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Maths
Section – A
Q.1. Find | AB |, if A =
Solution:
Q.2. Differentiate 
Solution:
Section – B
Q.6. If A = and A3 =
Solution:
Q.12. Find the general solution of the differential equation 
Solution:
Section – C
Q.21. If (a + bx)ey/x = x, then prove that
Solution: We have
Q.22. The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of its edge is 12 cm?
Solution: Let x be the length of side, V be the volume and S be the surface area of cube.
Then, V = x3 and S = 6x2, where x is a function of time t
Q.23. Find the cartesian and vector equations of the plane passing through the point A(2, 5, – 3), B(- 2, 3, 5) and C(5, 3, – 3).
Solution: We know that the general equation of the plane passing through three points (x1, y1, z1) (x1, y1, Z1), (x3, y3, z3)
Section – D
Q.24. Find the point on the curve y2 = 4x, which is nearest to the point (2, – 8).
Solution: Given curve is of the form, y2 = Ax and let p(x, y) is a point on the curve which is nearest to the point (2, – 8).
Q.25. Find as the limit of sums 
Solution: We have
OR
Q.24. Using integration, find the area of the triangular region whose sides have the equation y = 2x + 1, y = 3x + 1 and x = 4.
Solution: The equations of sides of triangle are
y = 2x + 1, …(i)
y = 3x + 1 …(ii)
and x = 4 …(iii)
The equation y = 2x + 1 meets x and y axes at 
Solving equation (i), (ii) and (iii) in pairs, we obtain the coordinates of vertices of ∆ABC are A(0, 1), B (4, 13) and C (4, 9).
Then, area of ∆ABC = Area (OLBAO) – Area (OLCAO)
Maths
SET-III
Section – A
Q.1.Find the differenital equation representing the family of curves y = ae2x + 5, where a is an arbitrary constant.
Solution:
Given, y = ae2x + 5
On differentiating w.r.t. x, we get
Q.2. If y = cos√3x, then find dy/dx.
Solution:
Section – B
Q.5. Show that the points 
Solution:
Q.6. Find:
Solution:
Section – C
Q.13. Solve for x: tan-1 + tan-1(x – 1) = tan-1(8/31).
Solution:
Q.14. If x = aet (sin t + cost t) and y = aet (sin t – cos t), then prove that
dy/dx= x+y/x-y.
Solution:
dy/dx= x+y/x-y Hence proved.
OR
Q.14. Differentiate xsinx + (sinx)cosx with respect to x.
Solution:
Q.15. Find:
Solution:
Section – D
Q.24.
Show that for the matrix A =
Solution:
OR
Q.24. Using matrix method, solve the following system of equations:
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
Solution:
The given equations are
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
These equations can be written in the form AX = B, where
Q.26. A bag contains 5 red and 4 black balls, a second bag contains 3 red and 6 black balls. One of the two bags is selected at random and two balls are drawn at random (without replacement) both of which are found to be red. Find the probability that the balls are drawn from the second bag.
Solution:
Let E1 be the event of choosing the bag I, E2 be the event of choosing the bag II and A be the event of drawing a red ball.
Then,
