Easy2Siksha.com
GNDU Question Paper-2024
B.A 3
rd
Semester
QUANTITATIVE TECHNIQUES
(Quantitative Technqiues-III)
Time Allowed: Three Hours Max. Marks: 100
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks
SECTION-A
1. (i) Find the first order and second order partial derivatives with respect to x and y of the
function: z=4x²y² + x + 3y².
(ii) Determine two positive numbers, whose sum is 15 and the sum of whose squares is
minimum.
2. (i) Find the maximum and minimum values of the function:
(x-1)
2
(x+3)
3
.
(ii) Divide 16 into two parts so that their product is maximum.
(iii) What is a decreasing function? Explain with the help of an example.
SECTION-B
3. (1) Find the integral of:
w.r.t.x.
Easy2Siksha.com
(ii) Evaluate: integrate
from 0 to 1
4. (i) If the market is given by p = 20 - 2x where p and x are respectively the price and
amount demanded of a commodity, find the consumer's surplus when p = 4 and p = 8
(ii) The supply curve for a commodity is given by: p =
and the quantity sold is 7
units. Find producer's surplus.
SECTION-C
5. With the help of hypothetical examples, explain the following:
(i) Transpose of a matrix
(ii) Symmetric matrix
(iii) Skew-symmetric matrix
(iv) Diagonal matrix.
6. Solve the following system of equations by Crammer's rule:
SECTION-D
7. The input-output coefficient matrix (A) and final demand vector (D) for an economy
with three sectors are given below:
A =
, D =
calculate the output level for the three sectors.
8. (i) What are the underlying assumptions of linear programming?
(ii) Solve the following linear programming problem by graphic method:
Minimize Z = 2x
1
+ 3x
2
Easy2Siksha.com
subject to the constraints:
GNDU Answer Paper-2024
B.A 3
rd
Semester
QUANTITATIVE TECHNIQUES
(Quantitative Technqiues-III)
Time Allowed: Three Hours Max. Marks: 100
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks
SECTION-A
1. (i) Find the first order and second order partial derivatives with respect to x and y of the
function: z=4x²y² + x + 3y².
(ii) Determine two positive numbers, whose sum is 15 and the sum of whose squares is
minimum.
Ans:
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
2. (i) Find the maximum and minimum values of the function:
(x-1)
2
(x+3)
3
.
(ii) Divide 16 into two parts so that their product is maximum.
(iii) What is a decreasing function? Explain with the help of an example.
Ans:
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
SECTION-B
3. (i) Find the integral of:
w.r.t.x.
(ii) Evaluate: integrate
from 0 to 1
Ans:
Easy2Siksha.com
Easy2Siksha.com
Story-Like Summary
• Part (i) is like breaking a big treasure map into smaller, manageable sections. Rewrite
the function, split it into simpler terms, and collect the “areas” piece by piece.
• Part (ii) is like walking along a path from 0 to 1 and measuring the total gain along
the way, taking both growth (positive) and decay (negative) into account.
Key Takeaways
1. Simplify first: Convert roots and fractions to exponents.
2. Integrate term by term: Power rule and exponential rule make the process
systematic.
Easy2Siksha.com
3. Definite integrals: Apply limits carefully to measure actual area.
4. Check with examples: Smaller, simpler integrals help solidify the method
4. (i) If the market is given by p = 20 - 2x where p and x are respectively the price and
amount demanded of a commodity, find the consumer's surplus when p = 4 and p = 8
(ii) The supply curve for a commodity is given by: p =
and the quantity sold is 7
units. Find producer's surplus.
Ans:
Easy2Siksha.com
Easy2Siksha.com
Higher market prices reduce consumer happiness because they pay closer to what
they were willing to pay.
Step 4: Visualization
Think of the demand curve as a downward-sloping line:
Easy2Siksha.com
Easy2Siksha.com
SECTION-C
5. With the help of hypothetical examples, explain the following:
(i) Transpose of a matrix
(ii) Symmetric matrix
(iii) Skew-symmetric matrix
(iv) Diagonal matrix.
Ans: A Walk Through the Matrix Museum
Imagine you are walking into a grand museum, but instead of paintings and sculptures, the
halls are filled with mathematical objects—matrices—each displayed like a masterpiece. A
friendly guide (that’s me!) takes you through the exhibits, explaining their beauty and
uniqueness.
Today, we’ll stop at four special exhibits:
1. The Transpose of a Matrix – the “mirror” exhibit.
2. The Symmetric Matrix – the “perfectly balanced” exhibit.
3. The Skew-Symmetric Matrix – the “opposite reflection” exhibit.
4. The Diagonal Matrix – the “minimalist masterpiece.”
Let’s begin our tour.
Exhibit 1: The Transpose of a Matrix
Our first stop is a giant mirror in the museum. On one side, you see a matrix, and on the
other, you see its reflection. This reflection is called the transpose.
Definition
The transpose of a matrix is obtained by interchanging its rows and columns.
• If the original matrix is AA, its transpose is written as ATA^T.
• The element in the i
th
row and j
th
column of A becomes the element in the j
th
row
and i
th
column of AT.
Hypothetical Example
Suppose we have a matrix:
Easy2Siksha.com
Story Analogy
Think of a classroom seating chart. If the teacher writes students’ names row by row, the
transpose is like rewriting the chart column by column. Same students, different
arrangement.
Exhibit 2: The Symmetric Matrix
Now we walk into a hall with a perfectly balanced sculpture. No matter how you look at it—
left or right, top or bottom—it looks the same. This is the symmetric matrix.
Definition
A matrix is called symmetric if it is equal to its own transpose.
A=A
T
This means the elements on one side of the main diagonal are mirror images of the other
side.
Hypothetical Example
Easy2Siksha.com
Story Analogy
Imagine folding a square paper along its diagonal. If the numbers on one side match exactly
with the numbers on the other side, you have a symmetric matrix.
Key Points
• Symmetric matrices are always square (same number of rows and columns).
• They often appear in physics, statistics, and computer science (like covariance
matrices).
Exhibit 3: The Skew-Symmetric Matrix
Next, we enter a room with a strange mirror. When you look into it, your reflection is the
opposite of you—your right hand looks like your left, but with a twist. This is the skew-
symmetric matrix.
Definition
A matrix is called skew-symmetric if its transpose is equal to the negative of the original
matrix.
A
T
=−A
This means:
• All diagonal elements must be zero.
• The elements above the diagonal are the negatives of those below it.
Hypothetical Example
Easy2Siksha.com
Story Analogy
Think of two friends who are exact opposites. If one says “yes,” the other says “no.” That’s
how the entries of a skew-symmetric matrix behave—mirror opposites across the diagonal.
Key Points
• Skew-symmetric matrices are always square.
• Diagonal entries are always zero.
• They are important in advanced mathematics, especially in vector cross products and
physics (like representing rotations).
Exhibit 4: The Diagonal Matrix
Finally, we reach a minimalist gallery. In the center is a sculpture made only of a straight line
of stones, with everything else empty. This is the diagonal matrix.
Definition
A matrix is called diagonal if all the elements outside the main diagonal are zero.
Here, only the diagonal elements (5, 7, 9) are non-zero. Everything else is zero.
Story Analogy
Think of a chessboard where only the squares from the top-left to the bottom-right corner
are occupied by pieces, and the rest are empty. That’s a diagonal matrix.
Key Points
• Diagonal matrices are always square.
• They are very easy to work with in calculations (multiplication, powers, etc.).
• Special cases:
o If all diagonal elements are 1 → Identity Matrix.
o If all diagonal elements are 0 → Zero Matrix.
Easy2Siksha.com
Comparative Glance
Let’s put all four exhibits side by side for clarity:
Why These Matrices Matter
These aren’t just abstract definitions—they appear everywhere:
• Transpose: Used in computer graphics, data science, and solving systems of
equations.
• Symmetric: Appear in statistics (covariance matrices), physics (inertia tensors), and
optimization problems.
• Skew-Symmetric: Used in mechanics, rotations, and vector algebra.
• Diagonal: Simplify computations, especially in linear algebra and eigenvalue
problems.
Conclusion: The Beauty of the Matrix Museum
As we leave the Matrix Museum, we realize that each type of matrix is like a character in a
story:
• The Transpose is the mirror, showing us a new perspective.
• The Symmetric Matrix is the
Easy2Siksha.com
6. Solve the following system of equations by Crammer's rule:
Ans:
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
SECTION-D
7. The input-output coefficient matrix (A) and final demand vector (D) for an economy
with three sectors are given below:
A =
, D =
calculate the output level for the three sectors.
Ans: The Story of Three Sectors and Their Balancing Act
Imagine a small country with only three industries—let’s call them Sector 1, Sector 2, and
Sector 3. These three sectors are like three friends who constantly exchange goods and
services with each other.
• Sector 1 produces some goods but also needs inputs from Sectors 2 and 3.
• Sector 2 produces its own goods but buys from Sectors 1 and 3.
• Sector 3 does the same, depending partly on Sectors 1 and 2.
On top of this internal exchange, they also have to meet the final demand of society—what
households, government, and exports require.
The big question is: How much should each sector produce so that everyone’s needs are
met—both the internal exchanges and the final demand?
This is exactly what the Input-Output Model of Leontief helps us calculate.
Step 1: Understanding the Data
We are given:
Input-Output Coefficient Matrix (A):
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
8. (i) What are the underlying assumptions of linear programming?
(ii) Solve the following linear programming problem by graphic method:
Minimize Z = 2x
1
+ 3x
2
subject to the constraints:
Ans:
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.”
