BCS-012 Solved Assignment with solved question paper

BCS-012

1.         Use Cramer’s Rule to solve the system of linear equation given below 2x – y + 3z = 0; x + 5y – 7z = 0; x – 6y + 10z = 0

2.         Find the inverse of  and verify that A^-1A = I₃.

3.         Solve the system of equation using matrix method 2x – y + 3z = 5; 3x + 2y – z = 7; 4x + 5y – 5z = 9.

4.         Reduce the Matrix to triangular form & hence determine its rank .

5.         Show that x(x + 1) (2x + 1) is a multiple of 6 for every natural number x.

6.         Find the sum of the series 1² + 3² + 5² + …. + (2n – 1)².

7.         Prove that (2 – w) (2 – w²) (2 – w¹⁰) (2 – w¹¹) = 49 where w², w . 1 are cube root of unity.

8.         If α, β, γ are roots of equation x³ + px + q = 0. Then show that (α⁷ + β⁷ + γ⁷) =  (α² + β² + γ²)  (α⁵ + β⁵ + γ⁵).

9.         Solve the inequality    and graph its solution.

10.      Show that f(x) = | x | is continuous at x = 0.

11.      Find derivative of the following (i) x² e^x (ii) ln x/x

12.      If Y = ln , Prove that .

13.      If a camphor ball evaporates at a rate proportional to its surface area 4πr² . Show that its radius decreases at a constant rate.

14.      Determine the intervals in which the function f(x) = e ^1/x . (x ≠ 0) is increasing or decreasing.

15.      Find local maximum and local minimum values for f(x) = x³ – 6x² + 9x +1. (xЄR).

16.      Evaluate the integral

            (i)     

            (ii)   I = ò x³ (log x)² dx

17.      Find the area bounded by curves Y = and Y = x.

18.      Find the length of the curve Y = 2x + 3.

19.      Prove that the straight line joining the mid points of two non parallel sides of a trapezium is parallel to the parallel sides and half of their sum.

20.      Find maximum values of 5 x + 2y, subject to the following constraints. – 2x – 3y ≤ – 6 ; x – 2y ≤ 2 ; 6x + 4y ≤ 24 ; – 3x + 2y ≤ 3; x ≥ 0, y ≥ 0.

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