C

Any intuitive explanation?
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C

soumya, what you have shown in your solution is not correct, if m is positive, then x^2 + 2x -15 will be negative according to the equation. So for m to be positive, x^2 +2x -15 has to be negative, i.e x^2 +2x -15 < 0. After factorizing the left side, it comes as (x+5)(x-3)<0. After solving this inequality, -5<x<3, so x can be 7 values. And probability of this x out of 21 is 7/21, i.e the answer is 1/3 (B).

There could be 21 values of x ranging from -10 to 10.The values of x i.e. -4,-3,-2,-1,0,1,2 (total 7 values of x) could result m greater than zero. Hence , the probability is 7/21 or 1/3.

Two ways..

(I) Also explained above..
\(x^2+2x-15=-m.....x^2+5x-3x-15=-m........(x+5)(x-3)=-m\)
Now x can take any value from -10 to 10, but m >0.

When x is -5 or 3, m is 0 as roots of quadratic equation \((x+5)(x-3)=0\) are -5 and 3.
Now if we take any value less than -5, both (x+5) and (x-3) will be negative and their product will be positive that is -m>0...m<0
Similarly, if we take any value more than 3, both (x+5) and (x-3) will be positive and their product will be positive that is -m>0...m<0

However for values between -5 and 3, (x+5) will be positive and (x-3) will be negative and their product will be negative.. Thus -m<0....m>0
So, the values -4, -3, -2, -1, 0, 1, 2 will fit in..

so 7 values out of 21 values will give a probability of \(\frac{7}{21}=\frac{1}{3}\)

(II) Next could be
\(x^2+2x-15=-m.....x^2+2x=15-m........(x)(x+2)=15-m......m=15-x(x+2)\)
So, sum of two consecutive even integers should be less than 15 for m>0..
Now, -5*-3 = 15 so x>-5 AND 3*5=15, so x<3
-5<x<3... Rest same as above for probability

B
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