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Author Archives: Nalin Pithwa
Annual Essay Contest: Mentoris Project: Aug 2021
Match making, STEM, math, algorithm by 18 year old
Derivatives: part 11: IITJEE maths tutorial problems for practice
Problem 1: Find .
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 ðŸ™‚ hahaha…lol ðŸ™‚
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.
A problem of log, GP and HP…
Question: If and , pyrove that:
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
Express a given integral number in any scale (radix)
Several scales (radix) have been used by mathematicians. Binary (2), Ternary (3), Quaternary (4), Quinary (5), Senary (6), Septenary (7), Octenary(8), Nonary (9), Denary (10/Decimal), Undenary(11), Duodenary (12) and of course, hexadecimal (16). Note that in any scale the base radix is “10”. Thus, “10” stands for 2 in “binary”, “ten” in “decimal”, 8 for “octal” radix respectively, etc.
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.
STEM for Australia
Derivatives: Part 10: IITJEE maths tutorial problems for practice
Problem 1: If , and , then is equal to:
(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.
Derivatives: Part 9: IITJEE maths tutorial problems practice
Problem 1: is equal to:
(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
Derivatives: part 8: IITJEE mains tutorial problems practice
Problem 1: If , then is equal to:
(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.