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MHTCET 26 Apr 2025 Shift 2 Paper

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Question : 117 of 150

Marks: +1, -0

If

p^{3}=q^{4}=r^{6}=t^{7}=s^{2}

\log _{t}(p q r s)=\ldots .

. .

$\frac{168}{5}$
28
$\frac{31}{4}$
$\frac{35}{4}$

Solution:

Concept:

Express each variable as a power of

t

using the common value

t^{7}

, then use the logarithm power rule.

Explanation:

Given

p^{3}=q^{4}=r^{6}=t^{7}=s^{2}

.

Take the common value as

t^{7}

.

So

p^{3}=t^{7}

,

q^{4}=t^{7}

,

r^{6}=t^{7}

,

s^{2}=t^{7}

.

Write each variable as a power of

t

:

p=t^{\frac{7}{3}}

,

q=t^{\frac{7}{4}}

,

r=t^{\frac{7}{6}}

,

s=t^{\frac{7}{2}}

.

Multiply them:

p q r s = t^{\frac{7}{3}} \cdot t^{\frac{7}{4}} \cdot t^{\frac{7}{6}} \cdot t^{\frac{7}{2}}

.

Add the exponents:

\frac{7}{3}+\frac{7}{4}+\frac{7}{6}+\frac{7}{2} = \frac{28+21+14+42}{12} = \frac{105}{12} = \frac{35}{4}

.

Therefore,

p q r s = t^{\frac{35}{4}}

.

Now compute the logarithm:

\log_{t}(p q r s) = \log_{t}\left(t^{\frac{35}{4}}\right) = \frac{35}{4} \log_{t} t = \frac{35}{4}

.

Answer:

\frac{35}{4}

Therefore, the correct option is D.

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