Choose (a) (b) (c) (d)

Solution 1:

Let . Hence, . Differentiating both sides w.r.t. x, we get the following:

But,

Hence, the answer is . Option c.

Problem 2: Find if

Choose (a) (b) (c) (d)

Solution 2:

The given equation is . Differentiating both sides wrt x,

is the answer. Option D.

Problem 3: If then is

choose (a) (b) (c) (d)

Solution 3:

Given that so that we have

so now differentiating both sides w.r.t. x,

Now, we also know that

But, note that by laws of logarithms, on simplification, we get

and so that on squaring, we get

so that now we get

, which all put together simplifies to

so that the answer is option C.

Problem 4: Find

Choose option (a) (b) (c) (d)

Solution 4:

Let us consider the first differential. Let us substitute . Hence,

and so we , and so also, we get so we get

required derivative

. Answer is option C.

Problem 5: Find

Choose option (a) zero (b) 26 (c) 26! (d) does not exist

Solution 5: the expression also includes a term so that the final answer is zero only.

Problem 6: Find .

Solution 6: Let

so

so so that differentiating both sides w.r.t. x, we get

we get

we get

we get

so the answer is option B.

Choose option (a): (b) (c) (d) none of these

Problem 7:

Find

Choose option (a) (b) (c) (d)

Solution 7: Let so that taking logarithm of both sides

so that

. Differentiating both sides w.r.t.x we get:

so that we get now

$latex\frac{1}{y(\log{y})} \times \frac{dy}{dx} = 1 + \log{x} $

so we get option a as the answer.

Problem 8:

Find

Choose option (a): (b) (c) (d) none of these.

Solution 8:

let taking logarithm of both sides we get

and now differentiating both sides w.r.t.x, we get

and now let and again take logarithm of both sides so that we get (this is quite a classic example…worth memorizing and applying wherever it arises):

The answer is option C.

Problem 9:

Find .

Choose option (a): (b) (c) (d) none of these

Solution 9:

Given that

Remark: Simply multpilying out thinking the symmetry will simplify itself is going to lead to a mess…because there will be no cancellation of terms …:-) The way out is a simple algebra observation…this is why we should never ever forget the fundamentals of our foundation math:-)

note that the above can be re written as follows:

Now, we are happy like little children because many terms cancel out hahaha…lol

and now differentiating both sides w.r.t.x we get

The answer is option A.

Problem 10:

If and then find the value of at

Choose option (a): (b) (c) (d)

Solution 10:

Answer is option D.

Regards,

Nalin Pithwa.

]]>Solution: This is same as proving: y is Harmonic Mean (HM) of x and z;

That is, to prove that is the same as the proof for :

Now, it is given that —– I

and —– II

Let say. By definition of logarithm,

; ;

; ; .

Now let us see what happens to the following two algebraic entities, namely, and ;

Now, …call this III

Now,

Hence, ….equation IV

but it is also given that …see equation II

Hence,

Take log of above both sides w.r.t. base N:

So, above is equivalent to

But now see relations III and IV:

Hence,

Hence,

Hence, as desired.

Regards,

Nalin Pithwa

]]>Let the digits used in a proposed scale(radix r) be . Let us express an integer in this scale. Let be unit’s digits. Analagous to the place value system (in decimal):

Now, let us say we want to express this number N in terms of these digits s.

Dividing N by , we get the unit’s digit as the remainder; and the quotient is:

.

Dividing the above quotient by r, we get as the remainder and the quotient as:

, and so on.

Example: Express the denary number 5213 in the scale of seven.

Solution: gives 5 as remainder and as quotient.

gives 2 as remainder and as remainder.

Continuing this way, we are able to express:

. That is . You can check the equivalence by converting both to decimal values.

Cheers,

Nalin Pithwa.

]]>(a) (b) (c) (d)

Problem 2: If , and , then is equal to:

(a) (b) (c) (d)

Problem 3: is equal to:

(a) (b) (c) cosec(x) (d)

Problem 4: , then is:

(a) (b) (c) (d)

Problem 5: is equal to:

(a) (b) (c) (d)

Problem 6: then is equal to :

(a) (b) (c) (d)

Problem 7: If then is

(a) (b) (c) (d)

Problem 8: is:

(a) (b)

(c) (d)

Problem 9: If then is:

(a) (b) 0 (c) 1 (d)

Problem 10: is:

(a) (b) (c) (d)

Regards,

Nalin Pithwa.

]]>(a) (b)

(c) (d)

Problem 2: is equal to:

(a) (b) (c) (d)

Problem 3: If where , then is given by :

(a) (b) (c) (d)

Problem 4: is equal to:

(a) (b) (c) (d)

Problem 5:

If , and , then is equal to:

(a) (b) (c) (d)

Problem 6: is equal to:

(a) (b) (c) (d)

Problem 7: is equal to

(a) 0 (b) (c) (d)

Problem 8: If , then is equal to:

(a) (b) (c) (d)

Problem 9: is equal to:

(a) (b) (c) 9d)

Problem 10: If then is equal to:

(a) (b) (c) (d)

Cheers,

Nalin Pithwa

]]>

(a) (b) -1 (c) 0 (d) b

Problem 2: If , then is:

(a) (b) (c) (d)

Problem 3: is equal to:

(a) (b) (c) (d)

Problem 4: If , then is equal to:

(a) (b) (c) (d)

Problem 5: is equal to:

(a) (b) (c) (d)

Problem 6: If then the value of is

(a) (b) (c) (d)

Problem 7: is equal to:

(a) (b) (c) (d) none

Problem 8: is equal to:

(a) (b)

(c) (d)

Problem 9: If and , then is :

(a) (b) (c) (d)

Problem 10: is equal to:

(a) (b) (c) (d)

Cheers,

Nalin Pithwa.

]]>