we have \tan (2y) = \frac{2\tan y }{1-\tan^2 y}
hence \tan (- 2y) = \frac{-2\tan y }{1-\tan^2 y}
hence if x = \tan y
-2y = \tan ^-1 \frac{-2x}{1-x^2}
or \tan ^{-1} \frac{-2x}{1-x^2}= -2 \tan ^{-1} x\cdots(1)
further \cos(2y) = \sin ^2 y - \cos^2 y = \frac{\sin ^2 y - \cos^2 y}{\sin ^2 y + \cos^2 y}
or \cos(2y) = \frac{\tan ^2 y - 1}{\tan ^2 y + 1}
or \cos(\pi - 2y) = \frac{1 - \tan ^2 y}{\tan ^2 y + 1}
or \pi - 2y = \cos ^{-1} \frac{1 - \tan ^2 y}{\tan ^2 y + 1}
putting \tan y = x we get \pi - 2 \tan ^{-1} x = \cos ^{-1} \frac{1 - x^2}{1 + x^2}\cdots(2)
putting values from (1) and (2) in given equation we get
\pi - 2\tan^{-1} x - (1/2)*2\tan^{-1} x = \frac{2\pi}{3}
or 3\tan^{-1}(x) = \frac{\pi}{3}
or \tan^{-1} (x) = \frac{\pi}{9}
Thus x = \tan\frac{\pi}{9}
some short and selected math problems of different levels in random order I try to keep the ans simple
Thursday, February 25, 2016
2016/019) If \theta = \frac{\pi}{2^{n+1}} then show that 2^n cos\theta \ cos 2 \theta \cos 2^2\theta \cdots cos 2^n \theta = 1
2^n cos\theta \cos 2 \theta \cos 2^2\theta \cdots cos 2^n \theta
= (2\cos\theta) (2\cos 2 \theta)(2 \cos 2^2\theta) \cdots (2cos 2^n \theta) = 1
= (\frac{\sin 2\theta}{\sin \theta}) (\frac{\sin 4\theta}{\sin 2\theta}) \cdots (\frac{\sin 2^n\theta}{\sin 2^{n-1}\theta}) = 1
=\frac{\sin 2^n \theta }{\sin \theta}
==\frac{\sin 2^n\frac{\pi}{2^n+1})}{\sin \frac{\pi}{2^n+1}}
==\frac{\sin (\frac{\pi}{2^n}-\frac{\pi}{2^n+1}) }{\sin \frac{\pi}{2^n+1}}
==\frac{\sin (\frac{\pi}{2^n+1}) }{\sin \frac{\pi}{2^n+1}}=1
Tuesday, February 23, 2016
2016/018) Show that function f(x) = | x+2 | is continuous at x = - 2 but not differentiable at x = - 2.
We have f(x) = | x+2 |
or f(x) = x + 2 for x >=-2 and f(x) = -x-2 for x < -2
at x = -2 the right hand limit is 0 and the left hand limit is 0
so it is continuous at x = -2
differentiating from left we get f'(x) = -1 and differentiating from right f'(x) = 1
as left hand derivative and right hand derivative are not same of it is
not differenctiable
or f(x) = x + 2 for x >=-2 and f(x) = -x-2 for x < -2
at x = -2 the right hand limit is 0 and the left hand limit is 0
so it is continuous at x = -2
differentiating from left we get f'(x) = -1 and differentiating from right f'(x) = 1
as left hand derivative and right hand derivative are not same of it is
not differenctiable
Monday, February 22, 2016
2016/017) In a quadrilateral ABCD ab is the smallest and CD is the largest side. Prove that angles -- (1) A > C and B > D
Join AC. In triangle ABC AB < BC so \angle BAC > \angle BCA
in triangle ADC CD > AD so \angle CAD > \angle DCA
adding above 2 we get the result. Similarly the 2nd part
in triangle ADC CD > AD so \angle CAD > \angle DCA
adding above 2 we get the result. Similarly the 2nd part
2016/016) Show that 4 * (29!) + 5! \equiv 0 (\,mod\, 31)
because 31 is prime we have as per wilson's theorem
30! \equiv -1 (\,mod\, 31)\cdots(1)
and also 30 * (-1) = -30 \equiv 1 (\,mod\, 31) =>30^{-1} = \equiv 1 (\,mod\, 31)\cdots(2)
from (1) and (2)
29! \equiv 1 (\,mod\, 31)
or 4 * 29! \equiv 4 (\,mod\, 31)
or 4 * 29! + 5! \equiv 4 + 120 (\,mod\, 31) \equiv 124 (\,mod\, 31) \equiv 0 (\,mod\, 31)
30! \equiv -1 (\,mod\, 31)\cdots(1)
and also 30 * (-1) = -30 \equiv 1 (\,mod\, 31) =>30^{-1} = \equiv 1 (\,mod\, 31)\cdots(2)
from (1) and (2)
29! \equiv 1 (\,mod\, 31)
or 4 * 29! \equiv 4 (\,mod\, 31)
or 4 * 29! + 5! \equiv 4 + 120 (\,mod\, 31) \equiv 124 (\,mod\, 31) \equiv 0 (\,mod\, 31)
Saturday, February 20, 2016
2016/015) Solve for x (5+2\sqrt 6)^{x^2-3} + (5-2\sqrt 6)^{x^2-3} = 10
We have (5+ 2 \sqrt6)(5-2\sqrt6) = 25 - 24 =1
So if t= 5 + 2 \sqrt 6 then \dfrac{1}{t} = 5 - 2\sqrt 6
So we get
t^{x^2-3} + \dfrac{1}{t^{x^2-3}} = 10
let t^{x^3-3} = p\cdots(1)
so we get
p + \dfrac{1}{p} = 10
or p^2 - 10p +1 = 0
or p = 5 + 2 \sqrt 6 or p= 5 - 2\sqrt 6= (5+ 2 \sqrt 6)^{-1}
Hence from (1) and above x^2-3 =1 => x = \pm 2
or x^2 -3 = -1 => x = \pm \sqrt 2
So if t= 5 + 2 \sqrt 6 then \dfrac{1}{t} = 5 - 2\sqrt 6
So we get
t^{x^2-3} + \dfrac{1}{t^{x^2-3}} = 10
let t^{x^3-3} = p\cdots(1)
so we get
p + \dfrac{1}{p} = 10
or p^2 - 10p +1 = 0
or p = 5 + 2 \sqrt 6 or p= 5 - 2\sqrt 6= (5+ 2 \sqrt 6)^{-1}
Hence from (1) and above x^2-3 =1 => x = \pm 2
or x^2 -3 = -1 => x = \pm \sqrt 2
2016/014) if x,y,z are in H.P then show that log(x+z)+ log (x+z-2y) = 2log(x-z)
We have x,y,z are in HP
so \dfrac{1}{x} + \dfrac{1}{z} = 2\dfrac{1}{y}
or y(x+z) = 2xz
Now (x+z)(x+z-2y) = (x+z) ( x + z - \dfrac{4xz}{x+z})
= (x+z)^2 - 4xz
= x^2 + z^2 + 2xz -4xz
= x^2 + z^2 -2xz = (x-z)^2
taking log of both sides we get the result
so \dfrac{1}{x} + \dfrac{1}{z} = 2\dfrac{1}{y}
or y(x+z) = 2xz
Now (x+z)(x+z-2y) = (x+z) ( x + z - \dfrac{4xz}{x+z})
= (x+z)^2 - 4xz
= x^2 + z^2 + 2xz -4xz
= x^2 + z^2 -2xz = (x-z)^2
taking log of both sides we get the result
Wednesday, February 17, 2016
2016/013) Prove that \cos(\tan^{-1}(\sin(\cot^{-1} x)))= \sqrt{\frac{x^2+1}{x^2+2}}
Let \cot^{-1} x = y
so \cot y = x
so \csc^2 y = x^2+1
so \sin y = \frac{1}{\sqrt{x^2+1}}
so \sin\cot^{-1} x = \frac{1}{\sqrt{x^2+1}}
similarly \cos \tan ^{-1}x = \sqrt{\frac{1}{x^2+1}}
Hence \cos(\tan^{-1}(\sin(\cot^{-1} x)))
=\cos(\tan^{-1}\frac{1}{\sqrt{x^2+1}})
=\frac{1}{1+\frac{1}{x^2+1}}
=\frac{1}{\sqrt{1+\frac{1}{x^2+1}}}
=\frac{1}{\sqrt{\frac{x^2+2}{x^2+1}}}
=\sqrt{\frac{x^2+1}{x^2+2}}
so \cot y = x
so \csc^2 y = x^2+1
so \sin y = \frac{1}{\sqrt{x^2+1}}
so \sin\cot^{-1} x = \frac{1}{\sqrt{x^2+1}}
similarly \cos \tan ^{-1}x = \sqrt{\frac{1}{x^2+1}}
Hence \cos(\tan^{-1}(\sin(\cot^{-1} x)))
=\cos(\tan^{-1}\frac{1}{\sqrt{x^2+1}})
=\frac{1}{1+\frac{1}{x^2+1}}
=\frac{1}{\sqrt{1+\frac{1}{x^2+1}}}
=\frac{1}{\sqrt{\frac{x^2+2}{x^2+1}}}
=\sqrt{\frac{x^2+1}{x^2+2}}
Tuesday, February 16, 2016
2016/012) Suppose that f(x+3)=3x^2+7x+4\cdots(1) and f(x)=ax^2+bx+c\cdots(2) What is a+b+c
we have from 2nd equation f(1) = a + b + c\cdots(3)
Now we need to evaluate f(1)
as f(x+3) = 3x^2 + 7x + 4
so f(x) = 3(x-3)^2 + 7(x-3) + 4\cdots(4)
putting x = 1 we get f(1) = 3 * (-2)^2 + 7 * (-2) + 4 = 2
so a + b+ c = 2
Now we need to evaluate f(1)
as f(x+3) = 3x^2 + 7x + 4
so f(x) = 3(x-3)^2 + 7(x-3) + 4\cdots(4)
putting x = 1 we get f(1) = 3 * (-2)^2 + 7 * (-2) + 4 = 2
so a + b+ c = 2
2016/011) Simplify a^2(b+c)+b^2(c+a)+c^2(a+b) , if a,b,c are in Arithmetic Progression
a^2(b+c)+b^2(c+a)+c^2(a+b)
= a^2(b+c+a) + b^2(c+a+b) + c^2(a+b+c) - (a^3+b^3+c^3)
= (a+b+c) (a^2+b^2+c^2) - ((a+b+c)(a^2+b^2 + c^2-ab-bc-ca) + 3abc)
(because a^3+b^3+c^3 - 3abc = (a+b+c) ( a^2+ b^2+ c^2 - ab- bc-ca))
= (a+b+c)( ab + bc + ca) - 3abc
a b c are in ap so a+c = 2b so a+b+c = 3b
now ab+bc+ ca = b(a+c) + ca = 2b^2 + ca
so sum = 3b(2b^2+ ca) - 3abc = 6b^3
= a^2(b+c+a) + b^2(c+a+b) + c^2(a+b+c) - (a^3+b^3+c^3)
= (a+b+c) (a^2+b^2+c^2) - ((a+b+c)(a^2+b^2 + c^2-ab-bc-ca) + 3abc)
(because a^3+b^3+c^3 - 3abc = (a+b+c) ( a^2+ b^2+ c^2 - ab- bc-ca))
= (a+b+c)( ab + bc + ca) - 3abc
a b c are in ap so a+c = 2b so a+b+c = 3b
now ab+bc+ ca = b(a+c) + ca = 2b^2 + ca
so sum = 3b(2b^2+ ca) - 3abc = 6b^3
Monday, February 8, 2016
2016/010) Solve |e^{it} - 1| = 2 for -\pi<\theta<=\pi
e^ {it} = \cos t + i \sin t
so e^{it} - 1 = (\cos t-1) + i \sin t
take mod and square
(\cos t-1)^2 + \sin ^2 t = 4
or \cos^2 t - 2 \cos t + 1 + \sin ^2 t = 2
or 2 - 2\cos t = 4 or cos t = - 1 and hence t = \pi
so e^{it} - 1 = (\cos t-1) + i \sin t
take mod and square
(\cos t-1)^2 + \sin ^2 t = 4
or \cos^2 t - 2 \cos t + 1 + \sin ^2 t = 2
or 2 - 2\cos t = 4 or cos t = - 1 and hence t = \pi
2016/009) What is the value of c such that a straight line exists which intersects f(x)=x^4+9x^3+cx2+9x+4 at 4 points
f(x)=x^4+9x^3+cx^2+9x+4
for a straight line to intersect at 4 distinct points
f^{''}(x) must have 2 roots
The reason
if it has no root f'(x) is either positive or -ve and so f(x) is monotonically
increasing or decreasing. as it is cubic polynomial with leading coefficient positive
it is increasing. hence no line can intersect at more than 2 points
if it has one zero then it does not have any point of inflection
now f^{''}(x)=12x^2+54x+2c=12(x+\frac{9}{4})^2+2(c-\frac{243}{8})
it has a double root when c<\frac{243}{8}
hence c<\frac{243}{8}
for a straight line to intersect at 4 distinct points
f^{''}(x) must have 2 roots
The reason
if it has no root f'(x) is either positive or -ve and so f(x) is monotonically
increasing or decreasing. as it is cubic polynomial with leading coefficient positive
it is increasing. hence no line can intersect at more than 2 points
if it has one zero then it does not have any point of inflection
now f^{''}(x)=12x^2+54x+2c=12(x+\frac{9}{4})^2+2(c-\frac{243}{8})
it has a double root when c<\frac{243}{8}
hence c<\frac{243}{8}
Tuesday, February 2, 2016
2016/008) Simplify \frac{x^4}{(x-y)(x-z)}+\frac{y^4}{(y-z)(y-x)}+\frac{z^4}{(z-x)(z-y)}
\dfrac{x^4}{(x-y)(x-z)}+\dfrac{y^4}{(y-z)(y-x)}+\dfrac{z^4}{(z-x)(z-y)}
= - (\dfrac{x^4}{(x-y)(z-x)}+\dfrac{y^4}{(y-z)(x-y)}+\dfrac{z^4}{(z-x)(y-z)})
= - (\dfrac{x^4(y-z) + y^4(z-x) + z^4(x-y)}{(x-y)(y-z)(z-x)})
now
x^4(y-z) + y^4(z-x) + z^4(x-y)
= x^4(y-z) + yz(y^3-z^3) - x (y^4-z^4)
= x^4(y-z) + yz(y-z)(y^2+yz+z^2) - x(y-z)(y^3 + y^2z + yz^2 + z^3)
= (y-z)(x^4 + yz(y^2 +yz+z^2) - xy(y^2 + yz + z^2) - xz^3)
= (y-z)(x^4 + (y^2+yz+z^2)(yz-xy) - xz^3)
= (y-z)(x(x^3-z^3) + y(z-x)(y^2 + yz + z^2)
=(y-z)(z-x)(y(y^2 + yz + z^2) - x(x^2 + zx + z^2)
= (y-z)(z-x)(y^3 + y (yz+ z^2) - x^3 - x(zx + z^2)
= (y-z)(z-x)(y^3-x^3 + (y^2z + yz^2 - zx^2 - z^2 x)
= (y-z)(z-x)((y-x) (x^2 + xy + y^2) + (z(y^2 - x^2) +z^2(y-x))
= (y-z)(z-x)((y-x)(x^2 + xy + y^2 + z(y+x) + z^2)
= (-(x-y)(y-z)(z-x)(x^2 + y^2 + z^2 + xy+yz+zx)
So the given expression
= x^2 + y^2 +z^2 + xy + yz+ xz
hence \dfrac{x^4}{(x-y)(x-z)}+\dfrac{y^4}{(y-z)(y-x)}+\dfrac{z^4}{(z-x)(z-y)}= \dfrac{1}{2}((x+y)^2 + (y+z)^2 + (x+z)^2
= - (\dfrac{x^4}{(x-y)(z-x)}+\dfrac{y^4}{(y-z)(x-y)}+\dfrac{z^4}{(z-x)(y-z)})
= - (\dfrac{x^4(y-z) + y^4(z-x) + z^4(x-y)}{(x-y)(y-z)(z-x)})
now
x^4(y-z) + y^4(z-x) + z^4(x-y)
= x^4(y-z) + yz(y^3-z^3) - x (y^4-z^4)
= x^4(y-z) + yz(y-z)(y^2+yz+z^2) - x(y-z)(y^3 + y^2z + yz^2 + z^3)
= (y-z)(x^4 + yz(y^2 +yz+z^2) - xy(y^2 + yz + z^2) - xz^3)
= (y-z)(x^4 + (y^2+yz+z^2)(yz-xy) - xz^3)
= (y-z)(x(x^3-z^3) + y(z-x)(y^2 + yz + z^2)
=(y-z)(z-x)(y(y^2 + yz + z^2) - x(x^2 + zx + z^2)
= (y-z)(z-x)(y^3 + y (yz+ z^2) - x^3 - x(zx + z^2)
= (y-z)(z-x)(y^3-x^3 + (y^2z + yz^2 - zx^2 - z^2 x)
= (y-z)(z-x)((y-x) (x^2 + xy + y^2) + (z(y^2 - x^2) +z^2(y-x))
= (y-z)(z-x)((y-x)(x^2 + xy + y^2 + z(y+x) + z^2)
= (-(x-y)(y-z)(z-x)(x^2 + y^2 + z^2 + xy+yz+zx)
So the given expression
= x^2 + y^2 +z^2 + xy + yz+ xz
hence \dfrac{x^4}{(x-y)(x-z)}+\dfrac{y^4}{(y-z)(y-x)}+\dfrac{z^4}{(z-x)(z-y)}= \dfrac{1}{2}((x+y)^2 + (y+z)^2 + (x+z)^2
Monday, February 1, 2016
2016/007) How many integer values of x and y are there such that 4x+7y=3 while |x| < 500 and |y| < 500
1st let us find one solution
this can be found by any method but as we see that 7- 4 = 3
so (-1,1) is a solution
as coeffcient of y is larger so we need to restrict x between - 500 to 500
general solution is x = -1+ 7t and y = 1 + 4t
Now - 500 < x < 500 or - 500 < 1 + 7t < 500 or
-501 < 7t < 499 or -71 <= t <= 71 so 143 values
this can be found by any method but as we see that 7- 4 = 3
so (-1,1) is a solution
as coeffcient of y is larger so we need to restrict x between - 500 to 500
general solution is x = -1+ 7t and y = 1 + 4t
Now - 500 < x < 500 or - 500 < 1 + 7t < 500 or
-501 < 7t < 499 or -71 <= t <= 71 so 143 values
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