HSC Commerce (Marathi Medium) इयत्ता १२ वी - Maharashtra State Board Important Questions for Mathematics and Statistics

Write the negation of the statement “An angle is a right angle if and only if it is of measure 90°”

Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p

Examine whether the statement pattern

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.

Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

Complete the truth table.

The given statement pattern is a `square`

If p ∨ q is true, then the truth value of ∼ p ∧ ∼ q is ______.

Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."

Determine whether the following statement pattern is a tautology, contradiction, or contingency:

[(∼ p ∧ q) ∧ (q ∧ r)] ∧ (∼ q)

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`

Find `("d"y)/("d"x)`, if y = [log(log(logx))]² 

Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`

Find `("d"y)/("d"x)`, if y = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`

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

Find `("d"y)/("d"x)`, if x = e^m, y = `"e"^(sqrt("m"))`

Solution: Given, x = e^m and y = `"e"^(sqrt("m"))`

Now, y = `"e"^(sqrt("m"))`

Diff.w.r.to m,

`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`

∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))`    .....(i)

Now, x = e^m

Diff.w.r.to m,

`("d"x)/"dm" = square`    .....(ii)

Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`

∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`

∴  `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`

If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`

If ax² + 2hxy + by² = 0, then prove that `(d^2y)/(dx^2)` = 0.

If the demand function is D = `((p + 6)/(p − 3))`, find the elasticity of demand at p = 4.

Choose the correct alternative:

Slope of the normal to the curve 2x² + 3y² = 5 at the point (1, 1) on it is 

Choose the correct alternative:

The function f(x) = x³ – 3x² + 3x – 100, x ∈ R is

The price P for the demand D is given as P = 183 + 120D − 3D², then the value of D for which price is increasing, is ______.