Mathematics and Statistics 2017-2018 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Draw a Venn diagram for the truth of the following statement : 

All rational number are real numbers. 

Draw Venn diagram for the truth of the following statements : 

Some rectangles are squares. 

Find the inverse of the matrix A=`[[1,2],[1,3]]` using elementry transformations.  

Examine the continuity of f(x)=`x^2-x+9  "for"  x<=3`

=`4x+3  "for"  x>3,  "at"  x=3` 

find `dy/dx, if y= cos ^-1 (sin 5x)` 

The price P for demand D is given as P = 183 + 120 D – 3D².
Find D for which the price is increasing.

Evaluate`int (1)/(x(3+log x))dx` 

Find cofactors of the elements of the matrix A = `[[-1,2],[-3,4]]`

Evaluate : `int 1/(9x^2+49) dx` 

Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0

                     = k                    for x = 0

is continuous at x = 0. 

Examine whether the following statement pattern is tautology, contradiction or contingency :
p ∨ – (p ∧ q)

If x = cos2 θ and y = cot θ then find `dy/dx  at  θ=pi/4` 

The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number we get 11. By adding first and third numbers we get a number, which is double than the second number. Use this information and find a system of linear equations. Find these three numbers using matrices.

Find the area of the region bounded by the parabola y² = 16x and the line x = 4. 

The consumption expenditure E_c of a person with the income x. is given by E_c = 0.0006x² + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.

Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4 

= 10 for x = 4  at x = 4

Find `dy/dx,if e^x+e^y=e^(x-y)`

Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q 

Evaluate :`int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx`

The total cost function of a firm is `C = x^2 + 75x + 1600` for output x. Find the output (x) for which average
cost is minimum. Is `C_A = C_M` at this output?

Evaluate : `int_1^2 1/((x+1)(x+3)) dx` 

A shop valued at ₹ 2,40,000 is insured for 75% of its value. If the rate of premium is 90 paise percent, find the premium paid by the owner of the shop. 

Find the Age-Specific Death Rate (Age-SDR) for the following data :

 If `Σd_i^2` = 25, n = 6 find rank correlation coefficient where d_i, is the difference between the ranks of i^th values.

The following table gives the ages of husbands and wives 

Find : (i) The marginal frequency distribution of the age of husbands.
(ii) The conditional frequency distribution of the age of husbands when the age of wives lies between
25 - 35.

The regression equation of Y on X is y = `2/9` x and the regression equation of X on Y is `x=y/2+7/6` 

Find:

1.  The correlation coefficient between X and Y. 
2.  `σ_y^2 if σ _x^2=4`

Identify the regression equations of X on Y and Y on X from the following equations :
2x + 3y = 6 and 5x + 7y – 12 = 0 

If X has Poisson distribution with parameter m = 1, find P[X ≤ 1]. (Use e–1 = 0.3679)

Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. 

Ramesh, Vivek, and Sunil started a business by investing capital in the ratio 4 : 5 : 6. After 3 months Vivek withdrew all his capital and after 6 months Sunil withdrew all his capital from the business. At the end of the year, Ramesh received ₹ 6,400 as profit. Find the profit earned by Vivek.

Solve the following minimal assignment problem and hence find the minimum value : 

Calculate from `e_0^0,e_1^0,e_2^0` from the following data : 

A bill was drawn on 12th April for 3,500 and was discounted on 4th July at 5% p.a. If the banker paid 3,465 for the bill. Find period of the bill. 

Find Karl Pearson's correlation coefficient for the following data : 

Solve the following using graphical method : 

Minimize :Z=3x+5y

`2x+3x>=12` 

`-x+y<=3` 

`x<=4,y>=3,x>=0,y>=0` 

Given the following information : 

Find X, if the CDR = 13.4 per thousand. 

The manager of a company wants to find a measure which he can use to fix the monthly wages of persons applying for a job in the production department. As an experimental project, he collected data of 7 persons from that department referring to years of service and their monthly income : 

Find regression equation of income on the years of service. 

Solve the following inequation :
– 8 < – (3x – 5) < 13. 

Find the probability of guessing correctly at most three of the seven answers in a True or False objective test. 

A person bought a television set paying ₹ 20,000 in cash and promised to pay ₹ 1,000 at the end of every month for the next 2 years. If the money is worth 12% p.a. converted monthly, what is the cash price of the television set? 
[(1.01)^-24 = 0.7884]

There are found jobs to be completed. Each job must go through machines M₁ , M₂ , M₃ in the order M₁ - M₂ - M₃.  Processing time in hours is given below. Determine the optimal sequesnce and idle time for machine M₁ .