Mathematics and Statistics 2015-2016 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

If y = ( sin x )^x , Find `dy/dx`

If A = `[(1, 3), (3, 1)]`, Show that A² - 2A is a scalar matrix.

Write the negation of the Following Statement :
∀ y ∈  N, y² + 3 ≤ 7

Write the negation of the following statement : 
If the lines are parallel then their slopes are equal.

The total revenue R = 720 - 3x² where x is number of items sold. Find x for which total  revenue R is increasing.

Evaluate : `int  (sec^2 x)/(tan^2 x + 4)` dx

Find `dy/dx, if y = cos^-1 ( sin 5x)`

Discuss the continuity of function f at x = 0.
Where f(X) = `[ [sqrt ( 4 + x ) - 2 ]/ ( 3x )]`, For x ≠ 0
                  = `1/12`,                      For x = 0

State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.

State if the following sentence is a statement. In case of a statement, write down the truth value :
√-4 is a rational number.

Solve the following equations by the inversion method :
2x + 3y = - 5 and 3x + y = 3.

Find x and y, if 
`{ 3[(1, 2, 0), (0, -1, 3)] - [(1, 5, -2), (-3, -4, 4)]}[(1), (2), (1)] = [(x), (y)]`

Evaluate : `int tan^-1x dx`

Express the truth of each of the following statements using Venn diagram.
(1) All teachers are scholars and scholars are teachers.
(2) If a quadrilateral is a rhombus then it is a parallelogram..

Write converse and inverse of the following statement :
"If Ravi is good in logic then Ravi is good in Mathematics."

Find the area of the region bounded by the lines 2y + x = 8, x = 2 and x = 4.

Evaluate : `int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]`

If f(x) = `("e"^(2"x") - 1)/"ax"`   , for x < 0 , a ≠ 0

         = 1                                    for x = 0

       = `("log" (1 + 7"x"))/"bx"` , for x > 0 , b ≠ 0

is continuous at x = 0, then find a and b.

If the function f is continuous at x = 0 then find f(0),
where f(x) =  `[ cos 3x - cos x ]/x^2`, `x!=0`

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours  `("Given"  sqrt(2) = 1.414)`

Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure E_c of a person with income I is given as E_c = ( 0.0003 ) I² + ( 0.075 ) I when I = 1000.

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.

If `cos^-1[(x^2 - y^2)/(x^2 + y^2)] = 2k    "show that y" dy/dx = xtan^2k`

Anandi and Rutuja invested Rs. 10,000 each in a business. Anandi withdrew her capital after 7 months. Rutuja continued for the year. After one year, the profit earned by them was Rs. 5,700. Find the profit earned by each person.

Calculate age specific death (A-SDR) rates for the following data :

For a bivariate data b_yx = -1.2 and b_xy = -0.3. Find the correlation coefficient between x and y.

A random variable x has the following probability distribution :

Find k.

The probability distribution function of continuous random variable X is given by
f( x ) = `x/4`,  0 < x < 2
        = 0,       Otherwise
Find P( x ≤ 1)

From the two regression equations y = 4x - 5 and 3x = 2y + 5, find `barx and bary`

Draw scatter diagram for the following data and interpret it :

If Σd² = 66 and n = 10 then find the rank correlation coefficient.

Determine `l_92 and l_93, "given that"  l_91 = 97, d_91 = 38 and q_92 = 27/59`

Calculate CDR for districts A and B and compare them. Also state which district is more healthy..

If for a bivariate data `barx = 10,  bary = 12, V(X) = 9, σ_y = 4` and r = 0.6, estimate y  when x = 5.

Calculate the coefficient of correlation between X and Y series from the following data :

`n = 15 ,bar x = 25, bary = 18, σ_x = 3.01, σ_y = 3.03,`

`sum ("x"_i - bar x) ("y"_i - bar y) = 122`

Solve the following minimal assignment problem and hence find minimum time where  '- ' indicates that job cannot be assigned to the machine : 

Solve the following maximal assignment problem :

Find the true discount, banker's discount and banker's gain on a bill of Rs. 36,600 due 4 months hence at 5% p.a.

Mr. Anil wants to invest at most Rs. 60,000 in Fixed Deposit (F.D.) and Public Provident Fund ( P.P.F. ). He wants to invest at least Rs. 20,000 in F.D. and at least Rs. 15,000 in P.P.F. The rate of interest on F.D. is 8% p.a. and that on P.P.F. is 10% p.a. Formulate the above problem as L.P.P. to determine maximum yearly income.

Find graphical solution for following system of linear inequations :
3x + 2y ≤ 180; x+ 2y ≤ 120, x ≥ 0, y ≥ 0
Hence find co-ordinates of corner points of the common region.

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 car valued at Rs. 4,00,000 is insured for Rs. 2,50,000. The rate of premium is 5% less 20%. How much loss does the owner bear including the premium if value of the car is reduced to 60% of its original value ?

If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)