Q14 of 43 Page 1

If show that.

OR


Differentiate with respect to x.

Given: x = and y =


To prove:


The formula to use:


Taking log of x -


log x = log


⇒ log x = 0.5 sin⁻1(t) log a


Differentiating both sides w.r.t t, we get -



⇒ ___(1)


Taking the log of y -


log y = log


⇒ log y = 0.5 cos⁻1(t) log a


Differentiating both sides w.r.t t, we get -



⇒ ___(2)


Now, Dividing equation 2 by 1,we get -




OR


Let, y = tan-1


Put x = tan β


∴ y = tan-1


∵ 1 + tan2 β = sec2 β


⇒ y = tan-1


⇒ y = tan-1


∵


⇒ y = tan-1


y = tan-1 (tan ) {∵ tan-1(tan x) = x }


⇒ y = = tan-1x


Differentiating both sides -


{∵ }


∴


More from this chapter

All 43 →
12

Using properties of determinants, show that

 13

Show that f : N → N, given by


is both one-one and onto.


OR


Consider the binary operations * : R × R → R and o : R × R → R defined as a * b =| a -b| and aob = a for all a, b ∈ R. Show that ‘*’ is commutative but not associative, ‘o’ is associative but not commutative.

 15

If x = a (cos t + t sin t) and y = a (sin t – t cos t), findand

 16

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?