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Monday, August 17, 2015

2015/083) The L.C.M and H.C.F of two numbers are 1760 and 32 respectively. If one of the number is 160 find the other.

1760 = 160 * 11 = 32 * 5 * 11
160 = 32 * 5

32 is the HCF and and one number is 32 * 5
the other number has to be 32m when 32m* 5 = 1760 and m is coprime to 5
so m = 11
so other number = 32 * 11 = 352

This can also be solved by using the rule that product of HCF and LCM is the product of numbers

2015/082) Evaluate (\dfrac{n}{n+1})^{1 + n} as n\rightarrow \infty

(\dfrac{n}{n+1})^{1 + n}

Let's remember one thing first: lim x\rightarrow\infty (1 + \frac{r}{x})^x = e^r
(\dfrac{n}{n+1})^{1 + n}
(1- \dfrac{1}{n+1})^{1 + n}

substitute x = n + 1

Now, as n goes to infinity, then n + 1 will also go to infinity, and x will go to infinity

lim n\rightarrow\infty (1 - \dfrac{1}{n + 1})^{n + 1} 
= lim x\rightarrow\infty (1 - \dfrac{1}{x})^x 
 
Which is e^{-1}, or \dfrac{1}{e}


2015/081) If a(y + z) = x, b(z + x) = y, c(x + y) = z show that bc + ca + ab + 2abc = 1

a(y + z) = x
Hence \dfrac{1}{a} = \dfrac{y+z}{x}

add 1 to both sides
\dfrac{a+1}{a} = \dfrac{x + y +z}{x}
so \dfrac{a}{a+1} = \dfrac{x}{x+y+z}
similarly
 
\dfrac{b}{b+1} = \dfrac{y}{x+y+z}
\dfrac{c}{c+1} = \dfrac{z}{x+y+z}

adding the above we get
\dfrac{a}{a+1}+\dfrac{b}{b+1} +\dfrac{c}{c+1}= 1
or a(b+1)(c+1) + b(a+1)(c+1) + c(a+1)(b+1) = (1+a)(1+b)(1+c)
or (abc + ab + ac + a) + (abc + bc + ca + b) + (abc + ca + cb + c) = 1 + a + b+ c + ab + bc+ ca +abc
hence 2abc + ab + bc + ca = 1

 

Saturday, August 15, 2015

2015/080) If log_4 5 =a and log_5 6 =b, then log_3 2 =

From the given condition
5= 4^a
6 = 5^b = (4^a)^b = (2^2)^{ab} = 2^{2ab}
so 3 *2 = 2^{2ab}
so 3 = \frac{2^{2ab}}{2} = 2^{2ab-1}
or 2 = 3^{\frac{1}{2ab-1}}

hence log_3 2=\dfrac{1}{2ab-1}

2015/079) if a^{x-1} = bc, b^{y-1} = ca, c^{z-1} = ab show that xy + yz + zx = xyz

a^{x-1} = bc


hence a^x = abc
or a = (abc)^{\frac{1}{x}}\cdots (1)
similarly
b = (abc)^{\frac{1}{y}}\cdots (2)
c = (abc)^{\frac{1}{z}}\cdots (3) 
hence abc = (abc)^{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}
hence 
\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1
or yz + zx + xy = xyz

proved

Saturday, August 8, 2015

2015/078) Find the equation of the sphere passing (a,0,0), (0,b,0), (0,0,c) and (0,0,0)

the general equation of sphere

x^2+y^2+z^2+mx +ny + rz = p^2


as it passes through (0,0,0) put (x,y,z) = (0,0,0) to get p^2 = 0 or p = 0


as it passes through (a,0,0) put (x,y,z) = (a,0,0) to get a^2 + ma = 0 or m = -a


similarly n = -b and r = -c and we get equation as

x^2+y^2+z^2 - ax - by - cz = 0

2015/077) Find the formula for the number pattern : 2,5,10,17 \cdots

The 1st order difference 3,5,7
The second order difference 2,2,2,
So the term is quadratic

t_n = an^2 + bn + c

put n = 1 to get

a + b+ c = 2\cdots(1)

put n =2 to get

4a + 2b + c = 5 \cdots(2)

put n = 3 to get

9a + 3b + c = 10 \cdots(3)

subtract (1) from (2) to get
 
3a + b = 3 \cdots(4)

subtract (2) from (3) to get
 
5a + b = 5 \cdots(5)

from (4) and (5) 2a = 2 or a = 1 so b = 0 and hence c = 1
so

t_n = n^2 + 1
 

Sunday, August 2, 2015

2015/076) Prove that: \tan\,x\tan(60^\circ-x)\tan(60^\circ+x) = \tan3x

LHS = \tan\,x\tan(60^\circ-x)\tan(60^\circ+x)
= \tan\,x\dfrac{\tan\,60^\circ-\tan\,x}{1+\tan\,60^\circ \tan\,x}\dfrac{\tan\,60^\circ+\tan\,x}{1-\tan\,60^\circ \tan\,x}
\tan\,x\dfrac{\tan^260^\circ-\tan^2x}{1- \tan^260^\circ \tan^2x}
\tan\,x\dfrac{3-\tan^2x}{1- 3 \tan^2x}
\dfrac{3\tan\,x-\tan^3x}{1- 3 \tan^2x}
= \tan 3x = RHS

Saturday, August 1, 2015

2015/075) find x^{63} + x^{44} + x^{37} + x^{31} + x^{26} + x^9 +6

what is the value of the above expression for (x + \dfrac{1}{x})^2 = 3

we have
x^2 + 2 + \dfrac{1}{x^2}= 3
or x^4 -x^2 + 1=0
or (x^4-x^2+1) (x^2+1)= 0 as x^2+1 is not zero
or x^6+1=0

so x^6= - 1 and x^{12} = 1

so   x^{63} + x^{44} + x^{37}+ x^{31} + x^{26} + x^9 +6
= x^3 + x^8 + x + x^7 + x^2 + x^9 + 6 taking mod 12 of exponent
= x^3(x^6+1) + x^2(x^6+1) + x ( 1 + x^6) + 6
= 6 
This I have picked from https://in.answers.yahoo.com/question/index?qid=20150801013839AAEmdz0

2015/074) If ax^2+bx+c=0 has complex roots and a + c \lt b, then prove 4a+c\lt 2b

because f(x) = ax^2+bx+c=0 has complex roots so this is positive for all x or -ve for all x

f(-1) = a – b + c
from given condition a + c – b \lt 0 so f(-1) \lt 0
so f(-2) = 4a – 2b + c \lt 0 or 4a + c \lt 2b
proved



2015/073) Show that the following is true for all real values of x.

\dfrac{a^2(x-b)(x-c)}{(a-b)(a-c) } + \dfrac{b^2(x-c)(x-a)}{(b-c)(b-a)} + \dfrac{c^2(x-a)(x-b)}{(c-a)(c-b)} = x^2
Solution 2015/073)
This can be done in 2 ways
method 1

expand
we have 1st term
= \dfrac{a^2(x-b)(x-c)}{(a-b)(a-c)}
= \dfrac{-a^2(x-b)(x-c)}{(a-b)(c-a)}
=\dfrac{-a^2(x-b)(x-c)(b-c)}{(a-b)(b-c)(c-a)}
= \dfrac{x^2(- a^2(b-c)) +xa^2(b^2-c^2) -a^2bc(b-c)}{(a-b)(b-c)(c-a)}
similarly we have 2nd term
= \dfrac{b^2(x-c)(x-a)}{(b-c)(b-a)}
= \dfrac{x^2(- b^2(c-a)) +xb^2(c^2-a^2) -b^2ca(c-a)}{(a-b)(b-c)(c-a)}
3rd term
= \dfrac{c^2(x-a)(x-b)}{(c-a)(c-b)}
= \dfrac{x^2(- c^2(a-b)) +x(c^2(a^2-b^2) -c^ab(a-b)}{(a-b)(b-c)(c-a)}

so the sum = 
\frac{(x^2(- a^2(b-c)-b^2(c-a) -c^2(a-b)) +x(a^2(b^2-c^2) + b^2(c^2-a^2) + c^2(a^2-b^2)) – (a^2bc(b-c) + b^2ca(c-a) + c^2ab(a^2-b^2)}{(a-b)(b-c)(c-a))}
= \dfrac{x^2(- a^2(b-c)-b^2(c-a) -c^2(a-b)}{(a-b)(b-c)(c-a)}
numerator = (x^2(- a^2(b-c)-b^2(c-a) -c^2(a-b))
= x^2(-a^2(b-c) – b^2c+ ab^2 -ac^2 +bc^2)
= x^2(-a^2(b-c) – b^2c+ bc^2 -ac^2 +ab^2)
= x^2(-a^2(b-c) – bc(b-c) + a(b^2-c^2))
= x^2(-a^2(b-c) – bc(b-c) + a(b+c)(b-c))
= x^2(b-c)(-a^2-bc+ab+ac)
= x^2(b-c)(a-b)(c-a)
so we have ratio = x^2
hence proved

Aliter (Method 2)
This can be proved as below

because this is quadratic this cannot have more than 2 root.
But x= a, x= b, x= c satisfy the condition.
Hence this has to be identity.
Proved