Pankaj S.

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Trigonometry

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Question:

sin^2 1 + sin^2 5 + sin^2 9 +..........+ sin^2 89. Find the sum.

Pankaj S.

Answer:

apply arithmetic equation, l = a + (n-1 )d 89=1 +(n-1)4 so, n=23/2 thus , the sum is equal to n, that is 23/2

Geometry

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Question:

If 10, 12 and 'x' are sides of an acute-angled triangle, how many integer values of 'x' are possible?

Pankaj S.

Answer:

For an acute-angled triangle, the square of the LONGEST side MUST BE LESS than the sum of squares of the other two sides. If 'a', 'b', and 'l' are the 3 sides of an acute triangle where 'l' is the longest side then l^2 < a^2 + b^2 Possibilities in scenario 1: When x ≤ 12, let us find the number of values for x that will satisfy the inequality 122 < 10^2 + x^2 i.e., 144 < 100 + x^2 The least integer value of x that satisfies this inequality is 7. The inequality will hold true for x = 7, 8, 9, 10, 11, and 12. i.e., 6 values. Possibilities in scenario 2: When x > 12, x is the longest side. Let us count the number of values of x that will satisfy the inequality x^2 < 10^2 + 12^2 i.e., x^2 < 244 x = 13, 14, and 15 satisfy the inequality. That is 3 more values. Hence, the values of x for which 10, 12, and x will form sides of an acute triangle are x = 7, 8, 9, 10, 11, 12, 13, 14, 15. A total of 9 values.

Algebra

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Question:

x= (sqrt 87 - sqrt 17)/(sqrt92 - sqrt22), y = (sqrt 87 - sqrt 17)/(sqrt 92 + sqrt 22) find 1/(x^2+1) + 1/(y^2+1)

Pankaj S.

Answer:

whenever a*b =1 , then 1/1+a^2 + 1/1+b^2=1 so, in this case, x*y=1 x*y = 87-17/ 92-22 = 70/70 =1 thus, 1/(x^2+1) + 1/(y^2+1) =1

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