# BCS-012 June 2022 Solution

**Q1 (a): **Solve the following system of linear equations using Cramer’s rule:

*x* + *y* = 0; *y* + *z* = 1; *z* + *x* = 3

Solution: Refer to* BCS-12 June 2021 Part 5 [Q4 (c)]*

Time Stamp : 13:57 (Must see the description)

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**Q1 (b): **If 1, ω and ω^{2} are cube roots of unity, show that (2 − ω)(2 − ω^{2})(2 − ω^{10})(2 − ω^{11}) = 49.

Solution: Refer to* BCS-12 December 2020 Part 4 [Q3 (b)]*

Time Stamp : 13:21 (Must see the description)

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**Q1 (c): **Evaluate the integral

Solution: Refer to* BCS-12 June 2021 Part 2 [Q1 (h)]*

Time Stamp : 49:03 (Must see the description)

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**Q1 (d): **Solve the inequality

Solution: Refer to* BCS-12 June 2021 Part 2 [Q1 (g)]*

Time Stamp : 33:23 (Must see the description)

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**Q1 (e): **Show that

Solution: Refer to* BCS-12 December 2020 Part 1 [Q1 (a)]*

Time Stamp : 00:16 (Must see the description)

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**Q1 (f): **Find the quadratic equation whose roots are (2 − √3) and (2 + √3).

Solution: Refer to* BCS-12 December 2021 [Q1 (g)]*

**Q1 (g): **Find the sum of an infinite G.P. whose first term is 28 and fourth term is 4/49.

Solution: Refer to* BCS-12 December 2020 Part 4 [Q3 (d)]*

Time Stamp : 29:34 (Must see the description)

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**Q1 (h): **If * Z* is a complex number such that |*Z* − 2*i*| = |*Z* + 2*i*|, show that *Im* (*Z*) = 0.

Solution: Refer to* BCS-12 December 2021 [Q1 (c)]*

**Q2 (a): **Evaluate

Solution: Refer to* BCS-12 December 2020 Part 2 [Q1 (d)]*

Time Stamp : 00:16 (Must see the description)

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**Q2 (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 of 2 : 1.

Solution:

**Q2 (c): **A young child is flying 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 ?

Solution:

**Q3 (a): **Using Principle of Mathematical Induction, show that *n*(*n* + 1)(2*n* + 1) is a multiple of **6** for every natural number * n*.

Solution:

**Q3 (b): **Find the points of local minima and local maxima for

Solution: Refer to* BCS-12 December 2021 [Q2 (b)]*

**Q3 (c): **Determine the 100^{th} term of the Harmonic progression

Solution: Refer to* BCS-12 December 2021 [Q3 (c)]*

**Q3 (d): **Find the length of the curve *y* = 2*x *^{3/2} from (1, 2) to (4, 16).

Solution: Refer to* BCS-12 December 2020 Part 5 [Q4 (b)]*

Time Stamp : 17:58 (Must see the description)

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**Q4 (a): **Determine the shortest distance between the lines

and

Solution: Refer to* BCS-12 December 2021 [Q2 (a)]*

**Q4 (b): **Find the area lying between two curves

*y* = 3 + 2*x*, *y* = 3 − *x*, 0 ≤ *x* ≤ 3

using integration.

Solution:

**Q4 (c): **If , prove that

Solution: Refer to* BCS-12 June 2021 Part 2 [Q1 (e)]*

Time Stamp : 07:30 (Must see the description)

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**Q4 (d): **Find the angle between the lines

and

Solution: Refer to* BCS-12 June 2021 Part 1 [Q1 (c)]*

Time Stamp : 23:18 (Must see the description)

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**Q5 (a): **If , show that *A*(adj *A*) = O.

Solution:

**Q5 (b): **Use De Moivre’s theorem to find (√3 + 1)^{3}.

Solution: Refer to* BCS-12 June 2021 Part 3 [Q2 (d)]*

Time Stamp : 28:24 (Must see the description)

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**Q5 (c): **Show that is perpendicular to , for any two non zero vectors and .

Solution: Refer to* BCS-12 December 2021 [Q1 (d)]*

**Q5 (d): **If , find .

Solution: Refer to* BCS-12 June 2021 Part 5 [Q4 (d)]*

Time Stamp : 25:48 (Must see the description)

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