Waste less time on Facebook — follow Brilliant.
×

JEE Algebra Contest

TOPICS for week 1+2.

Logarithms, Trigonometry(Ratios and Identies) , Quadratic Equations, Squence and series, Binomial Theorem

Rules

  1. I will post the first problem. If someone solves it, he or she can post a solution and then must post a new problem.

  2. A solution must be posted below the thread of the problem. Then, the solver must post a new problem as a separate thread.

  3. Please make a substantial comment.

  4. Make sure you know how to solve your own problem before posting it, in case no one else is able to solve it within 24 hours. Then, you must post the solution and you have the right to post a new problem.

  5. If the one who solves the last problem does not post a new problem in 2 hours, the creator of the previous problem has the right to post another problem.

  6. It is NOT compulsory to post original problems. But make sure it has not been posted on Brilliant.

  7. If a diagram is involved in your problem please make sure it is drawn by a computer program.

  8. Format your solution in \(\LaTeX\), picture solution will be accepted but only picture will not be, i.e. you can use picture for diagram or something, but not for the complete solution. Also make sure your solution is detailed and make sure to proof all claims.

  9. Do not post problems on same topic frequently like 3 continuous inequality problem are not allowed. 2 are sufficient.

  10. For those who are on slack Please post in #general that new problem is up along with problem number and the link to contest (https://brilliant.org/discussions/thread/algebra-contest/?sort=new)

  11. Handwritten picture solution are accepted only when they are easily understandable!!

Note by Rajdeep Dhingra
1 year, 3 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

PROBLEM 15

Find the value of summation:

\(\displaystyle \sum_{r=1}^{k} (-3)^{r-1} \binom{3n}{2r-1}\)

where \(k = \frac{3n}{2} \) and \(n\) is an even positive integer

This is an old JEE question.

Neelesh Vij - 1 year, 3 months ago

Log in to reply

\( {(1 + \sqrt{x})}^{3n} = 3nC0 + 3nC1 {x}^{0.5} + ........ \)

\( {(1 - \sqrt{x})}^{3n} = 3nC0 - 3nC1 {x}^{0.5} + ........\)

Subtract the equations and put \( x = -3 \)

You will get

\( \frac {{ (1 + \sqrt{3} i )}^{3n} - {( 1 - \sqrt{3} i )}^{3n} }{2\sqrt{3}i } = Required \) \( equation \)

\( \boxed {\frac{{2}^{3n}}{2\sqrt{3}i } ( cos\pi n + i sin\pi n - cos\pi n + sin \pi n ) = 0 }\)

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

Post problem 16

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Prince Loomba If u have one , then post na ! I am not getting what to post

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

Ans 0 , RIGHT?

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

ya bro!

Neelesh Vij - 1 year, 3 months ago

Log in to reply

PROBLEM 14

Define f= tan1tan2 + tan2tan3 ..............+tan88tan89

Define g = cot^2(1) -f

define h = log(g)/log(9) where all angles are measured in degrees

Then which of the following statements is true

A) g is prime number

B) g is composite number neither a perfect square nor a perfect cube

C) g is composite number and a perfect square

D) g is composite number and a perfect cube

E) h lies in (1,2)

F) h lies in (2,3)

G) h lies in (3,4)

H) h lies in (0,1)

One or more than one correct type

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

\(f\) can be written as \( \displaystyle \sum_{x=1}^{88} \tan(x) . \tan(x+1) \)

As \( \displaystyle \tan( (x+1) -x) ) = \dfrac{ \tan(x+1) - \tan(x) }{1 + \tan(x) . \tan(x+1) } \)

Using this we get the value of \( \displaystyle \tan(x) . \tan(x+1) = \dfrac{ \tan(x+1) - \tan(x)}{\tan(1) } -1 \)

Summing this we get \(f = \cot^2 (1) -89\)

So \(g = 89\)

And \(\displaystyle h = \dfrac{\log(89)}{\log(9)} \)

Neelesh Vij - 1 year, 3 months ago

Log in to reply

Yeah neatly done bro! +1

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Yes you can post the next question. Its right AF

Prince Loomba - 1 year, 3 months ago

Log in to reply

the answer is B right?

Rex Holmes - 1 year, 3 months ago

Log in to reply

He is gone. Tell complete answer then I will tell. Ik the answer but I cant post

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Prince Loomba the answer is B cause cot^2 doesn't give you a perfect square or a perfect cube so it must be B

Rex Holmes - 1 year, 3 months ago

Log in to reply

@Rex Holmes Wrong

Prince Loomba - 1 year, 3 months ago

Log in to reply

PROBLEM 8

Define

K = Tan(27x)-Tan(x)

and

P = sin(x)/cos(3x) +sin(3x)/cos(9x) + sin(9x)/cos(27x)

Assume That x lies in common domain of all the given functions.

Calculate the ratio K/P

Give 0 As your answer if you think that answer depends on x if you think answer is independent of x

And please solve it properly dont just put some value of x to get the ratio .

Type- Integer (0000 to 9999)

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Answer is 2

\(sinx/cos3x=1/2*(2 sinx cosx/cosx cos 3x)=1/2*(sin2x/cosx cos 3x)=1/2*(tan 3x-tanx)\)

Similarly \(sin3x/cos9x=1/2*(tan9x-tan3x)\)

\(sin9x/cos27x=1/2*(tan27x-tan9x)\)

Adding these 3 equations, we get \(P=1/2*K\) or \(K/P=2\)

Prince Loomba - 1 year, 3 months ago

Log in to reply

Yeah right!

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

@Prakhar Bindal Posted the next one

Prince Loomba - 1 year, 3 months ago

Log in to reply

PROBLEM 7

\(\sin(x)+\sin^{2}(x)=1\) and \(a\cos^{12}(x)+b\cos^{10}(x)+c\cos^{8}(x)+d\cos^{6}(x)-2=0\)

Find the value of \(\frac {a+c+d}{2}\)

NOTE a,b,c,d are non zero even integers

Type: Integer type

Prince Loomba - 1 year, 3 months ago

Log in to reply

SOLUTION 2 :

\( sinx = 1 - (sinx)^2 = (cosx)^2 \)

Now, putting this in the given eq gives

\( a (sinx)^6 + b (sinx)^5 + c (sinx)^4 + d (sinx)^3 - 2 = 0\)

Now , take cube of equation \( sinx + (sinx)^2 = 1 \) to get

\( (sinx)^6 + 3 (sinx)^5 + 3 (sinx)^4 + (sinx)^3 = 1 \)

\( (sinx)^6 + 3 (sinx)^5 + 3 (sinx)^4 + (sinx)^3 - 1 = 0\)

\( 2 (sinx)^6 + 6 (sinx)^5 + 6 (sinx)^4 + 2 (sinx)^3 - 2 = 0 \)

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

Thats the perfect example of elegance!

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

@Prakhar Bindal Yeah!! Thanks bro! :)

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

Answer to the problem is 5

Firstly Notice the fact that sinx = cos^2(x)

Putting in values we get

asin^6(x) +bsin^5(x)+csin^4(x)+dsin^3(x) = 2 1.....

Now sinx+sin^2(x) = 1

From this we will find values of successive powers of sinx which come to be

sin^2(x) = 1-sinx

sin^3(x) = 2sinx-1

sin^4(x) = 2-3sinx

sin^5(x) = 5sinx-3

sin^6(x) = 5-8sinx

Substituting these values in 1........

And then putting sinx = root 5 -1 /2

In 1..

We will get an expression in terms of a,b,c,d

Now a,b,c,d are integers , RHS is integer therefore irrational term on LHS should be Zero and Integral terms should be equal

Solving you will get

8a-5b+3c-2d = 0

and

18a-11b+7c-4d = 4

Eliminate b from both sides and solving you get a+c+d = 10

Actually the problem was flawed initially.

It took me and @Prince Loomba quite a lot of time to make the problem as it appears now.

Actually According to initial problem statement and answer that prince had only one quadruple was given as the answer. but then i proved that infinitely many quadruples exist. Then prince Told that a,b,c,d are integers . And then after quite a long discussion on slack the final agreement was done that a+c+d has a fixed value not a+b+c+d.

Although are many quadruples exist which satisfy that equation but value of a+c+d will be fixed for each quadruple

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Bro! You can solve this problem in much shorter way , just take cube of \( sinx + (sinx)^2 = 1 \) ( to get the eq you formed in sin by replacing cos^2 by sin ) .

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

Problem 5

Consider a polynomial P(x) = \(x^2 - 23569x + k\). It has 2 prime number roots. Find the value of k. (Give the answer as sum of digits like 57 = 5+7 = 12 = 1+2 = 3.)

Type:- Integer Type.

Rajdeep Dhingra - 1 year, 3 months ago

Log in to reply

Answer is 1

Harsh is right but his answer is wrong

Since 23569 is an odd number,and we know that sum of 2 primes is always even if none of them is 2,therefore two roots of this equation are 2 and 23567.

Thus k = 47134

SOD(k) =19=1+9=10=1+0=1

Prince Loomba - 1 year, 3 months ago

Log in to reply

Damn I edited my comment.

Who will post the next question?

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Decide who will post the next one

Prince Loomba - 1 year, 3 months ago

Log in to reply

I have a good problem so can I post because I answered wrong but due to calculation mistake?

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava Yeah you can. I have posted so many. I was just writing same solution as you posted and then I got a chance to get this question correct

Prince Loomba - 1 year, 3 months ago

Log in to reply

Since 23569 is an odd number,and we know that sum of 2 primes is always even if none of them is 2,therefore two roots of this equation are 2 and 23567.

Thus k = 47134

SOD(k) = 1

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Problem 1

If a,b,c \(\in\) R - {0}, such that both roots of the equation \(ax^2 + bx + c = 0\) lies in (2,3), then both roots of the equation \(a{(x+1)}^2 +b(x^2 -1) + c{(x-1)}^2 = 0\) always lies in
A) [1,4]
B)(2,3)
C)[1,3)
D)[1,2)

Type:- ONE OR MORE THAN ONE CORRECT.

Rajdeep Dhingra - 1 year, 3 months ago

Log in to reply

Let the first function be f (x). We know that f (2).f (3)>0 since the roots lie between (2,3) or (4a+2b+c) (9a+4b+c)>0

Let the second be g (x). We know that g (1)=4a and g (2)=9a+4b+c and g (3)=16a+8b+4c=4 (4a+2b+c)

From the above inequality, we figure out that g (3).g (2)>0. So the roots of equation always lie in (2,3) which is included in ABC.

Prince Loomba - 1 year, 3 months ago

Log in to reply

You may post the next problem.

Rajdeep Dhingra - 1 year, 3 months ago

Log in to reply

Problem 20

A=(1/1) arccot(1/1)+(1/2) arccot (1/2)+(1/3) arccot (1/3) and B= (1) arccot (1)+(2)arccot(2)+(3)arccot(3)

Find the value of |B-A|

Prince Loomba - 1 year, 2 months ago

Log in to reply

Answer has to be in which form ? @Prince Loomba

Devang Agarwal - 1 year, 2 months ago

Log in to reply

Post next problem anybody.

Prince Loomba - 1 year, 2 months ago

Log in to reply

(a/b)×pi+(c /d)arccot3

Prince Loomba - 1 year, 2 months ago

Log in to reply

Allen test paper question.

Archit Agrawal - 1 year, 2 months ago

Log in to reply

Ik that Bro

Prince Loomba - 1 year, 2 months ago

Log in to reply

Bro I know haha

Prince Loomba - 1 year, 2 months ago

Log in to reply

Problem 19

How many numbers between \(1\) and \(10000\) ,both inclusive contain different digits when represented in the scientific notation?

Example: \(3450=3.45×10^{3}\) is right and \(4010=4.01×10^{3}\) is right but \(4001=4.001×10^{3}\) is wrong as it contain 2 zeroes.

Prince Loomba - 1 year, 3 months ago

Log in to reply

1st make it clear , is number 4000 right? (Means can it be counted ?)

Aniket Sanghi - 1 year, 2 months ago

Log in to reply

Yes 4000 is right 4.0×10^3. I have given such example for 4010 bro

Prince Loomba - 1 year, 2 months ago

Log in to reply

PROBLEM 18

Evaluate

\( \displaystyle \sum_{r=1}^{n-1} \left( \sum_{k=1}^r \dfrac 1k \right) \times (-1)^{r-1} \binom{n}{r} \)

Neelesh Vij - 1 year, 3 months ago

Log in to reply

I Am attaching photos of my solution

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

I dont have any question so you can post next!

Prince Loomba - 1 year, 2 months ago

Log in to reply

Bro initially sigma should be from 1 to n not n-1 !!

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Are you sure that this series converges?

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Sorry for inconvenience, I have edited the question

Neelesh Vij - 1 year, 3 months ago

Log in to reply

problem 17

Find the value of \(\displaystyle \sum_{r=0}^{n}{\frac {^{n}C_{r}}{r+1}}\)

Prince Loomba - 1 year, 3 months ago

Log in to reply

we know that

\( \displaystyle (1+x)^n = \sum_{r=0}^{n} \binom{n}{r} \)

Simply integrating it from 0 to 1 gives the required result

So answer is \( \displaystyle \int_{0}^{1} (1+x)^n dx = \dfrac{2^{n+1} -1}{n+1} \)

Neelesh Vij - 1 year, 3 months ago

Log in to reply

Bro! It is \( \boxed{\frac{ {2}^{n+1} - 1 }{ n + 1}} \)

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

@Aniket Sanghi I saw the method and not the answer and thats why I thought it to be right!

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Aniket Sanghi Thanks for that I have edited it

Neelesh Vij - 1 year, 3 months ago

Log in to reply

Post next question

Prince Loomba - 1 year, 3 months ago

Log in to reply

problem 16

If x={\(3^{2n}/8\)} where {} denotes fractional part

Find value of \(sec^{-1}8x\)

Prince Loomba - 1 year, 3 months ago

Log in to reply

\( x = \dfrac{3^{2n}}{8}= \dfrac{9^{n}}{8} = \dfrac{(8+1)^{n}}{8} = \dfrac{\binom{n}{0}8^{n}+\binom{n}{1}8^{n-1}+....+ \binom{n}{n-1}8+ \binom{n}{n}}{8} = {\dfrac{1}{8}}=\dfrac{1}{8}\)

\(sec^{-1}(1) = 0 \)

There are supposed to be { } around every fraction which i was not able to add.

Aditya Chauhan - 1 year, 3 months ago

Log in to reply

Right post next

Prince Loomba - 1 year, 3 months ago

Log in to reply

Note:Everybody please note that new topics have been added

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 13

Let\(-1\leq p \leq 1\)

Show that the equation \(4x^{3}-3x-p=0\) has \(1\) unique root in the interval \([1/2,1]\) and identify it. This is JEE problem with exact wordings.

Prince Loomba - 1 year, 3 months ago

Log in to reply

Looking at the range you should definitely think of sine and cosine functions as there range is also the same.

Then recall the trigonometric identity cos(3x) = 4cos^3(x)-3cos(x)

So we carefully substitute x = cos(y)

Now problem is over!.

The root will cos[arccos(p)/3]

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Exact answer....

Prince Loomba - 1 year, 3 months ago

Log in to reply

oh then the answer must be p+1

Rex Holmes - 1 year, 3 months ago

Log in to reply

No and p+1 is out of the given range

Prince Loomba - 1 year, 3 months ago

Log in to reply

is the root 1

Rex Holmes - 1 year, 3 months ago

Log in to reply

No it is in terms of p

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 12 IF \(a\) and \(b\) are the roots of equation \(x+1=c x (1-cx)\) and \(c_{1},c_{2}\) be two values of \(c\), determined from the equation \(\frac {a}{b}+\frac {b}{a}=\pi -2\).

If \(\frac {c_{1}^{2}}{c_{2}^{2}}+\frac {c_{2}^{2}}{c_{1}^{2}}+2=k (\frac {\pi+1}{\pi-1})^{2}\), find the value of \(k\).

Type: Integer type

Prince Loomba - 1 year, 3 months ago

Log in to reply

\(c^{2}x^{2}-x(c-1)+1=0\)

\(\dfrac{a}{b}+\dfrac{b}{a} = \dfrac{a^{2}+b^{2}}{ab} = (c-1)^{2}-2 \)

\((c-1)^{2}-2=\pi-2\)

\(c^{2}-2c+1-\pi = 0\)

\(\dfrac{c_{1}^{2}}{c_{2}^{2}}+\dfrac{c_{2}^{2}}{c_{1}^{2}} +2 = \dfrac{(c_{1}^{2}+c_{2}^{2})^{2}}{c_1^{2}c_2^{2}}\)

\(c_{1}^{2}+c_{2}^{2} = 4 - 2(1-\pi) = 2\pi + 2\)

\(\dfrac{c_{1}^{2}}{c_{2}^{2}}+\dfrac{c_{2}^{2}}{c_{1}^{2}} +2 = \dfrac{(2\pi+2)^{2}}{(\pi-1)^{2}}\)

\(\boxed{k}=4\)

Someone else may post the next question. @Prince Loomba @Aniket Sanghi

Aditya Chauhan - 1 year, 3 months ago

Log in to reply

K = 4 ; I will edit sol after coming back from my class

Aniket Sanghi - 1 year, 3 months ago

Log in to reply

4 is right. Post solution and then the next question after coming.

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 10

Lets define a function named prince (x,n) such that prince (x,n)=[x]+[x/n]+...+[x/\(n^{\infty}\)]

Find the SOD (prince (10000,50)) without calculating each floor function value

Type: Integer type

Prince Loomba - 1 year, 3 months ago

Log in to reply

\(\lfloor \frac{a}{b}\rfloor = 0\) if b>a.

Therefore, the sum is effective only till \(i\) where \(i < 3 \).

Therefore answer = SOD(10000+200+4) = SOD(10204) = 7.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Is the answer Infinity?

Rex Holmes - 1 year, 3 months ago

Log in to reply

No and SOD stands for sum of digits, refer problem 9

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 9

Consider a rational function of the form of

\(\frac {a_{0}x^{0}+a_{1}x^{1}+.....+a_{m}x^{m}}{b_{0}x^{0}+b_{1}x^{1}+.....+b_{n}x^{n}}\)

As \(x\) tends to \(\infty\), the function tends to \(\infty\).

It is given that \(0\leq m,n \leq 10\)

If the number of ordered pairs \((m,n)\) that satisfy the given conditions is \(c\),

Find \(SOD (c)\)

Note \(SOD\) denotes repeated sum of digits for example

\(SOD(189)=SOD (1+8+9)=SOD (18)=SOD (1+8)=SOD (9)=9\).

Type: Integer type

Prince Loomba - 1 year, 3 months ago

Log in to reply

The given function will tend to \(\infty\) when m>n . So we have to find ordered pairs (m,n) in \([0,10]\) where m>n which is equal to \(\binom{11}{2} = 55 \).

SOD(55)=SOD(10)=1.

Aditya Chauhan - 1 year, 3 months ago

Log in to reply

Right. Are you posting next one or shall I?

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Prince Loomba I don't have any good question to post at the moment so you may post the next one.

Aditya Chauhan - 1 year, 3 months ago

Log in to reply

Should we also start a mechanics contest again?

@Rajdeep Dhingra @Prince Loomba

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

As soon as this contest or any one of the other ends we will immediately start up with mech .... btw I also wanted to start up with Analytical and regular NT contest ,,, ;)

Chinmay Sangawadekar - 1 year, 3 months ago

Log in to reply

After mechanics, we will start NT :)

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava Bro try Problem 7

Prince Loomba - 1 year, 3 months ago

Log in to reply

Hey guys @Harsh Shrivastava @Chinmay Sangawadekar @Rajdeep Dhingra the problem is corrected now it was a bit flawed initially

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

3 contests are running now. Later. Now: geometry,algebra,jee algebra

Prince Loomba - 1 year, 3 months ago

Log in to reply

PROBLEM 4

If \(x^{2}+ax+b\) is an integer for every integer x, then

A)a is always an integer but b need not be an integer

B)b is always an integer but a need not be an integer

C)a and b are always integers

D)none of these

Type: Single correct

Prince Loomba - 1 year, 3 months ago

Log in to reply

Put x=0 Then we have b must be an integer Put x=1 Then we have 1+a+b is an innteger Since 1,b both are integers a is also integer

Kaustubh Miglani - 1 year, 3 months ago

Log in to reply

Ok you can post next question

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Prince Loomba Sorry I am a little busy.So I cant.Please u do post

Kaustubh Miglani - 1 year, 3 months ago

Log in to reply

@Kaustubh Miglani No I wont. Its your turn.

Prince Loomba - 1 year, 3 months ago

Log in to reply

C)

Rajdeep Dhingra - 1 year, 3 months ago

Log in to reply

Please post solution and next question

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 3

In a triangle ABC, \(-a\leq sin3A+sin3B+sin3C \leq b\). That is this can be minimum -a and maximum b.

Find the value of [a+2b]

Type: Integer type

Prince Loomba - 1 year, 3 months ago

Log in to reply

Minimum -2 and maximum \(3\sqrt 3/2\)

Prince Loomba - 1 year, 3 months ago

Log in to reply

Please post solution also.

Rajdeep Dhingra - 1 year, 3 months ago

Log in to reply

@Rajdeep Dhingra Let y = sin3A + sin3B + sin3C for sin3A to be non positive

we have 2pi < 3A < 3pi

2pi/3< A <pi

since A + B + C = pi all of sin3A, sin3B, sin3C can’t be negative.

Let us take sin3A = – 1 or A = pi/2

sin3A = – 1, sin3B = – 1 and sin3C = 0 is possible So the minimum value is – 2.

Prince Loomba - 1 year, 3 months ago

Log in to reply

It's more than 24 hours. Post the solution and the next problem.

Rajdeep Dhingra - 1 year, 3 months ago

Log in to reply

Is a=2 and b=2.

Archit Agrawal - 1 year, 3 months ago

Log in to reply

No wrong

Prince Loomba - 1 year, 3 months ago

Log in to reply

Is b=4?

Kaustubh Miglani - 1 year, 3 months ago

Log in to reply

No wrong b is not 4

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 2

Find the minimum value of

|sinx+cosx+tanx+cotx+secx+cosecx|

If the answer is \(\theta\), find [\(\theta\)], where [.] Denotes floor function

TYPE: Integer type

Prince Loomba - 1 year, 3 months ago

Log in to reply

We make the careful substitution sinx+cosx = t

Converting all in terms of sinx and cosx and writing sinxcosx =(t^2-1)/2

and simplifying i got the expression as

y = t + 2/(t-1)

Just remember we need to find minimum value in (-root 2 , root 2) as range of sinx+cosx is that only

Differentiate this expression to get that function increases from (-infinite,1-root 2) and (1+root 2 , infinite) and decreases in the rest.

So for minimum value of modulus we check that value of t= 1-root 2 and -root 2 the smaller one will be the answer

On solving i got minimum value of modulus = 2root2 -1

And hence box theta = 1

I Dont have much time and good algebra problem so i give the right to @Prince Loomba to post the next

Prakhar Bindal - 1 year, 3 months ago

Log in to reply

Problem 11

Given that the maximum value of \(\displaystyle \prod_{i=1}^{n}(1+\sin ^2 x_{i})(1 + \cos ^2 x_{i} )\) is the minimum value of \(\dfrac{9^{10}}{2}y^{10} + \dfrac{2}{4^{10}y^{10}} \).

Find the value of n.

Type : Integer Type.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Answer is 5

The product is max when all terms equal 3/2. Or we can write it as \((3/2)^{2n}\)

The min value of the second as calculate by AM-GM is \((3/2)^{10}\).

From these 2 we can conclude \(n=5\)

Prince Loomba - 1 year, 3 months ago

Log in to reply

Justify your assumption why max occurs when all terms are equal?

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Your solution is wrong though answer is correct.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava Whaat is wrong. Differentiate. 1 more thing, the min value is 2 times of that I have written. This impies that n is not integer

Prince Loomba - 1 year, 3 months ago

Log in to reply

Problem 6

Let number of pairs of positive integers that satisfy the equation \( x^3 +y^3 = (x+y)^2\) be \(n\).

Find the value of \(n/3\).

Type-: Integer Type.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Prince Loomba - 1 year, 3 months ago

Log in to reply

Correct post the next question.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Aftr the calculations, answer is 1 and the 3 pairs are 2,2 , 1,2 , 2,1

Prince Loomba - 1 year, 3 months ago

Log in to reply

Now its 0, 3xy (x+y)=0 implies x=0 or y=0 or x=-y. Any of the case is not possible

Prince Loomba - 1 year, 3 months ago

Log in to reply

(2,2).

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava Try your luck again. I am right

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Prince Loomba I have edited the question.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava 2^3+2^3=16 and 4^3=64. Wrong

Prince Loomba - 1 year, 3 months ago

Log in to reply

@Prince Loomba Bro I am really sorry.

I have mistyped the question.

I have edited the question one last time.

Sorry.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

You committed a common mistake.

Answer is wrong.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava Then tell me one of the pairs so that I get my answer is incorrect

Prince Loomba - 1 year, 3 months ago

Log in to reply

Please see typo has been corrected.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Now its 0

Prince Loomba - 1 year, 3 months ago

Log in to reply

Its infinite. x=0 y=any nonnegative integer

Prince Loomba - 1 year, 3 months ago

Log in to reply

Sorry correction done.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Post other. Its out of the topics list

Prince Loomba - 1 year, 3 months ago

Log in to reply

If I tell the topic. the problem will become very easy

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

Well I have read the note and I very well know what to post.

The problem is alright.

It is within topic.

Harsh Shrivastava - 1 year, 3 months ago

Log in to reply

@Harsh Shrivastava Its infinite

Prince Loomba - 1 year, 3 months ago

Log in to reply

See the first line of the note

Prince Loomba - 1 year, 3 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...