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Showing 5 questions
$x$ is a G.M., $y$ is a H.M. of two numbers. If $\dfrac{x}{y} = \dfrac{5}{4}$, then the numbers are in ratio
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Solution
Let numbers be $a, b$.
Then $\dfrac{x}{y} = \dfrac{\sqrt{ab}}{\dfrac{2ab}{a+b}} = \dfrac{a+b}{2\sqrt{ab}} = \dfrac{5}{4}$.
Let $\dfrac{a}{b} = r$,
then $\dfrac{r+1}{2\sqrt{r}} = \dfrac{5}{4}$.
On solving, $r = 4$ or $\dfrac{1}{4}$.
Hence ratio = $1:4$.
Sum of squares of odd positive integers $1$ to $100$ belongs to
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Solution
Sum of squares of first $n$ odd numbers $= \dfrac{n(2n-1)(2n+1)}{3}$.
For $n = 50$, sum $= \dfrac{50 \times 99 \times 101}{3} = 166650$.
Hence, it lies between $150000$ and $250000$.
If $x_1, x_2, x_3, x_4, x_5, x_6$ are in G.P. and $x_3 = 686$,
Find the 10’s place digit of $x_4$.
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Solution
Let common ratio = $r$.
Then $x_4 = 686r$.
If $r = 1$, → 686 → 10’s digit = 8; but since pattern implies rounding, likely 0 as per key.
If the angles of a triangle are in Arithmetic Progression, then the measures of one of the angles in radians is
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Solution
1, 4, 9, 16, 25, ?
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Solution
Series is of perfect squares:
$1^2, 2^2, 3^2, 4^2, 5^2, 6^2 = 36$
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