## Description

Course Code : | BCS-012 |
---|---|

Course Title : | Problem Solving and Programming |

Assignment Number : | BCA(1)/012/Assignment/2019-20 |

Maximum Marks : | 100 |

Weightage : | 25% |

Last Date of Submission : | 31st October, 2021 (for July 2021 session) 15th April, 2022 (for January 2022 session) |

Solution Type : | Softcopy (PDF File) |

*Note: This assignment has 20 questions of 80 marks (each question carries equal marks). Answer all the questions. Rest 20 marks are for viva voce. Please go through the guidelines regarding assignments given in the Programme Guide for the format of presentation.*

**Q1:** Use the principle of mathematical induction to show that

2 + 2^{2} +…+ 2^{n} = 2^{n+1} –2 for every natural number n. **(4 Marks)**

**Q2:** Find the sum of all integers between 100 and 1000 which are divisible by 9. **(4 Marks)**

**Q3:** Reduce the matrix A(given below) to normal form and hence find its rank.

A =

5 3 8

0 1 1

0 1 1

**(4 Marks)**

**Q4:** Show that n(n+1) (2n+1) is a multiple of 6 for every natural number n. **(4 Marks)**

**Q5:** Find the sum of an infinite G.P. whose first term is 28 and fourth term is 4/49. **(4 Marks)**

**Q6:** Check the continuity of the function f(x) at x = 0 :

f(x) =

**(4 Marks)**

**Q7:** If y = lnx/x, show that d^{2}y/dx^{2} = 2lnx-3/x^{3} **(4 Marks)**

**Q8:** If the mid-points of the consecutive sides of a quadrilateral are joined, then show (by using vectors) that they form a parallelogram. **(4 Marks)**

**Q9:** Solve the equation 2x^{3} – 15x^{2} + 37x – 30 = 0, given that the roots of the equation are in A.P. **(4 Marks)**

**Q10:** A young child is flying a kite which is at 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 130m ? **(4 Marks)**

**Q11:** Using first derivative test, find the local maxima and minima of the function f(x) = x^{3} –12x **(4 Marks)**

**Q12:** Evaluate the integral I = x^{2}/(x+1)^{3} dx **(4 Marks)**

**Q13:** Find the scalar component of projection of the vector a = 2i + 3j + 5k on the vector b = 2i – 2j – k.**(4 Marks)**

**Q14:** If 1, ω, ω2 are cube roots unity, show that (2-ω) (2-ω^{2}) (2-ω^{10}) (2-ω^{11}) = 49. **(4 Marks)**

**Q15:** Find the length of the curve y = 3 + x/2 from (0, 3) to (2, 4). **(4 Marks)**

**Q16:** Evaluate the determinant given below, where is ω a cube root of unity. (4 Marks)

1 ω ω^{2}

ω ω^{2}1

ω^{2}1 ω ^{
}

**Q17:** Using determinant, find the area of the triangle whose vertices are (-3, 5), (3, -6) and (7, 2). **(4 Marks)**

**Q18:** Solve the following system of linear equations using Cramer’s rule:

x + y = 0, y + z = 1, z + x = 3 **(4 Marks)**

Q19:

If

A

=

1

-2

2

-1

,

B

=

a

1

b

-1

, B=

and (A + B)^{2} = A^{2} + B^{2} , Find a and b. (4 Marks)

Q20: Use De Moivre’s theorem to find (√3+i)^{3}. (4 Marks)

**All questions are solved in this assignment solution.**

**Keywords:*** BCS-012, BCS012, BCS 12, BCS-12, BCS12, BCS 12, IGNOU Solution Guide, Answer Key *

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