HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions

Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941:

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

Which of the following is not a statement?

The complicated but efficient method of measuring trend of time series is ______.

The following table shows gross capital information (in Crore ₹) for years 1966 to 1975:

Obtain trend values using 5-yearly moving values.

The publisher of a magazine wants to determine the rate of increase in the number of subscribers. The following table shows the subscription information for eight consecutive years:

Fit a trend line by graphical method.

The management of a large furniture store would like to determine sales (in thousands of ₹) (X) on a given day on the basis of number of people (Y) that visited the store on that day. The necessary records were kept, and a random sample of ten days was selected for the study. The summary results were as follows:

`sumx_i = 370 , sumy_i = 580, sumx_i^2 = 17200 , sumy_i^2 = 41640, sumx_iy_i = 11500, n = 10`

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`

For a bivariate data `barx = 10`, `bary = 12`, V(X) = 9, σ_y = 4 and r = 0.6
Estimate y when x = 5

Solution: Line of regression of Y on X is

`"Y" - bary = square ("X" - barx)`

∴ Y − 12 = `r.(σ_y)/(σ_x)("X" - 10)`

∴ Y − 12 = `0.6 xx 4/square ("X" - 10)`

∴ When x = 5

Y − 12 = `square(5 - 10)`

∴ Y − 12 = −4

∴ Y = `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.