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

Evaluate : `int  "e"^(3"x")/("e"^(3"x") + 1)` dx

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

Write the Truth Value of the Negation of the Following Statement :

The Sun sets in the East. 

Write the truth value of the negation of the following statement : 

cos² θ + sin² θ = 1, for all θ ∈ R 

Simplify the following :

`{3 [(1,2,0),(0,-1,3)] - [(1,5,-2),(-3,-4,4)]} [(1),(2),(1)]`

Examine the continuity off at x = 1, if

f (x) = 5x - 3 , for 0 ≤ x ≤ 1

       = x² + 1 , for 1 ≤ x ≤ 2

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

If A = `[(1,2,3),(2,"a",2),(5,7,3)]` is a singular matrix , find the value of 'a'.

Evaluate :  `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`

Show that the following statement pattern in contingency : 

(~p v q) → [p ∧ (q v ~ q)] 

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.

x^y = e^x-y, then show that  `"dy"/"dx" = ("log  x")/("1 + log x")^2`

Evaluate : `int_0^1 "x" . "tan"^-1  "x"  "dx"`

If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` ,

verify that A (adj A) = (adj A) A = |A| . I

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

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

Differentiate : log (1 + x²)  w.r.t. cot^-1 x. 

Using the Venn diagram, examine the logical equivalence of the following statements: 

(a) Some politicians are actors.

(b) There are politicians who are actors.

(c) There are politicians who are not actors.

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

Evaluate  : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`

The total cost of manufacturing x articles is C = 47x + 300x² − x⁴.  Find x, for which average cost is increasing.

The total cost of manufacturing x articles C = 47x + 300x² – x⁴ . Find x, for which average cost is decreasing

Let X = amount of time for which a book is taken out of a college library by a randomly selected student and suppose X has p.d.f. 

`f(x)={(0.5x",",0≤x≤2,,),(0",","otherwise",,):}`

Calculate (a) P (X ≤ 1) (b) P (0.5 ≤ X ≤ 1.5)   

If the correlation coefficient between X and Y is 0.8, what is the correlation coefficient between:

`"X"/2` and Y

If the correlation coefficient between X and Y is 0.8, what is the correlation coefficient between:

`"X - 5"/7` and  `"Y - 3"/8`

Raghu, Madhu and Ramu started a business in a partnership by investing ₹ 60,000, ₹ 40,000 and ₹ 75,000 respectively.  At the end of the year they found that they have incurred a loss of ₹ 24,500. Find that how much loss each one had to bear.  

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.

For an immediate annuity paid for 3 years with interest compounded at 10% p.a.  its present value is ₹ 10,000. What is its accumulated value after 3 years. 

[Given: (1.1)³ = 1.331)].

Calculate CDR for district A and B. 

Compute Age - Specific Death rate tor the following data 

The ratio of prices of two houses was 2 : 3. Two years later when price of first house has increased by 30% and that of the second by ₹  90,000 the ratio of prices becomes 5 : 7. Find the original prices of two houses. 

The income of an agent remains unchanged though the rate of commission is increased from 5% to 6.25%. Find the percentage reduction in the value of business.

A new treatment for baldness is known to be effective in 70% of the cases treated. Four bald members from different families are treated. Find the probability that exactly two members are successfully treated.

A new treatment for baldness is known to be effective in 70% of the cases treated. Four bald members from different families are treated. Find the probability that at least  one member is  successfully treated.

Find the graphical solution for following system of linear inequations

`"x"_1/60 + "x"_2/90 <= 1 ; 5x_1 + 8x_2 ≤ 600 , x_1 ≥ 0 , x_2 ≥ 0`

Bring out the inconsistency, if any in the following : 

b_YX + b_XY = 1.30 and r = 0.75

Bring out the inconsistency, if any in the following : 

b_YX  = b_XY = 1.50 and r = -0.9

Bring out the inconsistency, if any in the following : 

b_YX  = 1.9 and b_XY = -0.25

Bring out the inconsistency, if any in the following : 

b_YX = 2.6 and b_XY  = `1/2.6`

A pharmaceutical company has four branches, one ea.ch at city A, B, C, D. A branch manager is to be appointed one at each city, out of four candidates P, Q, R and S. The monthly business deyending upon the city and the effectiveness of the branch manager in that city is given below :

Which manager should be appointed at which city so as to get the maximum total monthly business·? 

Fill in the blanks which are marked with a question mark in the following marked from the life table : 

For a bivariate data,

`bar x = 53 , bar y = 28 , "b"_"xy" = -1.5 and "b"_"xy"=- 0.2` Find 

Correlation coefficient between X and Y 

For a bivariate data, 

`bar x = 53 , bar y = 28 , "b"_"xy" = - 0.2` , `"b"_"yx" = -1.5` Find 

 Estimate of Y , When X = 50. 

For a bivariate data, 

`bar x = 53 , bar y = 28 , "b"_"yx"=-1.5 and "b"_"xy"=- 0.2` Find Estimate of X for y = 25. 

If X has a Poisson distribution with variance 2, find P (X = 4) 

[Use e^-2 = 0.1353] 

If X has a Poisson distribution with variance 2, find P(X ≤ 4) 

[Use e^-2 = 0.1353] 

If X has a Poisson distribution with variance 2, find 

Mean of X [Use e^-2 = 0.1353] 

There are 5 jobs each of which is to be processed through three machines A, B and C in the order ABC. Processing times in hours are as shown in the following table : 

Determine the optimum sequence for the five jobs and the minimum elapsed time. 

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 )

 Minimize : Z = 3x₁ + 2x₂

Subject to constraints 

5x₁ + x₂ ≥ 10

2x₁ + 2x₂ ≥ 12

x₁ + 4x₂ ≥ 12

x₁ , x₂ ≥ 0