Mathematics and Statistics Set 1 2018-2019 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

Write converse and inverse of the following statement: 
“If a man is a bachelor then he is unhappy.” 

Discuss the continuity of f at x = 1
Where f(X) = `[ 3 - sqrt ( 2x + 7 ) / ( x - 1 )]`           For x ≠ 1
                    = `-1/3`                                                 For x = 1

Find k, if the function f is continuous at x = 0, where

`f(x)=[(e^x - 1)(sinx)]/x^2`,      for x ≠ 0

     = k                             ,        for x = 0

Find the marginal revenue if the average revenue is 45 and elasticity of demand is 5.

Find `dy/dx if x^3 + y^2 + xy = 7`

Find the area bounded by the curve y = x⁴, x-axis and lines x = 1 and x = 5.

Evaluate :  `int_-2^3 dx/(x+5)`

Evaluate 
`int  1/(16 - 9x^2) dx`

Prove that the following statement pattern is a tautology : ( q → p ) v ( p → q )

Find `dy/dx` if y = x^x + 5^x

Evaluate : 
`int xcos^-1x dx`

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

If f is continuous at x = 0 then find f(0) where f(x) = `[5^x + 5^-x - 2]/x^2`, x ≠ 0

A manufacturer can sell x items (x > 0) at a price of Rs.(280 - x) each. The cost of producing x items is Rs. (x² + 40x + 35). Find the number of items to be sold so that the manufacturer can make maximum profit.

If p and q are true statements and r and s are false statements, find the truth value of the following :
( p ∧  ∼ r ) ∧ ( ∼ q ∧ s )

Differentiate e^{4x + 5} w.r..t.e^3x

Evaluate :
`int [e^x ( 1 + x )]/[ cos^2 (xe^x)] dx`

If A = `[(2, 3), (1, 2)], B = [(1, 0),(3, 1)]`, Find (AB)^-1

For manufacturing x units, labour cost is 150 – 54x and processing cost is x². Price of each unit is p = 10800 – 4x². Find the value of x for which Total cost is decreasing.

For manufacturing x units, labour cost is 150 – 54x and processing cost is x². Price of each unit is p = 10800 – 4x². Find the values of x for which Revenue is increasing.

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

Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).

If the correlation coefficient between X and Y is 0.8, what is the correlation
coefficient between.
(a) X and 3Y
(b) X - 5 and Y - 3

Find the premium on property worth Rs.12,50,000 at 3% if the property is insured to the extent of 80% of its value.

If the sum of squares of differences of ranks for 10 pairs of observations is 66, find the rank correlation coefficient.

If the present worth of a bill due 6 months hence is Rs. 2,500 at 10% per annum, what is the true discount ?

From the following table find q₀ :

Compute CDR using the information given below :

Total number of deaths in a year is given to be 900.

What must be subtracted from each of the numbers 5, 7 and 10, so that the
resulting numbers are in continued proportion?

An article is marked at Rs. 1500. A trader allows a discount at 3% and still gains 20% on the cost. Find the cost price of the article.

For a Binomial distribution n = 6 and p = 0.3, find the probability of getting exactly 3 successes.

Diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1500 calories Two foods F₁ and F₂ cost Rs. 50 and Rs. 75 per unit respectively. Each unit of food F₁ contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas each unit of food F₂ contains 100 units of  vitamins, 2 units of minerals and 30 calories. Formulate the above problem as L.P.P. to satisfy the sick person's requirements at minimum cost.

Two samples from bivariate populations have 15 observations each. The sample mean of X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from means are 136 and 148. The sum of product of deviations from respective means is 122. Obtain the equation of line of regression of X on Y.

Suggest optimum solution to the following assignment. Problem, also find the total minimum service time.
                                             Service Time ( in hrs.)

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

For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is 3y – 5x + 180 = 0. The mean marks in accountancy is 44 and the variance of marks in statistics is `(9/16)^(th)` of the variance of marks in accountancy. Find the mean marks in statistics and the correlation coefficient between marks in the two subjects.

The p.d.f. of a random variable X is given by :
f(x) = 2x,                  0 ≤ x ≤ 1
      = 0,                    otherwise
Find `P(1/3 < x < 1/2)`

Find the sequence that minimizes the total elapsed time required to complete the following task. The table below gives the processing time in hours. Also, find the minimum elapsed time and idle times for both machines.

A bill of Rs.7,500 was discounted for Rs. 7,290 at a bank on 28^th October 2006. If the rate of interest was 14% p.a., what is the legal due date ?

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.
Also prepare marginal distributions :

(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 )

Find the feasible solution for the following system of linear inequations:
0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4