Mathematics and Statistics 2016-2017 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Find x , y , z , w if `[("x+y","x-y"),("y+z+w","2w-z")]` = `[(2,-1),(9,5)]`

Express the truth of the following statements with the help of Venn diagram:
(a) No circles are polygon
(b) If a quadrilateral is rhombus , then it is a parallelogram .

Find the points of discontinuity , if any for the function : f(x) = `(x^2 - 9)/(sinx - 9)`

Write negation of the following statements : 
(a) The number 6 is an even number or the number 25 is a perfect square. 
(b) If x ∈ A ∩ B , then x ∈ A and x ∈ B.

Evaluate : ∫ `cos^2`x dx

Find `(d^2y)/(dx^2)` , if y = log x

Evaluate : `∫(e^x + 1)/(e^x + x)` dx

Find `(dy)/(dx) , "If"   x^3 + y^2 + xy = 10`

Find the inverse of the matrix `[(1      2     3),(1    1     5),(2    4     7)]` by adjoint method

If f(x) = `(e^(2x) - 1)/(ax)` .                for x < 0 , a ≠ 0
         = 1.                             for x = 0
         = `(log(1 + 7x))/(bx)`.        for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b

Demand function x, for a certain commodity is given as x = 200 - 4p where p is the unit price. Find :
(a) elasticity of demand as function of p.
(b) elasticity of demand when p = 10 , interpret your result.

Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).

If the demand function is D = 150 - p² - 3p, find marginal revenue, average revenue and elasticity of demand for price p = 3. 

Evaluate : `∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x)`.dx

Solve the following equations by reduclion method 

x+3y+3z= 16 ,  x+4y+4z=21 , x+3y+4z = 19 

 If the function f (x) = `(15^x - 3^x - 5^x + 1)/(x tanx)`,  x ≠ 0 is continuous at x = 0 , then find f(0).

Examine the function f(x) = `x + 25/x ` for maxima and minima 

Find the volume of a solid obtained by the complete revolution of the ellipse `x^2/36 + y^2/25 = 1` about X-axis.

If `x^3y^5 = (x + y)^8` , then show that `(dy)/(dx) = y/x`

Evaluate : `∫((1 + logx))/(x(2 + logx)(3 + logx)`dx

The ratio of number of boys and girls in a school is 3 : 2. If 20% of the boys and 30% of the girls are scholarship holders, find the percentage of students who are not scholarship holders.

Calculate crude death rates (CDR) for district A.

What is the sum due of ₹5,000, due 4 months, hence at 12.5% p.a. simple interest? 

The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

A wholesaler allows 25% trade discount and 5% cash discount, what will be the net price of an article marked at ₹1,600? 

Verify the following function, which can be regarded as p.m.f. for the given values of X : 

Solve the following minimal assignment problem : 

 If X has Poisson distribution with parameter m = 1, find P[X ≤ 1]  [Use `e^-1 = 0.367879`]

Find the present value of annuity immediate of ₹ 18000 p.a. for 3 years at 9% p.a. compounded annually. [(Given : `(1.09)^-3 = 0.7722`]

Complete the following life table : 

Given that r = 0.4 , `Σ(x - barx)(y - bary) = 108 , σ_y = 3 and Σ(x - barx)^2 = 900` . Find the number of pairs of observations.

Find mean and standard deviation of the continuous random variable X whose p.d.f. is given by f(x) = 6x(1 - x);= (0);      0 < x < 1(otherwise)

Solve the following L.P.P. graphically Minimize Z = 3x + 5y Subject to 2x + 3y ≥ 12 
-x + y ≤ 3 
x ≤ 4 
y ≥ 3 

We have seven jobs each of which has to go through two machines M₁ and M₂ in the order  M₁ - M_{2 .} Processing times (in hours) are given as :

Determine a sequence of these jobs that Will minimize the total elapsed time 'T', and idle time for each machine .

Compute the age specific death rate for the following data : 

If the rank correlation coefficient is 0.6 and the sum of squares of differences of ranks is 66, then find the number of pairs of observations.

The equations of the two regression lines are 2x + 3g - 6 = 0 and 5x + 7g - 12 = 0
Find: (a) Correlation coefficient. 
        (b) `sigma_x/sigma_y`

John and Mathew started a business with their capitals in the ratio 8 : 5. After 8 months, john added 25% of his earlier capital as further investment. At the same time, Mathew withdrew 20% of bis earlier capital. At the end of the year, they earned ₹ 52000 as profit. How should they divide the profit between them? 

A departmental store gives trafnfng to the salesmen in service followed by a test. It is experienced that the performance regarding sales of any salesman is linearly related to the scores secured by him. The following data gives the test scores and sales made by nine (9) salesmen during a fixed period. 

(a) Obtain the line of regression of Y on X.
(b) Estimate Y when X = 17. 

Three different aeroplanes are to be assigned to carry three cargo consignments with a view to maximize profit. The profit matrix (in lakhs of ₹) is as follows : 

How should the cargo consignments be assigned to the aeroplanes to maximize the profit?