HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Solve the following differential equation:

(x² – y²)dx + 2xy dy = 0

Let p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). Then, this law is known as ______.

Without using truth table, show that

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

Without using truth table, show that

p ∧ [(~ p ∨ q) ∨ ~ q] ≡ p

Without using truth table, show that

~ [(p ∧ q) → ~ q] ≡ p ∧ q

Without using truth table, show that

~r → ~ (p ∧ q) ≡ [~ (q → r)] → ~ p

Without using truth table, show that

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

Using the algebra of statement, prove that

[p ∧ (q ∨ r)] ∨ [~ r ∧ ~ q ∧ p] ≡ p

Using the algebra of statement, prove that

(p ∧ q) ∨ (p ∧ ~ q) ∨ (~ p ∧ ~ q) ≡ (p ∨ ~ q)

Using the algebra of statement, prove that (p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ (~ p ∧ q).

For the following bivariate data obtain the equations of two regression lines:

From the data of 20 pairs of observations on X and Y, following results are obtained.

`barx` = 199, `bary` = 94,

`sum(x_i - barx)^2` = 1200, `sum(y_i - bary)^2` = 300,

`sum(x_i - bar x)(y_i - bar y)` = –250

Find:

1. The line of regression of Y on X.
2. The line of regression of X on Y.
3. Correlation coefficient between X and Y.

Given the following data, obtain a linear regression estimate of X for Y = 10, `bar x = 7.6, bar y = 14.8, sigma_x = 3.2, sigma_y = 16` and r = 0.7

b_YX is ______.

Obtain the trend line for the above data using 5 yearly moving averages.

The data obtained on X, the length of time in weeks that a promotional project has been in progress at a small business, and Y, the percentage increase in weekly sales over the period just prior to the beginning of the campaign.

Find the equation of the regression line to predict the percentage increase in sales if the campaign has been in progress for 1.5 weeks.

If for bivariate data `bar x = 10, bar y = 12,` v(x) = 9, σ_y = 4 and r = 0.6 estimate y, when x = 5.

Fit a trend line to the data in Problem 4 above by the method of least squares. Also, obtain the trend value for the index of industrial production for the year 1987.

The equation of the line of regression of y on x is y = `2/9` x and x on y is x = `"y"/2 + 7/6`.
Find (i) r,  (ii) `sigma_"y"^2 if sigma_"x"^2 = 4`

Identify the regression equations of x on y and y on x from the following equations, 2x + 3y = 6 and 5x + 7y − 12 = 0