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

Solve the following differential equation:

`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`

Solve the following differential equation:

`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`

Solve the following differential equation:

x dx + 2y dx = 0, when x = 2, y = 1

Solve the following differential equation:

`x^2.  dy/dx = x^2 + xy + y^2`

Solve the following differential equation:

(9x + 5y) dy + (15x + 11y)dx = 0

Solve the following differential equation:

(x² + 3xy + y2)dx - x² dy = 0

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 ______.