Instructions

For the following questions answer them individually

Question 91

The number of solutions $$(x, y, z)$$ to the equation $$x - y - z = 25$$, where x, y, and z are positive integers such that $$x\leq40,y\leq12$$, and $$z\leq12$$ is

Question 92

For how many integers n, will the inequality $$(n - 5) (n - 10) - 3(n - 2)\leq0$$ be satisfied?

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Question 93

If $$f_{1}(x)=x^{2}+11x+n$$ and $$f_{2}(x)=x$$, then the largest positive integer n for which the equation $$f_{1}(x)=f_{2}(x)$$ has two distinct real roots is

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Question 94

If $$a, b, c,$$ and $$d$$ are integers such that $$a+b+c+d=30$$ then the minimum possible value of $$(a - b)^{2} + (a - c)^{2} + (a - d)^{2}$$ is

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Question 95

Let AB, CD, EF, GH, and JK be five diameters of a circle with center at 0. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle?

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Question 96

The shortest distance of the point $$(\frac{1}{2},1)$$ from the curve y = I x -1I + I x + 1I is

Question 97

If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference is

Question 98

In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

Question 99

$$f(x) = \frac{5x+2}{3x-5}$$ and $$g(x) = x^2 - 2x - 1$$, then the value of $$g(f(f(3)))$$ is

Question 100

Let $$a_1$$, $$a_2$$,............., $$a_{3n}$$ be an arithmetic progression with $$a_1$$ = 3 and $$a_{2}$$ = 7. If $$a_1$$+ $$a_{2}$$ +...+ $$a_{3n}$$= 1830, then what is the smallest positive integer m such that m($$a_1$$+ $$a_{2}$$ +...+ $$a_n$$) > 1830?

CAT Averages, Ratios & Proportions

CAT Logarithms, Surds & Indices