# CAT 2021 Set-3 | Quantitative Aptitude | Question: 3

1 vote
66 views

For a real number $a,$ if $\dfrac{\log_{15}a + \log_{32}a}{(\log_{15}a)(\log_{32}a)} = 4$ then $a$ must lie in the range

1. $a>5$
2. $3<a<4$
3. $4<a<5$
4. $2<a<3$

retagged

1 vote

Given that, $\dfrac{\log_{15}{a} + \log_{32}{a}}{(\log_{15}{a})(\log_{32}{a})} = 4$

$\Rightarrow \log_{15}{a} + \log_{32}{a} = 4\left[(\log_{15}{a})(\log_{32}{a})\right]$

$\Rightarrow \dfrac{\log_{c}{a}}{\log_{c}{15}} = 4 \times \dfrac{\log_{c}{a}}{\log_{c}{15}} \times \dfrac{\log_{c}{a}}{\log_{c}{32}}$

$\Rightarrow \require{cancel} \cancel{\log_{c}{a}} \left[\dfrac{\log_{c}{32} + \log_{c}{15}}{\cancel{(\log_{c}{15})} \cdot \cancel{(\log_{c}{32})}}\right] = 4 \times \dfrac{\cancel{\log_{c}{a}}}{\cancel{\log_{c}{15}}} \times \dfrac{\log_{c}{a}}{\cancel{\log_{c}{32}}}$

$\Rightarrow \log_{c}{32} + \log_{c}{15} = 4\log_{c}{a}$

$\Rightarrow \log_{c}{480} = \log_{c}{a}^{4}$

$\Rightarrow \boxed{a^{4} = 480}$

We know that,

• $4^{4} = 256$
• $5^{4} = 625$

$\therefore \boxed{4<a<5}$

Correct Answer $:\text{C}$

10.1k points 4 8 30
edited

## Related questions

1 vote
1
65 views
Consider a sequence of real numbers $x_{1}, x_{2}, x_{3}, \dots$ such that $x_{n+1} = x_{n} + n – 1$ for all $n \geq 1.$ If $x_{1} = -1$ then $x_{100}$ is equal to $4950$ $4850$ $4849$ $4949$
1 vote
2
130 views
Anil can paint a house in $12 \; \text{days}$ while Barun can paint it in $16 \; \text{days}.$ Anil, Barun, and Chandu undertake to paint the house for $₹ \; 24000$ and the three of them together complete the painting in $6 \; \text{days}.$ If Chandu is paid in proportion to the work done by him, then the amount in $\text{INR}$ received by him is
1 vote
In a triangle $\text{ABC}, \angle \text{BCA} = 50^{\circ}. \text{D}$ and $\text{E}$ are points on $\text{AB}$ and $\text{AC},$ respectively, such that $\text{AD = DE}.$ If $\text{F}$ is a point on $\text{BC}$ such that $\text{BD = DF},$ then $\angle \text{FDE, in degrees},$ is equal to $96$ $72$ $80$ $100$
Bank $\text{A}$ offers $6 \%$ interest rate per annum compounded half yearly. Bank $\text{B}$ and Bank $\text{C}$ offer simple interest but the annual interest rate offered by Bank $\text{C}$ is twice that of Bank $\text{B}.$ ... same amount in Bank $\text{A}$ for one year. The interest accrued, in $\text{INR},$ to Rupa is $3436$ $2436$ $2346$ $1436$
Let $\text{ABCD}$ be a parallelogram. The lengths of the side $\text{AD}$ and the diagonal $\text{AC}$ are $10 \; \text{cm}$ and $20 \; \text{cm},$ respectively. If the angle $\angle \text{ADC}$ is equal to $30^{\circ}$ then the area of the parallelogram, in sq. cm, is $\frac{25(\sqrt{5} + \sqrt{15})}{2}$ $25 (\sqrt{5} + \sqrt{15})$ $\frac{25 (\sqrt{3} + \sqrt{15})}{2}$ $25 (\sqrt{3} + \sqrt{15})$