1. Investigate for what values of k the equations

x + y + z = 1, 2x + y + 4z = k, 4x + y + 10z = k2 have infinite solutions.
2. Investigate for what values of λ and μ, the system of linear equations
x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ
have (i) no solution, (ii) unique solution and (iii) infinite number of solutions.
3. If the following system
ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0
has a nontrivial solution, then prove that a + b + c = 0 or a = b = c and hence find the solution in each case.
4. Find the eigenvalues and eigenvectors of the matrix A = (2 −1 1−1 2 −11 −1 2). Is A
diagonalizable? If yes, then find the matrix P such that P−1AP = D.

5. Show that the matrix A = (2 1 −22 3 −41 1 −1) is diagonalizable and find the modal matrix P such that P−1AP = D.

6. Let a 3 × 3 matrix A have eigenvalues 1, −1, 2. Find the determinant and trace of the matrix B = A − A−1 + A2.

7. Verify the Cayley-Hamilton theorem for the matrix A = (1 2 32 4 53 5 6). Find A−1andexpress A8 − 11A7 − 4A6 + A5 + A4 − 11A3 − 3A2 + 2A + I as quadratic polynomial in A.

8. Find n th derivative of the following functions:
(i) cos x cos 2x cos 3x (ii) e2x cos2 x sin x

9. If y =sin−1 x√1−x2 then prove that (1 − x2)yn+1 − (2n + 1)xyn − n2yn−1 = 0.Hence find the value of yn+1 when x = 0.
10. If Vn =ddxn(xnlog x) then show that Vn = nVn−1 + (n − 1)! .Hence show that Vn = n! (log x + 1 +12+13+ ⋯ +1n.
11. Expand the following functions using Maclaurin’s Theorem
(i) log(1 + sin x) (ii) sin−1 x (iii) log[1 − log(1 − x)].
12. Expand the function f (x21+xin powers of (x21+x).
13. Name the figure formed by the asymptotes of the curvex2y2 − a2(x2 + y2) − a3(x + y) + a4.

14. Check the convergence of the following series212x +3223x2 +4334x3 + ⋯

15. Trace the curve x2 = y2(x + 1)3.