Pre RMO Practice Sheet 2

Question 1:

A man walks a certain distance and rides back in 3.75 hours, he could ride both ways in 2.5 hours. How many hours would it take him to walk both ways?

Answer 1:

Let the man walk distance x km in time t hours. His walking speed is \frac{x}{t} kmph.

Let him ride distance x km in time T hours. His riding speed is \frac{x}{T} kmph.

First journey:

\frac{x}{t} + \frac{x}{T} = 3.75

Second journey:

\frac{2x}{T} = 2.5

Hence, \frac{x}{T} = 1.25

Using above in first equation: \frac{x}{t} + 1.25 = 3.75

Hence, \frac{x}{t} = 2.50 = his speed of walking. Hence, it would take him 5 hours to walk both ways.

Question 2:

Positive integers a and b are such that a+b=\frac{a}{b} + \frac{b}{a}. What is the value of a^{2}+b^{2}?

Answer 2:

Given that a and b are positive integers.

Given also \frac{a}{b} + \frac{b}{a} = a+b.

Hence, a^{2}(b-1)+b^{2}(a-1)=0

As a and b are both positive integers, so a-1 and b-1 both are non-negative.

So, both the terms are non-negative and hence, sum is zero if both are zero or a=1 and b=1.

Hence, a^{2}+b^{2}=2

Question 3:

The equations x^{2}-4x+k=0 and x^{2}+kx-4=0 where k is a real number, have exactly one common root. What is the value of k?

Answer 3:

x^{2}-4x+k=0…equation I

x^{2}+kx-4=0…equation II

Let k \in \Re and let \alpha is a common root.

Hence, \alpha satisfies both the equations. So, by plugging in the value of \alpha we get the following:

\alpha^{2} -4\alpha + k =0 and x^{2}+kx-4=0 and so using these two equations, we get the following:

(k+4)(1-\alpha)=0.

Case 1: \alpha \neq  1 then k=-4. But hold on, we havent’t checked thoroughly if this is the real answer. We got to check now if both equations with these values of alpha and k have only one common root.

Equation I now goes as : x^{2}-4x-4=0 so this equation has irrational roots. On further examination, we see that if \alpha=1, then k can be any value. So, what are the conditions on k? We get that from equation I: plug in the value of alpha:

1-4+k=0 so k-3=0 and k=3.

So, we now recheck if both equations have only one common root when alpha is 1 and k is 3:

x^{2}-4x+3=0 so the roots of first equation are 3 and 1.

x^{2}+3x-4=0 so the roots of second equation are -4 and 1.

Clearly so k=3 is the final answer. 🙂

Cheers,

Nalin Pithwa

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