 7 Q 5. 8,24,72,216,648 Q 6. (a) C (b) B Q 7. (a) B (b) D

Q 8. r = 1/9 ; n = 2 ; a = 144/180 OR r = 1/3 ; n = 4 ; a= 108 OR r = 1/81 ; n = 1 ; a= 160

Q9. (a) D (b) A Q 10. A = 3 ; B = 8

Q l l . A.P. Q 12. x = 2V2 andy = 3

Q13. (a) A, (b)C, (c) D, (d)[(A,, A2, An) (Hl5 H2, Hn)] Q14. (a) D Q.16 B Q.18 n = 7

fa Ban sal Classes Sequence & Progression [11]

BANSAL CLASSES

MATHEMATICS I TARGETIITJEE 2007

XI (P> Q, R, S)

* -yfi

r i i c csi I Inn Lae

CONTENTS KEY- CONCEPTS

EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY

KEY CONCEPTS S T A N D A R D R E S U L T S :

1. EQUATION OF A CIRCLE IN VARIOUS FORM : (a) The circle with centre (h, k) & radius 'r' has the equation;

(x-h) 2 + (y-k) 2 =r2. (b) The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 with centre as :

(-g, -f) & radius = ^g 2 +f 2 -c . Remember that every second degree equation in x & y in which coefficient of x2 = coefficient of y2 & there is no xy term always represents a circle. If g2 + f 2 - c > 0 => real circle.

g2 + f 2 - c = 0 => point circle. g2 + f 2 - c < 0 => imaginary circle.

Note that the general equation of a circle contains three arbitrary constant s, g, f & c which corresponds to the fact that a unique circle passes through three non collinear points. (c) The equation of circle with (x,, y}) & , y2) as its diameter is :

( X - X l ) ( x - x ^ + ( y - y i ) ( y - y 2 ) = 0. Note that this will be the circle of least radius passing through , yj) & (xj, y2).

2. INTERCEPTS MADE BY A CIRCLE ON THE AXES : The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the co-ordinate axes are

2 vg2 - c & 2 ^ / f ^ c respectively. NOTE : If g2 - c >0 => circle cuts the x axis at two distinct points. If g2 = c circle touches the x-axis. If g2 circle lies completely above or below the x-axis.

3. POSITION OF A POINT w.r.t. A CIRCLE : The point (x t, yj) is inside, on or outside the circle x2 + y2 + 2gx + 2fy+ c~ U. according as Xj2 + y 2 + 2gx, + 2fv, + c O 0 .

Note : The greatest & the least distance of a point Afrom a circle with centre C & radius r is AC + r & AC - r respectively. (Xi,yi) p\

4. LINE & A CIRCLE: Let L = 0 be a line & S = 0 be a circle. If r is the radius of the circle & p is the length of the perpendicular from, the centre on the line, then: (i) p > r the line does not meet the circle i. e. passes out side the circle. (ii) p = r o the line touches the circle. (iii) p < r o the line is a secant of the circle. (iv) p = 0 => the line is a diameter of the circle.

5. PARAME TRIC EQUATIONS OF A CIRCLE: The parametric equations of (x - h)2 + (y - k)2 = r2 are: x = h + rcos9 ; y = k + rsin9 ; -TC < 0 < TC where (h, k) is the centre, r is the radius & 9 is a parameter. Note that equation of a straight line joining two point a & (3 on the circle x2 + y2 = a2 is

a+B , . a+B a-B x cos + y sin = a cos - .

(!%Bansal Classes Circles [12]

6. TANGENT & NORMAL: (a) The equation of the tangent to the circle x2+y2 = a2 at its point (x t , yt) is,

xxj + y y, = a2. Hence equation of a tangent at (a cos a, a sina) is; x cos a + y sin a = a. The point of intersection of the tangents at the points P(a) and Q(f3) is

a+ff . a + 6 acos 2 asrn^-11 a - f l a - p COS x- cos v u o - 2

(b) The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at its point (x t, yj) is XX, + yyj + g (x + Xj ) + f (y + yj) + c = 0.

(c) y = mx + c is always a tangent to the circle x2+y2=a2 if c2 = a2 (1 + m2) and the point of contact

is ( 2 2\ a m a

c c J (d) If a line is normal/orthogonal to a circle then it must pass through the centre of the circle. Using

this fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at(x ;, y,) is y_y = Zi (X-Xl).

x,+g 7. A FAMILY OF CIRCLES : (a) The equation of the family of circles passing through the points of intersection of two circles

St =0 & S2 = 0is : S!+KS 2 = 0 (K*- l ) . (b) The equation of the family of circles passing through the point of intersection of a circle

S = 0 & a line L = 0 is given by S+KL = 0. (c) The equation of a family of circles passing through two given points (x}, yt) & , y2) can be written

in the form:

(x-x1)(x-x2) + (y-y 1 ) (y-y 2 )+K = 0 where K is a parameter. y i Yi 1

x2 y2 i (d) The equation of a family of circles touching a fixed line y - yL = m (x - x}) at the fixed point (xL, yj) is

(x - xt)2 + (y - yx)2 + K [y - yj - m (x - Xj)] = 0, where K is a parameter. In case the line through (xj, yj) is parallel to y - axis the equation of the family of circles touching it at (Xj, yt) becomes (x - x,)2 + (y - y^2 + K (x - Xj) = 0. Also if line is parallel to x - axis the equation of the family of circles touching it at (xiYi) becomes ( x - X j ) 2 + ( y - y ^ 2 + K ( y - y i ) = 0.

(e) Equation of circle circumscribing a triangle whose sides are given by Lj = 0 ; L2 = 0 & L3 = 0 is given by; LjL2 + A. L2L3 + \x L3Lj = 0 provided co-efficient of xy = 0 & co-efficient of x2 = co-efficient of y2.

(f) Equation of circle circumscribing a quadrilateral whose side in order are represented by the lines Lj = 0, L2 = 0, L3 = 0 & L4 = 0 is L,L3 + A L2L4 = 0 provided co-efficient of x2 = co-efficient of y2 and co-efficient of xy=0.

8. LENGTH OF A TANGENT AND POWER OF A POINT : The length of a tangent from an external point (x t, y^ to the circle S = x2 + y2 + 2gx + 2fy + c = 0 is given by L= Jx12+y12+2gx,+2f1y+c = Js^. Square of length of the tangent from the point P is also called THE POWER OF POINT w.r.t. a circle. Power of a point remains constant w.r.t. a circle. Note that : power of a point Pis positive, negative or zero according as the point 'P'is outside, inside or on the circle respectively.


9 . D I R E C T O R C I R C L E :

The locus of the point of intersection of two perpendicular tangents is called the DIRECTOR CIRCLE ofthe given circle. The director circle of a circle is the concentric circle having radius equal to V2 times the original circle.

1 0 . E Q U A T I O N O F T H E C H O R D W I T H A G I V E N M I D D L E P O I N T :

The equation of the chord of the circle S = x2 + y2 + 2gx + 2fy + c = 0 in terms of its mid point Xi 2

M(xj, yj) is y - yj = - -1- (x - Xj). This on simplication can be put in the form Yj+r xxj + yyj + g (x + Xj) + f (y + yj) + c = Xj2 + Y!2 + 2gx} + 2fyj + c which is designated by T = S,. Note that : the shortest chord of a circle passing through a point 'M' inside the circle,

is one chord whose middle point is M. 11. CHORD O F C O N T A C T :

If two tangents PTj & PT2 are drawn from the point P (x]; yt) to the circle S = x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact TtT2 is: xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.

REMEMBER : (a) Chord of contact exists only if the point 'P' is not inside.

2LR (b) Length of chord of contact T, T2 =

RL3 (c) Area ofthe triangle formed by the pair of the tangents & its chord of contact = |> 2+T2

Where R is the radius of the circle & L is the length of the tangent from (x1; y}) on S = 0.

(d) Angle between the pair of tangents from (xt, yj) = tan 1 ' 2RL ^ vL2-*2/

where R=radius ; L = length of tangent. (e) Equation of the circle circumscribing the triangle PTj T2 is:

(x-Xj) (x + g) + ( y - y i ) (y + f) = 0. (f) The joint equation of a pair of tangents drawn from the point A(xj,yj)to the circle

x2 + y2 + 2gx + 2fy + c = 0 is : S S ^ T 2 . Where S s x2 + y2 +2gx +2fy+c ; Sj =Xj2 + y2 + 2gXj + 2fyj + c

T= xxj + yyl + g(x + XJ) + f(y + y^ + c. 1 2 . P O L E & P O L A R :

(i) If through a point P in the plane of the circle, there be drawn any straight line to meet the circle in Q and R, the locus of the point of intersection of the tangents at Q & R is called the POLAR O F THE POINT P ; also P is called the POLE O F THE POLAR.

(ii) The equation to the polar of a point P (xj, y,) w.r.t. the circle x2 + y2 = a2 is given by xxj + yy t =s a2, & if the circle is general then the equation of the polar becomes xx1 + yy,+g(x + Xj) + f (y + y^ + c = 0. Note that if the point (x t, yj) be on the circle then the chord of contact, tangent & polar will be represented by the same equation.

(iii) Pole of a given line Ax + By + C = 0 w.r.t. any circle x2 + y2 = a2 is Aa2 Ba2^


(iv) If the polar of a point P pass through a point Q, then the polar of Q passes through P. (v) Two lines L, & L2 are conjugate of each other if Pole of Lj lies on L2 & vice versa Similarly two points

P & Q are said to be conjugate of each other if the polar of P passes through Q & vice-versa.

13. COMMON TANGENTS TO TWO CIRCLES: (i) Where the two circles neither intersect nor touch each other, there are FOUR common tangents,

two of them are transverse & the others are direct common tangents. (ii) When they intersect there are two common tangents, both of them being direct. (iii) When they touch each other:

(a) EXTERNALLY : there are three common tangents, two direct and one is the tangent at the point of contact.

(b) INTERNALLY: only one common tangent possible at their point of contact. (iv) Length of an external common tangent & internal common tangent to the two circles is given by:

L e W d 2 - ( r . - r 2 ) 2 & L i n t = A / d 2 - ( r 1 + r 2 ) 2 .

Where d = distance between the centres of the two circles. ^ & r2 are the radii of the two circles. (v) The direct common tangents meet at a point which divides the line joining centre of circles

externally in the ratio of their radii. Transverse common tangents meet at a point which divides the line joining centre of circles internally in the ratio of their radii.

14. RADICAL AXIS & RADICAL CENTRE : The radical axis of two circles is the locus of points whose powers w.r.t. the two circles are equal. The equation of radical axis of the two circles S j = 0 & S2 = 0 is given; S 1 - S 2 = 0 i.e. 2 (g 1 -g 2 )x + 2( f 1 - f 2 )y + (c1-c2) = 0. NOTE THAT:

(a) If two circles intersect, then the radical axis is the common chord of the two circles. (b) If two circles touch each other then the radical axis is the common tangent of the two circles at

the common point of contact. (c) Radical axis is always perpendicular to the line joining the centres of the two circles. (d) Radical axis need not always pass through the mid point of the line joining the centres of the two

circles. (e) Radical axis bisects a common tangent between the two circles. (f) The common point of intersection of the radical axes of three circles taken two at a time is

called the radical centre of three circles. (g) A system of circles, every two which have the same radical axis, is called a coaxal system. (h) Pairs of circles which do not have radical axis are concentric. 15. ORTHOGONALITY OFTWO CIRCLES:

Two circles St = 0 & S2=0 are said to be orthogonal or said to intersect orthogonally if the tangents at their point of intersection include a right angle. The condition for two circles to be orthogonal is : 2 g[ g2 + 2 f j f2 = Ci + C2 .

Note : (a) Locus of the centre of a variable circle orthogonal to two fixed circles is the radical axis between the

two fixed circles. (b) . If two circles are orthogonal, then the polar of a point 'P' on first circle w.r.t. the second circle passes

through the point Q which is the other end of the diameter through P . Hence locus of a point which moves such that its polars w.r.t. the circles S j = 0, S2 = 0 & S3 = 0 are concurrent in a circle which is orthogonal to all the three circles.


EXERCISE-I Q 1. Find the equation of the circle circumscribing the triangle formed by the lines ;

x + y = 6,2x + y = 4& x + 2y = 5, without finding the vertices of the triangle.

Q 2. If the lines a, x + b, y + Cj = 0 & a2X + b2y + c2=0 cut the coordinate axes in concyclic points. Prove that aj bj b2.

Q 3. One of the diameters of the circle circumscribing the rectangle ABCD is4y=x + 7. If A& B are the points (-3,4) & (5,4) respectively. Then find the area of the rectangle.

Q 4. Lines 5x + 12y - 10 = 0 & 5 x - 12y-40 = 0 touch a circle Cj of diameter 6. If the centre of Cj lies in the first quadrant, find the equation of the circle C2 which is concentric with C} & cuts interceptes of length 8 on these lines.

Q 5. Find the equation of the circle inscribed in a triangle formed by the lines 3x + 4y = 12; 5x + 12y = 4 & 8y = 15x + 10 without finding the vertices of the triangle.

Q 6. Find the equation of the circles passing through the point (2,8), touching the lines 4x - 3y - 24 = 0 & 4x + 3y - 42 = 0 & having x coordinate of the centre of the circle less than or equal to 8.

Q 7. Find the equation of a circle which is co-axial with circles 2x2 + 2y2 - 2x + 6y - 3 = 0 & x2 + y2 + 4x + 2y + 1 = 0. It is given that the centre of the circle to be determined lies on the radical axis of these two circles.

Q 8. Let A be the centre of the circle x2 + y2 - 2x - 4y - 20 = 0. Suppose that the tangents at the points B(1,7) & D(4,-2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

Q 9. The radical axis of the circles x2 + y2 + 2gx + 2fy + c = 0 and 2x2 + 2y2 + 3x + 8y + 2c = 0 touches the circle x2 + y2 + 2x - 2y + 1 = 0. Show that either g = 3/4 or f=2.

Q. 10 Find the equation of the circle through the points of intersection of circles x2+y2-4x-6y-12=0 and x2 + y2 + 6x + 4y - 12 = 0 & cutting the circle x2 + y2 - 2x - 4 = 0 orthogonally.

Q 11. Consider a curve ax2+2 hxy + by2 = 1 and a point P not on the curve. Aline is drawn from the point P intersects the curve at points Q & R. If the product PQ. PR is independent of the slope of the line, then show that the curve is a circle.

Q 12, Find the equations of the circles which have the radius Vl3 & which touch the line 2x-3y+1 = 0at(l, 1).

Q 13. A circle is described to pass through the origin and to touch the lines x = 1, x + y = 2. Prove that the

radius of the circle is a root of the equation 3^ - 2-j2j t 2 - 2 j 2 t + 2 = 0.

Q 14. The centre of the circle S = 0 lie on the line 2x-2y + 9 = 0 & S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through two fixed points & find their coordinates.

Q 15. Show that the equation x2 + y2 - 2x - 2 Ay - 8 = 0 represents, for different values of A, a system of circles passing through two fixed points A B on the x - axis, and find the equation of that circle of the system the tangents to which at A & B meet on the line x + 2y + 5 = 0.


Q 16. Find the equation ofthe circle which passes through the point (1, 1) & which touches the circle x2 + y2 + 4x - 6y - 3 = 0 at the point (2, 3) on it.

Q 17. Find the equation of a circle which touches the lines 7x2 - 18xy + 7y2 = 0 and the circle x2 + y2 - 8x- 8y = 0 and is contained in the given circle.

Q 18. Find the equation of the circle which cuts the circle x2 + y2 -14x - 8y + 64 = 0 and the co-ordinate axes orthogonally.

Q 19. Obtain the equations ofthe straight lines passing through the point A(2,0)&making 45 angle with the tangent at A to the circle (x + 2)2 + (y - 3)2 = 25. Find the equations of the circles each of radius 3

whose centres are on these straight lines at a distance of 5 V2 from A.

Q 20. Find the equations of the circles whose centre lie on the line 4x + 3y - 2 = 0 & to which the lines x + y + 4 = 0 & 7 x - y + 4 = 0 are tangents.

Q 21. Find the equations to the four common tangents to the circles x2 + y2 = 25 and (x-12)2 + y2 = 9.

Q 22. If 4/2 - 5m2 + 6/ + 1 = 0. Prove that /x + my + 1 = 0 touches a definite circle. Find the centre & radius of the circle.

Q 23. Find the condition such that the four points in which the circle x2 + y2 + ax + by + c = 0 and x2 + y2 + a'x + b'y + c' = 0 are intercepted by the straight lines Ax + By + C = 0 & A'x + B'y + C' = 0 respectively, lie on another circle.

Q 24. Show that the equation of a straight line meeting the circle x2 + y2 = a2 in two points at equal distances

d2 'd' from a point (x}, yj) on its circumference is xxt + yy} - a2 + = 0.

Q 25. If the equations of the two circles whose radii are a & a' be respectively S = 0 & S - 0, then prove that S S'

the circles + = 0 will cut each other orthogonally, a a

Q 26. Let a circle be given by 2x (x - a) + y (2y - b) = 0, (a * 0, b ^ 0). Find the condition on a & b if

two chords, each bisected by the x-axis, can be drawn to the circle from v ^ j

Q 27. Prove that the length of the common chord of the two circles x2 + y2 = a2 and

(x - c)2 + y2 = b2 is -7(a+b+c)(a-b+c)(a+b-c)(-a+b+c) . c

Q 28. Find the equation of the circle passing through the points A (4,3) & B (2, 5) & touching the axis of y. Also find the point P on the y-axis such that the angle APB has largest magnitude.

Q 29. Find the equations of straight lines which pass through the intersection of the lines x - 2y - 5 = 0, 7x + y = 50 & divide the circumference of the circle x2 + y2 = 100 into two arcs whose lengths are in the ratio 2:1.

Q 30. Find the equation of the circle which cuts each of the circles x2 + y2 = 4, x2 + y 2 -6x-8y+ 10=0 & x2 + y2 + 2 x - 4 y - 2 = 0 at the extremities of a diameter.


EXER CISE-II

Q 1. A point moves such that the sum of the squares of its distance from the sides of a square of side unity is equal to 9. Show that the locus is a circle whose centre coincides with centre of the square. Find also its radius.

Q 2. A triangle has two of its sides along the coordinate axes, its third side touches the circle x2 + y2 - 2ax - 2ay + a2 = 0. Prove that the locus of the circumcentre of the triangle is :

a2 - 2a (x + y) + 2xy = 0.

Q 3. A variable circle passes through the point A (a, b) & touches the x-axis; show that the locus ofthe other end of the diameter through A is (x - a)2 = 4by.

Q 4. (a) Find the locus of the middle point of the chord of the circle, x2 + y2 + 2gx + 2fy + c = 0 which subtends a right angle at the point (a, b). Show that locus is a circle.

(b) Let S= x2+y2 + 2gx+2fy+c=0 be a given circle. Find the locus of the foot of the perpendicular drawn from the origin upon any chord of S which sustends a right angle at the origin.

Q 5. A variable straight line moves so that the product of the perpendiculars on it from the two fixed points (a, 0) & (- a, 0) is a constant equal to c2 . Prove that the locus of the feet of the perpendiculars from each of these points upon the straight line is a circle, the same for each.

Q 6. Showthat the locus of the centres of a circle which cuts two given circles orthogonally is a straight line & hence deduce the locus of the centers of the circles which cut the circles x2 + y2 + 4x - 6y + 9 = 0 & x2 + y2 - 5x + 4y + 2 = 0 orthogonally .

Q 7. Afixed circle is cut by afamilyofcirclespassingthroughtwofixedpointsA(x1,y1)andB(x2,y2). Show that the chord of intersection of the fixed circle with any one of the circles of family passes through a fixed point.

Q 8. The sides of a variable triangle touch the circle x2 + y2 = a2 and two of the vertices are on the line y2 - b2 = 0 (b > a > 0) . Show that the locus of the third vertex is;

(a2 - b2) x2 + (a2 + b2) y2 = (a(a2 + b2))2.

Q 9. Show that the locus of the point the tangents from which to the circle x2 + y2 - a2 = 0 include a constant angle a is (x2 + y2 - 2a2)2 tan2 a = 4a2 (x2 + y2 - a2).

Q 10. ' O' is a fixed point & P a point which moves along a fixed straight line not passing through O; Q is taken on OP such that OP. OQ=K(constant) . Prove that the locus of Q is a circle. Explain how the locus of Q can still be regarded as a circle even if the fixed straight line passes through 'O'.

Q 11. P is a variable point on the circle with centre at C. CA & CB are perpendiculars from C on x-axis & y-axis respectively. Show that the locus of the centroid of the triangle PAB is a circle with centre at the centroid of the triangle CAB & radius equal to one third of the radius of the given circle.

Q 12. A(-a, 0) ; B(a,0) are fixed points. C is a point which divides AB in a constant ratio tana. If AC & CB subtend equal angles at P, prove that the equation ofthe locus of P is x2 + y2 + 2ax sec2a + a2 = 0.

Bansal Classes Circles

Q 13. The circle x2 + y2 +2ax- c2 = 0 and x2 + y2 + 2bx- c2 = 0 intersect at Aand B. Aline through Ameets one circle at P and a parallel line through B meets the other circle at Q. Show that the locus of the mid point of PQ is a circle.

Q 14. Find the locus of a point which is at a least distance from x2 + y2 = b2 & this least distance is equal to its distance from the straight line x = a.

Q 15. The base of a triangle is fixed. Find the locus of the vertex when one base angle is double the other. Assume the base of the triangle as x-axis with mid point as origin & the length ofthe base as 2a.

Q 16. An isosceles right angled triangle whose sides are 1, 1, V2 lies entirely in the first quadrant with the ends of the hypotenuse on the coordinate axes. If it slides prove that the locus of its centroid is

(3x-y)2 + (x-3y) 2 =- 3 ^.

Q 17. The circle x2 + y2 - 4x - 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the coordinate axes. The locus of the circumcentre of the triangle is:

x + y - xy + K ^ / P + y 1 = 0. Find K. *

Q 18. Find the locus of the point of intersection of two perpendicular straight lines each of which touches one of the two circles (x - a)2 + y2 = b2, (x + a)2 + y2 = c2 and prove that the bisectors of the angles between the straight lines always touch one or the other of two other fixed circles.

Q.19 Find the locus of the mid point of the chord of a circle x2 + y2 = 4 such that the segment intercepted by the chord on the curve x2 - 2x - 2y = 0 subtends a right angle at the origin.

Q 20. TheendsAB ofafixedstraightlineoflength'a'&endsA'&B'ofanotherfixedstraightlineoflength 'b' slide upon the axis ofx&the axis ofy (one end on axis of x& the other on axis of y). Find the locus of the centre of the circle passing through A, B, A' & B'.

Q 21. The foot of the perpendicular from the origin to a variable tangent of the circle x2+y2 - 2x = 0 is N. Find the equation ofthe locus ofN.

Q 22, Find the locus of the mid point of all chords of the circle x2+y2 - 2x - 2y = 0 such that the pair of lines joining (0,0) & the point of intersection of the chords with the circles make equal angle with axis of x.

Q 23. P (a) & Q (p) are the two points on the circle having origin as its centre & radius 'a' & AB is the diameter along the axis of x. If a - p = 2 y, then prove that the locus of intersection of AP & BQ is x2 + y2 - 2 ay tany = a2.

Q 24. Show that the locus of the harmonic conjugate of a given point P (xl5 yt) w.r.t. the two points in which any line through P cuts the circle x2 + y2 = a2 is xxj + yy} = a2.

Q 25. Find the equation of the circle which passes through the origin, meets the x-axis orthogonally & cuts the circle x2 + y2 = a2 at an angle of 45.


EXERCISE-III Q.l (a) The intercept on the line y=x bythe circle x 2 +y 2 -2x=0 isAB . Equation ofthe circle with

AB as a diameter is . (b) The angle between a pair of tangents drawn from a point P to the circle

x2 + y2 + 4x - 6y + 9 sin2a + 13 cos2a = 0 is 2 a. The equation of the locus of the point P is : (A) x2 + y2 + 4 x - 6 y + 4 = 0 (B) x2 + y2 + 4x - 6y - 9 = 0 (C) x2 + y2 + 4 x - 6 y - 4 = 0 (D) x2 + y2 + 4 x - 6y + 9 = 0

(c) Find the intervals of values of a for which the line y + x = 0 bisects two chords drawn from a

' l + V2a 1-V2a 1 r-point , - to the circle; 2x2+2y2 - (1+V2 a) x - (1 - V2 a) y = 0.

V 2 J

[JEE'96, 1+1+5]

Q.2 Atangent drawn from the point (4,0)tothecircle x2+y2 = 8 touches it at apointAin the first quadrant. Find the coordinates of the another point B on the circle such that AB = 4. [ REE '96, 6 ]

Q.3 (a) The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to the circle x2 + y2 = 1 pass through the point .

(b) Let C be any circle with centre (o, 42 ) Prove that at the most two rational point can be there on C. (A rational point is a point both of whose co-ordinate are rational numbers).

[JEE '97, 2+5] Q.4 (a) The number of common tangents to the circle x2 + y2 = 4 & x2 + y2 - 6x - 8y = 24 is :

(A) 0 (B) 1 (C) 3 (D) 4 (b) Cj & C2 are two concentric circles, the radius of C2 being twice that of Cj. From a point P on

C2, tangents PA & PB are drawn to C}. Prove that the centroid of the triangle PAB lies on C j. [ JEE '98, 2 + 8 ]

Q. 5 Find the equation of a circle which touches the line x + y = 5 at the point (-2, 7) and cuts the circle x2 + y2 + 4x - 6y + 9 = 0 orthogonally. [ REE '98, 6 ]

Q.6 (a) If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq q) are bisected by the x-axis, then: (A) p2 = q2 (B) p2 = 8q2 (C) p2 < 8q2 (D)p 2 >8q 2

(b) Let Lj be a straight line through the origin and L2 be the straight line x+y = 1. If the intercepts made by the circle x2 + y2 - x + 3y = 0 on L} & L2 are equal, then which of the following equations can represent Lj? (A) x + y = 0 ( B ) x - y = 0 (C)x + 7y = 0 ( D ) x - 7 y = 0

(c) Let Tj, T2 be two tangents drawn from (- 2,0) onto the circle C: x2 + y2 = 1. Determine the circles touching C and having T t , T2 as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.

[ JEE '99, 2 + 3 + 10 (out of200) ] Q.7 (a) The triangle PQR is inscribed in the circle, x2+ y2 = 25. IfQ and Rhave co-ordinates (3,4) &

( - 4,3) respectively, then Z QPR is equal to :

(A) (B) f (C) f (D) f

(b) If the circles, x2 + y2 + 2x + 2ky + 6 = 0 & x2 + y2 + 2 ky + k = 0 intersect orthogonally, then 'k' is:

(A) 2 or - | (B) - 2 or - | (C) 2 or | (D) - 2 or \ [ JEE '2000 (Screening) 1 + 1 ]

^Bansal Classes Circles [10]

Q.8 (a) Extremities of a diagonal of a rectangle are (0,0) & (4,3). Find the equation of the tangents to the circumcircle of a rectangle which are parallel to this diagonal.

(b) A circle of radius 2 units rolls on the outerside of the circle, x2 + y2 + 4 x = 0, touching it externally. Find the locus of the centre of this outer circle. Also find the equations of the common tangents of the two circles when the line joining the centres ofthe two circles makes on angle of 60 with x-axis. [REE '2000 (Mains) 3 + 5]

Q.9 (a) 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

[JEE'2001 (Screening) 1 out of 35]

(b) Let 2x2 + y2 - 3xy = 0 be the equation of a pair of tangents drawn from the origin 'O' to a circle of radius 3 with centre in the first quadrant. IfAis one of the points of contact, find the length of OA

[JEE '2001 (Mains) 5 out of 100]

Q. 10 (a) Find the equation of the circle which passes through the points of intersection of circles x2 + y2 - 2x - 6y + 6 = 0 and x2 + y2 + 2x - 6y + 6 = 0 and intersects the circle x2 + y2 + 4x + 6y + 4 0 orthogonally. [ REE '2001 (Mains) 3 out of 100 ]

(b) Tangents TP and TQ are drawn from a point T to the circle x2 + y2 = a2. If the point T lies on the line px + qy = r, find the locus of centre of the circumcircle of triangle TPQ.

[ REE '2001 (Mains) 5 out of 100 ]

Q. 11 (a) If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the length of PQ is (A) 4 (B)2V5 (C)5 (D)3V5

(b) If a > 2b > 0 then the positive value of m for which y = mx-b-Jl + m is a common tangent to x2 + y2 = b2 and (x - a)2 + y2 = b2 is

2b v'a2 -4b 2 2b b

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

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

(A)l (B)2 (C)3 (D)V3 [JEE'2004 (Scr)]

Q.13 Line2x + 3y+ 1 = 0 is a tangent to a circle at (1,-1). This circle is orthogonal to a circle which is drawn having diameter as a line segment with end points (0,-1) and (- 2,3). Find equation of circle.

[JEE'2004, 4 out of 60] Q.14 A circle is given by x2 + (y -1)2 = 1, another circle C touches it externally and also the x-axis, then the

locus of its centre is (A){(x,y):x2 = 4y}u{(x,y):y = 0} (B) {(x, y) : x2 + (y - 1)2 = 4} u {x, y): y = 0} (C) {(x, y): x2 = y} VJ {(0, y): y = 0} (D*) {(x, y): x2 = 4y} u {(0, y): y = 0} [JEE '2005 (Scr)]


ANSWER KEY EXERCISE-I

Q 1. x2 + y2 - 17x - 19y + 50 = 0 Q 3. 32 sq. unit Q 4. x2 + y2 - lOx-4y + 4 = 0

Q 5. x2 + y2 - 2x - 2y + 1 - 0 Q 6. centre (2 ,3), r = 5 ; centre f 182 " 2 0 5

Q 7. 4x2 + 4y2 + 6x + lOy- 1 = 0 Q 8. 75 sq.units Q 10. x2 + y2 + 16x+ 14y- 12 = 0

Q 12. x2 + y2 - 6x + 4y = 0 OR x2 + y2 + 2 x - 8 y + 4 = 0 Q 14. ( - 4, 4) ; f _ 1 V I 2' 2 J

Q 15. x2 + y2 - 2x - 6y - 8 = 0 Q 16. x2 + y2 + x - 6y + 3 = 0 Q 17. x2 + y2 - 12x -12y + 64 = 0 Q18. x2 + y2 = 64 Q 19. x - 7y = 2, 7x + y = 14 ; (x - l)2 + (y - 7)2 = 32 ; (x -3) 2 + (y + 7)2 = 32 ;

(x - 9)2 + (y - l)2 = 32 ; (x + 5)2 + (y + I)2 = 32

Q 20. x2 + y2 - 4x + 4y = 0 ; x2 + y2 + 8x - 12y + 34 = 0 Q 21. 2 x - V 5 y - 1 5 = 0, 2 x + V 5 y - 1 5 = 0 , x - ^ I J y - 3 0 = 0, x + ^ /35 y - 3 0 = 0

Q 22. Centre == (3, 0), 1 (radius) = S Q 23. a - a ' b - b ' c - c '

A B C A' B' C

Q 26. (a2 > 2b2)

Q 28. x2 + y2 - 4x - 6y + 9 = 0 OR x2 + y 2 -20x-22y+121 =0, P(O,3),0 = 45 Q 29. 4x - 3y - 25 = 0 OR 3x + 4y-25 = 0 Q 30. x2 + y 2 - 4 x - 6 y - 4 = 0

EXER CISE-II Q l . r = 2 Q 4. (b) 2(x2 + y2) + 2gx + 2fy + c = 0 Q 6. 9 x - 10y + 7 = 0 Q 10. aline Q 14. y2 = (b + a) (b + a - 2x) OR y2 = (b - a) (b - a + 2x) Q 15. 3 x 2 - y 2 2 a x - a 2 = 0 Q 17. K = 1 Q 18. { cy + b(x + a)}2 + { - by c(x - a)}2 = (a2 - x2 - y2)2

Q 19. x2 + y2 - 2x - 2y = 0 Q 20. (2ax - 2by)2 + (2bx - 2ay)2 = (a2 - b2)2

Q 21. (x2 + y2 - x ) 2 = x2 + y2 Q 22. x + y = 2 Q 25. x2 + y2 aV2 x = 0

EXER CIS E-II I

( n2 f if 1 Q.l (a) x + yI = - , (b) D, (c) ( - 00, -2) u (2,00) Q.2 (2, -2) or (-2, 2) Q.3 (a) (1/2, 1/4) V 2 y V 2 J 2

2 4- 2 . Q.4 (a) B Q.5 x2 + y2 + 7 x - l l y +38 = 0

Q.6 (a) D (b) B, C (c) Cl : ( x - 4 ) 2 + y2 = 9 ; c2 : + +y 2 = I

common tangent between c & Cj : Tj = 0; T2 = 0 and x - 1 = 0 ; common tangent between c & c2 : Tt = 0; T2 = 0 and x + 1 = 0 ;

common tangent between c, & c2 : T, = 0 ; T, = 0 and y = -jL= ^x +

where ^ : x - V 3 y + 2 = 0 and T2 : x + v / 3 y + 2 = 0 Q.7 (a) C (b) A Q.8 (a) 6 x - 8 y + 25 = 0 & 6 x - 8 y - 2 5 = 0

(b) x2 + y2 + 4 x - 12= 0, T. : V3x-y + 2^3 +4 = 0, T2 : V3x-y + 2V3-4 = 0(D.C.T.)

T3: x + V 3 y - 2 = 0, T4 : x + V3y + 3 = 0 (T.C.T.)

Q.9 (a) A ; (b).OA=3(3 +V10) Q.IO (a) x2 + y2 + 14x-6y + 6 = 0 ; (b) 2px + 2qy = r Q.ll (a) C ; (b). A Q.12 C Q.13 2x2 + 2 y 2 - 1 0 x - 5 y + 1 = 0


BANSAL CLASSES

MATHEMATICS TARGET IIT JEE 2007

XI(PQRS)

PERMUTATION AND COMBINATION

CONTENTS KEY- CONCEPTS

EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY

KEY CONCEPTS D E F I N I T I O N S : 1. P E R M U T A T I O N : Each of the arrangements in a definite order which can be made by taking some or

all ofa number of things is called a P E R M U T A T I O N . 2. C O M B I N A T I O N : Each of the groups or selections which can be made by taking some or all of a

number of things without reference to the order of the things in each group is called a C O M B I N A T I O N . F U N D A M E N T A L P R I N C I P L E O F C O U N T I N G : If an event can occur in'm' different ways, following which another event can occur in'/?' different ways, then the total number of different ways of simultaneous occurrence of both events in a definite order is m x n. This can be extended to any number of events.

R E S U L T S : (i) A Useful Notation :n! = n ( n - l ) ( n - 2 ) 3. 2. 1 ; n! =n. (n - 1) !

0! = 1! = 1 ; (2n)! = 2n. n ! [1. 3. 5. 7...(2n- 1)] Note that factorials of negative integers are not defined.

(ii) If nPr denotes the number of permutations of n different things, taking r at a time, then n! nPr = n (n - 1) (n - 2) ( n - r + 1)= ( n _ r ) j Note that, nPn = n !.

(iii) If nCr denotes the number of combinations of n different things taken r at a time, then n! n p nCr = . = L where r < n ; n e N and r e W . r!(n-r)! rj

(iv) The number of ways in which (m+n) different things can be divided into two groups containing m & n

things respectively is : ( m + n ) - if m=n, the groups are equal & in this case the number of subdivision m!n!

is ; for in any one way it is possible to interchange the two groups without obtaining a new n!n!2!

distribution. However, if 2n things are to be divided equally between two persons then the number of (2n)! ways = n!n!

(v) Number of ways in which (m + n + p) different things can be divided into three groups containing m, n

& p things respectively is 5 m ^ n ^ p. m! n!p! /A \ I

If m = n = p then the number of groups^ n!n!n!3!'

(3n)! However, if 3 n things are to be divided equally among three people then the number of ways = - .

(n!) (vi) The number of permutations of n things taken all at a time whenp of them are similar & of one type, q of

them are similar & of another type, r of them are similar & of a third type & the remaining I

n - (p + q + r) are all different is: -. p!q!r!

(vii) The number of circular permutations ofn different things taken all at a time is; (n-1)!. Ifclockwise&

anti-clockwise circular permutations are considered to be same, then it is Note : Number of circular permutations ofn things when p alike and the rest different taken all at a time

distinguishing clockwise and anticlockwise arrangement is^. p!

(!i Bansal Classes Permutation and Combination [7]

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

(ix) Total number of ways in which it is possible to make a selection by taking some or all out of p + q + r + things, where p are alike of one kind, q alike of a second kind, r alike of third kind & so on is given by: (p+ l ) (q+ l ) ( r+ 1) -1.

(x) Number of ways in which it is possible to make a selection ofm + n + p = N things, where p are alike of one kind, m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr

in the expansion of (1 + X + X 2 + + X ? ) ( 1 + X + X 2+ + Xm) (1 + X + X2 + + x n ) .

Note : Remember that coefficient ofxr in (1 -x)_n=n+r_1Cr (n e N). For example the number ofways in which a selection of four letters can be made from the letters of the word PROPORTION is given by coefficient of x4 in (1 + x + x2 + x3) (1 + x + x2) (1 + x + x2) (1 + x) (1 + x) (1 + x).

(xi) Number ofways in which n distinct things can be distributed to p persons if there is no restriction to the number of things received by men = pn.

(xii) Number of ways in which n identical things may be distributed among p persons if each person may receive none, one or more things is;n+p_1 Cn.

(xiii) a. nCr = nCn_r; nC0 = nCn = 1 ; b. nCx = nCy =>x = y orx + y = n

c. nc r + nCr_! = n+1Cr

(xiv) nCr is maximum if: (a) r = y if n is even, (b) r = o r - y - if n is odd.

(xv) Let N = p8- qb- r- where p, q, r. are distinct primes & a, b, c are natural numbers then: (a) The total numbers of divisors of N including 1 & N is = (a + 1 )(b + 1 )(c + 1) (b) The sum ofthese divisors is

- (p + p1 + p2+.... +pa)(q+q1 + q2+.... + qb) (r +r1 + r2+....+r).... (c) Number of ways in which N can be resolved as a product of two

. 4(a + l)(b + l)(c +1).... if N is not a perfect square factors is

j [(a + l)(b + l)(c +1).... +1] if N is a perfect square

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

(xvi) Grid Problems and tree diagrams.

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

envelope is = n! 1 1 1 / I V . 1 + + +(-1)

2! 3! 4! v ' n! (xvii) S ome times students find it difficult to decide whether a problem is on permutation or combination or

both. Based on certain words / phrases occuring in the problem we can fairly

Bansal Classes Mathematics Study Material for IIT JEE

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