Wed. Aug 7th, 2024

Aug 30, 2022

# 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)

Q1 (b): If  1, ω  and  ω2  are cube roots of unity, show that  (2 − ω)(2 − ω2)(2 − ω10)(2 − ω11) = 49.

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

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

Q1 (c): Evaluate the integral  $I=\int&space;\frac{x^{2}}{\left&space;(&space;x+1&space;\right&space;)^{3}}\:&space;dx.$

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

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

Q1 (d): Solve the inequality  $\frac{5}{\left&space;|&space;x-3&space;\right&space;|}<7.$

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

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

Q1 (e): Show that  $\left&space;|&space;\begin{matrix}&space;1&space;&a&space;&a^{2}&space;\\1&space;&b&space;&b^{2}&space;\\1&space;&c&space;&c^{2}&space;\end{matrix}&space;\right&space;|=\left&space;(&space;b-a&space;\right&space;)\left&space;(&space;c-a&space;\right&space;)\left&space;(&space;c-b&space;\right&space;).$

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

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

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)

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

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

Q2 (a): Evaluate  $\lim_{x\rightarrow&space;0}\;&space;\frac{\sqrt{1+2x}-\sqrt{1-2x}}{x}\,&space;.$

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

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

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)(2n + 1)  is a multiple of  6  for every natural number  n.

Solution:

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

$f\left&space;(&space;x&space;\right&space;)=\frac{3}{4}x^{4}-8x^{3}+\frac{45}{2}x^{2}+2015\,&space;.$

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

Q3 (c): Determine the 100th term of the Harmonic progression  $\frac{1}{7},\frac{1}{15},\frac{1}{23},\frac{1}{31},\:&space;....$

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

Q3 (d): Find the length of the curve  y = 2x 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)

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

$\vec{r_{1}}=\left&space;(&space;1+\lambda&space;\right&space;)\hat{i}+\left(&space;2-\lambda&space;\right&space;)\hat{j}+\left&space;(&space;1+\lambda&space;\right&space;)\hat{k}$ and

$\vec{r_{2}}=2\left&space;(&space;1+\mu&space;\right&space;)\hat{i}+\left&space;(&space;1-\mu&space;\right&space;)\hat{j}+\left&space;(&space;-1+2\mu&space;\right&space;)\hat{k}.$

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

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

y = 3 + 2xy = 3 − x,  0 ≤ x ≤ 3

using integration.

Solution:

Q4 (c): If  $y=1+ln\left&space;(&space;x+\sqrt{x^{2}+1}&space;\right&space;)$ , prove that  $\left&space;(&space;x^{2}+1&space;\right&space;)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}=0.$

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

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

Q4 (d): Find the angle between the lines

$\vec{r_{1}}=2\hat{i}+3\hat{j}-4\hat{k}+t\left&space;(&space;\hat{i}-2\hat{j}+2\hat{k}&space;\right&space;)$  and

$\vec{r_{2}}=3\hat{i}-5\hat{k}+s\left&space;(&space;3\hat{i}-2\hat{j}+6\hat{k}&space;\right&space;).$

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

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

Q5 (a): If  $A=\begin{bmatrix}&space;-1&space;&2&space;&3&space;\\4&space;&5&space;&7&space;\\5&space;&3&space;&4&space;\end{bmatrix}$ , 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)

Q5 (c): Show that   $\left&space;|&space;\vec{a}&space;\right&space;|\vec{b}+\left&space;|&space;\vec{b}&space;\right&space;|\vec{a}$   is perpendicular to  $\left&space;|&space;\vec{a}&space;\right&space;|\vec{b}-\left&space;|&space;\vec{b}&space;\right&space;|\vec{a}$ ,  for any two non zero vectors  $\vec{a}$  and  $\vec{b}$ .

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

Q5 (d): If  $y=ln\left&space;[&space;e^{x}\left&space;(&space;\frac{x-2}{x+2}&space;\right&space;)^{\frac{3}{4}}&space;\right&space;]$ , find  $\frac{dy}{dx}$ .

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

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

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