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

If p ↔ q and p → q both are true, then find truth values of the following with the help of activity

p ˅ q

If p ↔ q and p → q both are true, then find truth values of the following with the help of activity

p ˄ q

Given following statements
p: 9 × 5 = 45
q: Pune is in Maharashtra
r: 3 is the smallest prime number

Write truth values by activity

If y = `1/sqrt(3x^2 - 2x - 1)`, then `("d"y)/("d"x)` = ?

Choose the correct alternative:

If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?

Choose the correct alternative:

If y = `x^(sqrt(x))`, then `("d"y)/("d"x)` = ?

If y = (5x³ – 4x² – 8x)⁹, then `("d"y)/("d"x)` is ______

If y = x¹⁰, then `("d"y)/("d"x)` is ______

If y = `("e")^((2x + 5))`, then `("d"y)/("d"x)` is ______

If y = x², then `("d"^2y)/("d"x^2)` is ______

State whether the following statement is True or False:

If y = e^x, then `("d"y)/("d"x)` = e^x 

State whether the following statement is True or False:

If x² + y² = a², then `("d"y)/("d"x)` = = 2x + 2y = 2a

State whether the following statement is True or False:

If y = e^x, then `("d"^2y)/("d"x^2)` = e^x 

Find `("d"y)/("d"x)`, if y = (6x³ – 3x² – 9x)¹⁰ 

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

Find `("d"y)/("d"x)`, if y = `root(5)((3x^2 + 8x + 5)^4`

y = (6x⁴ – 5x³ + 2x + 3)⁶, find `("d"y)/("d"x)`

Solution: Given,

y = (6x⁴ – 5x³ + 2x + 3)⁶ 

Let u = `[6x^4 - 5x^3 + square + 3]`

∴ y = `"u"^square`

∴ `("d"y)/"du"` = 6u^6–1

∴ `("d"y)/"du"` = 6(  )⁵ 

and `"du"/("d"x) = 24x^3 - 15(square) + 2`

By chain rule,

`("d"y)/("d"x) = ("d"y)/square xx square/("d"x)`

∴ `("d"y)/("d"x) = 6(6x^4 - 5x^3 + 2x + 3)^square xx (24x^3 - 15x^2 + square)`

The slope of the tangent to the curve y = x³ – x² – 1 at the point whose abscissa is – 2, is ______.

Choose the correct alternative:

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