HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Use quantifiers to convert the given open sentence defined on N into a true statement.

3x – 4 < 9

Use quantifiers to convert the given open sentence defined on N into a true statement.

Y + 4 > 6

Choose the correct alternative:

If y = (x )^x + (10)^x, then `("d"y)/("d"x)` = ?

If x^y = 2^{x – y}, then `("d"y)/("d"x)` = ______

If y = `"a"^((1 + log"x"))`, then `("d"y)/("d"x)` is ______

If u = 5^x and v = log x, then `("du")/("dv")` is ______

If u = e^x and v = log_e x, then `("du")/("dv")` is ______

State whether the following statement is True or False:

If y = log(log x), then `("d"y)/("d"x)` = logx

State whether the following statement is True or False:

If y = 4^x, then `("d"y)/("d"x)` = 4^x  

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

Find `(dy)/(dx)`, if x^y = y^x 

Find `("d"y)/("d"x)`, if xy = log(xy)

Find `("d"y)/("d"x)`, if x = `sqrt(1 + "u"^2)`, y = log(1 +u²)

If x = t.logt, y = t^t, then show that `("d"y)/("d"x)` = t^t 

Find `("d"y)/("d"x)`, if y = (log x)^x + (x)log^x

Find `("d"y)/("d"x)`, if y = x^x + (7x – 1)^x 

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

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

If x^a .y^b = `(x + y)^((a + b))`, then show that `("d"y)/("d"x) = y/x`

Find `("d"y)/("d"x)`, if y = x^(x) + 20^(x) 

Solution: Let y = x^(x) + 20^(x) 

Let u = `x^square` and v = `square^x`

∴ y = u + v

Diff. w.r.to x, we get

`("d"y)/("d"x) = square/("d"x) + "dv"/square`   .....(i)

Now, u = x^x

Taking log on both sides, we get

log u = x × log x

Diff. w.r.to x,

`1/"u"*"du"/("d"x) = x xx 1/square + log x xx square`

∴ `"du"/("d"x)` = u(1 + log x)

∴ `"du"/("d"x) = x^x (1 +  square)`    .....(ii)

Now, v = 20^x

Diff.w.r.to x, we get

`"dv"/("d"x") = 20^square*log(20)`     .....(iii)

Substituting equations (ii) and (iii) in equation (i), we get

`("d"y)/("d"x)` = x^x(1 + log x) + 20^x.log(20)