End-of-Chapter Exercises

Introduction to Linear Polynomials • Class 9 Mathematics • Ganita Manjari

Practice Mode: पहले हर question खुद solve करने की कोशिश करें। समझ न आए तो हिंदी में समझें दबाएँ, फिर Solution देखें से answer check करें।

यह section Chapter 2 के official End-of-Chapter Exercises को cover करता है।

Question 1

Write a polynomial of degree 3 in the variable x in which the coefficient of the x² term is −7.

हिंदी में समझें:

आपको degree 3 का कोई polynomial बनाना है। उसकी highest power 3 होनी चाहिए और x² का coefficient −7 होना चाहिए। ऐसे कई सही answers हो सकते हैं।

Step-by-Step Solution:

Example: x³ − 7x² + 2x + 5

Degree = 3 और x² का coefficient = −7. इसलिए यह एक valid answer है।

Question 2

Find the value of each polynomial at the indicated value:

(i) 5x² − 3x + 7, when x = 1.

(ii) 4t³ − t² + 6, when t = a.

हिंदी में समझें:

Variable की दी हुई value को polynomial में substitute करना है। पहले x = 1 रखें। दूसरे में t की जगह a रखें।

Step-by-Step Solution:

(i) 5(1)² − 3(1) + 7 = 5 − 3 + 7 = 9

(ii) 4(a)³ − (a)² + 6 = 4a³ − a² + 6

Answers: (i) 9 (ii) 4a³ − a² + 6

Question 3

When a number is multiplied by 5/2 and 23 is added, the result is −7/12. Find the number.

हिंदी में समझें:

Unknown number को x मानें। फिर question के अनुसार (5/2)x + 23 = −7/12 equation बनेगी और x निकालेंगे।

Step-by-Step Solution:

(5/2)x + 23 = −7/12

(5/2)x = −7/12 − 23
= −7/12 − 276/12
= −283/12

x = (−283/12) × (2/5) = −283/30

Answer: x = −283/30

Question 4

A positive number is 5 times another number. If 21 is added to both numbers, one new number becomes twice the other. Find the two numbers.

हिंदी में समझें:

छोटी संख्या x मानें। दूसरी संख्या 5x होगी। दोनों में 21 जोड़ने के बाद बड़ी/छोटी नई संख्या के बीच दिए गए relation से equation बनेगी।

Step-by-Step Solution:

Let smaller number = x
Larger number = 5x

5x + 21 = 2(x + 21)

5x + 21 = 2x + 42
3x = 21
x = 7

Larger number = 5 × 7 = 35

Answer: 7 and 35

Question 5

You have ₹800 and save ₹250 every month. Find the amount after (i) 6 months and (ii) 2 years. Express this as a linear pattern.

हिंदी में समझें:

शुरुआत में ₹800 हैं और हर महीने ₹250 बढ़ रहे हैं। इसलिए n months बाद amount = 800 + 250n। 2 years को 24 months में बदलना होगा।

Step-by-Step Solution:

A(n) = 800 + 250n

(i) 6 months: 800 + 250(6) = 800 + 1500 = ₹2300

(ii) 2 years = 24 months: 800 + 250(24) = 800 + 6000 = ₹6800

Linear pattern: A(n) = 800 + 250n

Question 6

The digits of a two-digit number differ by 3. If the digits are interchanged and the resulting number is added to the original number, the sum is 143. Find both numbers.

हिंदी में समझें:

दहाई का digit x और इकाई का digit y मानें। दोनों numbers का sum 143 है। Digit difference 3 है। पहले x + y निकालकर digits पहचान सकते हैं।

Step-by-Step Solution:

Original = 10x + y
Interchanged = 10y + x

(10x + y) + (10y + x) = 143
11x + 11y = 143
x + y = 13

Digits differ by 3.
x − y = 3
x + y = 13

2x = 16 → x = 8
y = 5

Numbers: 85 and 58

Question 7

Draw the graphs of the four given linear equations, identify their slopes and y-intercepts, and state which lines are parallel:

(i) y = −3x + 4

(ii) 2y = 4x + 7

(iii) 5y = 6x − 10

(iv) 3y = 6x − 11

हिंदी में समझें:

हर equation को पहले y = ax + b form में लाएँ। यहाँ a slope और b y-intercept है। फिर equal slopes वाली lines parallel होंगी।

Step-by-Step Solution:

(i) y = −3x + 4 → slope = −3, y-intercept = 4, point (0,4)

(ii) 2y = 4x + 7 → y = 2x + 7/2 → slope = 2, y-intercept = 7/2, point (0,7/2)

(iii) 5y = 6x − 10 → y = 6/5x − 2 → slope = 6/5, y-intercept = −2, point (0,−2)

(iv) 3y = 6x − 11 → y = 2x − 11/3 → slope = 2, y-intercept = −11/3, point (0,−11/3)

Parallel lines: (ii) and (iv), because both have slope 2.

Question 8

The temperature relation is y = (9/5)(x − 273) + 32, where x is Kelvin and y is Fahrenheit. Find (i) y when x = 313 K and (ii) x when y = 158°F.

हिंदी में समझें:

बस equation में given temperature substitute करें। दूसरी part में y = 158 रखकर x solve करें।

Step-by-Step Solution:

(i) y = (9/5)(313 − 273) + 32 = (9/5)(40) + 32 = 72 + 32 = 104°F

(ii) 158 = (9/5)(x − 273) + 32
126 = (9/5)(x − 273)
x − 273 = 70
x = 343 K

Answers: 104°F and 343 K

Question 9

The work done by a body under a constant force equals force × distance. Express the relation in work w and distance d, take the constant force as 3 units, draw the graph, and find the work done when d = 2 units.

हिंदी में समझें:

Work = Force × Distance. Force 3 है, इसलिए w = 3d। d = 2 रखने पर work 6 आएगा।

Step-by-Step Solution:

w = Fd
F = 3
Therefore w = 3d

When d = 2:
w = 3(2) = 6 units

Graph: origin से गुजरने वाली straight line with slope 3.

Question 10

The graph of a linear polynomial p(x) passes through (1,5) and (3,11). Find p(x), the points where its graph cuts the axes, and verify using the graph.

हिंदी में समझें:

Linear polynomial को p(x) = ax + b मानें। दोनों given points को equation में रखकर a और b निकालें।

Step-by-Step Solution:

a + b = 5
3a + b = 11

Subtract: 2a = 6 → a = 3
b = 2

p(x) = 3x + 2

y-intercept: x = 0 → (0,2)
x-intercept: 0 = 3x + 2 → x = −2/3 → (−2/3,0)

Graph should pass through (1,5) and (3,11) and cut the axes at (0,2) and (−2/3,0).

Question 11

Let p(x) = ax + b and q(x) = cx + d satisfy p(0) = 5, p(x) − q(x) cuts the x-axis at (3,0), and p(x) + q(x) = 6x + 4 for all real x. Find p(x) and q(x).

हिंदी में समझें:

p(0)=5 से b=5 मिलता है। Sum condition से a+c=6 और b+d=4। Difference वाला condition x=3 पर p−q=0 देगा।

Step-by-Step Solution:

p(0)=5 → b=5

p+q = 6x+4 → a+c=6 and b+d=4 → d = −1

p(3)−q(3)=0
3(a−c)+(5−(−1))=0
3(a−c)+6=0
a−c = −2

a+c=6, a−c=−2
→ 2a=4 → a=2
→ c=4

p(x) = 2x + 5
q(x) = 4x − 1

Question 12

Consider a growing pattern of hexagons made with matchsticks, where each new hexagon shares one side with the previous hexagon. Draw the next two stages, complete the stage/matchstick table, find the nth-stage rule, find the 15th-stage number, and decide whether 200 matchsticks can form a complete stage.

हिंदी में समझें:

पहले hexagon में 6 sticks हैं। हर नया hexagon previous hexagon के साथ एक side share करता है, इसलिए नया hexagon 5 नई sticks जोड़ता है। Sequence 6, 11, 16, 21, ... बनेगी।

Step-by-Step Solution:

(i) Next two stages: Stage 4 में 21 sticks और Stage 5 में 26 sticks।

(ii) Table:

Stage	1	2	3	4	5	n
Sticks	6	11	16	21	26	5n + 1

(iii) nth stage rule = 5n + 1.

(iv) 15th stage = 5(15)+1 = 76.

(v) 5n + 1 = 200 → 5n = 199 → n = 39.8, not a whole stage number. इसलिए 200 sticks से complete stage नहीं बनेगा।

Question 13

Let p(x) = ax + b and q(x) = cx + d. The graph of p passes through (2,3) and (6,11); q passes through (4,−1) and is parallel to p. Find p(x), q(x), and the x-axis intersection of each line.

हिंदी में समझें:

p का slope दो points से निकलेगा। Parallel lines की slopes equal होती हैं। फिर q के दिए point से उसका y-intercept निकलेगा।

Step-by-Step Solution:

Slope of p = (11−3)/(6−2) = 8/4 = 2

p(x)=2x+b. Using (2,3): 3=4+b → b=−1.
p(x)=2x−1

q parallel है → slope = 2.
q(x)=2x+d. Using (4,−1): −1=8+d → d=−9.
q(x)=2x−9

p x-intercept: 0=2x−1 → x=1/2 → (1/2,0)
q x-intercept: 0=2x−9 → x=9/2 → (9/2,0)

p(x)=2x−1 and q(x)=2x−9

Question 14

What do all linear functions of the form f(x) = ax + a, where a > 0, have in common?

हिंदी में समझें:

a common है, इसलिए f(x) को factor कर सकते हैं। इससे common zero सीधे दिख जाता है।

Step-by-Step Solution:

f(x)=ax+a=a(x+1)

f(x)=0 when x+1=0 → x=−1

All these functions pass through the common point (−1,0). Their slopes are positive because a > 0.