#### Half Yearly Model Question Paper 1

11th Standard

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Do not write anything on question paper

Time : 02:30:00 Hrs
Total Marks : 90

Part A

20 x 1 = 20
1. If $\triangle=\begin{vmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{vmatrix}$ then $\begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix}$ is

(a)

$\triangle$

(b)

-$\triangle$

(c)

3$\triangle$

(d)

-3$\triangle$

2. If any three-rows or columns of a determinant are identical, then the value of the determinant is

(a)

0

(b)

2

(c)

1

(d)

3

3. The number of ways to arrange the letters of the word "CHEESE" is

(a)

120

(b)

240

(c)

720

(d)

6

4. The number of permutation of n different things taken r at a time, when the repetition is allowed is

(a)

rn

(b)

nr

(c)

$\frac { n! }{ (n-r)! }$

(d)

$\frac { n! }{ (n+r)! }$

5. The x - intercept of the straight line 3x + 2y - 1 = 0 is

(a)

3

(b)

2

(c)

1/3

(d)

1/2

6. Combined equation of co-ordinate axes is

(a)

x2-y2=0

(b)

x2+y2=0

(c)

xy=c

(d)

xy=0

7. The equation of the circle with centre (3,-4) and touches the x - axis

(a)

(x - 3)2 +(y - 4)2 = 4

(b)

(x - 3)2 +(y + 4)2 =16

(c)

(x-3)2+(y- 4)2=16

(d)

x2+y2=16

8. The value of $\sin(-420^o)$ is

(a)

$\frac{\sqrt3}{2}$

(b)

$-\frac{\sqrt3}{2}$

(c)

$\frac{1}{2}$

(d)

$\frac{-1}{2}$

9. The value of $cosec^{-1}\left(\frac{2}{\sqrt{3}}\right)$ is

(a)

$\frac{\pi}{4}$

(b)

$\frac{\pi}{2}$

(c)

$\frac{\pi}{3}$

(d)

$\frac{\pi}{6}$

10. $\tan\left(\frac{\pi}{4}-x\right)$ is

(a)

$\left(\frac{1+\tan x}{1-\tan x}\right)$

(b)

$\left(\frac{1-\tan x}{1+\tan x}\right)$

(c)

1-tan x

(d)

1+tan x

11. The graph of the line y = 3 is

(a)

Parallel to x-axis

(b)

Parallel to y-axis

(c)

Passing through the origin

(d)

Perpendicular to x-axis

12. If y = x and z = $\frac{1}{x}$ then $\frac{dy}{dz}=$

(a)

x2

(b)

1

(c)

-x2

(d)

$-\frac{1}{x^2}$

13. if u = 4x2 + 4xy + y2 + 32 + 16 , then $\frac { \partial ^{ 2 }u }{ \partial y\partial x }$ is equal to

(a)

8x + 4y + 4

(b)

4

(c)

2y + 32

(d)

0

14. If u = ${ e }^{ { x }^{ 2 } }$ then $\frac { \partial u }{ \partial x }$ is equal to

(a)

${ 2x }e^{ { x }^{ 2 } }$

(b)

${ e }^{ { x }^{ 2 } }$

(c)

${2 e }^{ { x }^{ 2 } }$

(d)

0

15. A man purchases a stock of Rs 20,000 of face value 100 at a premium of 20%, then investment is

(a)

Rs 20,000

(b)

Rs 25,000

(c)

Rs 22,000

(d)

Rs 30,000

16. The annual income on 500 shares of face value 100 at 15% is

(a)

Rs 7,500

(b)

Rs 5,000

(c)

Rs 8,000

(d)

Rs 8,500

17. Example of contingent annuity is

(a)

(b)

An endowment fund to give scholarships to a student

(c)

Personal loan from a bank

(d)

All the above

18. When calculating the average growth of economy, the correct mean to use is?

(a)

Weighted mean

(b)

Arithmetic mean

(c)

Geometric mean

(d)

Harmonic mean

19. If two events A and B are dependent then the conditional probability of P(B/A) is

(a)

$P(A)P(B/A)$

(b)

$\frac { P(A\cup B) }{ P(B) }$

(c)

$\frac { P(A\cap B) }{ P(A) }$

(d)

$P(A)P(A/B)$

20. Probability that at least one of the events A, B occur is

(a)

$P(A\cup B)$

(b)

$P(A\cap B)$

(c)

P(A/B)

(d)

$(A\cup B)$

21. Part B

Answer all the questions. Each question carries two marks.

7 x 2 = 14
22. In the expansion of ${ \left( x+\frac { 1 }{ x } \right) }^{ 6 }$, find the third terms.

23. Find the center and radius of the circle 5x2 + 5y2 +4x - 8y - 16 = 0

24. Show that $\tan^{-1}\left(\frac{1}{2}\right)+\tan^{-1}\left(\frac{2}{11}\right)=\tan^{-1}\left(\frac{3}{4}\right)$

25. If $f\left( x \right) =\frac { 1 }{ 2x+1 } ,x\neq -\frac { 1 }{ 2 }$ then show that $f\left( f\left( x \right) \right) =\frac { 2x+1 }{ 2x+3 }$ provided that $x\neq -\frac { 3 }{ 2 }$

26. If f(x,y) = 3x2 + 4y3 + 6xy - x2y3 + 6. Find fx(1, -1)

27. Gopal invested Rs 8,000 in 7% of Rs 100 shares at Rs 80. After a year he sold these shares at Rs 75 each and invested the proceeds (including his dividend) in 18% for Rs 25 shares at Rs 41. Find
(i) his dividend for the first year
(ii) his annual income in the second year
(iii) The percentage increase in his return on his original investment

28. An investor buys Rs. 1,500 worth of shares in a company each month. During the first four months he bought the shares at a price of Rs. 10, 15, 20 and 30 per share. What is the average price paid for the shares bought during these four months? Verify your result.

29. Part C

Answer all the questions. Each question carries three marks.

7 x 3 = 21
30. Show that $\begin{vmatrix} a & a+b&a+b+c \\2a &3a+2b &4a+3b+2c\\3a&6a+3b&10a+6b+3c \end{vmatrix}=a^3.$

31. Show that the middle term in the expansion of (1 +x)2n is $\frac { 1.3.5....(2n-1){ 2 }^{ n }.{ x }^{ n } }{ n! }$

32. Find the locus of the point which is equidistant from (2, –3) and (3, –4).

33. Prove that $\frac { sin(-\theta )tan({ 90 }^{ o }-\theta )sec\left( { 180 }^{ o }-\theta \right) }{ sin(180+\theta )cot(360-\theta )cosec({ 90 }^{ o }-\theta ) } =1$

34. If $y=\sqrt { x } +\frac { 1 }{ \sqrt { x } }$ show that $2x\frac { dy }{ dx } +y=2\sqrt { x }$.

35. Verify Euler’s theorem for the function $u=\frac{1}{\sqrt{x^2+y^2}}$

36. Find the amount of an ordinary annuity of Rs 500 payable at the end of each year for 7 years at 7% per year compounded annually.

37. Part D

Answer all the questions. Each question carries five marks.

7 x 5 = 35
38. Prove that the term independent of x in the expansion of ${ \left( x+\frac { 1 }{ x } \right) }^{ 2n }is\quad \frac { 1.3.5.....,(2n-1){ 2 }^{ n } }{ n! }$

39. The average variable cost of a monthly output of x tonnes of a firm producing a valuable metal is  Rs. $\frac { 1 }{ 5 } { x }^{ 2 }-6x+100$ Show that the average variable cost curve is a parabola. Also find the output and the  average cost at the vertex of the parabola

40. Prove that ${ cot }^{ -1 }\left[ \frac { \sqrt { 1+sinx } +\sqrt { 1-sinx } }{ \sqrt { 1+sinx } -\sqrt { 1-sinx } } \right] =\frac { x }{ 2 }$ where $x\in \left( 0,\frac { \pi }{ 4 } \right)$

41. Calculate the geometric mean of the data given below giving the number of families and the income per head of different classes of people in a village of Kancheepuram District.

 Class of people No. of Families Income per head in 1990 (Rs) Landlords 1 1000 Cultivators 50 80 Landless labourers 25 40 Money- lenders 2 750 School teachers 3 100 Shop-keepers 4 150 Carpenters 3 120 Weavers 5 60
42. In a Shooting test, the probabilities of hitting the target are $\frac { 1 }{ 2 }$ for A, $\frac { 2 }{ 3 }$ for B and $\frac { 3 }{ 4 }$  for C. If all of them fire at the same target, calculate the probabilities that only one of them hit the target.

43. Without expanding show that $\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0$

44. Examine the following functions for continuity at indicated points
$f(x)=\begin{cases} \frac { { x }^{ 2 }-9 }{ x-3 } ,if\quad x\neq 0 \\ \quad \quad \quad \quad 6,\quad \quad \quad if\quad x=3 \end{cases}$ at x =3