Mathematics and Statistics 2013-2014 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

If `"A" = [(1,2,-3),(5,4,0)] , "B" = [(1,4,3),(-2,5,0)]`, then find 2A + 3B.

 If the function f is continuous at x = I, then find f(1), where f(x) = `(x^2 - 3x + 2)/(x - 1),` for x ≠ 1

If x = tan^-1t and y = t³ , find `(dy)/(dx)`.

Evaluate : `∫sin^2x` dx

Write the negation of the following statements : 

(a) Chetan has black hair and blue eyes. 
(b) ∃ x ∈ R such that x² + 3 > 0. 

If A = `[(1,1),(2,2)] , "B" = [(1,2),(3,4)]` then find |AB|.

Evaluate : `∫dx/(4 - 9x^2)`

If the function f is continuous at x = 2, then find 'k' where

f(x) = `(x^2 + 5)/(x - 1),` for  1< x ≤ 2 
      = kx + 1 , for x > 2

If `x^y = e^(x - y)` , show that `(dy)/(dx) = logx/(1 + logx)^2`

If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`

Discuss extreme values of the function f(x) = x.logx

Discuss the continuity of the function f at x = 0, where
f(x) = `(5^x + 5^-x - 2)/(cos2x - cos6x),` for x ≠ 0
      = `1/8(log 5)^2,`  for x = 0

The expenditure E_c of a person with income I is given by E_c = (0.000035) I² + (0. 045) I. Find marginal propensity to consume (MPC) and average propensity to consume (APC) when I = 5000.

Evaluate : `∫x  cot^-1x` dx

If p : It is a day time , q : It is warm 
Give the verbal statements for the following symbolic statements : 
(a) p ∧ ∼ q (b) p v q (c) p ↔ q 

Using the truth table statement, examine whether the statement pattern (p → q) ↔ (∼ p v q) is a tautology, a contradiction or a contingency.

The cost C of producing x articles is given as C = x³-16x² + 47x.  For what values of x, with the average cost is decreasing'?  

If A = `[(1,0,0),(2,1,0),(3,3,1)]` then find A^-1 by using elementary transformation .

Evaluate : `∫_0^3dx/(x + sqrt(9 - x^2))`

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

Alex spends 20% of his income on food items and 12% on conveyance. If for the month of June 2010, he spent ₹900 on conveyance, find his expenditure on food items during the same month. 

Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured. 

The following table gives the age of the husbands and of the wives : 

Find the marginal frequency distribution of the age of husbands. 

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. 

Values of two regression coefficients between the variables X and Y are `b_"yx" = - 0.4` and `b_"xy"` = - 2.025 respectively. Obtain the value of correlation coefficient. 

Verify whether the following function can be regarded as probability mass function (p.m.f.) for the given values of X : 

The p.m.f. of a random variable X is
`"P"(x) = 1/5` , for x = I, 2, 3, 4, 5 
        = 0 , otherwise.
Find E(X).

The time (in hours) required to perform the printing and binding operations (in that order) for each book is given in the following table : 

Find the sequence that minimizes the total elapsed time (in hours) to complete the work .

Find the present value of an annuity immediate.of ₹18,000. p.a. for 3 years at 9% p.a. compounded annually. [Given `(1.09)^-3 = 0.7722`]

Compute rank correlation coefficient for the following data : 

If the rank correlation coefficient is `2/3` and `Σ"d"_1^2` = 55` , then find the number of  pairs of observations. (Assume that no rank is repeated.) 

From the following data, find the crude death rates (C.D.R.) for Town I and Town II, and comment on the results : 

Calculate quantities indicated by'?' for the following part of a life table : 

The probability that a bomb dropped from an aeroplane will strike a target is `1/5`, If four bombs are dropped, find the probability that : 

(a) exactly two will strike the target,
(b) at least one will strike the target. 

Amit and Rohit started a business by investing ₹20,000 each. After 3 months Amit withdrew ₹5,000 and Rohit put in ₹5,000 additionally. How should a profit of ₹12,800 be divided between them at the end of the year? 

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 ?

Let X be the number of matches played by the player and Y he the number of matches in which he scored more thun 50 runs. The following data is obtained for 5 players : 

Find the regression line of X on Y. 

Find the sequence that minimizes total elapsed time (in hours) required to complete the following jobs on two machines M₁ and M₂ in order M₁-M₂. Also, find the minimum elapsed time T and idle time for the two machines. 

Solve the following LPP:

Minimize z = 4x + 2y

Subject to 3x + y ≥ 27, x + y ≥ 21, x + 2y ≥ 30, x ≥ 0, y ≥ 0

Solve the following L.P.P. : 
Minimize : Z = 4x + 10y, 
Subject to : 2x + 5y ≤10 , 5x + 3y ≤ 15, 
                  x + 2y  ≥ 30, x ≥ 0, y ≥ 0.