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#### Model question-Binomial Theorem, Sequences and Series

11th Standard

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Maths

Use blue pen Only

Time : 01:00:00 Hrs
Total Marks : 70

Part A

Answer all the questions

10 x 1 = 10
1. The HM of two positive numbers whose AM and GM are 16,8 respectively is

(a)

10

(b)

6

(c)

5

(d)

4

2. The nth term of the sequence $\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 8 } ,\frac { 15 }{ 6 }$......is

(a)

2- n - 1

(b)

1 - 2-n

(c)

2-n + n - 1

(d)

2n-1

3. The sum up to n terms of the series $\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +$.....is

(a)

$\frac { n(n+1) }{ 2 }$

(b)

2n(n+1)

(c)

$\frac { n(n+1) }{ \sqrt { 2 } }$

(d)

1

4. If $\frac { { T }_{ 2 } }{ { T }_{ 3 } }$is the expansion of (a+b)n and $\frac { { T }_{ 3 } }{ { T }_{ 4 } }$ is the expansion of (a+b)n+3 are equal, then n=

(a)

3

(b)

4

(c)

5

(d)

6

5. If the first, second and last term of an A. P. are a, b and 2a respectively, then its sum is

(a)

$\frac { ab }{ 2(b-a) }$

(b)

$\frac { ab }{ b-a }$

(c)

$\frac { 3ab }{ 2(b-a) }$

(d)

none of these

6. The value of ${ 9 }^{ \frac { 1 }{ 3 } }$ ,${ 9 }^{ \frac { 1 }{ 9 } }$${ 9 }^{ \frac { 1 }{ 27 } }$ ,$\infty$is

(a)

1

(b)

3

(c)

9

(d)

none of these

7. If a is the arithmetic mean and g is the geometric mean of two numbers, then

(a)

$\le$ g

(b)

$\ge$ g

(c)

a = g

(d)

a > g

8. The value of $1-\frac{1}{2}(\frac{3}{4})+\frac{1}{3}(\frac{3}{4})^2-\frac{1}{4}(\frac{3}{4})^3+...$is:

(a)

$\frac{3}{4}log(\frac{7}{4})$

(b)

$\frac{4}{3}log(\frac{7}{4})$

(c)

$\frac{1}{3}log(\frac{7}{4})$

(d)

$\frac{4}{3}log(\frac{4}{7})$

9. The ratio of the coefficient of x 15 to the term independent of x in $[x^2+(\frac{2}{x})]^{15}$ is

(a)

1:16

(b)

1:8

(c)

1:32

(d)

1:64

10. 21/4 41/8 81/16 161/32 . . . =

(a)

1

(b)

2

(c)

$\frac{3}{2}$

(d)

$\frac{5}{2}$

11. Part B

Answer all the questions

10 x 2 = 20
12. Compute 1024

13. Find the Co-efficient of x4 in the expansion (1+x3)50 $\left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 3 } } \right) ^{ 5 }$

14. Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
${ \left( 5+{ x }^{ 2 } \right) }^{ \frac { 2 }{ 3 } }$

15. If n is a postive integer, show that 9n+1 - 8n - 9 is always divisible by 64

16. If p - q is small compared to either p or q, then show that $n\sqrt { \frac { p }{ q } } =\frac { \left( n+1 \right) p+\left( n-1 \right) q }{ \left( n-1 \right) p+\left( n+1 \right) q }$
Hence find $8\sqrt { \frac { 15 }{ 16 } }$

17. Find the sum of first n terms of the series 12+32+52+...

18. Find the 5th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

19. Find the coefficient of x6 in the expansion of (3 + 2x)10.

20. Find $\sum_{k=1}^{n}{1\over k(k+1)}.$

21. Find the $\sqrt [ 3 ]{ 126 }$ approximately to two decimal places.

22. Part C

Answer all the questions

5 x 3 = 15
23. If a, b, c are respectively the pth qth and rth terms of a GP. show that (q - r) log a + (r - p) log b + (p - q) log c = 0.

24. Write the first 6 terms of the sequences whose nth term an given below
${ a }_{ n }=\begin{cases} n+1\quad if\quad n\quad is\quad odd \\ n\quad \quad if\quad n\quad is\quad even \end{cases}$

25. Write the nth term of the following sequences
​​​​​​​6,10,4,12,2,14,0,16,-2...

26. Sum the series $\frac { 2 }{ 5 } +\frac { 2 }{ { 3.5 }^{ 3 } } +\frac { 2 }{ { 5.5 }^{ 5 } } ....\infty$

27. The sum of two members is$\frac { 13 }{ 6 }$ .An even number A.M.S are being inserted between them and their sum exceeds their number by 1.Find the number of A.M.S inserted.

28. Part D

Answer all the questions

5 x 5 = 25
29. A man repays an amount of Rs.3250 by paying Rs.20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

30. In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day, the virus which caused the disease spread in Geometric Progression. The amount of infectious virus particle gets doubled each day, being 5 particles on the first day. Find the day when the infectious virus particles just grow over 1,50,000 units?

31. If x=0.001, prove that $\frac { { \left( 1-2x \right) }^{ \frac { 2 }{ 3 } }{ \left( 4+5x \right) }^{ \frac { 3 }{ 2 } } }{ \sqrt { 1-x } }$ =8.01 up to two places of decimals

32. If $\alpha ,\beta$are the roots of the equation x2-px+q=0, then prove that $\log { (1+px+q{ x }^{ 2 }) } =(\alpha +\beta )x=\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } { x }^{ 2 }+\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 3 } { x }^{ 3 }-....\infty$

33. If sum of the n terms of a G.P be S, their product P and the sum of their reciprocals R, then prove that $P^{2}=(\frac{S}{R})^{n}$