Mathematics and Statistics Set 1 2017-2018 HSC Commerce (English Medium) 12th Standard Board Exam Question Paper Solution

If p : It is a day time

q : It is warm

Give the verbal statement for the following symbolic statement :

(a) p ^ - q  (b) P → q

Express the truth of each of the following statements using Venn diagrams: 

(a) No circles are polygons

(b) Some quadratic equations have equal roots 

Find the values of x and y if

`2 [(x,5),(7,y-3)] [(3,-4),(1,2)] = [(7,6),(15,14)]`

Find `"dy"/"dx"` ; if x = sin³θ , y = cos³θ

Find `"dy"/"dx"` ; if y = cos^-1 `("2x" sqrt (1 - "x"^2))`

Evaluate: ∫ x . log x dx

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'?  

 Evaluate : `int _0^(pi/4) 1/(1 + "x"^2)  "dx"`

Solve the following equations by reduction method: 

x+ y+z = 6,

3x-y+3z = 10

5x+ y-4z = 3 

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

Evaluate `int _0^l "x" (1 - "x")^(3/2)  "dx"`

Using the rules of negation, write the negatlon of the following: 

(a) p ∧ (q → r)

(b)  ~P ∨ ~q

If the function f is continuous at x = 2 and x = 4 then find the values of a and b.

Where f(x) = x² + ax + b, x < 2  

= 3x + 2, 2 ≤ x ≤ 4 

=  2ax + 5b, 4 < x 

A manufacturing company produces x items at the total cost of Rs (180 + 4x). The demand function of this product is P = (240 − x). Find x for which profit is increasing.

If A = `[(1,2),(3,-1)] , "B" = [(7,1),(2,5)]`

Verify that |AB| = |A|.|B|

Evaluate : `int 1/("x" [("log x")^2 + 4])  "dx"`

Find the volume of solid generated by rotating the area bounded by x²+y² =36 and the lines x = 0, x = 3 about X -axis. 

If f is continuous at x = 0, then find f (0). 

Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`

If x⁷ . y⁹ = (x + y)¹⁶ then show that `"dy"/"dx" = "y"/"x"`

In a firm the cost function for output x is given as C = `"x"^3/3 - 20"x"^2 + 70 "x"`.  Find the 3 output for which marginal cost  (C_m) is minimum.

The price of a T.V. set is Rs 17,000. An agent charges at 3% and earns Rs. 25,500. Find the number of T.V. sets sold by him. 

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

The regression equation of y on x is given by 3x + 2y - 26 = O. Find b_yx.

Verify whether the following function can be regarded as p.m.f. of the random variable X: 

\[ P(X) = \begin{cases} \frac{x - 1}3 & \quad \text{x } = \text{ 1,2,3}\\ 0 & \quad \text{} , \text{otherwise} \end{cases} \] 

If X has a binomial distribution with n = 20 , p = `1/10` , find E (X) and V (X).

Bring out the inconsistency; if any: b_yx + b_xy = 1.30. and r = 0.75 

A train travelled between two stations. The distance and time were recorded as below:

Draw scatter diagram and identify the type of correlation. 

If r = 0.5, σ_x = 1 and σ_y = 4, then find Cov.(X,Y).

Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following: 

Calculate the CDR for District A and B and compare them: 

The equation of the line of regression of Y on X is 3x + 2y = 26 and X on Y is 6x + y = 31. Find Var. (X) if Var. (Y) = 36.

Calculate Spearman's Rank Correlation Coefficient between the following marks given by 'two' judges (A) and (B) to •eight' contestants in the elocution competition:

Solve the following assignment problem to minimize the cost: 

Find the sequence of the following five jobs to be processed on three machines M₁ M_2, , M₃ that will minimize the total elapsed time to complete the tasks . Each job is to be processed in the order M₁ - M₂ - M₃ :

Minimize: Z = 2x + y 

Subject to: x + y ≤ 5 

x + 2y ≤  8 

4x + 3y ≥ 12 

x ≥  o,  y ≥ o 

Solve graphically.

Find the graphical solution for the following system of linear equations :

3x + 4y ≥ 12 , 4x + 7y ≤ 28 , y ≥ 1 , x ≥ 0 , y ≥ 0 ,

Hence find the co-ordinates of comer points of the common region. 

A wholesaler allows 25% trade discount and 5% cash discount.  Find the list price of an article, if it was sold for the net amount of Rs 1,140.

Find the accumulated value after 3 years of an immediate annuity of Rs 2,000 p.a. with interest compounded at 10% p.a. [Given (1.1)³ = 1.331] 

If the difference between true discount and banker's discount on a sum due 4 months hence is Rs 20,  find true discount, banker's discount and amount of the bill, the rate of simple interest charged being 5% p.a .

 If a random variable X follows Poisson distribution such that P(X = l) =P(X = 2), then find P(X ≥ 1).  [Use e^-2 = 0.1353]