Mathematics and Statistics 2014-2015 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Find x and y if `x + y = [(7,0),(2,5)] , x - y[(3,0),(0,3)]`

Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`

Use the quantifiers to convert the following open sentence defined on N into true statement
5x - 3 < 10

Use the quantifiers to convert the following open sentence defined on N into true statement:
x² ≥ 1

Examine the continuity of the following function :

`{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4`

Find the adjoint of the matrix `"A" = [(2,-3),(3,5)]`

Find the elasticity of demand if the marginal revenue is ₹ 50 and price is ₹ 75.

Evaluate:`int(tansqrtx)/sqrtxdx`

Evaluate : `int  1/("x"^2 + 8"x" + 20)` dx

Write converse, inverse contrapositive of the statement "If two triangles are not congruent then their areas are not equal.

Examine the continuity of the followin function : 

  `{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`   

Find `"dy"/"dx"` if y = x^x + 5^x

Solve the following equations by reduction method : 

x + 2y + z = 8 

2x+ 3y - z = 11 

3x - y - 2z = 5

Find the volume of the solid obtained by revolving about the X-axis, the region bounded by the curve `"x"^2/4 - "y"^2/9 = 1` and the lines x = 2 , x = 4.

If the demand function is D = 50 - 3p - p², find the elasticity of demand at (a) p = 5 (b) p = 2 ,  Interpret your result. 

By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or . contingency.  (p →  q) ∧  (p ∧ ~ q ).

If f (x) = `(1 - "sin x")/(pi - "2x")^2` , for x ≠ `pi/2` is continuous at x = `pi/4` , then find `"f"(pi/2) .`

If `"x"^(5/3) . "y"^(2/3) = ("x + y")^(7/3)` , the show that `"dy"/"dx" = "y"/"x"`

Cost of assembling x wallclocks is `( x^3/3 - 40x^2)` and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.

Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx

Evaluate :  ∫ log (1 + x²) dx

Alex spends 20% of his income on food items and 12% on conveyance. If for the month of June 2010, he spent ~ 900 on conveyance, find his expenditure on food items during he same month.

Find the age · specific Death Rates for the following data : 

Compute the coefficient of correlation for the following data : 

n = 100 ; `bar x` = 62 , ` bar y` = 53 . σx = 10 ; σ y = 12 ,

`sum ("x"_1 - bar x) ("y"_1 - bar y) = 8000`

A building is insured for 80% of its value. The annual premium at 70 paise percent amounts to ₹ 2,800.  Fire damaged the building to the extent of 60% of its value. How much amount for damage can: be claimed under the policy? 

A random variable X has the following probability distribution : 

Find the value of k and calculate mean. 

A, B and C are in partnership. A's capital was  ₹ 65,000, C's capital was  ₹ 50,000. The total profit is ₹ 38,000, out of which B's profit is ₹ 15,000. What was B's capital ?

If the Crude Death Rate (CDR) for the following data is 13.4 per thousand, find x:

Draw scatter diagram for the following data and identify the type of correlation. 

Mrs. Menon plans to save for her daughter's marriage. She wants to accumulate a sum of ₹ 4,00,000 at the end of 4 years. How much should she invest at the end of each year from now, if she can get interest compounded at 10% p.a. ?

 [Given (1.1)⁴ = 1.4641]

A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .

A fair coin is tossed 12 times. Find the probability of getting  at least 2 heads .

For the following problem find the sequence that minimizes total elapsed time (in hrs) required to complete the jobs on 2 machines M₁ and M₂ in the order M₁ - M₂ · Also find the minimum elapsed time T.

A bill of ₹ 2,000 drawn on 15th February 2003 for 10 months was discounted on 13th May 2003 at  `3 3/4` % p.a.  Calculate the banker's discount.

From the following table which relates to the number of animals of a certain
species at age x. complete the life table :

A person makes two types of gift items A and B requiring the services of a cutter and a finisher.  Gift item A requires 4 hours of the cutter's time and 2 hours of finisher's time. Fifth item B requires 2 hours of the cutter's time and 4 hours of finisher's time. The cutter and finisher have 208 hours and 152 hours available time respectively every month. The profit on one gift item of type A is ₹ 75 and on one gift item of type, B is ₹ 125. Assuming that the person can sell all the gift items produced, determine how many gift items of each type should he make every month to obtain the best returns?

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible : 

Variable of X = 9 

Regression equations : 8x - 10g + 66 = 0 and 40x- 18g = 214

Find Mean values of X and Y on the basis of the above information .

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible : 

Variable of X = 9 

Regression equations : 8x - 10g + 66 = 0 and  40x- 18g = 214

Find Correlation coefficient between X and Y on the basis of the above information.

Mr. Rajesh has ₹ 1800 to spend on fruits for a meeting. Grapes cost ₹ 160/kg llnd peaches ₹ 200/kg. Let x and g represent the number of kilogrames of grapes and peaches he can buy. Write the graph of an inequation to model the amounts of grapes and peaches he can buy within his budget. 

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:
(a) the mean and
(b) P(X = 0). [Given: e^-2 = 0.1353)

Ranking of 8 trainees at the beginning (X) and at the end (Y), of a certain course are given below : 

Calculate Spearman's rank correlation coefficient. 

A computers centre has four expert programmers . The centre needs four application programmes to be developed. The head of the computer centre , after stying carefully the programmes to be developed , estimates the computer time in minutes required by the respective experts to develop the application programmes as follows :

How should the head of the computer centre assign the programmes to the programmers so that the total time required is minimum ? 

From the data 7 pairs of observations on X and Y following results are obtained :

∑(x_i - 70) = - 38 ; ∑ (y_i - 60) = - 5 ;

∑ (x_i - 70)2 = 2990 ; ∑ (y_i - 60)2 = 475 ;

∑ (x_i - 70) (y_i - 60) = 1063

Obtain :

(a) The line of regression of Y on X.

(b) The line of regression of X on Y.