HSC Commerce: Marketing and Salesmanship 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Without using truth table, prove that:

[p ∧ (q ∨ r)] ∨ [∼r ∧ ∼q ∧ p] ≡ p

Fit a trend line to the following data by the method of least square :

The processing times required for four jobs A, B, C and D on machine M₁ are 5, 8, 10 and 7 hours and on machine M₂ it requires 7, 4, 3 and 6 hours respectively. The jobs are processed in the order M_l, M₂. The sequence that minimises total elapsed time is ______

A publisher produces 5 books on Mathematics. The books have to go through composing, printing and binding done by 3 machines A, B, C. The time schedule for the entire task in proper unit is as follows :

Determine the optimum time required to finish the entire task. Also, find idle time for machines A, B, C.

Complete the following activity to fit a trend line to the following data by the method of least squares.

Solution:

Here n = 9. We transform year t to u by taking u = t - 1979. We construct the following table for calculation :

The equation of trend line is x_t= a' + b'u.

The normal equations are,

`sumx_t = na^' + b^' sumu`              ...(1)

`sumux_t = a^'sumu + b^'sumu^2`      ...(2)

Here, n = 9, `sumx_t = 47, sumu= 0, sumu^2 = 60`

By putting these values in normal equations, we get

47 = 9a' + b' (0)       ...(3)

40 = a'(0) + b'(60)      ...(4)

From equation (3), we get a' = `square`

From equation (4), we get b' = `square`

∴ the equation of trend line is x_t = `square`

Without using truth table, prove that : [(p ∨ q) ∧ ∼p] →q is a tautology.

Complete the following activity to find, the equation of line of regression of Y on X and X on Y for the following data:

Given:`n=8,sum(x_i-barx)^2=36,sum(y_i-bary)^2=40,sum(x_i-barx)(y_i-bary)=24`

Solution:

Given:`n=8,sum(x_i-barx)=36,sum(y_i-bary)^2=40,sum(x_i-barx)(y_i-bary)=24`

∴ `b_(yx)=(sum(x_i-barx)(y_i-bary))/(sum(x_i-barx)^2)=square`

∴ `b_(xy)=(sum(x_i-barx)(y_i-bary))/(sum(y_i-bary)^2)=square`

∴ regression equation of Y on :

`y-bary=b_(yx)(x-barx)` `y-bary=square(x-barx)`

`x-barx=b_(xy)(y-bary)`  `x-barx=square(y-bary)`

Following table gives the number of road accidents (in thousands) due to overspeeding in Maharashtra for 9 years. Complete the following activity to find the trend by the method of least squares.

Solution:

We take origin to 18, we get, the number of accidents as follows:

The equation of trend is x_t =a'+ b'u.

The normal equations are,

`sumx_t=na^'+b^'sumu             ...(1)`

`sumux_t=a^'sumu+b^'sumu^2      ...(2)`

Here, n = 9, `sumx_t=68,sumu=0,sumu^2=60,sumux_t=-44`

Putting these values in normal equations, we get

68 = 9a' + b'(0)     ...(3)

∴ a' = `square`

-44 = a'(0) + b'(60)          ...(4)

∴ b' = `square`

The equation of trend line is given by

x_t = `square`

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers. 

If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`

If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`

Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`

Draw a Venn diagram for the truth of the following statement : 

All rational number are real numbers. 

Draw Venn diagram for the truth of the following statements : 

Some rectangles are squares. 

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

(a) No circles are polygons

(b) Some quadratic equations have equal roots 

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

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 .

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.

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

Express the truth of each of the following statements by Venn diagram :
(a) Some hardworking students are obedient.
(b) No circles are polygons.
(c) All teachers are scholars and scholars are teachers.

Express the truth of each of the following statements using Venn diagram.
(1) All teachers are scholars and scholars are teachers.
(2) If a quadrilateral is a rhombus then it is a parallelogram..