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Thursday, September 23, 2021

2021/078) What are two numbers whose HCF is 20 and LCM is 300?

Because HCF is 20 so the 2 numbers are 20m,20n where m or n = 1 or they are co-primes. without loss of generality let us assume that that m < n.

So we have LCM = 20mn = 300 or mn= 15

as mn = 15 and m and n are co-primes so m= 1 , n= 15 or m =3 nd n = 5 giving the 2 numbers (20,300) , (60,100) 

 

Tuesday, September 21, 2021

2021/077) Let a>b>c be the real roots of the equation \sqrt{2014}x^3-4029x^2+2=0. Find b(a+c).

To avoid radicals let \sqrt{2014}=p

So we get px^3-(2p^2+1)x^2 +2 = 0

Or factoring we get (px-1)(x^2-2px-2) = 0

So one root is x= \frac{1}{p} and  other two roots are roots of x^2-2px-2=0

For the equation x^2-2px-2=0 sum of the roots is 2p and product is -2. so one root has to be -ve and

The positive root shall be above 2p

So b=\frac{1}{p}\cdots(1)

And c is the -ve root and a> 2p

a,c are roots of x^2-2px-2=0 so a+c = 2p\cdots(2)

Hence b(a+c) = 2 using (1) and (2) 


Sunday, September 19, 2021

2021/076) Evaluate \sum^{89}_{n=0}\frac{1}{1+\tan^3(k^\circ)}

We have \sum^{89}_{n=0}\frac{1}{1+\tan^3(k^\circ)}
 = \sum^{89}_{n=0}\frac{\cos^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)}
=\frac{\cos^3(0^\circ)}{\cos^3(0^\circ)+\sin ^3(0^\circ)} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} + \frac{\sin^3(45^\circ)}{\cos^3(45^\circ)+\sin ^3(45^\circ)}+ \sum^{89}_{n=46}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} splitting into 4 parts
=\frac{1}{1+0} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} + \frac{\sin^3(45^\circ)}{\sin^3(45^\circ)+\sin ^3(45^\circ)}+ \sum^{89}_{n=46}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} as \sin\,45^\circ = \cos \, 45^\circ
=\frac{1}{1+0} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} + \frac{1}{2}+ \sum^{89}_{n=46}\frac{\cos^3(90^\circ- k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} putting values and using  \sin\,x^\circ = \cos \,(90^\circ-x)
=\frac{3}{2} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} +  \sum^{89}_{n=46}\frac{\cos^3(90^\circ- k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)}
=\frac{3}{2} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} +  \sum^{46}_{n=89}\frac{\cos^3(90^\circ- k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} reversing order of sum of cos terms
=\frac{3}{2} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)} +  \sum^{44}_{n=1}\frac{\cos^3( k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)}
=\frac{3}{2} + \sum^{44}_{n=1}\frac{\sin^3(k^\circ)+\cos^3( k^\circ)}{\cos^3(k^\circ)+\sin ^3(k^\circ)}
=\frac{3}{2} + \sum^{44}_{n=1} 1= 1.5 + 44 = 45.5

Friday, September 17, 2021

2021/075) If x^2-x-1 divides ax^{17}+bx^{16} + 1 find a-b

If x^2-x-1 divides ax^{17}+bx^{16} + 1 then x^2=x+1 =>  ax^{17}+bx^{16} + 1 = 0
The above is so because x =t which is root of x^2= x+1 is also a root of ax^{17}+bx^{16} + 1= 0

We have x^2=x+1
Putting y=\frac{1}{x} we get \frac{1}{y^2} =  \frac{1}{y} +1
or y^2 = 1 - y
 =>y^4 = (1-y)^2 = 1-2y +y^2 = (1-2y) + (1-y) = 2-3y
=>y^8 = (2-3y)^2 = 4-12y +9y^2 = (4-12y) + 9(1-y) =13-21y
=>y^{16} = (13-21y)^2 = 169-546y +441y^2 = (169-546y) + 441(1-y) = 610-987y
=>y^{17} = 610y-987y^2 = 610y - 987(1-y) = 1597y - 987
Putting back x=\frac{1}{y} we get
\frac{1}{x^{17}} =  \frac{1597}{x} - 987
Or 987x^{17} - 1597x^{16} +1 = 0

comparing with above we get a = 987, b = -  1597 so a - b = 2584

Wednesday, September 15, 2021

2021/074) A number n has sum of digits 100, while 44n has sum of digits 800. Find the sum of the digits of 3n.

The number can contain only the digits 1,2 besides 0. 1 * 44 = 44 and there is no overflow( if in the number the sum is one then it becomes 8, and if it is 2 the the sum of digits is 16) so the sum of digits is 8 times. if any digit is 3 to 9 then sum of digits less than 8 times so this shall give a lesser sum


Again 1 may be preceded/succeeded  by 0 or 1 as 11 * 44 = 484 . but 2 has to be    preceded/succeeded  by 0 as 21 * 44 = 924 and 12 * 44 = 538 and the sum of digits become less

So the number shall have 0 1 and 2 meeting above conditions so that sum of digits 100 and when we multiply by 3 (that is 3n) the digits shall be 0,3,6 and there is no overflow and sum of digits 300.

Sunday, September 12, 2021

2021/073)Consider polynomials P(x) of degree at most 3 each of whose coefficients is an element of { 0,1,2,3,4,5,6,7,8,9}. How many such polynomials satisfy P(-1) = 9.

Because the polynomial is of degrees at most 3 with above coefficients we have 

P(x) = ax^3+bx^2 + cx +d where a, b ,c, d are  element of { 0,1,2,3,4,5,6,7,8,9}.

now P(-1) = -a + b -c + d = (b+d) - (a+c)

now maximum sum of b+d can be 18 and as b+d - (a+c) = 9 so we have

a+c =k and b+d = 9+k and hence 0 <=<=9

as the digits are from 0 to 9 for the sum to be k <= 9 one of the numbers can be x = 0 to k and another number can be k -x so there are k+1 possibilities

for the sum to be  k >=9 one of the numbers a should be k-9 to 9 and another number c shall be k-9 and there are (9+1) - (k-9) or 19-k choices.

for the sum to be  k >=9= 9 + m there are  19-(9+m) or 10-m choices.

For the value  b+d - (a+c) = 9 a + c can be k and b+d can be k+9 so there are (k+1)(10-k) ways 

now the value k can be from 0 to 9 so we have sum

\sum_{k=0}^9(k+1)(10-k) =\sum_{n=1}^{10}n(11-n) = 11\sum_{n=1}^{10}n - \sum_{n=1}^{10}n^2 = 11 * \frac{10*11}{2} - \frac{10 * 11 * 21}{6} = 605- 385 = 220


So number of polynomials = 220  

Thursday, September 9, 2021

2021/072) if in a triangle a^2+b^2= \frac{19}{9}c^2 evaluate \frac{\cot\,C}{\cot\, A + \cot \, B}

We are given
a^2+b^2= \frac{19}{9}c^2
Using the above and law of cosine we get
2ab\cos\, C = a^2+b^2-c^2 = \frac{19}{9}c^2 - c^2 = \frac{10}{9}c^2
or ab\cos\, C = \frac{5c^2}{9}\cdots(1)

Further
\cot\, A +\cot\, B= \frac{\cos\, A}{\sin\, A} +  \frac{\cos\, B}{\sin\, B}
= \frac{\cos\, A\sin\, B +  \sin\,A\cos\, B}{\sin \, A\sin\, B}
= \frac{\sin(A+B)}{\sin \, A\sin\, B}
= \frac{\sin(\pi-C)}{\sin \, A\sin\, B} as A+B+C=\pi
= \frac{\sin\,C}{\sin \, A\sin\, B}

Hence \frac{\cot\,C}{\cot\, A + \cot \, B} = \frac{\cos\, C \sin\, A\sin\, B}{\sin ^2C}
=\frac{ab\cos\,C}{c^2} (using law of sin)
=\frac{5}{9} 

Tuesday, September 7, 2021

2021/071) For the triangle with angles A,B,C, the following trigonometric equality holds. \sin^2B+\sin^2C−\sin^2A=\sin\,B\sin\,C Find the measure of the angle A.

Using law of sin's \sin A = ka, \sin B= kb, \sin C = kc

We get

b^2+c^2 - a^2 = bc

Or a^2 = b^2 + c^2 + bc\cdots(1)

By law of cos

a^2 = b^2 + c^2 - 2bc \cos A \cdots(2)

from (1) and (2)

2 \cos A = - 1 or \cos A = \frac{-1}{2} or A = 12^circ

Sunday, September 5, 2021

2021/070) Evaluate \dfrac{1}{1-\cos \dfrac{\pi}{9}}+\dfrac{1}{1-\cos \dfrac{5\pi}{9}}+\dfrac{1}{1-\cos \dfrac{7\pi}{9}}.

We know that \cos \frac{\pi}{9},\cos \frac{5\pi}{9},\cos \frac{7\pi}{9} are different and they are

roots of equation \cos 3x = \cos \frac{\pi}{3} = \frac{1}{2}

or 4\cos^3 x - 3\cos\,x =\frac{1}{2}

or

so \cos\frac{\pi}{9}, \cos\frac{5\pi}{9}, \cos\frac{7\pi}{9} are roots of equation

x^3 - \frac{3}{4}x - \frac{1}{8}= 0

let x_1= \cos\frac{\pi}{9}, x_2 = \cos\frac{5\pi}{9}, x_3=\cos\frac{7\pi}{9}

Now x_1,x_2,x_3 are roots of equation

f(x) = x^3 - \frac{3}{4}x - \frac{1}{8}= 0\cdots(1)

by Vieta's formula we have

x_1 + x_2 + x_3 = 0\cdots(2)

x_1 x_2 + x_2x_3 + x_3 x_1 = \frac{-3}{4}\cdots(3)

Further f(1) = (1-x_1)(1-x_2)(1-x_3) =  1- \frac{1}{4} - \frac{1}{8} = \frac{1}{8}\cdots(3)

And we need to evaluate \frac{1}{1-x_1 } + \frac{1}{1-x_2} + \frac{1}{1-x_3}

Now

\frac{1}{1-x_1 } + \frac{1}{1-x_2} + \frac{1}{1-x_3}

= \frac{(1-x_2)(1-x_3) + (1-x_1)(1-x_3) + (1-x_1)(1-x_2)}{(1-x_1)(1-x_2)(1-x_3)}

= \frac{1-x_2 - x_3 + x_2x_3 + 1-x_1 - x_3  + x_1x_3  + 1-x_1 - x_2 + x_1x_2}{(1-x_1)(1-x_2)(1-x_3)}

= \frac{3 - 2(x_1 + x_2 + x_3)  + (x_2x_3 + x_3x_1  + x_1x_2)}{(1-x_1)(1-x_2)(1-x_3)}

= \frac{3 - 2 * 0 + \frac{-3}{4}}{\frac{1}{8}} putting the values using (2) , (3) and (4)

= 18

Hence \frac{1}{1-\cos \frac{\pi}{9}} + \frac{1}{1-\cos \frac{5\pi}{9}} + \frac{1}{1-\frac{7\pi}{9}}= 18


 

Thursday, September 2, 2021

2021/069) Prove that there are no integers a,\,b,\,c and d such that the polynomial ax^3+bx^2+cx+d equals 1 at x=19 and 2 at x=62.

We have  f(62)−f(19)=a(62^3−19^3)+b(62^2−19^2)+c(62−19)=1

Or  (62−19)(a(62^2+62∗19+19^2)+b(62+19)+c)=1 

LHS is a multiple of 43 and RHS is 1 so this does not have integer solution