The equation X 3-4log 2 x 2-log 2 x -5-4-2 hasA- atleast one real solutionB- exactly three real solutionsC- exactly one irrational solutionD- complex roots

The correct option is B exactly three real solutionsGiven, x^3/4(log_2 x)^2+log_2 x 5/4=√(2) ⇒3/4(log_2 x)^2+log_2x 5/4=log_x √(2) ⇒3/4(log_2 x)^2+log_2 x 5/4 ...

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Standard XII

Mathematics

Higher Order Equations

The equation ...

Question

The equation $x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}$ has

atleast one real solution

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exactly three real solutions

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exactly one irrational solution

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complex roots

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Solution

The correct option is B exactly three real solutions
Given, $x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}$
$\Rightarrow \frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4} = l o g_{x} \sqrt{2} \Rightarrow \frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4} = \frac{1}{2 l o g_{2} x} \Rightarrow 3 (l o g_{2} x)^{3} + 4 (l o g_{2} x)^{2} - 5 (l o g_{2} x) - 2 = 0$
Put $l o g_{2} x = y$
$∴ 3 y^{3} + 4 y^{2} - 5 y - 2 = 0 \Rightarrow (y - 1) (y + 2) (3 y + 1) = 0 \Rightarrow y = 1, - 2, - \frac{1}{3} \Rightarrow l o g_{2} x = 1, - 2, - \frac{1}{3}$
$x = 2, \frac{1}{2^{\frac{1}{3}}}, \frac{1}{4}$

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The equation x34(log2x)2+log2x−54=√2 has

The correct option is B exactly three real solutionsGiven, x34(log2x)2+log2x−54=√2 ⇒34(log2x)2+log2x−54=logx√2⇒34(log2x)2+log2x−54=12log2x⇒3(log2x)3+4(log2x)2−5(log2x)−2=0 Put log2x=y ∴3y3+4y2−5y−2=0⇒(y−1)(y+2)(3y+1)=0⇒y=1,−2,−13⇒log2x=1,−2,−13 x=2,1213,14

The equation $x^{\frac{3}{4} (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = \sqrt{2}$ has