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GNDU Question Paper-2021
B.A 1
st
Semester
QUANTITATIVE TECHNIQUES-I
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. (1) Solve
(ii) Solve:
(iii) Demand equations are given as p2 + q
2
= 20 and 2p + q = 8 respectively where p is the
price and q is the quantity. Find equilibrium price and quantity.
2. (i) Insert 6 Arithmetic Means between 3 and 24.
(ii) Which term of the series 1 + 1/2 + 1/4 + 1/8 1/512
(iii) Find sum up to n terms of the series
SECTION-B
3. (i) Prove that y = 5x - 7 and 2y = 10x + 5 are parallel.
(ii) Find the equation of st. line which passes through (1, 3, 5) and sum of its intercepts on
the coordinates axes is 9.
(ii) The demand for milk is given by:
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Price Rs/Liter
Demand in Liters
1
100
2
50
3
0
Find linear demand function.
4. (i) Define sets. Explain various types of sets.
(ii) Explain union, intersection, difference and symmetric difference of sets.
(iii) A class of 70 students, out of which 30 have Maths and 20 have taken Maths but not
Statistics. Find no. of students who have taken Maths and Statistics and those who have
taken Statistics but not Maths.
SECTION-C
5. (1) Explain the concept of function and various types of functions.
(ii) Prove that
6. (i) Distinguish between a continuous function and discontinuous function.
(ii) Prove that
is continuous at x = 2
SECTION-D
7. (i) Differentiate (7x - 8)
4
(5x - 1)
3
w.r.t. x.
(ii) Find the derivative of
w.r.t. x.
(iii) Differentiate
wr.t.x.
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8. (i) Given the demand function p = 50 - 3q find elasticity of demand p = 5
(ii) Prove that elasticity of demand function is =
where demand p = 50 - 3x and p = 5
(iii) Given the total cost function C = 60 - 12q + 2q
2
find AC, MC and show that slope of AC
=
(MC - AC)
GNDU Answer Paper-2021
B.A 1
st
Semester
QUANTITATIVE TECHNIQUES-I
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. (1) Solve
(ii) Solve:
(iii) Demand equations are given as p2 + q
2
= 20 and 2p + q = 8 respectively where p is the
price and q is the quantity. Find equilibrium price and quantity.
Ans: A Gentle Beginning
Imagine one fine morning in a peaceful village, three friends Aryan, Bella, and Zoya went
to their local maths tutor under a big banyan tree. They had just returned from school, and
their heads were spinning with numbers, symbols, and equations.
Aryan was the most puzzled. Sir, he said, I got this question in our assignment. It's like
solving three puzzles at once!
Their tutor, wise and kind, smiled and said, Lets solve these puzzles together, one by one.
Remember, math is like untying a knot one small thread at a time.
And thus, began their step-by-step adventure into solving three different types of
mathematical problems. Lets walk along with them and learn the same way slow, clear,
and with purpose.
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(i) Solving the System of Equations:
We are given the following:
Let us take each equation and make things easier by inverting both sides. This trick is often
useful when the variable expression is in the denominator.
Step 1: Use the Identity
Recall a helpful identity:
Lets use this logic for the equations.
So,
Lets name:
A = 1/x
B = 1/y
C = 1/z
Now substitute:
A + B = 9 (i)
A + C = 11 (ii)
C + A = 10 (iii)
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Step 2: Subtract the Equations
Lets subtract (ii) (iii):
Oops! This seems confusing both (ii) and (iii) are about A + C, but they give different
results: 11 and 10. That means one of these must be different.
Lets double-check:
From original:
Wait! But both (2) and (3) are the same in value, just written differently.
So how can they be different? Thats not possible unless there's a mistake in question or
value.
Lets fix this. Lets suppose:
Lets take:
A = 1/x
B = 1/y
C = 1/z
From:
A + B = 9 → (1)
A + C = 11 → (2)
C + A = 10 → (3)
Now clearly, (2) and (3) say:
A + C = 11
A + C = 10
Contradiction again.
So maybe the third equation is supposed to be something else. But in such questions, two
equations should be different. Let's assume there's a typo, and proceed with any two:
From:
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A + B = 9 (1)
A + C = 11 (2)
Now subtract (2) (1):
(A + C) (A + B) = 11 9
C B = 2 B = C 2
Now from (1):
A + B = 9 A + (C 2) = 9 A + C = 11
This matches equation (2), so good!
Lets assign a value:
Suppose A = 4 (just guessing),
then C = 7 (from A + C = 11),
then B = C 2 = 5
So we have:
A = 1/x = 4 → x = 1/4
B = 1/y = 5 → y = 1/5
C = 1/z = 7 → z = 1/7
Now plug back into the original equations and verify:
xy / (x + y) = (1/4 * 1/5) / (1/4 + 1/5)
= (1/20) / (9/20) = 1/9
xz / (x + z) = (1/4 * 1/7) / (1/4 + 1/7)
= (1/28) / (11/28) = 1/11
zx / (z + x) = (1/7 * 1/4) / (1/7 + 1/4)
= (1/28) / (11/28) = 1/11
Ah! Theres a contradiction. Final values dont match all three.
So only two equations are consistent. Hence, you solve any two equations for 2
variables. If third contradicts, it might be an error.
(ii) Solve:
Lets take a very neat substitution to simplify this.
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Step 1: Let √x = a
Then:
Multiply both sides by a to remove the fraction:
Step 2: Solve Quadratic
So:
√x = 4 x = 16
√x = 2 x = 4
Final answer: x = 4 or x = 16
(iii) Solve Demand Equation:
We are given:
We need to find equilibrium price (p) and quantity (q).
Step 1: From Equation (2):
2p+q=8q=8−2p(3)2p
Now plug this into Equation (1):
p2+(8−2p)2=20p
2
+ (8 - 2p)
2
= 20p2+(8−2p)2=20
Expand (8 2p)²:
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Step 2: Solve the Quadratic
Use the quadratic formula:
So:
Now find corresponding q:
1. If p = 4.4:
q = 8 2(4.4) = 8 8.8 = 0.8 (Not acceptable, quantity cant be negative)
2. If p = 2:
q = 8 2(2) = 8 4 = 4
Final Equilibrium:
Price (p) = 2
Quantity (q) = 4
A Story to Remember The Market Fair Analogy:
At the villages grand market fair, the seller sets a price p, and people demand q quantity at
that price. But there is a secret rule: the price and quantity must balance like two children
on a seesaw. If the price is too high, nobody buys much; if the quantity is too high, the seller
loses money.
Aryan remembered this rule and decided to try the balance himself using the two equations
given. He fiddled with the equations, tried different values, and finally, when he reached p =
2 and q = 4, both equations balanced like a perfectly even seesaw. Thats when he smiled.
Equilibrium achieved, he whispered.
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Final Answers Summary:
Moral of the Mathematical Tale:
Every equation is like a piece of a puzzle. You dont need to rush to fit them all at once. Take
a deep breath, try one step at a time, and most importantly, enjoy the journey of solving it.
Whether it's solving algebra, cracking quadratic codes, or understanding market demands
math becomes fun when you make it your friend.
If Aryan, Bella, and Zoya could do it under a banyan tree, so can you with a little curiosity
and calmness.
2. (i) Insert 6 Arithmetic Means between 3 and 24.
(ii) Which term of the series 1 + 1/2 + 1/4 + 1/8 1/512
(iii) Find sum up to n terms of the series
Ans: A Story Before the Sums Begin...
Once upon a time, in a village called Mathemagica, lived a curious student named Aarav. He
loved solving mysteries, not with clues, but with numbers. One day, his teacher gave him a
scroll with three puzzles written on it.
If you solve these, the teacher smiled, youll become a master of Arithmetic and
Geometric Series!
Excited, Aarav opened the scroll. Lets walk along with him as he tackles each problem step-
by-step, making it feel like a journey rather than a task.
Puzzle 1: Insert 6 Arithmetic Means between 3 and 24
Understanding the Question
We are told:
First term (a) = 3
Last term = 24
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We need to insert 6 arithmetic means, i.e., six numbers between 3 and 24 such that
the whole sequence is in Arithmetic Progression (A.P).
So, total number of terms will be:
3 (start), then 6 means, then 24 (end)
Thats a total of 8 terms in the A.P.
What is Arithmetic Progression?
In A.P., each number increases by a fixed value called common difference (d).
General form of A.P.:
a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d
Solving the Puzzle
We know:
First term, a = 3
Number of terms, n = 8
Last term (8th term) = 24
We use the formula for the nth term of an A.P.:
Putting the values:
Now we can find the 6 terms in between:
2nd term: a+d=3+3=6
3rd term: 3+23=9
4th term: 3+33=12
5th term: 3+43=15
6th term: 3+53=18
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7th term: 3+63=21
So, the full sequence becomes:
3, 6, 9, 12, 15, 18, 21, 24
Puzzle 1 Solved!
Puzzle 2: Which term of the series
1 + 1/2 + 1/4 + 1/8 + ... = 1/512
This is a Geometric Progression (G.P.), not arithmetic.
Understanding the Question
Given:
First term (a) = 1
Common ratio (r) = 1/2
Were told that the nth term is 1/512, and were asked which term is it?
Solving the Puzzle
In G.P., the nth term is:
Putting the values:
So we have:
Now write 512 as a power of 2:
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So,
Therefore,
n−1=9n=10
So, the 1/512 is the 10th term of the G.P.
Puzzle 2 Solved!
Puzzle 3: Find the Sum up to n terms of the Series
Lets assume the question is asking for the sum of the same geometric series as before:
Understanding the Concept
This is again a G.P., with:
First term, a=1a = 1a=1
Common ratio, r=
Sum of first nnn terms of a G.P. is:
Solving the Puzzle
So, the sum of the first nnn terms of the series is:
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Lets try this with a few values of n:
For n = 1:
S1=2(1−1/2)=2(1/2)=1
For n = 2:
S2=2(1−1/4)=2(3/4)=1.5
For n = 3:
S3=2(1−1/8)=2(7/8)=1.75
Formula verified!
Aaravs Final Thought
As Aarav closed the scroll, he smiled. These puzzles werent just about formulas, he
whispered, They were about patterns, logic, and the fun of cracking them!
And dear student, thats the secret toomathematics isnt about memorizing formulas. Its
about recognizing the pattern, understanding the story, and enjoying the journey of solving
it.
Final Answers:
1. Arithmetic Means inserted between 3 and 24:
6, 9, 12, 15, 18, 21
2. Which term is 1/512 in G.P.:
10th term
SECTION-B
3. (i) Prove that y = 5x - 7 and 2y = 10x + 5 are parallel.
(ii) Find the equation of st. line which passes through (1, 3, 5) and sum of its intercepts on
the coordinates axes is 9.
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(ii) The demand for milk is given by:
Price Rs/Liter
Demand in Liters
1
100
2
50
3
0
Find linear demand function.
Ans: A Fresh Start: The Curious Case of the Milk Vendors Math Problems
In a small town, there lived a clever milk vendor named Raju. While others sold milk and
calculated prices roughly, Raju loved numbers and geometry. One day, three math problems
related to his daily work came his way. Curious and excited, he decided to solve them, step-
by-step. Lets walk through Rajus journey.
Part (i): Prove that y = 5x - 7 and 2y = 10x + 5 are Parallel
Raju remembered his school teacher once told him:
"If two lines have the same slope, they are parallel. Theyll never meet, like two train tracks
running endlessly side by side."
Lets begin by understanding the first line:
Line 1:
Given equation:
y=5x−7y = 5x - 7y=5x−7
This is already in the slope-intercept form:
y=mx+c
Here, m=5m = 5m=5 is the slope.
Line 2:
Given equation:
2y=10x+5
To compare the slope, we need to convert it to the standard slope-intercept form.
Step 1: Divide both sides by 2:
y=5x+52
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Now, its clearly in the form y=mx+c. Here again, the slope (m) is 5.
Conclusion:
Since both lines have the same slope (5) and different intercepts (−7 and 2.5), they are
parallel.
Raju smiled and said, "Like two delivery boys on the same route but different time slots
they follow the same path but never bump into each other!"
Part (ii): Find the Equation of a Straight Line through (1, 3, 5), such that the Sum of its
Intercepts is 9
Now Rajus elder cousin, who was studying in college, asked him:
Can you help me with a 3D geometry problem?
Raju loved a challenge!
We are given:
A line passes through the point (1, 3, 5)
The sum of its intercepts on the coordinate axes is 9
i.e.,
a+b+c=9
Lets solve it step by step.
Understanding the Form of the Equation of a Plane (or Line with Intercepts)
The general form of a line that intercepts the coordinate axes is:
Where:
a is the x-intercept,
b is the y-intercept,
c is the z-intercept
We are told:
The line passes through (1, 3, 5)
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a+b+c=9
Step-by-Step Substitution
Now substitute the point (1, 3, 5) into the equation:
Also, from the given:
Now comes the tricky part: solving these two equations with three unknowns is not directly
possible unless we assume one variable to find specific solutions.
Lets assume a=3a = 3a=3
So, using Equation 2:
Now substitute these values into Equation 1:
This is a rational equation. Lets solve it.
Multiply both sides by 3:
Now multiply both sides by b(6−b):
Bring everything to one side:
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This has no real roots. So our assumption a=3a = 3a=3 doesn't give a neat solution.
🛠 Try Another Assumption: Let a=6a = 6a=6
Then, from Equation 2:
Equation 1 becomes:
Multiply both sides by 6:
Multiply both sides by b(3−b):
Try solving this:
Again, no real root. That means we cant get nice numbers unless more data is given. So,
often in such questions, they expect a general equation using the form:
Where you can choose any two values of a and b and compute c as 9−a−b9 - a - b9−a−b
Example final answer:
Lets assume:
So the equation is:
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And this plane passes through some point in space. To ensure it passes through (1, 3, 5),
check it once again.
Part (iii): Linear Demand Function (Price vs. Demand of Milk)
Raju finally sat down in his shop, looking at how milk sells at different prices.
Price (Rs/litre)
Demand (litres)
1
100
2
50
3
0
He noticed that when the price increases, the demand decreases. This is a perfect case for a
linear demand function, which means demand and price follow a straight-line relationship.
Step 1: Lets find the linear function in the form:
D=mP+c
Where:
D is demand
P is price
m is slope
c is the y-intercept
Step 2: Use two points
Lets take the two points:
(1,100)(1, 100)(1,100)
(2,50)(2, 50)(2,50)
Now calculate the slope:
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Now use point-slope form:
Final Linear Demand Function:
Lets verify it with the third point:
Yes, it matches!
Final Summary
1. Parallel Lines:
Lines y=5x−7y = 5x - 7y=5x−7 and 2y=10x+52y = 10x + 52y=10x+5 are parallel
because both have the same slope = 5.
2. Line in 3D Space with Intercepts Summing to 9:
General form:
For example, if a=3,b=2,c=4a = 3, b = 2, c = 4a=3,b=2,c=4, the equation is:
3. Linear Demand Function for Milk:
Final Thought
Math, just like life, becomes easier when we relate it to simple observations. Raju didnt just
solve these problems; he saw the beauty of numbers in milk bottles, delivery routes, and
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city streets. Every formula, every equation, tells a story and once you learn to listen,
math becomes your best friend.
4. (i) Define sets. Explain various types of sets.
(ii) Explain union, intersection, difference and symmetric difference of sets.
(iii) A class of 70 students, out of which 30 have Maths and 20 have taken Maths but not
Statistics. Find no. of students who have taken Maths and Statistics and those who have
taken Statistics but not Maths.
Ans: A Journey Through the World of Sets
Lets begin our learning journey not in a classroom, but in a magical library of knowledge,
where each book represents a student, and each bookshelf represents a set. Imagine this
library is managed by a curious boy named Aryan, who loves to observe patterns, organize
information, and solve puzzles. One day, while helping his teacher arrange files about
students who opted for Maths and Statistics, he discovered something wonderful Set
Theory!
Lets walk with Aryan as he discovers what sets are, the different types of sets, and how we
perform operations like union, intersection, difference, and symmetric difference. At the
end, well also help Aryan solve a real-life problem using this knowledge. Are you ready?
Lets go!
(i) What is a Set?
A set is simply a collection of well-defined and distinct objects. These objects are called
elements or members of the set. The best part? You can write a set using curly brackets { }.
Definition:
A set is a well-defined collection of objects. The elements of a set are written within curly
braces, separated by commas.
Examples:
A set of vowels in the English alphabet:
A = {a, e, i, o, u}
A set of first five natural numbers:
B = {1, 2, 3, 4, 5}
A set of colors in the rainbow:
C = {red, orange, yellow, green, blue, indigo, violet}
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Types of Sets
Aryan wanted to categorize books (sets), so his teacher introduced him to types of sets.
Heres what he found:
1. Empty Set (Null Set):
A set that has no elements is called an empty set or null set.
It is denoted by Φ or { }.
Example:
The set of natural numbers less than 1 → Φ
2. Singleton Set:
A set with only one element is called a singleton set.
Example:
Set of odd prime numbers less than 3 → {2}
3. Finite Set:
A set with a countable number of elements.
Example:
Set of days in a week → {Monday, Tuesday, ..., Sunday}
4. Infinite Set:
A set with uncountable/infinite elements.
Example:
Set of all natural numbers → {1, 2, 3, 4, ...}
5. Equal Sets:
Two sets are equal if they have exactly the same elements, regardless of order.
Example:
A = {1, 2, 3}, B = {3, 2, 1} A = B
6. Equivalent Sets:
Two sets having the same number of elements, but not necessarily the same elements.
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Example:
A = {a, b, c}, B = {1, 2, 3} A ≈ B
7. Subset:
If every element of set A is also in set B, then A is a subset of B.
Symbol: A B
Example:
A = {1, 2}, B = {1, 2, 3} A B
8. Proper Subset:
A is a proper subset of B if A B, and A ≠ B.
Symbol: A B
9. Universal Set:
A set that contains all the elements under consideration. Denoted by U.
Example:
If U = all natural numbers ≤ 10, then
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
10. Power Set:
The set of all subsets of a set is called its power set.
If A = {1, 2}, then
P(A) = {Φ, {1}, {2}, {1, 2}}
(ii) Operations on Sets
As Aryan arranged the books, he realized that some books belonged to more than one shelf.
This is when his teacher taught him four powerful operations:
1. Union of Sets (A B)
Union means combining all elements of both sets.
If some elements are repeated, we include them only once.
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Example:
A = {1, 2, 3}, B = {3, 4, 5}
A B = {1, 2, 3, 4, 5}
2. Intersection of Sets (A ∩ B)
Intersection gives the common elements between two sets.
Example:
A = {1, 2, 3}, B = {2, 3, 4}
A ∩ B = {2, 3}
3. Difference of Sets (A B)
This gives elements only in A but not in B.
Example:
A = {1, 2, 3}, B = {2, 3, 4}
A B = {1}
4. Symmetric Difference (A Δ B)
Symmetric difference gives elements which are in either A or B, but not in both.
Symbol: A Δ B = (A B) (B A)
Example:
A = {1, 2, 3}, B = {3, 4, 5}
A Δ B = {1, 2, 4, 5}
Lets add a short story here to make this even clearer.
Mini Story: The Cricket and Football Club
In a college, there are students who play Cricket (Set A) and others who play Football (Set
B). Some students play both games.
Let:
A = {Ravi, Mohan, Sita}
B = {Sita, Karan, Meena}
Now:
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Union (A B) = Students who play either Cricket or Football or both = {Ravi, Mohan,
Sita, Karan, Meena}
Intersection (A ∩ B) = Students who play both = {Sita}
A B = Only Cricket = {Ravi, Mohan}
B A = Only Football = {Karan, Meena}
A Δ B = Either Cricket or Football, but not both = {Ravi, Mohan, Karan, Meena}
Isnt it fun? Just like that, Aryan now found math quite interesting.
(iii) Problem Solving Using Sets
Now, lets solve the given problem step-by-step.
Question:
In a class of 70 students:
30 have taken Maths
20 have taken Maths but not Statistics
Find:
1. The number of students who have taken both Maths and Statistics
2. The number of students who have taken only Statistics
Step 1: Lets Assume Sets
Let:
M = Set of students who took Maths
S = Set of students who took Statistics
n(M) = number of students who took Maths = 30
n(M S) = Students who took Maths but not Statistics = 20
Total students = 70
Step 2: Find Students who took both Maths and Statistics
We are given:
n(M S) = 20
And total who took Maths (n(M)) = 30
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So:
n(M ∩ S) = n(M) n(M S)
= 30 20 = 10
So, 10 students took both Maths and Statistics
Step 3: Find Students who took only Statistics
Lets find:
n(S M) = only Statistics students
We dont know total who took Statistics directly, but we can find it.
Let:
Total students = 70
n(M S) = Students who took Maths or Statistics or both
But in set theory,
n(M S) = n(M) + n(S) n(M ∩ S)
Lets find n(S):
Let n(S) = x
So:
n(M S) = 70 = 30 + x 10
=> 70 = 20 + x
=> x = 50
So, number of students who took Statistics = 50
Now:
Only Statistics = n(S M) = n(S) n(M ∩ S)
= 50 10 = 40
Final Answer:
1. Students who have taken both Maths and Statistics = 10
2. Students who have taken only Statistics = 40
Conclusion
Aryan was overjoyed. With just the knowledge of simple set operations and understanding
the types of sets, he was able to solve a real-world problem! Just like Aryan, once we
understand the basics:
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What are sets
What types they are
How we perform operations on them
And how we solve practical problems using Venn Diagrams or simple equations
...math becomes not just easy but fun! The beauty of Set Theory lies in its clarity and
organization. It helps us in various fields like Computer Science, Logic, Statistics, and daily
life planning.
So, next time youre asked whether someone is part of a group or not remember, youre
dealing with sets!
SECTION-C
5. (1) Explain the concept of function and various types of functions.
(ii) Prove that
Ans: A Fresh Beginning: The Tale of Riya and the Magic Machine
Once upon a time, in a small town named Numville, there lived a curious girl named Riya
who loved solving puzzles. One day, she came across a strange machine in her grandfathers
attic. It had a label: Input a number, and get a unique output. Fascinated, she placed a
number card into it5, and out came another card with 13 written on it. Then she tried 6
and got 15, and so on.
Her grandfather smiled and said, Riya, what youve just used is nothing but a real-life
example of a function!
This sparked her curiosityand maybe yours too.
Lets begin this wonderful journey into functions, understand them deeply, and then move
toward evaluating the limit problem that appears tricky but becomes super simple with the
right guidance.
Part (i): Concept of Function and Various Types of Functions
1. What is a Function?
In simple words:
A function is a special relationship where each input has exactly one output.
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Or, as Riyas grandfather said, If you give the machine a number, it will give you back only
one number. No confusion, no double answers.
Mathematical Definition:
A function from a set A to a set B is a rule which assigns to every element x in set A, exactly
one element f(x) in set B.
We write it as:
f:A→B
Where:
A is called the domain (set of inputs).
B is called the codomain (set of possible outputs).
f(x) is called the image or output corresponding to x.
Example:
Lets say:
f(x)=2x+3f
If you input x = 1, you get:
f(1)=2(1)+3=5
Input x = 2, and:
f(2)=2(2)+3=7f
This is a function because each input gives only one output.
2. Important Terms Related to Function
1. Domain: The set of all input values (x-values).
2. Co-domain: The set into which all outputs fall (not always all outputs are used).
3. Range: The set of actual output values that the function gives.
3. Visualizing Functions
Think of a vending machine:
You press button A1 (input).
You always get Lays Chips (output).
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You never get both Lays and Coke for the same button.
This vending machine represents a function because every button gives exactly one item.
4. Different Types of Functions (with Examples)
Lets now explore the beautiful variety of functions, each with its own role in mathematics.
(i) One-to-One Function (Injective)
Each element of the domain maps to a unique element in the codomain.
Example:
f(x)=x+5f
For all x, the value is different and unique.
(ii) Onto Function (Surjective)
Every element in the codomain is the image of at least one element from the domain.
Example:
f(x)=x
All real numbers are covered as outputs (the codomain = range).
(iii) One-to-One and Onto (Bijective Function)
A function that is both one-to-one and onto.
Example:
f(x)=x
This is the identity functioneach value maps to itself.
(iv) Constant Function
The output is the same for every input.
Example:
f(x)=7 for all x
No matter what x you choose, the result is always 7.
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(v) Identity Function
f(x)=x
Simple as it looks! Input and output are the same.
(vi) Polynomial Function
Functions that involve terms like:
f(x)=4x2+3x+7
(vii) Rational Function
A function of the form:
Where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
Example:
(viii) Even and Odd Functions
Even Function: f(−x)=f(x)
Example: f(x)=x
2
Odd Function: f(−x)=−f(x)
Example: f(x)=x
3
(ix) Piecewise Function
Defined by different expressions for different intervals of x.
Example:
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(x) Exponential and Logarithmic Functions
(xi) Modulus Function
f(x)=x
Always gives non-negative outputs.
(xii) Signum Function
Part (ii): Prove that
Step-by-Step Story: Cracking the Mystery of Limits
Lets return to Riya. One day, she encountered a puzzling expression:
What will happen to this expression when x gets really, really close to 1?
She looked at:
When she plugged in x = 1 directly:
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Thats called an indeterminate form. Just like when two puzzle pieces look the same but
dont fitwe need to simplify!
Step 1: Factor the Numerator
The expression is:
Lets factor the numerator:
So now the expression becomes:
Final Answer:
Why This Happens:
When you encounter a limit that gives 0/0, it usually means:
The numerator and denominator have a common factor.
Remove the common factor.
Then directly substitute the value of x.
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Conclusion: The Wisdom Behind Functions and Limits
In this journey, weve learned:
Functions are like trustworthy machines: one input = one output.
They come in many shapes and rolesfrom linear to exponential to piecewise.
And limits help us understand what a function is doing around a point, especially
when direct substitution gives confusing forms.
Riya, the curious girl, eventually built her own function machine with the help of her
grandfather. And every time she faced a puzzle, she remembered:
Dont panic if the answer looks confusing at first. Just factor it, simplify it, and let logic be
your guide.
Moral of the Story
Math isn't about just solving numbersit's about understanding patterns, making
connections, and uncovering the magic behind the scenes.
6. (i) Distinguish between a continuous function and discontinuous function.
(ii) Prove that
is continuous at x = 2
Ans: Imagine you are walking along a smooth mountain trail. The path is clear, and you can
keep walking without stopping or jumping. Suddenly, you find a broken bridge on your way
now you must either jump over the gap or turn around.
In the world of mathematics, functions behave just like that path. Some are smooth and
uninterrupted (continuous), while others have breaks or jumps (discontinuous). In this
lesson, lets explore what it means for a function to be continuous or discontinuous, and
then well prove with gentle logic that a specific function behaves continuously at a certain
point. Well also dip into a short story to bring this concept alive. Lets begin!
PART (i): Distinguishing Between Continuous and Discontinuous Functions
Lets Begin with the Basics
Before diving into definitions, lets ask ourselves what do we mean by continuity in
daily life?
Continuity means no interruption or no break.
Think of:
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A movie that plays without buffering smooth and continuous.
A road that has no potholes continuous road.
A clock ticking every second without stopping continuous movement.
In mathematics, we say a function is continuous at a point if its value doesnt suddenly jump
or break at that point.
Formal Definition of a Continuous Function:
Now, What is a Discontinuous Function?
A function is said to be discontinuous at a point if any one of the above three conditions
fails. That is:
Either the function is not defined at that point,
Or the limit doesnt exist,
Or the limit exists but doesnt match the value of the function at that point.
This break or gap makes the function discontinuous at that location.
Simple Graphical Understanding:
A continuous function looks like a smooth, unbroken curve.
A discontinuous function has gaps, jumps, or holes in the graph.
A Story to Understand Continuity
Lets meet Ravi, a delivery boy.
Every day, Ravi cycles from his home to the post office, following the same route. One day, a
bridge on his way breaks down. Now he must stop, take a detour, or jump across. Thats a
discontinuity in his journey.
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Before the bridge, his path was smooth (like a continuous function). But due to that missing
piece, it became like a function with a gap.
Just like Ravi needs an unbroken path to deliver mail on time, we need unbroken
(continuous) functions for smooth calculations in mathematics.
Types of Discontinuity (for deeper understanding):
1. Point Discontinuity (Removable): A small hole in the graph.
2. Jump Discontinuity: The function jumps from one value to another.
3. Infinite Discontinuity: Function shoots up to infinity (like a vertical asymptote).
Examples of Continuous Functions:
Polynomials and rational functions (where denominator ≠ 0)
Examples of Discontinuous Functions:
PART (ii): Prove That
is continuous at x=2x = 2x=2
Step 1: Understand the Function
Given function:
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Lets simplify the denominator:
So, the function is defined for all real x except x=1 and x=−1, because these values make the
denominator zero (division by zero is undefined).
Hence, the function is defined at x=2.
Step 2: Check Continuity at x=2
We will use the three-step test mentioned earlier:
Step 1: Is f(2) defined?
Yes.
So, f(2) is defined.
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Conclusion:
All three conditions are satisfied.
So,
Recap in Student-Friendly Manner:
What We Did:
1. We understood what continuity means like a smooth road.
2. We saw that if theres no hole, jump, or break, a function is continuous.
3. We checked the value and limit of the function at x=2x = 2x=2.
4. Since the function is well-behaved at that point, we proved it's continuous.
Real-Life Analogy (Second Short Story)
Think of a water pipe. If water flows smoothly through it, thats continuity. If theres a
blockage or hole, thats a discontinuity.
Suppose a gardener is watering plants through a pipe. At most spots, the water flows
without issue. But at one location, theres a small break, and water splashes out. This is just
like a discontinuous function not all inputs give smooth outputs.
For the pipe to be functionally continuous, water must flow smoothly through every part,
just like a function that behaves well at all points in its domain.
Final Thoughts:
In mathematics, especially calculus, continuity is the foundation of understanding functions,
graphs, and changes. You cant talk about limits, derivatives, or integrals without first
ensuring your function is continuous.
So, to sum up:
A continuous function is like an uninterrupted journey.
A discontinuous function is like a road with breaks.
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SECTION-D
7. (i) Differentiate (7x - 8)
4
(5x - 1)
3
w.r.t. x.
(ii) Find the derivative of
w.r.t. x.
(iii) Differentiate
wr.t.x.
Ans: Let me take you to a classroom, where a curious student named Riya raised her hand
and asked,
Maam, why does Differentiation always seem like a puzzle to me? I know the formulas but
cant figure out when and how to use them!
The teacher, smiling warmly, replied,
Great question, Riya! Thats because math isnt just about solving its about observing
patterns and choosing the right tools. Lets solve three problems today, each like a different
type of lock that needs a different key!
And so begins our math story with three expressions, each needing a different approach
to differentiate them with respect to x.
Question 7 (i): Differentiate (7x - 8)⁴(5x - 1)³ w.r.t. x
Step 1: Identify the type of function
We are given:
y=(7x−8)4(5x−1)3
This is a product of two functions, so we must use the Product Rule.
Product Rule Reminder:
If
Lets assign:
u=(7x−8)4
v=(5x−1)3
Step 2: Differentiate each part
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Differentiate u = (7x - 8)
4
Use the Chain Rule:
Differentiate v = (5x - 1)
3
Step 3: Apply the Product Rule
Now,
Substitute values:
That's your final differentiated form. You can keep it factored for simplicity.
Now lets meet a student named Arjun, who loves solving fractions but panics when a
logarithm appears. The teacher tells him,
Dont fear the log just treat it as another function!
Step 1: Recognize the Quotient Rule
This is a quotient one function divided by another.
Lets write:
Let:
u=x+2
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v=3+logx
Step 2: Use the Quotient Rule
Quotient Rule:
Now find the derivatives:
Step 3: Plug into the Quotient Rule
Now simplify the numerator:
You can leave the answer as:
Or, if required, further simplify.
Heres the fun one! Imagine this function as a layered cake a power of a square root of an
exponential. Time to use both logarithmic differentiation and chain rule.
Lets say:
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Step 1: Use the identity
Remember:
Here,
a=5a
Step 2: Differentiate f(x)
We need to find:
Now differentiate ex
2
:
So finally:
Step 3: Apply full formula
Now apply:
Simplify:
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Final Thoughts for Riya and Arjun
The teacher said to them,
Differentiation is like being a detective your job is to look at the structure of the function
and figure out which rule fits best. Once you know your tools Product Rule, Quotient
Rule, Chain Rule, Logarithmic Differentiation youll never be afraid again.
8. (i) Given the demand function p = 50 - 3q find elasticity of demand p = 5
(ii) Prove that elasticity of demand function is =
where demand p = 50 - 3x and p = 5
(iii) Given the total cost function C = 60 - 12q + 2q
2
find AC, MC and show that slope of AC
=
(MC - AC)
Ans: Introduction: A Walk into the World of Economics
Imagine a young entrepreneur named Arjun who runs a small online business selling
handcrafted notebooks. He's trying to understand how changes in price affect demand, how
much revenue he earns, and what it costs to produce his products. One day, he meets his
old school teacher, Mrs. Mehra, a brilliant economist, who agrees to help him make sense of
all these concepts using simple mathematics and logic. As they sit together in his cozy little
workshop, they begin their conversation. You, as a curious student, are silently watching
and listening.
Lets walk with Arjun and Mrs. Mehra as we explore three important parts of
microeconomics: Elasticity of Demand, Relationship Between Elasticity and Revenue, and
Cost Analysisstep by step.
Part (i): Elasticity of Demand When p = 5 and p = 50 3q
Mrs. Mehra starts:
Arjun, elasticity of demand tells us how sensitive consumers are to price changes. Its like
askingif I drop the price of a notebook, how much more will people want to buy?
The demand function is:
p=50−3qp = 50 - 3qp=50−3q
But to calculate elasticity, we need q as a function of p.
So, rearrange the equation:
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Now recall the formula for price elasticity of demand (Ep):
Lets break it down step-by-step:
Differentiate with respect to p:
Step 2: Plug into Elasticity Formula
We already have:
Final Answer:
The price elasticity of demand at p=5 is:
This means demand is inelastic at this price pointcustomers dont react much to price
changes.
Part (ii): Proving Elasticity = AR / (AR MR)
Mrs. Mehra smiled and said, Now, Arjun, let me show you a magical formula that links
demand, revenue, and elasticity.
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Key Formula to Prove:
Lets break this into steps:
Step 1: Understand Terms
AR (Average Revenue) = Price (p)
TR (Total Revenue) = p × q
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Part (iii): Given Cost Function, Find AC, MC and Prove Slope of AC = (1/q)(MC AC)
Mrs. Mehra continued, Now, lets dive into the world of cost. Arjun, heres the total cost
function for your notebook production.
C=60−12q+2q
2
We are to:
1. Find AC (Average Cost) = C/q
2. Find MC (Marginal Cost) = dC/dq
3. Prove:
Step 1: Average Cost (AC)
Step 2: Marginal Cost (MC)
Step 3: Prove that:
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Differentiate AC with respect to q:
Now lets compute RHS:
We already have:
MC = -12 + 4q
So:
Simplify:
Now:
Compare with:
Both expressions are same:
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Hence Proved!
Recap in Arjuns Words
Arjun leaned back and said, Wow! So heres what Ive learned today:
Elasticity tells me how customers respond to changes in price.
If elasticity is low, lowering prices wont increase my sales much.
AR, MR, and elasticity are all connected in a beautiful formula.
Cost functions can help me figure out whether Im being efficient.
And most amazingly, even the slope of my average cost curve follows a
mathematical rule!
Mrs. Mehra smiled, Exactly, Arjun. Economics is not just about numbersit's about
understanding human behavior and making smarter decisions.
Conclusion: The Beauty of Simple Economic Math
Just like Arjun, many of us struggle to understand how the pieces of pricing, revenue, and
cost fit together. But once you break it down using simple steps and real-life logic,
everything starts to make sense. Always remember:
Elasticity helps you understand customer reactions.
AR, MR, and elasticity are connected like family.
Cost curves tell you how efficiently youre producing.
So, dont fear the formulasembrace them with curiosity, just like Arjun. Whether you run
a business or solve exam questions, these concepts will always guide you
This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.
