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

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).

Find `"dy"/"dx"`, if x = at², y = 2at

Find `(dy)/(dx)`, if x = 2at², y = at⁴.

Find `"dy"/"dx"`, if x = e^3t, y = `"e"^((4"t" + 5))`

Find `"dy"/"dx"`, if x = `("u" + 1/"u")^2, "y" = (2)^(("u" + 1/"u"))`

Find `"dy"/"dx"`, if x = `sqrt(1 + "u"^2), "y" = log (1 + "u"^2)`

Find `"dy"/"dx"`, if Differentiate 5^x with respect to log x

Solve the following.

If x = `"a"(1 - 1/"t"), "y" = "a"(1 + 1/"t")`, then show that `"dy"/"dx" = - 1`

If x = `(4t)/(1 + t^2),  y = 3((1 - t^2)/(1 + t^2))` then show that `dy/dx = (-9x)/(4y)`.

If x = t . log t, y = t^t, then show that `dy/dx - y = 0`.

If x = 2at² , y = 4at, then `dy/dx = ?`

If x = `y + 1/y`, then `dy/dx` = ____.

Find `"dy"/"dx"` if x = 5t², y = 10t.  

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 ______.