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 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)

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

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 $\frac{5}{\left | x-3 \right |}<7.$

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 $\left | \begin{matrix} 1 &a &a^{2} \\1 &b &b^{2} \\1 &c &c^{2} \end{matrix} \right |=\left ( b-a \right )\left ( c-a \right )\left ( c-b \right ).$

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)]

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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 − 2i| = |Z + 2i|, show that Im (Z) = 0.

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

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

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:

BCS-012 June 2022 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:

BCS-012 June 2022 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:

BCS-012 June 2022 Solution

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

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

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

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

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

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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)

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

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

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

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

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

y = 3 + 2x, y = 3 − x, 0 ≤ x ≤ 3

using integration.

Solution:

BCS-012 June 2022 Solution

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

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

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

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Q4 (d): Find the angle between the lines

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

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

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 $A=\begin{bmatrix} -1 &2 &3 \\4 &5 &7 \\5 &3 &4 \end{bmatrix}$ , show that A(adj A) = O.

Solution:

BCS-012 June 2022 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 $\left | \vec{a} \right |\vec{b}+\left | \vec{b} \right |\vec{a}$ is perpendicular to $\left | \vec{a} \right |\vec{b}-\left | \vec{b} \right |\vec{a}$ , for any two non zero vectors $\vec{a}$ and $\vec{b}$ .