#### Half Yearly Model Question Paper 2019

11th Standard

Reg.No. :
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Time : 02:30:00 Hrs
Total Marks : 90

Part I

Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

20 x 1 = 20
1. The co-factor of -7 in the determinant $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$ is

(a)

-18

(b)

18

(c)

-7

(d)

7

2. Which of the following matrix has no inverse

(a)

$\begin{pmatrix} -1 & 1 \\ 1 &-4 \end{pmatrix}$

(b)

$\begin{pmatrix} 2 & -1 \\ -4 &2 \end{pmatrix}$

(c)

$\begin{pmatrix} cos\ a & sin\ a \\ -sin\ a & cos\ a \end{pmatrix}$

(d)

$\begin{pmatrix} sin\ a & cos\ a \\ -cos\ a & sin\ a \end{pmatrix}$

3. If A = $\begin{vmatrix}cos \theta & sin \theta \\ -sin \theta&cons\theta \end{vmatrix}$ then |2A| is equal to

(a)

4 cos 2 $\theta$

(b)

4

(c)

2

(d)

1

4. The possible out comes when a coin is tossed five times

(a)

25

(b)

52

(c)

10

(d)

$\frac { 5 }{ 2 }$

5. The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for n $\in$ N is

(a)

2

(b)

6

(c)

20

(d)

24

6. The angle between the pair of straight lines x2 - 7xy + 4y2 = 0

(a)

$tan^{-1}\left(1\over 3\right)$

(b)

$tan^{-1}\left(1\over 2\right)$

(c)

$tan^{-1}\left(\sqrt{33}\over 5\right)$

(d)

$tan^{-1}\left(5\over\sqrt{33}\right)$

7. If kx2 + 3xy - 2y2 = 0 represent a pair of lines which are perpendicular then k is equal to

(a)

1/2

(b)

-1/2

(c)

2

(d)

-2

8. The value of $\sin15^o$ is

(a)

$\frac{\sqrt{3}+1}{2\sqrt{2}}$

(b)

$\frac{\sqrt{3}-1}{2\sqrt{2}}$

(c)

$\frac{\sqrt3}{\sqrt2}$

(d)

$\frac{\sqrt3}{2\sqrt2}$

9. The value of 1-2sin245o is

(a)

1

(b)

$\frac{1}{2}$

(c)

$\frac14$

(d)

0

10. 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)

$\frac{\pi}{6}$

11. For what value of x, f(x) = $\frac{x+2}{x-1}$ is not continuous?

(a)

-2

(b)

1

(c)

2

(d)

-1

12. $\frac{d}{dx}(5e^x-2logx)$ is equal to

(a)

5ex - $\frac{2}{x}$

(b)

5ex - 2x

(c)

5ex - $\frac{1}{x}$

(d)

2 logx

13. Marginal revenue of the demand function p= 20–3x is

(a)

20–6x

(b)

20–3x

(c)

20+6x

(d)

20+3x

14. If demand and the cost function of a firm are p= 2–x and c = 2x2 +2x +7 then its profit function is

(a)

x2 + 7

(b)

x2 - 7

(c)

- x2 + 7

(d)

- x2 - 7

15. 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

16. The dividend received on 200 shares of face value Rs.100 at 8% dividend value is

(a)

1600

(b)

1000

(c)

1500

(d)

800

17. The brokerage paid by a person on this sale of 400 shares of face value Rs.100 at 1% brokerage

(a)

Rs 600

(b)

Rs 500

(c)

Rs 200

(d)

Rs 400

18. A person brought a 9% stock of face value Rs 100, for 100 shares at a discount of 10%, then the stock purchased is

(a)

Rs 9000

(b)

Rs 6000

(c)

Rs 5000

(d)

Rs 4000

19. The geometric mean of two numbers 8 and 18 shall be

(a)

12

(b)

13

(c)

15

(d)

11.08

20. Harmonic mean is better than other means if the data are for

(a)

Speed or rates.

(b)

Heights or lengths.

(c)

Binary values like 0 and 1.

(d)

Ratios or proportions.

21. Part II

Answer any 7 questions. Question no. 30 is compulsory.

7 x 2 = 14
22. If $A=\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$ then show that |2A| = 4 |A|.

23. Using the property of determinants show that $\begin{vmatrix} x &a &x+a \\ y & b &y+b \\z & c & z+c \end{vmatrix}=0.$

24. Resolve into partial fractions for the following
$\frac { 3x+7 }{ { x }^{ 2 }-3x+2 }$

25. In a railway compartment, 6 seats are vacant on a bench. In how many ways can 3 passengers sit on them?

26. In how many ways can 10 beads of different colours form a necklace?

27. Convert the equation of the parabola x2+y=6x-14 into the standard form.

28. Find the parametric equations of the circle x2+y2=25

29. Evaluate: cos 20° + cos 100° + cos 140°

30. Show that the function f(x) = 5x -3 is continous at x = +3

31. If the dividend received from 9% of Rs 20 shares is Rs 1,620, find the number of shares.

32. Part III

Answer any 7 questions. Question no. 40 is compulsory.

7 x 3 = 21
33. Prove that $\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right|$ is independent of $\theta$

34. If ${ A }^{ -1 }=\left[ \begin{matrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{matrix} \right]$ then, find A.

35. Solve : $\frac { (2x+1)! }{ (x+2)! } .\frac { (x-1)! }{ (2x-1)! } =\frac { 3 }{ 5 }$

36. For what value of $\lambda$ are the three lines 2x-5y+3 = 0, 5x-9y+$\lambda$=0 and x-2y+1=0 are concurrent?

37. Convert the following into the product of trigonometric functions.
sin9A+sin7A

38. Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

39. Find the elasticity of demand in terms of x for the demand law $p={(a-bx)^{1\over 2}}.$ Also find the values of x when elasticity of demand is unity.

40. Separate the intervals in which the function x3 + 8x2 + 5x - 2 is increasing or decreasing.

41. Find the purchase price of Rs.9300,8 3/4% stock at 4% discount?

42. Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 4. What is the probability that it is an even number?

43. Part IV

7 x 5 = 35
1. a bank pays 8% interest compounded quarterly. Determine the equal deposits to be made at the end of each quarter for 3 years so as to receive Rs.300 at the end of 3 years.

2. A factory has 3 machines A1, A2, A3 producing 1000, 2000, 3000 bolts per day respectively. A1 produces 1% defectives, A2 produces 1.5% and A3 produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A1?

1. A firm has revenue function R = 8x and production cost function $C = 150000 + 60\left(x^2\over 900\right)$ Find the total profit function and the number of units to be sold to get the maximum profit.

2. The first of three urns contains 7 White and 10 Black balls, the second contains 5 White and 12 Black balls and third contains 17 White balls and no Black ball. A person chooses an urn at random and draws a ball from it. And the ball is found to be White. Find the probabilities that the ball comes from
(i) the first urn
(ii) the second urn
(iii) the third urn

1. if y = 2 sin x + 3 cos x, then show that y2 + y = 0

2. The demand function of a commodity is p = 200-$\frac { x }{ 100 }$ and its cost is C=40x+12000 where p is a unit price in rupees and x is the number of units produced and sold. Determine (i) profit function (ii) average profit at an output of 10 units (iii) marginal profit at an output of 10 units and (iv) marginal average profit at an output of 10 units.

1. Find the value of tan $\left( {{\pi}\over{8}}\right).$

2. Differentiate: xy + y2 = tan x + y.

1. Find the equation of the parabola which is symmetrical about x-axis and passing through (-2, -3).

2. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

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

2. If the fourth term in the expansion of ${ \left( ax+\frac { 1 }{ x } \right) }^{ n }$ is $\frac { 5 }{ 2 }$  then find the values of a and n.

1. Let a, b and c denote the sides BC, CA and AB repectively of $\Delta$ ABC. If $\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0$, then find the value of sin2 A+sin2B+sin2C.

2. Find adjoint of $A=\left[ \begin{matrix} 1 & -2 & -3 \\ 0 & 1 & 0 \\ -4 & 1 & 0 \end{matrix} \right]$