Q22) Find from first principle, the derivative of (sin x) w.r.t. x

Q23) Show that the curve x = y² and xy = K cut at right angle, if 8K² = 1

Q24) Find the intervals in which the function f(x) is (a) increasing; (b) decreasing:
f(x) = 2x³ + 9x² + 12x + 20

Q25) For observations of pairs (x, y) of the variables x and y, the following results are obtained.
x = 100, y = 150, x² = 1500, xy = 1200 and n = 50. Find the equation of the line of regression of y on x. Estimate the value of y when x = 15

Q26) Using integration find the area of the region bounded between the line x = 2 and the parabola y² = 8x

Q27) Evaluate

State that property of definite which you have used

Q28) Evaluate 

dx     x(x4 + 1)

Q29) Find the value of

2   (x + 4)dx  as limit of sums 0

Q30) Solve the differential equation x(1 + y²)dx - y(1 + x²)dy = 0. Given that y = 0 when x = 1

Section C

Q31) Define the line of shortest distance between two skew lines. Find the shortest distance and the vector equation of the line of shortest distance between the lines given by:
= (8 + 3) - (9 + 16) + (10 + 7)
and = 15 + 29 + 5 + (3 + 8 - 5)

Q32) Solve the following system of equation using matrix method
3x - 4y + 2z = -1
2x + 3y + 5z = 7
x + z = 2

Q33) Solve the differential equation (x + y)dy + (x - y)dx = 0, given that y = 1 when x = 1

Q34) A wire of length 25 m is to be cut in to two pieces one of the pieces is to be made into a square and other into a circle. What should be the lengths of the two pieces so that the combined area of square and the circle is minimum?

Q35) Calculate the co-efficient of correlation between x and  y

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