Model Paper - II
[(Vikram Series - 2009)]
Time: 3 Hours Maximum Marks: 75
Instructions:
* Attempt all questions.
* Each question carries equal marks.
1. (a) Evaluate: $$tan^{-1} \sqrt {3}- sec^{-1}(\frac{2}{\sqrt{3}})$$
(b) Solve the equation: $$sin^{2} x+cos^{2} x=1$$
(c) Prove that: $$1+cot^{2} \theta = cosec^{2} \theta$$
2. (a) Find the equation of the straight line passing through the points (2, -3) and (5, 7).
(b) Find the slope and y-intercept of the straight line 3x - 4y = 12.
(c) Find the equation of the straight line parallel to the x-axis and passing through the point (4, -5).
3. (a) Find the derivative of the function f(x) = x^3 - 2x^2 + 3x - 4.
(b) Find the tangent to the curve y = x^2 + 4x - 5 at the point (1, 0).
(c) Find the maximum and minimum values of the function f(x) = x^2 - 4x + 3.
4. (a) Integrate the function f(x) = x^3 - 2x^2 + 3x - 4.
(b) Find the area under the curve y = x^2 between x = 0 and x = 2.
(c) Find the volume of the solid generated by revolving the region bounded by the curve y = x^2, the x-axis, and the lines x = 0 and x = 2 about the x-axis.
5. (a) Find the inverse of the matrix A = $$\begin{pmatrix} 1 & 2 \\\ 3 & 4 \end{pmatrix}$$.
(b) Solve the system of linear equations:
$$x+2y=5$$ $$3x+4y=10$$
(c) Find the eigenvalues and eigenvectors of the matrix A = $$\begin{pmatrix} 1 & 2 \\\ 3 & 4 \end{pmatrix}$$