Q13/100) 2 ^ sin x + 2 ^ cos x is greater than

A> 1/2

B> 2^(1/(2^(1/2)))

C> 2^(1/2)

D> 2^(1-(1/(2^(1/2))))

Solution

by AM GM inequality we have

(a+b) /2 >= sqrt(ab)

so 2^ sin x + 2^ cos x >= 2 sqrt(2^( sinx + cos x))

sin x + cos x = sqrt(2) ( sin x cos pi/4 + cos x sin pi/4) = sqrt(2) sin (x+pi/4)

lowest value is - sqrt(2)

so 2^ sin x + 2^ cos x >= 2 sqrt(2^-(sqrt(2)) = 2^(1-(1/2^(1/2))

hence D

 

Q13/099) If /(z - 5i)/ (z +5i)/ = 1 then prove that z is a real number

let z = x + iy

|z + 5i| = | z - 5i|

or | x + iy + 5i | = | x + iy - 5i|

or | x + iy + 5i |^2 = | x + iy - 5i|^2

or x^2 + (y + 5)^2 = x^2 + (y- 5)^2

or (y + 5)^2 = (y- 5)^2 or y^2 + 10y + 25 = y^2 - 10y + 25 => y =0 or z is real

Q13/098) If α and β are two different values of θ ( between 0 and 2π ) which satisfy the equation .................? . 6 cos θ + 8 sin θ = 9, Find the value of ..... sin ( α + β )

α and ß are the roots of the equation : 6 cosΘ + 8 sinΘ = 9
=> 6 cosα + 8 sinα = 9 ------------- (1) and
.....6 cos ß + 8 sin ß = 9 ------------- (2)
Subtracting (2) from (1), we have :
6(cosα - cos ß) + 8(sin α - sin ß) = 0
=> 4(sinα - sin ß) = 3(cos ß - cos α)
=> (sin α - sin ß)/(cos ß - cos α) = 3/4
=> [2 cos{(α + ß)/2} sin {(α - ß)/2}] / [2 sin {(α + ß)/2} sin {(α - ß)/2}] = 3/4
=> cot {(α + ß)/2} = 3/4
=> cot²{(α + ß)/2} = 9/16
=> 1 + cot²{(α + ß)/2} = 1 + 9/16 = 25/16
=> cosec²{(α + ß)/2} = 25/16
=> sin² {(α + ß)/2} = 16/25
=> sin {(α + ß)/2} = ± 4/5 --------- (3)
=> 1 - sin²{(α + ß)/2} = 1 - 16/25 = 9/25
=> cos²{(α + ß)/2} = 9/25
=> cos {(α + ß)/2} = ± 3/5 ---------- (4)
=> sin (α + ß) = 2 sin {(α + ß)/2} cos {(α + ß)/2} = ± 2(4/5)*(3/5) = ± 24/25

However as 6 cosα + 8 sinα = 9

so cosα and sinα both > 0 ( as if one is -ve then sum < 8) and so α is in 1st quadrant and similarly β

so ( α + β ) is either in 1st or 2nd quadrant so sin ( α + β )is positive and hence -24/25 need to be ruled out and hence it is 24/25.

Another solution

6 cos θ + 8 sin θ = 9
=> (3 / 5) cos θ + (4 / 5) sin θ = 9 / 10
=> sin (θ + φ) = 9/10

Clearly if
sin x = sin a
=> x = a or π - a
for given range

=> θ + φ = α + φ or π - (α + φ)
Clearly one of them is equal to ß + φ
=> π - (α + φ) = ß + φ
=> α + ß = π - 2φ
=> sin (α + ß) = sin (π - 2φ)
=> sin (α + ß) = sin 2φ
=> sin (α + ß) = 2 sin φ cos φ = 24 / 25

Q13/097) The LCM of 2numbers is 45 times their HCF.Sum of HCF and LCM is 1150, find the numbers?

LCM = 45 *HCF
LCM + HCF = 46 * HCF = 1150

so HCF = 25 and LCM = 25 * 45

now both x and y ( the 2 numbers are) multiple of 25 and coprime

so we x = 25m, y = 25n and LCM(m,n) = 45, HCF(m,n) = 1

without loss of generality let m >= n

mn = 45 = 1 * 45 = 5 * 9
so m = 9, n =5 or m = 45 , n= 1

numbers are (225,125) or (1125,25)

Q13/096) Find sum of n terms ---- 1 + ( 1 + x ) + ( 1 + x + x² ) + ( 1 + x + x² + x³ ) + ..... + up to nth term.?

y = 1 + (1+ x) + (1+ x + x^2) + ...
y ( 1-x) = (1-x) + 1(1-x^2) + .. ( 1- x^(n))
= n - ( x + x^2 + .. x^(n)) as
= n - x ( 1 + x ... x^(n-1))
= n - x ( 1- x^(n))/(1-x)

or y = n/(1-x) - x (1-x^(n)/(1-x)^2

the above ans is correct if x is not 1

if x = 1 then we have 1 + 2 + 3 + .. n = n(n+1)/2

Q13/095) Find coefficient of xⁿ in the expansion of ( 1 + 2x + 3x² + 4x³ + ........ inf. )^(1/2)....?

let y = ( 1 + 2x + 3x² + 4x³ + ........ inf. ...(1)
y converges for |x| < 1

now xy = x + 2x^2 + 3x^3 ....(2)

subtract (2) from (1)

y - xy = 1 + x + x^2 ... = 1/(1-x)

so y(1-x) = 1/(1-x)

or y = 1/(1-x)^2

so ( 1 + 2x + 3x² + 4x³ + ........ inf. )^(1/2)..= 1/(1-x) = 1 + x + x^2 ...

so coefficient of x^n = 1 for all n

Q13/094) Solve |x+1|+|x-2| < 5

As -1 < 2 and in a line the distance between -1 and 2 is 3 so all the points from -1 to 2 satisfy the relation as the sum is the distance from -1 to 2

Now if it to the left of -1 the distance from -1 shall be added twice once for |x+1| and another time for |x-2| the sum < 2 and hence x > - 2

For the point to the right of 2 similarly x < 3

So -2 < x < 3