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Values of a parameter for which a point lies inside a region bounded by lines

This type of question, asked 2 times: 27 Jan 2024, 24 Jan 2025. All 2 questions below, oldest first.

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  1. Question 1

    27 Jan 2024

    154
    Let RR be the interior region between the lines 3x−y+1=03x - y + 1 = 0 and x+2y−5=0x + 2y - 5 = 0 containing the origin. The set of all values of aa, for which the points (a2,a+1)\left( a^{2},a + 1 \right) lie in RR, is:

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  2. Question 2

    24 Jan 2025

    133
    Let the points (112,α)\left( \frac{11}{2},\alpha \right) lie on or inside the triangle with sides x+y=11,x+2y=16x + y = 11,x + 2y = 16 and 2x+3y=292x + 3y = 29. Then the product of the smallest and the largest values of α\alpha is equal to :

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