Class 9 Mathematics • Ganita Manjari •
Chapter 2 — Introduction to Linear Polynomials
Exam Focus:
ये questions Chapter 2 के important concepts पर आधारित
अतिरिक्त exam-oriented questions हैं।
Important:
ये page textbook exercises का replacement नहीं है।
Textbook के exact exercises अलग
📝 Exercises
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✓ Solution देखें
से पूरा तरीका check करें।
A. Algebraic Expressions & Polynomials
Important Question 1
For the polynomial
7x³ − 4x² + 9x − 12,
identify its variable, terms, coefficients, constant term
and degree.
आसान हिंदी में:
Polynomial के अलग-अलग parts को पहचानना है।
x variable है। Plus/minus से अलग होने वाले parts
terms हैं। हर variable term के आगे का numerical number
उसका coefficient है। Variable के बिना term constant होता है।
सबसे बड़ी power degree बताती है।
Step-by-Step Solution:
7x³ − 4x² + 9x − 12
Variable = x
Terms = 7x³, −4x², 9x, −12
Coefficient of x³ = 7
Coefficient of x² = −4
Coefficient of x = 9
Constant term = −12
Highest power = 3
✅ Degree = 3
Important Question 2
Classify the following polynomials as constant, linear,
quadratic or cubic:
(i) 8
(ii) 5x + 2
(iii) x² − 7x + 4
(iv) 3x³ + x − 1
आसान हिंदी में:
सबसे पहले हर polynomial की highest power देखनी है।
Power 0 → constant, power 1 → linear,
power 2 → quadratic और power 3 → cubic।
Solution:
Polynomial
Degree
Type
8
0
Constant
5x + 2
1
Linear
x² − 7x + 4
2
Quadratic
3x³ + x − 1
3
Cubic
Important Question 3
What is the coefficient of y² and the constant
term in the polynomial
−5y³ + 8y² − 3y + 11?
आसान हिंदी में:
y² के साथ लगा number उसका coefficient है।
जिस term में y नहीं है, वही constant term है।
−5y³ + 8y² − 3y + 11
y² वाला term = 8y²
इसलिए coefficient of y² = 8
Variable के बिना term = 11
✅ Coefficient of y² = 8
✅ Constant term = 11
Important Question 4
Is
4x² + 3x + 7
a linear polynomial? Give a reason.
आसान हिंदी में:
Linear polynomial की degree 1 होती है।
यहाँ highest power 2 है, इसलिए इसे linear नहीं कहेंगे।
Given polynomial:
4x² + 3x + 7
Highest power of x = 2
Therefore degree = 2.
✅ No. It is a quadratic polynomial, not a linear polynomial.
B. Linear Polynomials & Values
Important Question 5
Find the value of
3x + 8
when
x = 5.
आसान हिंदी में:
x की जगह 5 रखो और फिर calculation करो।
3x + 8
3(5) + 8 = 15 + 8 = 23
✅ Answer = 23
Important Question 6
For the linear polynomial
4p − 7,
find its value for
p = −3
and
p = 6.
आसान हिंदी में:
दोनों बार p की दी हुई value को expression में put करना है।
Negative value होने पर brackets का ध्यान रखें।
For p = −3:
4(−3) − 7 = −12 − 7 = −19
For p = 6:
4(6) − 7 = 24 − 7 = 17
✅ Values are −19 and 17.
Important Question 7
A square has side x cm.
Write a linear polynomial for its perimeter.
Find the perimeter when x = 7 cm.
आसान हिंदी में:
Square की चारों sides बराबर होती हैं।
इसलिए perimeter = 4 × side.
Perimeter = 4x
जब x = 7:
4(7) = 28 cm
✅ Linear polynomial = 4x
✅ Perimeter = 28 cm
Important Question 8
A club charges a joining fee of ₹120 and ₹40 for each match played.
Write a linear polynomial for the total amount paid after x matches.
Find the amount for 7 matches.
आसान हिंदी में:
शुरुआत की fixed fee ₹120 है।
इसके अलावा हर match पर ₹40 लगता है।
इसलिए x matches के लिए ₹40x और fixed fee ₹120 होगी।
Total = 120 + 40x
For x = 7:
120 + 40(7)
= 120 + 280
= 400
✅ Amount for 7 matches = ₹400
C. Linear Patterns
Important Question 9
Consider the sequence:
6, 10, 14, 18, 22, ...
Is it a linear pattern?
Find the next three terms and write an expression for
the nth term.
आसान हिंदी में:
Consecutive terms का difference check करो।
अगर difference same है, तो pattern linear है।
Differences:
10 − 6 = 4
14 − 10 = 4
18 − 14 = 4
Difference constant = 4.
Next three terms:
26, 30, 34
First term = 6 and common difference = 4.
इसलिए nth term:
aₙ = 6 + (n − 1)4
= 4n + 2
✅ Linear pattern: Yes
✅ nth term = 4n + 2
Important Question 10
The number of matchsticks in a growing pattern is:
5, 8, 11, 14, ....
Find the number of matchsticks at the 10th stage.
आसान हिंदी में:
हर stage में 3 matchsticks बढ़ रही हैं।
इसलिए difference 3 है और pattern linear है।
पहले stage पर 5 sticks हैं।
aₙ = 5 + (n − 1)3
n = 10:
a₁₀ = 5 + 9(3)
= 5 + 27
= 32
✅ 10th stage = 32 matchsticks
Important Question 11
A sequence has nth term
3n − 2.
Find its first five terms and the difference between consecutive terms.
आसान हिंदी में:
n की जगह 1, 2, 3, 4 और 5 रखकर terms निकालें।
फिर consecutive terms subtract करें।
n = 1 → 3(1) − 2 = 1
n = 2 → 3(2) − 2 = 4
n = 3 → 3(3) − 2 = 7
n = 4 → 3(4) − 2 = 10
n = 5 → 3(5) − 2 = 13
Sequence:
1, 4, 7, 10, 13
4 − 1 = 3
7 − 4 = 3
10 − 7 = 3
✅ Constant difference = 3
Important Question 12
A pattern is represented by the rule
y = 2x + 5.
Find y when x = 0, 1, 2, 3 and explain why this is a linear pattern.
आसान हिंदी में:
x की अलग-अलग values रखकर y निकालो।
अगर x में 1 की बढ़ोतरी पर y हमेशा 2 से बढ़ता है,
तो pattern linear है।
x
y = 2x + 5
0
5
1
7
2
9
3
11
Differences are:
+2, +2, +2.
✅ Therefore it is a linear pattern.
D. Linear Growth & Linear Decay
Important Question 13
A plant is 2.4 m tall and grows 0.3 m every month.
Find its height after 8 months and write a linear expression
for its height after n months.
आसान हिंदी में:
शुरुआती height 2.4 m है और हर month 0.3 m बढ़ रही है।
इसलिए n months के लिए:
Initial + Growth × n.
h(n) = 2.4 + 0.3n
n = 8:
h(8) = 2.4 + 0.3(8)
= 2.4 + 2.4
= 4.8 m
✅ Height after 8 months = 4.8 m
Important Question 14
A tank initially contains 900 litres of water.
If 35 litres are used every day, write an expression
for the water left after n days and find the amount left
after 12 days.
आसान हिंदी में:
यहाँ water हर दिन घट रहा है।
इसलिए यह linear decay है:
Initial amount − Daily decrease × days.
W(n) = 900 − 35n
W(12) = 900 − 35(12)
= 900 − 420
= 480 litres
✅ Amount left after 12 days = 480 litres
Important Question 15
State whether each situation represents linear growth
or linear decay:
(i) A salary increases by ₹1500 every month.
(ii) A machine loses ₹2000 of value every year.
(iii) A plant grows 4 cm every week.
(iv) A water level falls by 0.2 m every month.
आसान हिंदी में:
Fixed amount से increase = linear growth.
Fixed amount से decrease = linear decay.
Situation
Type
(i)
Linear Growth
(ii)
Linear Decay
(iii)
Linear Growth
(iv)
Linear Decay
E. Linear Relationships — y = ax + b
Important Question 16
In the equation
y = 6x + 4,
identify the slope and y-intercept.
आसान हिंदी में:
Equation को y = ax + b के standard form से compare करो।
x के साथ number = a और constant = b.
y = 6x + 4
Compare with:
y = ax + b
✅ Slope a = 6
✅ y-intercept b = 4
✅ Point on y-axis = (0, 4)
Important Question 17
Find the linear relationship
y = ax + b
passing through the points
(2, 11)
and
(5, 23).
आसान हिंदी में:
दोनों points को y = ax + b में डालकर two equations बनाओ।
फिर subtract करके a निकालो और किसी एक equation से b निकालो।
Point (2, 11):
11 = 2a + b
Point (5, 23):
23 = 5a + b
Subtract:
12 = 3a
a = 4
Put a = 4:
11 = 2(4) + b
11 = 8 + b
b = 3
✅ Linear relationship:
y = 4x + 3
Important Question 18
A linear relationship is
y = 3x − 7.
Find y when x = 0, 2 and 5.
Also state the y-intercept.
आसान हिंदी में:
हर value of x को equation में put करो।
y-intercept constant b होता है।
y = 3x − 7
x = 0 → y = −7
x = 2 → y = 6 − 7 = −1
x = 5 → y = 15 − 7 = 8
Positive slope वाली lines में जिस line का slope बड़ा होगा,
वह अधिक steep होगी।
Slope of A = 4
Slope of B = 1
✅ Line y = 4x + 1 is steeper.
Important Question 23
For the line
y = −3x,
find two points through which the line passes.
Does it pass through the origin?
आसान हिंदी में:
y = −3x में x की convenient values चुनो।
x = 0 और x = 1 रखने पर two points मिलेंगे।
b = 0 होने के कारण line origin से गुजरती है।
x = 0 → y = 0 → (0, 0)
x = 1 → y = −3 → (1, −3)
✅ Points = (0,0) and (1,−3)
✅ Yes, the line passes through the origin.
G. Mixed Exam Questions
Important Question 24
A sequence is given by
9, 13, 17, 21, ....
Find its nth term and determine the 25th term.
आसान हिंदी में:
First term 9 है और common difference 4 है।
Linear pattern का nth term:
first term + (n − 1) × difference.
aₙ = 9 + (n − 1)4
= 4n + 5
For n = 25:
a₂₅ = 4(25) + 5
= 105
✅ nth term = 4n + 5
✅ 25th term = 105
Important Question 25
A taxi charges a fixed starting amount of ₹80 and ₹18
for every kilometre travelled.
Write the linear relation between distance x and cost y.
Find the cost for 12 km.
आसान हिंदी में:
Fixed amount = ₹80.
Per kilometre amount = ₹18.
इसलिए total cost = 18x + 80.
y = 18x + 80
For x = 12:
y = 18(12) + 80
= 216 + 80
= 296
✅ Cost for 12 km = ₹296
Important Question 26
Two lines are
y = 2x + 7
and
y = 3x + 7.
Compare their slopes and y-intercepts.
Are they parallel?
आसान हिंदी में:
दोनों lines में b = 7 same है, लेकिन slopes 2 और 3 हैं।
Parallel होने के लिए slopes same होनी चाहिए।
Line
Slope
y-intercept
y = 2x + 7
2
7
y = 3x + 7
3
7
✅ They are not parallel because their slopes are different.
Important Question 27
A linear pattern starts with 12 and increases by 7 at every step.
(i) Write its nth-term expression.
(ii) Find the 20th term.
(iii) Is the corresponding graph increasing or decreasing?
आसान हिंदी में:
First term 12 और difference 7 है।
इसलिए nth term 12 + (n−1)7 होगी।
Positive difference होने से graph increasing होगा।
aₙ = 12 + (n − 1)7
= 7n + 5
20th term:
a₂₀ = 7(20) + 5
= 145
✅ nth term = 7n + 5
✅ 20th term = 145
✅ Graph direction = Increasing
Important Question 28
A quantity is represented by
y = −4x + 20.
Find the value of y when x = 0, 3 and 5.
Explain what the negative slope tells us.
आसान हिंदी में:
x की values equation में रखकर y निकालें।
Negative slope का मतलब है कि x बढ़ने पर y घटता है।
x = 0 → y = 20
x = 3 → y = −12 + 20 = 8
x = 5 → y = −20 + 20 = 0
Slope = −4, which is negative.
✅ Values = 20, 8, 0
✅ Negative slope means the quantity decreases as x increases.
Important Question 29
The graph of a linear polynomial passes through
(1, 7) and
(4, 16).
Find the polynomial and its y-intercept.
आसान हिंदी में:
Polynomial को y = ax + b मानो।
दोनों points डालकर two equations बनेंगी।
उनसे a और b मिलेंगे।