BCS-12.pdf

No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination June, 2022 BCS-012 : BASIC MATHEMATICS Time : 3 hours Ma...

No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination June, 2022 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the remaining questions. 1. (a) Solve the following system of linear equations using Cramer’s rule : 5 x + y = 0; y + z = 1; z + x = 3 (b) If 1,  and 2 are cube roots of unity, show that (2 – ) (2 – 2) (2 – 10) (2 – 11) = 49. 5 x2 (c) Evaluate the integral I =  (x  1)3 dx. 5 5 (d) Solve the inequality < 7. 5 | x  3| BCS-012 1 P.T.O.  1 a a2 (e) Show that 1 b b2 = (b – a) (c – a) (c – b). 5 2 1 c c (f) Find the quadratic equation whose roots are (2 – 3 ) and (2 + 3 ). 5 (g) Find the sum of an Infinite G.P., whose 4 first term is 28 and fourth term is. 5 49 (h) If z is a complex number such that |z – 2i| = |z + 2i|, show that Im(z) = 0. 5 1  2x  1  2x 2. (a) Evaluate Lim. 5 x0 x (b) Prove that the three medians of a triangle meet at a point called centroid of the triangle which divides each of the medians in the ratio 2 : 1. 7 (c) A young child is flying a kite which is at a height of 50 m. The wind is carrying the kite horizontally away from the child at a speed of 6·5 m/s. How fast must the kite string be let out when the string is 130 m ? 8 3. (a) Using Principle of Mathematical Induction, show that n(n + 1) (2n + 1) is a multiple of 6 for every natural number n. 5 BCS-012 2  (b) Find the points of local minima and local maxima for 3 4 45 2 f(x) = x – 8x3 + x + 2015. 5 4 2 (c) Determine the 100th term of the Harmonic 1 1 1 1 Progression , , , ,.... 5 7 15 23 31 (d) Find the length of the curve y = 2x3/2 from the point (1, 2) to (4, 16). 5 4. (a) Determine the shortest distance between ^ ^ ^ r1 = (1 + ) i + (2 – ) j + (1 + ) k and ^ ^ ^ r2 = 2(1 + ) i + (1 – ) j + (– 1 + 2) k. 5 (b) Find the area lying between two curves y = 3 + 2x, y = 3 – x, 0  x  3, using integration. 5 (c) If y = 1 + ln (x + x 2  1 ), prove that d 2y dy (x2 + 1) 2 x = 0. 5 dx dx (d) Find the angle between the lines ^ ^ ^ ^ ^ ^ r1 = 2 i + 3 j – 4 k + t ( i – 2 j + 2 k ) and ^ ^ ^ ^ ^ r2 = 3 i – 5 k + s (3 i – 2 j + 6 k ). 5 BCS-012 3 P.T.O.   1 2 3   5. (a) If A =  4 5 7  , show that A(adj A) = 0. 5    5 3 4 (b) Use De-Moivre’s theorem to find ( 3 + i)3. 5 (c) Show that |a |b +|b |a is perpendicular to | a | b – | b | a , for any two non-zero vectors a and b. 5  3/4  x  2 (d) If y = ln e x    , find dy. 5   x  2   dx   BCS-012 4