BCA Semester 1 – Mathematics I – Unit 3: Sequence and Series

Question 1

Find the Maclaurin series of the function $f(x)=sin⁡x.$

Marks: 5

Year: 2019 Final TU FOHSS

Given function: $ f(x) = \sin x $ Maclaurin series formula: $ f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!} f''(0) + \dfrac{x^3}{3!} f'''(0) + \cdots $ Derivatives of $ f(x) $: \( f(x) = \sin x \Right

Question 2

If H is the harmonic mean between a and b, prove that: \[\frac{1}{H−a}+\frac{1}{H−b}=\frac{1}{a+b} \]

Marks: 5

Year: 2020 Final TU FOHSS

$\textbf{Given:} $ $H$ is the harmonic mean between $a$ and $b$. $\textbf{By definition of harmonic mean:} $ \[ H = \frac{2ab}{a + b} \] $\textbf{To prove:} $ \[ \frac{1}{H - a} + \frac{1}{H - b}

Question 3

A class consists of boys whose ages are in A.P. (common difference = $4$ months). The youngest boy is $8$ years old, and the total age of the class is $168$ years. Find the number of boys.

Marks: 5

Year: 2021 Final TU FOHSS

$\textbf{Given:}$ $\text{A class consists of boys whose ages are in A.P. with common difference } d = 4 \text{ months } = \frac{1}{3} \text{ years.}$ $\text{Youngest boy's age } a = 8 \text{ years

Question 4

a) Three numbers in A.P. sum to $15$. If $1, 3, 9$ are added to them respectively, they form a G.P. Find the original numbers. b) Find the sum to $n$ terms of the series \[ \frac{1}{2} + \fr

Marks: 10

Year: 2021 Final TU FOHSS

$\textbf{Problem a:}$ Three numbers in A.P. sum to $15$. If $1, 3, 9$ are added to them respectively, they form a G.P. Find the original numbers. $\textbf{Step 1:}$ Let the three numbers in A.P. be $a

Question 5

If $A$ is the A.M. and $H$ is the H.M. between two numbers $a$ and $b$, show that: \[ \frac{a - A}{a - H} \times \frac{b - A}{b - H} = AH \]

Marks: 5

Year: 2022 Final TU FOHSS

$\textbf{Given:}$ $A$ is the arithmetic mean (A.M.) and $H$ is the harmonic mean (H.M.) of two numbers $a$ and $b$. $\textbf{Step 1:}$ Recall the formulas: \[ A = \frac{a + b}{2}, \quad H = \fra

Question 6

a) Find the Taylor series expansion of $ f(x) = x^3 - 2x + 4 $ at $a = 2$. b) In how many ways can the letters of the word $\text{'CALCULUS'}$ be arranged so that the two $\text{C'}$s do not c

Marks: 10

Year: 2022 Final TU FOHSS

$\textbf{Problem a:}$ Find the Taylor series expansion of \[ \quad f(x) = x^3 - 2x + 4 \quad \text{at } a = 2 \] $\textbf{Step 1:}$ Recall the Taylor series formula: \[ \quad f(x) = f(a) + f'(a)(x

Question 7

Expand $e^x$ about $x = 0$ using the Maclaurin series.

Marks: 5

Year: 2023 Final TU FOHSS

$\textbf{Given function:}$ \[ f(x) = e^x \] $\textbf{Step 1:}$ Recall the Maclaurin series formula: \[ f(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots =

Question 8

If $a, b, c, d$ are in G.P., prove that$ a^2 - b^2, \; b^2 - c^2, \; c^2 - d^2 $ are also in G.P.

Marks: 5

Year: 2024 Final TU FOHSS

$\textbf{Given:}$ $a, b, c, d$ are in G.P. $\textbf{Step 1:}$ Let the common ratio of the G.P. be $r$: \[ b = ar, \quad c = ar^2, \quad d = ar^3 \] $\textbf{Step 2:}$ Consider the terms: \[ a^2

Question 9

a) Find the Maclaurin series of the function: $ f(x) = \cos x $\ b) Take any matrix of order $3 \times 3$ and express it as a sum of a symmetric and a skew-symmetric matrix.

Marks: 10

Year: 2024 Final TU FOHSS

$\textbf{Problem a:}$ Find the Maclaurin series of $f(x) = \cos x$. $\textbf{Step 1:}$ Recall Maclaurin series formula: \[ f(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3