1) let a >= b >= c

mean = (a+b+c) /3 = 50

a+b+c = 150

median is the middle one , b = 35

hence, a + c = 115 .

if we know one quantity we can find the other quantity.

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2. Sin Ф = 3/5 2nd quadrant.

Cos² Ф = 1 - sin² Ф = 1 - 9/25 = 16/25

cos Ф = - 4/5 we put a minus sign because in 2nd quadrant Cosine is negative.

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3. AP series: 11 , a₁ , a₂ , a₃ ..... a_n , 53

number of terms = n + 2 first term = 11

common difference : d

53 = 11 + (n+1) d

* (n+1) d = 42 = 7 * 6 = *21 * 2 = 3 * 14 = 1 * 42

*So there are 8 combinations possible. then there could be 8 different series,*

all terms are integers, means that common difference d is an integer.

n is an integer .

case 1: d = 1 and n+1 = 42 so n =41, AP: 11,12,13,14....

case 2: d = 42 and n+1 = 1 so n = 0, AP: 11, 53, .....

case 3: d = 2 and n+1 = 21 , n = 20, AP: 11,13,15,..51, 53

case 4: d = 21 and n +1 = 2, n = 1, AP : 11, 32, 53 , ....

case 5: d = 3 and n+1 = 14 , n = 13, AP : 11,14, 17,...., 50, 53

case 6: d= 14 and n+1 = 3 n = 2, AP = 11, 25, 39, 53, ...

case 7 : d = 7 and n+1 = 6 so n = 5, AP = 11,18,25,32,39,46, 53,...

case 8: d = 6, n+1 = 7,... so n = 6, AP : 11, 17,23,29,35,41,47,53,

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4. four bells: toll at intervals of 10, 15 , 20 , 30 seconds ....

the interval between two times that they will toll all togehter will be the LCM of the time intervals of each.

LCM of 10, 15 , 20 , 30 seconds = 2 * 5 * 3 * 2 = 60

So it is one minute.

They will all together at 10 : 01 AM,

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5. a x² + b x + c = 0 is the general form of quadratic equation.

sum of roots : - b/a product : c/a --- (3)

[x - (a - b) ] [ x - (b - c) ] = 0 is the quadratic equation.

=> x² + (c-a) x + (a-b)(b-c) = 0 is the quadratic equation.

** It means that product of roots is (a - b) (b - c)**

** sum of the roots = a - b + b - c = - (c - a)**

** **

- b / a = - (c - a) => b = a (c - a) ---- (1)

(a - b) (b - c) = c/a

a b - a c - b² + b c = c/a --- (2)

** given (a-b)(b-c) / (c-a) = - product / sum = - c / -b = c/b ,,,,, by using (3)**