🏛️ UPSC Advanced Examination Practice

📌 Q13 (b) C Program to Check Whether a Given Year is a Leap Year

📝 Problem Statement

Write a C program to determine whether a given year is a leap year or not.

📖 Introduction

A leap year is a year that contains 366 days instead of 365 days. The extra day is added to the month of February.

A year is considered a leap year if it satisfies the following conditions:

• The year is divisible by 4.
• If the year is divisible by 100, then it must also be divisible by 400.

Examples: 2000 is a leap year. 1900 is not a leap year. 2024 is a leap year.

⚙️ Algorithm

1. Start the program.
2. Read the input year from the user.
3. Check if the year is divisible by 400. If true, it is a leap year.
4. Otherwise check if the year is divisible by 4 but not by 100. If true, it is a leap year.
5. Otherwise the year is not a leap year.
6. Display the result.
7. Stop the program.

💻 C Program

#include <stdio.h>

int main()
{
    int year;

    printf("Enter a year: ");
    scanf("%d", &year);

    if((year % 400 == 0) || (year % 4 == 0 && year % 100 != 0))
        printf("The year is a Leap Year");
    else
        printf("The year is Not a Leap Year");

    return 0;
}

📊 Meaning of Variables Used

• year: Stores the year entered by the user.

📌 Q13 (c) C Program to Calculate the Sum of the Expression

📝 Problem Statement

Write a C program to calculate the sum of the following series:

📖 Introduction

The given series follows a specific arithmetic pattern where:

• The numerator starts at 1 and increases by 4 each time.
• The denominator starts at 2 and increases by 4 each time.

The last term of the series is $\frac{99}{100}$. The program calculates the cumulative sum using a single loop, ensuring proper typecasting to handle floating-point division.

⚙️ Algorithm

1. Start the program.
2. Initialize the sum variable to zero (as a float).
3. Use a for loop initializing i = 1 and j = 2.
4. For each step, calculate the fraction: divide i by j.
5. Add the calculated value to the sum.
6. Increment both i and j by 4.
7. Continue the loop until the numerator i exceeds 99.
8. Display the final sum.
9. Stop the program.

💻 C Program

#include <stdio.h>

int main()
{
    int i, j;
    float sum = 0;

    /* Loop steps numerator and denominator by 4 simultaneously */
    for(i = 1, j = 2; i <= 99; i = i + 4, j = j + 4)
    {
        /* Explicitly cast 'i' to float to prevent integer division */
        sum = sum + (float)i / j;
    }

    printf("Sum of the series = %f", sum);

    return 0;
}

📊 Meaning of Variables Used

• i : Represents the numerator values (1, 5, 9, ..., 99).
• j : Represents the denominator values (2, 6, 10, ..., 100).
• sum : Stores the cumulative sum of the fractional series.

📌 Q13 (d) Explain Any Five File Operations in C

📖 Introduction

File handling in C allows programs to store data permanently in external files. This enables data to be saved, processed, and retrieved even after the program's execution is completed.

Instead of storing data temporarily in RAM (which is volatile), file operations allow direct interaction with the hard drive. The C standard library provides several functions to perform these critical operations.

⚙️ Important File Operations

1. fopen() - Open a File

This function is used to open an existing file or create a new one. It returns a file pointer that is used to access the file in subsequent operations.

• r : Read mode
• w : Write mode (overwrites existing data)
• a : Append mode (adds data to the end)

FILE *fp;
fp = fopen("data.txt", "r");

2. fclose() - Close a File

Once all operations are completed, it is mandatory to close the file to free up memory resources and ensure all buffers are flushed to the disk.

fclose(fp);

3. fprintf() - Write Formatted Data

Similar to printf(), but instead of writing to the console screen, it writes formatted text directly into the specified file.

fprintf(fp, "Student Name: %s, Marks: %d", name, marks);

4. fscanf() - Read Formatted Data

Similar to scanf(), it reads formatted data from the file instead of the standard input (keyboard).

int num;
fscanf(fp, "%d", &num);

5. fgetc() and fputc() - Character I/O

These functions are used to read (fgetc) and write (fputc) a single character from or to a file. They are often used in loops to process files character by character.

char ch;
/* Read a character */
ch = fgetc(fp);

/* Write a character */
fputc(ch, fp);

📌 Q13 (e) R Code for Vector and Matrix Operations

📝 Problem Statement

Write a set of R codes to perform the following operations:

1. Read a vector of values.
2. Create a matrix of dimensions $r_1 \times c_1$.
3. Compute the MinMax and MaxMin values.
4. Print whether these two values are equal or not.

📖 Introduction

In matrix theory, two highly important values can be computed from a matrix to determine optimal strategies or saddle points:

• MinMax : The minimum of the column maximum values.
• MaxMin : The maximum of the row minimum values.

If these two values are equal, that specific value is called the saddle point of the matrix.

To demonstrate computational proficiency, this program constructs the matrix from a given vector and computes the MinMax and MaxMin values manually using nested loops, entirely avoiding R's built-in min() and max() functions.

💻 R Program (100% Manual Implementation)

# Step 1: Input vector
v <- c(4, 7, 2, 9, 5, 6)

# Step 2: Matrix dimensions
r1 <- 2
c1 <- 3

# Step 3: Create matrix manually
m <- matrix(0, r1, c1)
k <- 1

for(i in 1:r1) {
  for(j in 1:c1) {
    m[i,j] <- v[k]
    k <- k + 1
  }
}

cat("Matrix is:\n")
print(m)

# Step 4: Compute row minimums
row_min <- numeric(r1)

for(i in 1:r1) {
  min_val <- m[i,1]
  
  for(j in 2:c1) {
    if(m[i,j] < min_val) {
      min_val <- m[i,j]
    }
  }
  row_min[i] <- min_val
}

# Step 5: Compute column maximums
col_max <- numeric(c1)

for(j in 1:c1) {
  max_val <- m[1,j]
  
  for(i in 2:r1) {
    if(m[i,j] > max_val) {
      max_val <- m[i,j]
    }
  }
  col_max[j] <- max_val
}

# Step 6: Find MinMax (Minimum of column maximums)
minmax <- col_max[1]

for(j in 2:c1) {
  if(col_max[j] < minmax) {
    minmax <- col_max[j]
  }
}

# Step 7: Find MaxMin (Maximum of row minimums)
maxmin <- row_min[1]

for(i in 2:r1) {
  if(row_min[i] > maxmin) {
    maxmin <- row_min[i]
  }
}

cat("MinMax =", minmax, "\n")
cat("MaxMin =", maxmin, "\n")

# Step 8: Check equality
if(minmax == maxmin) {
  cat("MinMax and MaxMin are Equal (Saddle Point Exists)\n")
} else {
  cat("MinMax and MaxMin are Not Equal\n")
}

📊 Meaning of Variables Used

• v : Vector containing the raw input values.
• m : Matrix formed iteratively from the vector.
• r1 / c1 : Number of rows and columns in the matrix.
• row_min : Array storing the minimum value of each row.
• col_max : Array storing the maximum value of each column.
• minmax : The minimum value among the computed column maximums.
• maxmin : The maximum value among the computed row minimums.

📚 Paper Summary & Key Focus Areas

🎯 Core Concepts Tested in This Paper

Based on the questions in this year's examination, aspirants should heavily focus on the following domains:

• Low-Level Operations in C: A deep understanding of bitwise operators (&, |, ^, ~) and shift operators is crucial. The examiner expects clarity on how these operate at the memory level.
• Control Structures & Logic Building: Problems like the Leap Year check and Series Summation test your ability to construct nested conditions and manage loop counters (especially manipulating multiple variables in a single for loop).
• File Handling in C: Fundamental file I/O operations (fopen, fclose, fprintf, fscanf) remain a high-yield topic for theory and short coding questions.
• Algorithmic Implementation in R: The paper emphasizes writing logic from scratch. You must know how to traverse and manipulate data structures like matrices using nested loops without relying on R's powerful built-in vectorization.

📌 Points to Remember

• Zero-Inbuilt Rule: Always prepare to write algorithms manually. If asked to find a maximum, minimum, or sort an array, write the explicit loop logic rather than using max(), min(), or sort().
• Data Types matter: In series summation, always ensure proper typecasting (e.g., (float)i/j) to prevent unintended integer division leading to zero or truncated results.

🎯 Try These: Practice Questions

To solidify the concepts learned in this paper, attempt the following questions using the exact same constraints (manual logic, no built-in functions):

📝 Practice Question 1 (C Programming)

Write a C program to check if a given positive integer is a power of 2 using only bitwise operators (no loops or division).

📝 Practice Question 2 (R Programming)

Write an R script to compute the transpose of a given $m \times n$ matrix. You must create an empty matrix of dimensions $n \times m$ and use nested loops to populate it. Do not use the t() function.