MATHEMATICS 2024 PYQ BCA 2ND SEMESTER

MATHEMATICAL FOUNDATIONS FOR COMPUTER SCIENCE–II
Question Paper with Answers

Compulsory Question

1(a). Write integration of basic trigonometric functions.

∫ sin x dx = –cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ csc²x dx = –cot x + C
∫ sec x tan x dx = sec x + C
∫ csc x cot x dx = –csc x + C

1(b). Differentiate between grouped and ungrouped data.

Ungrouped Data	Grouped Data
Raw data, each value is listed individually.	Data organized into classes/intervals with frequencies.
Suitable for small datasets.	Suitable for large datasets.
Detailed, but harder to analyze for trends.	Easier to analyze, but loses some detail.

1(c). What is meant by correlation?

Correlation is a statistical relationship between two variables, indicating how one variable changes in relation to another. The strength and direction of this relationship are often measured by the Pearson correlation coefficient.

1(d). What do you understand by linear regression?

Linear regression is a statistical technique to model the relationship between a dependent variable and one or more independent variables using a linear equation, typically of the form y = mx + c.

UNIT–I

2. Solve the integral ∫ x² dx / (a + bx)²

Let u = a + bx, so du = b dx and x = (u – a)/b.
Substitute and simplify:

∫ x² dx / (a + bx)² =
(1/b³) [ (a + bx) – 2a ln|a + bx| – a²/(a + bx) ] + C

3. What are various ways to represent data using graphs?

• Bar Graphs
• Histograms
• Pie Charts
• Line Graphs
• Scatter Plots
• Frequency Polygons
• Box-and-Whisker Plots

UNIT–II

4. The average salary of male employees in a firm was Rs. 520 and the number of females was Rs. 420. The mean salary of all the employees was Rs. 500. Find the percentage of male and female employees.

Let number of males = M, number of females = F.
520M + 420F = 500(M + F)
⇒ 20M = 80F ⇒ M/F = 4
Total employees = 5F
Percentage of males = 80%
Percentage of females = 20%

5. Discuss the various measures of dispersion using suitable examples.

• Range: Difference between highest and lowest values.
• Variance: Average of squared differences from the mean.
• Standard Deviation: Square root of variance.
• Mean Deviation: Average of absolute differences from the mean.
• Quartile Deviation: Half the difference between Q3 and Q1.

Example: For data 2, 4, 6, 8, 10:

• Range = 8
• Mean = 6
• Variance = 8
• Standard deviation ≈ 2.83

UNIT–III

6. Calculate the coefficient of correlation between x and y:

x	22	24	25	27	21	22	23
y	41	44	45	48	40	42	44

The Pearson correlation coefficient formula is:

r = [ n∑xy – (∑x)(∑y) ] / sqrt{ [ n∑x² – (∑x)² ] [ n∑y² – (∑y)² ] }

Substitute the values from the table to compute r.

7. The ranks of the same 16 students in Mathematics and Physics are as follows... Calculate the rank correlation coefficient for proficiencies of this group in Mathematics and Physics.

Use Spearman’s rank correlation coefficient:

ρ = 1 – (6∑d²) / (n(n²–1))

Where d = difference between ranks, n = 16.
Calculate d for each pair, sum d², and substitute in the formula to get ρ.

UNIT–IV

8. Fit a straight line to the following data:

X	1	2	3	4	6	8
Y	2.4	3	3.6	4	5	6

Use least squares method for Y = a + bX:

b = [ n∑XY – (∑X)(∑Y) ] / [ n∑X² – (∑X)² ]
a = [ ∑Y – b∑X ] / n

Substitute the values to get a and b.

9. Calculate the regression coefficient and obtain the lines of regression for the following data:

X	1	2	3	4	6	7	8	9
Y	9	10	8	11	13	12	11	14

Regression coefficient (b_yx):

b_yx = ∑(x – x̄)(y – ȳ) / ∑(x – x̄)²

Regression line: Y – ȳ = byx(X – x̄)
Compute means, substitute, and write the regression equation.