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Equation of a Plane : Intercept Form

Let a denote ...

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Let a denote the number of non negative values of p for which the equation p(2^x) + 2^-x = 5. Find the value of a.

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a

denote the number of non-negative values of

p

for which the equation

p . 2^{x} + 2^{2 - x} = 5

passes a unique solution.

\frac{1}{a}, \frac{1}{α_{1}}, \frac{1}{α_{2}}, . . ., \frac{1}{α_{20}}, \frac{1}{6}

1, β_{1}, β_{2}, . . . ., β_{20}, 6

are in AP. Then, the number of digits in value of

α_{18} β_{3}

Q. For what value of

p

does the equation

p 2^{x} + 2^{- x} = 5

possess a unique solution.

A

denote the number of integers satisfying the inequality

3 \sqrt{6 + x - x^{2}} > 4 x - 2

B

denote the value of

a

for which the minimum value of the function

P (x) = x^{2} - 4 a x - a^{4}

assume the greatest value.

f (x) = x^{n}

n

being a non - negative integer . The number of values of

n

f^{'} (p + q) = f^{'} (p) + f^{'} (q)

is valid for all

p, q > 0

Q. Number of non-negative integral values of

^{'} k^{'}

for which roots of the equation

x^{2} + 6 x + k = 0

are rational is-

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