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#### +2 Public Exam March 2019 Important Creative 3 Mark Questions and Answers

12th Standard

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Time : 02:30:00 Hrs
Total Marks : 150
50 x 3 = 150
1. Given $A=\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}$ , $B=\begin{bmatrix} -1 & 0 \\ 2 & 1 \end{bmatrix}$  verify that Adj (AB) = (Adj B) (Adj A)

2. Verify $(AB)^{-1}=B^{-1}A^{-1}$, when $A=\begin{pmatrix} 3 & 1 \\ 2 & -1 \end{pmatrix}$and $B=\begin{pmatrix} -6 & 0 \\ 0 & 9 \end{pmatrix}$

3. If  $\begin{bmatrix} 4 & -3 \\ 5 & 2 \end{bmatrix} \ \ X=\begin{bmatrix} 14 \\ 29 \end{bmatrix}$ find the matrix X.

4. Solve by Cramer’s rule : $x+y=2,y+z=6,z+x=4$.

5. Find the rank of the matrix $\begin{pmatrix} -2 & 1 & 3\quad 4 \\ 0 & 1 & 1\quad 2 \\ 1 & 3 & 4\quad 7 \end{pmatrix}$

6. If $A=\begin{vmatrix} 1 &2 \\1 &1 \end{vmatrix}$and $B=\begin{vmatrix} 0 &-1 \\1 &2 \end{vmatrix}$then verify that (AB)-1 = B-1A-1

7. Using determinnts, find the value of k so that the points (k, 2 - 2k), (-k + I, 2k) and (-4 - k, 6 - 2k) may be collinear.

8. Solve: 2x - y = 17, 3x + 5y = 6 using Cramer's rule.

9. Find the minors and Co-factors of elements of the matrix $A=\begin{bmatrix} 1 & 3&-2 \\ 4&-5&6\\3&5&2 \end{bmatrix}$

10. Find the equation to the hyperbola which has 3x − 4 y + 7 = 0 and 4x + 3y + 1 = 0 for asymptotes and which passes through  the origin.

11. Find the eccentricity, foci and latus rectum of the ellipse 9x2 + 16y2 = 144.

12. If the total cost C of making x units is  $C = 50+10x+5x^2$ . Find the average cost and marginal cost When x = 1.3 .

13. Find the elasticity of demand When the demand is $q = {20 \over p + 1}$ and p = 3. Interpret the result .

14. The supply of certain items is given by the supply function $x = a \sqrt {p -b }$, Where p is the price, a and b are positive constants. (p> b) . Find an expression for elasticity of supply $\eta _{s}$ Show that it becomes unity When the price is 2b.

15. Find the slope of the tangent line at the point (0 , 5) of the curve $y = {1\over 3} {(x ^2 +10x -15)} .$ At what point of the curve the slope of the tangent line is $8\over 5$ ?

16. The demand curve for a monopolist is given by x = 100-4p
(i) Find the total revenue, average revenue and marginal revenue.
(ii) At what value of x, the marginal revenue is equal to zero

17. At what points on the curve 3y = x3, the tangents are inclined at 45$°$  to the x - axis.

18. Find the stationary points and the stationary values of the function $f(x)={ 2 }x^{ 3 }+{ 3x }^{ 2 }-12x+7$.

19. A certain manufacturing concern has the total cost function  $C={ \frac { 1 }{ 5 } x }^{ 2 }-6x+100$. Find when the total cost is minimum.

20. Find the optimum output of a firm whose total revenue and total cost functions are given by $R=30x-x^{2}$ and $C=20+4x$, x being the output of the firm.

21. If $u={ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }-3xyz$, prove that  $x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } +z\frac { \partial u }{ \partial z } =3u$

22. If $u={ x }^{ 3 }+3{ xy }^{ 2 }+{ y }^{ 3 }$prove that $\frac { { \partial }^{ 2 }u }{ \partial x\partial y } =\frac { { \partial }^{ 2 }u }{ \partial y\partial x }$

23. What is the maximum slope of the tangent to the curve y = -x3+3x2+9x-27 and at what point is it?

24. If u = log$\left( \sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } } \right)$, find $\frac { \partial ^{ 2 } }{ \partial x^{ 2 } } +\frac { \partial ^{ 2 }u }{ \partial y^{ 2 } } +\frac { \partial ^{ 2 }u }{ \partial z^{ 2 } }$

25. Evaluate the following using the properties of definite integral :   $\overset { \frac { \pi }{ 3 } }{ \underset { \frac { \pi }{ 6 } }{ \int } } \frac { dx }{ 1+\sqrt { cotx } }$

26. Evaluate the following using the properties of definite integral : $\overset { \frac { \eth }{ 2 } }{ \underset { 0 }{ \int } } \frac { asinx+bcosx }{ sinx+cosx } dx$

27. The marginal cost function is $MC=20-0.04x+0.003{ x }^{ 2 }$ where $x$ is the number of units produced. The fixed cost of production is Rs.7,000. Find the total cost and the average cost.

28. If the marginal revenue function is $R'(x)=15-9x-3{ x }^{ 2 }$, find the revenue function and average revenue function.

29. Evaluate $\overset { \frac { \pi }{ 3 } }{ \underset { \frac { \pi }{ 6 } }{ \int } } \frac { dx }{ 1+\sqrt { \tan { x } } }$

30. Find the area of one loop of the curve x2 = 12 (4 - x2) between x = 0 and x = 2.

31. Solve : $x(y^2+1)dx+y(x^2+1)dy=0$

32. Solve : $\frac { dy }{ dx } +y\ cot\ x=cosec\ x$

33. Solve $\left( 4D^{ 2 }-8D+3 \right) y={ e }^{ \frac { 1 }{ 2 } x }$

34. Solve: $\frac { dy }{ dx } +\frac { 2xy }{ 1+{ x }^{ 2 } } =\frac { 1 }{ (1+{ x }^{ 2 })^{ 2 } }$ given that y= 0 when x= 1

35. Fit the line of best fit if  $Σx=75$$ΣY=115$$Σ{ x }^{ 2 }=1375$, $ΣXY=1875$ ,  and  $n=6$

36. Using Lagrange’s formula find y(11) from the following table

 x : 6 7 10 12 y : 13 14 15 17
37. From the following data find f(x).

 x 1 2 3 4 5 f(x) 2 5 - 14 32
38. A random variable X has the following probability distribution .

 Values of X,x  : 0 1 2 3 4 5 6 7 8 p(x)   : a 3a 5a 7a 9a 11a 13a 15a 17a

(i)  Determine  the value of a
(ii) Find p(x < 3), p(x >3) and p(0 < x < 5).

39. An unbiased coin is tossed six times. What is the probability of obtaining four or more heads?

40. Let X be a continuous random variable with p.d.f.
$f(x)\begin{cases}{1\over 2}\ for\ -1<x<1 \\0\quad otherwise \end{cases}$
Find (i)E(x), (ii)E(x2), (iii)var(x)

41. Ten coins are thrown simultaneously. Find the probability of getting atleast 7 heads.

42. A random sample of 600 eggs were taken from large consignment and 75 of them were found to be bad. Find the limits at which the bad eggs lie at 99% confidence level.

43. Out of 1000 soldiers, a sample of 50·selected random to test the accuracy of gun shouting and of them 5 did a mistake. Find the limits within which the number of soldiers who did wrongly in whole universe of 1000 soldiers at 95% confidence level. .

44. A random sample of 30 super market out of .150 super markets in Chennai showed a annual profit of 50 lakhs and a standard deviation of 10 lakhs . Pind the 95% confidence limits for the estimate of mean profit of 150 super market?

45. Out of 1000 mathematics note books, a sample of 150 note books were taken to test the accuracy of solving problem where in 15 note books with mistakes were found; Find 95% confidence limits within which tile number of defective note books can be expected to be?

46. The mean I.Q. of a sample of 2000 children was 71. It is likely that this was a random sample from the population with mean I.Q. 80 and std deviation 10. (Test at 5% level of significance).

47. Calculate the 3-yearly moving averages of the production figures (in metnc tonnes ) Given Below.

48. calculate the cost of living index number for the following information

 Expenses on items Food 35% fuel 10% Clothes20% Rent 15% Misc 20% Price in 2004 1500 250 750 300 400 Price in 1995 1400 200 50 200 250
49. Calculate weighted aggregative price index from the following data using (i) Laspeyre's Method ii) Paasche's method

 Item Base Period Current Period price Quantity Price Quantity A 2 10 4 5 B 5 12 6 10 C 4 20 5 15 D 2 15 3 10
50. Calculate the correlation Co-effcient from the Data given below

 X 1 2 3 4 5 6 Y 7 6 5 3 8 4