1. The vertices of a triangle are A(3,4), B(7,2),
C(-2,-5).Find the length of the median through the vertex A.
Solution
Vertices of triangle ABC are A(3,4), B(7,2),
C(-2,-5)
Using mid-point formula, coordinates of mid point of line joining two points
= [(x1 + x2)/2, (y1 + y2)/2]
Let D be the mid point of BC.
Therefore, coordinates of D = [(7 - 2)/2, (2 - 5)/2] = (5/2, -3/2)
Now, length of median AD =  (3 - 5/2)2 + (4 + 3/2)2
 1/4 + 121/4 = 55/2 units Ans.
2. The ratio of the incomes of two persons is 9:7 and the ratio of their expenditure is 4:3. if each of them saves Rs.200/month, find their monthly income?
Solution
Let the incomes of two persons be 9x and 7x and Let their expenditures be 4y and 3y
Now, Savings = Income - Expenditure
Hence 9x - 4y =200 and 7x - 3y = 200
Solving by elimination method, we get x = 200
Therefore, Income of first person = 9x = Rs.1800 Ans
and Income of second person = 7x = Rs.1400 Ans
3. While covering a distance of 30km Shelly takes 2 hours more than Vikas.if Shelly doubles her speed, she would take 1 hour less than Vikas. find their ratios of walking?
Solution Let the speeds of shelly and vikas be x km/hr and y km/hr
Here, Distance = 30 km We know Time = distance/speed
Now, Time taken by shelly - time taken by vikas = 2 hrs.
30 / x - 30 / y = 2 and 30 / y - 30 / 2x = 1
Solving the equations x = 5 km/hr and y = 7.5 km/hr
Ratios of walking = 5 : 7.5 = 2 : 3 Ans
4. Explain why 7x11x13+13 and 7x 6x5x4x3x2x1+5 are composite numbers?
We know that composite numbers are those, which can be written in the form of product of two or more factors. In simple words, composite numbers are divisible by at least one prime number.
In case one i.e. 7x11x13+13 can also be written as
13(7x11 + 1). This proves that the given number is divisible by 13 and therefore not a prime number, hence it is a composite number.
Similarly, 7x6x5x4x3x2x1+5 = 5(7x6x4x3x2 + 1), which is divisible by 5 and therefore a composite number.
5. Without using trigonometric tables evaluate:
cos 70'
-------- + cos 57'cosec33' - 2cos 60'
sin 20'
sin (90 - 70)
---------------- + sin(90 - 57) cosec33 - 2 (1/2) [using cos 60 = 1/2]
sin 20
sin 20 1 1
-------- + sin33. ---------- - 1 [using sin A = ----------- ]
sin 20 sin 33 cosec A
1 + 1 - 1 = 1 Ans
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