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

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 

The slope of the tangent to the curve x = `1/"t"`, y = `"t" - 1/"t"`, at t = 2 is ______

State whether the following statement is True or False:

The equation of tangent to the curve y = x² + 4x + 1 at (– 1, – 2) is 2x – y = 0 

Find the equations of tangent and normal to the curve y = 3x² – x + 1 at the point (1, 3) on it

Find the equation of tangent to the curve x² + y² = 5, where the tangent is parallel to the line 2x – y + 1 = 0

Find the equation of tangent to the curve y = x² + 4x at the point whose ordinate is – 3

Choose the correct alternative:

The value of `int ("d"x)/sqrt(1 - x)` is

Choose the correct alternative:

`int(("e"^(2x) + "e"^(-2x))/"e"^x) "d"x` =

`int 5^(6x + 9) "d"x` = ______ + c

State whether the following statement is True or False:

If f'(x) = 3x² + 2x, then by definition of integration, we get f(x) = x³ + x² + c 

If f(x) = k, where k is constant, then `int "k"  "d"x` = 0

State whether the following statement is True or False:

y² = 4ax is the standard form of parabola when curve lies on X-axis

State whether the following statement is True or False:

Standard form of parabola is x² = – 4by, when curve lies in the positive Y-axis

Find the area between the parabolas y² = 5x and x² = 5y

Choose the correct alternative:

Solution of the equation `x("d"y)/("d"x)` = y log y is