161 - Mathematics Questions Answers

2x⁸⁸ + 3x⁸⁷ − 13x² + 5x + 9 = 0

implies that 2x⁸⁸ + 3x⁸⁷ = 13x² - 5x - 9

or x⁸⁷(2x+3) = 13x² - 5x - 9

now plot the left and right graphs for getting the intersecting points

Hint: Parabola will face in upward direction as a is positive, take x=0 we get -9 so 1 root will be positive and 1 negative,

x⁸⁷(2x+3) will be 0 for x = -3/2 and x = 0 besides it consider some values as x = 1 and -1

You will get one root between 0 and -1 

now check both the graph at x= -2, you will get the idea about the second intersection

According to the given condition p is a negative number

consider p = -k so cube root of p will be α = -k^1/3, β = -k^1/3ω, γ = -k^1/3ω²

On applying given values you will get the answer as  ω or ω²

you should use ω³ = 1

According to the given condition p is a negative number

consider p = -k = (k^1/3)³(-1) so cube root of p will be α = -k^1/3, β = -k^1/3ω, γ = -k^1/3ω² , here -1, -ω, -ω² are the cube roots of -1

On applying given values on the given expression, you will get  

x(-k^1/3)+y(-k^1/3ω)+z(-k^1/3ω²)/x(-k^1/3ω)+y(-k^1/3ω²)+z(-k^1/3)

=x+yω+zω²/xω+yω²+z

first multiply ω in numerator and denominator we get answer as 1/ω = ω²

second interchange the value of α, β, γ for getting answer as ω

you should use ω³ = 1

Let p(A) = x and p(B) = y

we know that p(A∩B) + p(A∩B^') = p(A)

and                  p(A∩B) + p(B∩A^') = p(B)

and for independent event p(A∩B) = p(A)p(B)

so on applying all the given conditions relation etween x and y will be

xy + 1/5 = x

xy + 2/7 = y

solve now

You can solve this question by the following method

possile algebric expression for the given equation is

(x⁰+x²+x⁴+........+x²⁰)(x⁰+x¹+x²+..................+x²¹)(x⁰+x¹+x²+x³+....................x²¹)                 (i)

Here three brackets are for x , y and z, and powers are based on the range of least to greatest possile values of x, y, z

on solving eq (i) will become (x⁰+x²+x⁴+...........+x²⁰)(1-x²²)²(1-x)^-2

now we have to calculate coefficient of x²¹ in this expression so (1-x²²)² can be omitted

general term of (1-x)^-2 is (r+1)x^r

so required coefficient = 22+20+18+16+14+12+10+8+6+4+2 = 132

You can solve this question by the following method

possile algebric expression for the given equation is

(x⁰+x²+x⁴+........+x²⁰)(x⁰+x¹+x²+..................+x²¹)(x⁰+x¹+x²+x³+....................x²¹)                 (i)

Here three brackets are for x , y and z, and powers are based on the range of least to greatest possile values of x, y, z

on solving eq (i) will become (x⁰+x²+x⁴+...........+x²⁰)(1-x²²)²(1-x)^-2

now we have to calculate coefficient of x²¹ in this expression so (1-x²²)² can be omitted

general term of (1-x)^-2 is (r+1)x^r

so required coefficient = 22+20+18+16+14+12+10+8+6+4+2 = 132

let y = lim n->∞ (xⁿ/n!)

or y = x lim n->∞ (1/n) lim n->∞ x^n-1/(n-1)!

or y = 0

I think n is printed by mistake, so considering it as x

the question will be limit x tends to ∞

x{ [tan^-1 (x+1/x+4)] - (π/4)}

consider [tan^-1 (x+1/x+4)] - (π/4) = θ

so [tan^-1 (x+1/x+4)] = (π/4)+θ

so (x+1/x+4) = tan(π/4+θ)

or x = (-3-5tanθ)/2tanθ

so converted question will be limθ->0  (-3-5tanθ)θ/2tanθ

solve

I think n is printed by mistake, so considering it as x

the question will be limit x tends to ∞

x{ [tan^-1 (x+1/x+4)] - (π/4)}

consider [tan^-1 (x+1/x+4)] - (π/4) = θ

so [tan^-1 (x+1/x+4)] = (π/4)+θ

so (x+1/x+4) = tan(π/4+θ)

or x = (-3-5tanθ)/2tanθ

so converted question will be limθ->0  (-3-5tanθ)θ/2tanθ

now we know that limθ->0 tanθ/θ = limθ->0 θ/tanθ = 1

so next line will be limθ->0 (-3-5tanθ)/2 = -3/2 

We have to calculate limit n -> ∞ [{³√(n²-n³) }+ n ]

Use the formula a³+b³ = (a+b)(a²+b²-ab) in the format 

(a+b) = (a³+b³)/(a²+b²-ab)

here a = ³√(n²-n³)  and b = n

on solving we get 1/(1+1+1) = 1/3

f(x) = x|x|  => f(x) = -x²  for negative real values of x  and f(x) = x² for positive real values of x

so f^-1(x) = -√|x|  for negative real values and = +√|x| for positive real values of x

Total number of terms in the expansion = 4n-2+1 = 4n-1

for a negative real number i² is compulsory so associated terms are 3rd, 7th, 11th ..........

Total number of terms = n

sin(|x|)+|x| is differentiable at x = 0

LHD of first = {cos|0+h|+|0+h|-cos|0|-|0|}/0+h-0  = (cosh+h-1)/h = (1-2sin²h+h-1)/h = 1 (on taking limits)

now check RHD, its value will be -1

similarly in second both values are 0 so it is differentiable