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SEQUENCES AND SERIES&QUADRATIC EQUATION

Maximum marks 120 Time :30 minutes

1. The solution of the equation 3 + 1

x = 2 are

a) 0, -1, - 1 b) 2, -1

5

c) -1, - 1 d) None of these

5

2. One root of the equation 5x^{2 + 13x + K = 0 is the reciprocal of the other, if}

a) K = 0 b) K = 5 c) K = 1/6 d) 6

3. The roots of the equation 4x ? 3.2x +3 + 128 = 0 are

a) 1 and 2 b) 2 and 3 c) 3 and 4 d) 4 and 5.

4. The greatest value of a non ? negative real number l for which both the equations

2 x2 + (l - 1) x + 8 = 0 and x2 ? 8x + l + 4 = 0 have real roots is :

a) 9 b) 12 c) 15 d) 16.

5. If the roots of a_{1x2 + b1x +c1 = 0 and a2x2 +b2x + c2 = 0 are the same, then}

a) a1 = a2, b1 = b2, c1 = c2 b) c1 = c2 = 0

c) a1 = b1 = c1 d) a1 = b1 = c1; a2 = b2 = c2.

a2 b2 c2

6. For what value of K, the roots of the equation x2 ? 2(1 + 3 K) x + 7 (3 + 2K) = 0 are equal?

a) 1, - 10/9 b) 2, - 10/9 c) 3, - 10/9 d) 4, - 10/9

7. If b > a, then the equation (x ? a) (x ? b) ? 1 = 0 has

a) both roots in [a, b] b) both roots in (- ?. a)

c) both roots in ( b, +? ) and other in (b, +?) d) one root in (-?, a).

8. If roots of an equation xn ? 1 = 0 are 1, a1, a2 ......., an-1, then the value of

(1 ? a1) (1 ? a2) (1 ? a3) .... (1 ? an-1) will be

a) n b) n2 c) nn d) 0

9. If the roots of ax2 + ax + c = 0 are in the ratio p : q, then ?p/q + ?q/p =

a) ?a2 b) ?a c) ?a d) None of these

?c ?2c ?c

10. If the equation x2 ? bx = m ? 1 has roots equal in magnitude but opposite in sign then

ax -c m + 1

m equals

a) a + b b) a ? b c) b ? a d) None of these

a ? b a + b b + a

11. Let a and ? be the roots of the equation x2 + x + 1 = 0. The equation whose roots are a19, ?7 is

a) x2 ? x ?1 = 0 b) x2 ? x + 1 = 0

c) x2 + x ? 1 = 0 d) x2 + x +1 = 0

12. If the roots of x2 - bx + c = 0 are two consecutive integers, then b2 ? 4 c is

a) 0 b) 1 c) 2 d) None of these

13. If x2 ? x +1 = 0, then value of x3n is

a) -1, 1 b) 1 c) -1 d) 0.

14. If (?2)x + (?3)x = (?13)x/2, then the number of values of x is

a) 2 b) 4 c) 1 d) None of these.

15. The product of all the roots of the equation x2 - ½x ½ - 6 = 0 is

a) -9 b) 6 c) 18 d) 36.

16. a and c are two distinct positive real numbers such that a, b, c are in G.P. If b-c, c ? a,

a ? b are in H.P, then a + 4b + c is

(a) a constant (b) independent of a (c) independent of b (d) independent of c.

17. Sum of the series 1 + 2.2 + 3.22 + 4.23 + ?.. + 100.299 is

(a) (100)2100 + 1 (b) (99)2100 + 1 (c) (99)299-1 (d) 100(2100) -1.

18. If 1 + 1 = 1 + 1 , and b ? a + c, then a, b, c are in

b ? a b ? c a c

(a) A. P. (b) G.P. (c) H.P. (d) none of these.

19. The sum of (x + 2)n ? 1 + ( x + 2)n ? 2 (x + 1) + (x + 2)n ? 3 (x + 1)2 + ?.+ (x + 1)n ?1

is equal to

(a) (x + 2)n-2 - (x + 1)n (b) (x + 2)n-1 - (x + 1)n-1

(c) (x +2 )n ? (x + 1)n (d) none of these.

20. The sum of first three terms of a G.P. is to the sum of the first six terms as 125 : 152. The common ratio of the G.P is

(a) 1/5 (b) 2/5 (c) 3/5 (d) 4/5.

21. Let n (> 1) be a positive integer. Then the largest integer m such that (nm +1) divides

(1 + n + n2 +??+ n127) is

(a) 127 (b) 63 (c) 64 (d) 32

22. Two A.M.s A1 and A2, two G.M.s G1 and G2 and two H.M.s H1 and H2 are inserted between any two numbers, then H1 -1 + H2 -1 equals

(a) A1-1 +A2-1 (b) G1-1 + G2-1 (c) G1G2 (d) A1 + A2

A1 +A2 G1G2

23 The next term of the sequence 1, 5, 14, 30, 55,?? is

a) 95 b) 90 c) 85 d) 91

24 12 + 1 + 22 + 2 + 32 + 3 + ??? + n2 + n is equal to

a) n(n + 1) b) n (n + 1) 2

2 2

c) n(n+1) (n +2) d) n(n+1) (n +2) (n + 3)

3 4

25. The sum of the infinite series 1 + (1 + a)x + (1 + a + a2)x2 + (1 + a + a2 + a3)x3 + ?.. where 0 < a, x < 1 is

a) 1 b) 1

(1 ? x) (1 ? a) (1 ? x) (1 ? ax)

c) 1 d) 1

(1 ? a) (1 ? ax) (1 ? x) (1 + a)

26. Let a, b, c be in A.P. and ½a ½ < 1, ½b½ < 1, ½c½ < 1.

If x = 1 + a + a2 + ?. ¥

y = 1 + b + b2 + ? ¥

z = 1 + c + c2 + ?. ¥, then x, y, z are in

a) A.P. b) G.P. c) H.P. d) None of these.

27. If a > 0, b > 0, c > 0 and the minimum value of a (b2 + c2) + b(c2 + a2) + c(a2 + b2) is l abc, then l is

a) 2 b) 1 c) 6 d) 3.

28. The sum of the series 1 + 2x + 3x2 + 4x3 + ?.. upto infinity when x lies between

0 and 1 i.e., 0 < x < 1 is

a) 1 b) 1 c) 1 d) 1

1 + x 1 ? x 1 ? 2x (1 ? x)2

29 Let a1, a2, ??, a10 be in A.P. and h1, h2, ??, h10 be in H.P. If a1 = h1 = 2 and

a10 = h10 =3, then a4 h7 is

a) 2 b) 3 c) 5 d) 6

30. If a, b, c are in H.P., then straight line x + y + 1 = 0 always passes thro? a fixed

a b c

point and that point is

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

2

HINTS AND SOLUTION

1. ... 3 + 1/x = ± 2 \1/x = ± 2 ?3 = -1, -5

\ x = -1, -1/5.

2. ... product of the roots = 1

Þ K/5 = 1 Þ K = 5.

3. 4x ? 3.2x+3 + 128 = 0 Þ 22x ? 3. 2x23 + 128 = 0

Þ 22x ? 24.2x + 128 = 0 Þ y2 ? 24y + 128 = 0 where 2x = y

Þ (y ? 16) (y ? 8) = 0

Þ y = 16 , 8 = 0 Þ 2x = 16 or 2x = 8

Þ x = 4 or 3.

4. For real roots (l - 1)2 ? 64 ³ 0 and

64 ? 4 (l +4 ) ³ 0

Þ (l - 1)2 ³ 64 and 48 - 4l ³

Þ l -1 ³ 8 or l -1 £ -8 and 12 ³ l

Þ l £ 9 or l £ - 7 and l £ 12

Hence greatest value of l = 12.

5. Sum of the roots in both the equations are the same and product is also the same.

\ -b1 = -b2 ; c1 = c2 \ (c) is true

a1 a2 a1 a2

6. Since the roots are equal \4(1 +3K)2 ? 28 (3 +2K) = 0

ie. 1 + 9K2 + 6K ? 7 ( 3 +2K) = 0

ie. 9K2 ? 8K ? 20 = 0

\K = 8 ± ?64 + 720 = 8 ± ? 784

18

= 8 ± 28 = 36, -20

18 18 18

= 2, -10

9

7 The given equ. can be written as x2 ? (a + b) x + ab ?1 = 0

Let a, b be its root. \ a + b = a + b ab = ab ? 1

\ If one root is less than a, then the other root is greater than b.

\ we have one root in (-?, a) and other in (b, +?).

8 Clearly,

xn ? 1 = (x ? 1) (x ?a1) (x ? a2) ?.. (x ? a n ? 1)

Diff. both sides w.r.t x

nx n-1 = 1. (x ? a1) (x ? a2) ?. (x ? a n-1) + (n ? 1) d [(x ? a1) (x ? a2) ?(x ? a n ?1)]

Putting x = 1 on both sides, we get dx

n = (1- a1) (1 ? a2) ?. (1 ? a n-1).

9 Since roots are in the ratio p : q

\pq a2 = (p + q)2 ac Þ (p + q)2 = a Þ (p + q) = ?a

pq c ?pq ?c

Þ ?p + ?q = ?a

?q ?p ?c

10 Given equ. becomes

(m + 1) x2 ? (m + 1)bx = ( m ? 1) ax ? (m ?1)c

i.e., (m + 1)x2 ? [mb + b + ma ? a] x + (m + 1)c = 0

Since its roots are equal in magnitude and opposite in sign \sum of the roots = 0

\ mb + b + ma ? a = 0

\ m (a + b) = a ?b \ m = a - b

a + b

11 Roots of x2 + x + 1 = 0 are

x = -1 ± ?1 ?4 = ?, ?2.

Take a = ?, b = ?2 \ a19 = ?19 = ?, b7 = (?2)7 = ?14 = ?2

\ reqd. eqn. is x2 + x+1 = 0

12 Let the roots be m, m +1

Sum of the roots = 2m +1 = b

Product of the roots = m(m + 1) = c

\ b2 ? 4c = 4m2 + 4m + 1 ? 4 m2 ? 4m =1.

13 Since x2 ? x + 1 = 0

\(x + 1) (x2 ? x + 1) = 0

\ x3 ? 1 = 0 \ x3 = 1 \ x3n = 1

14Given equation is 2x/2 + 3 x/2 = (?13) x/2

Þ 2 x/2 + 3 x/2 = 1

?13 ?13

which is of the form cosx/2 a + sinx/2 a = 1

x = 2 Þ x = 4

2

15 x2 - ½ x ½ - 6 = 0

Þ ½ x ½2 - ½ x ½-6 = 0 Þ ½ x ½= -2, 3

Þ ½ x ½ = 3 Þ x = -3, 3.

\product of roots = (-3) (3) = -9.

16 We have b2 = ac ?? (1)

and 2 = 1 + 1 ?... (2)

c-a b ?c a ?b

(2) Þ 2(b-c) (a-b) = ( a ? b + b ?c) (c ? a)

Þ 2(ba ? ac ? b2 + bc) = - (a-c)2

Þ 2 (ba ? 2b2 + bc) = - (a ? c)2

Þ 2b(a ? 2 Öac + c) = -(a-c)2 (... b = Öac)

Þ 2b (Öa - Öc)2 = - (Öa - Öc)2 (Öa+Öc)2

Þ 2b = - (Öa + Öc)2 = - (a +c +2Öac)

Þ 2b = - a ?c -2b or a + 4b + c = 0

\ a + 4b+c is independent of a, b, c.

17 Let S = 1+2.2 + 3.22 + 4.23 +??+100.299

\ 2S= 2 + 2.22 + 3.23 + 4.23+??+100.2100

Subtracting, we get -S = 1+ (2 + 22 + 23 +??.+ 299) - 100.2100.

Þ -S = 1 + 2(299-1) - 100.2100

2 -1

\S = -1 - 2100 + 2 + 100.2100 = 99.2100 +1.

18. We have 1 - 1 = 1 - 1

b-a c a b-c

Þ -b+a = b-c-a

(b-a)c a(b-c)

Þ 1 = - 1_ (... b ? c + a)

(b-a)c a(b-c)

Þ ab ? ac = -bc + ac

Þ b(a+c) = 2ac Þ b = 2a

a+c

\ a, b, c are in H . P.

19. Since xn ? yn = (x ? y)(xn-1 + x n-2 y + x n-3 y2 + ??+ y n-1), the given expression is

equal to

(x+2)n-(x+1)n = (x+2)n ? (x+1)n.

(x+2) ? (x+1)

20. Let the G.P. be a, ar, ar2, ?

We have S3 = 125

S6 152

\ a(1-r3) ´ 1-r = 125

1 ?r a(1-r6) 152

Þ 1 = 125 Þ r3 = 152 -1 = 27

1+ r3 152 125 125

21. 1+ n + n2 + ??..+n127

=1(n128 ? 1) = (n64 ? 1)(n64 + 1)

n ? 1 n ? 1

=(n63 + n62 +??+1)(n64 + 1)

\ Required value of m is 64

22. Let the Numbers be a and b.

a,A1, A2,b in A.P. Þ A1-a = b-A2

Þ A1 + A2 = a + b

a, G1, G2, b in G.P.

Þ G1= b Þ G1G2 = ab

a G2

a,H1,H2, b in H.P. Þ 1 ? 1 = 1 ? 1

H1 a b H2

Þ 1 + 1 = 1 + 1

H1 H2 a b

Þ H1-1 + H2-1 = a + b A1 + A2 .

ab G1G2

23 5 = 22 + 1, 14 = 32 + 5, 30 = 4 + 14, 55 = 52 + 30

\ Next term = 62 + 55 = 36 + 55 = 91.

24 Given sum = å (n2 + n) = ån2 + ån

= n(n + 1) (2n + 1) + n(n + 1)

2

= n(n + 1) [2n + 1 + 3]

6

= 2n(n + 1) (n + 2)

6

= n(n + 1) (n + 2)

3

25. Let S = 1 + (1 + a) x + (1 + a + a2) x2 +?. ¥

\xS = x + (1 + a) x2 + ?? ¥

(1 ? x) S = 1 + ax + a2x2 + ?? ¥

= 1

1 ? ax

Þ S = 1

(1 ? x) (1 ? ax).

26. x = 1 , y = 1 , z = 1

1 ? a 1 ? b 1 ? c

Now, a, b, c are in A.P.

Þ 1 ? a, 1 ? b, 1 - c., are in A.P.

Þ 1 , 1 , 1 are in H.P.

1 ? a 1 ? b 1 ? c

Þ x, y, z are in H.P.

27. Consider the numbers ab2, ac2, bc2, ba2, ca2, cb2

Since A.M. ? G.M.

\ab2 + ac2 + bc2 + ba2 + ca2 + cb2 ? (ab2. ac2. bc2. ba2. ca2. cb2) 1/ 6

6

Þ a(b2 + c2) + b(c2 + a2) + c(a2 + b2) ? 6 (a6 b6 c6) 1/ 6 = 6abc

\ Min. value of a(b2 + c2) + b(c2 + a2) + c(a2 + b2) = 6abc

\l = 6

28. Let S = 1 + 2x + 3x2 + 4x3 + ??

xS = x + 2x2 + 3x3 + ??

\S (1 ? x) = 1+ x+ x2 + x3 + ??. = 1

1 - x

\S = 1

(1- x)2

29. a10 = a1 + (10 ? 1)d Þ 3 = 2 + 9d

\9d = 1 or d = 1/9

Again h1, h2, ??. h10 are in H.P.

\1 , 1 , 1 ??. 1 are in A.P.

h1 h2 h3 h10

\ 1 = 1 + (10 ? 1)D (Where D is the common difference)

h10 h1

Þ 1 = 1 + 9D Þ 9D = 1 ? 1 = -1 Þ D = -1

3 2 3 2 6 54

Now a4 = a1 + 3d = 2 + 3 = 2 + 1 = 7

9 3 3

1 = 1 + 6D = 1 + 6 (-1/54) = 1 ? 1 = 7

h7 h1 2 2 9 18

\h7 = 18 . Thus a4h7 = 7 ´ 18 = 6

7 3 7

30. Since a, b, c are in H.P. \ 1 , 1 , 1 are in A.P

a b c

\ 2 = 1 + 1 Þ 1 - 2 + 1 = 0

b a c a b c

This shows that the st. line x + y + 1 = 0 always passes thro? a fixed point ( 1, -2)

a b c

SOLUTION

1)C 2)B 3)D 4)B 5)C 6)B 7)C 8)A 9)C 10)B 11)B 12)D 13) B 14)B 15)A 16)C 17)B 18)C 19)C 20)C 21)C 22)D 23)D 24)C 25)B 26)C 27)C 28)D 29)D 30)D

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