Pokhara 2082 - Question Paper solution

Pokhara BLE Class 8 Math Exam Paper 2082 – Math Mantra Nepal

Pokhara Metropolitan City – Class 8 BLE Model Exam

Pokhara Metropolitan City, Kaski

पोखरा महानगरपालिका, कास्की

Basic Level Examination – 2082 (आधारभूत तह उत्तीर्ण परीक्षा २०८२)

Subject: Compulsory Mathematics (विषय : अनिवार्य गणित)

Class: 8 (कक्षा : ८)

Full Marks: 50 (पूर्णाङ्क : ५०)

Time: 2 Hours (समय : २ घण्टा)

Attempt all the questions. (सबै प्रश्नहरू अनिवार्य छन्।)

1. Study the given Venn-diagram and answer the following questions. १. दिइएको भेन चित्र अध्ययन गर्नुहोस् र तलका प्रश्नहरूको उत्तर दिनुहोस् ।

a. Identify and write, whether the sets A and B are overlapping or disjoint.

क. समूह A र B खप्टिएका वा अलगिएका समूह के हुन् ? लेख्नुहोस् ।

[1]

b. Write the improper subset of set B.

ख. समूह B को अनुपयुक्त उपसमूह लेख्नुहोस् ।

[1]

c. Which one element is to be added in set B so that it becomes overlapping with set A?

ग. समूह B मा कुन एउटा सदस्य थप गरेमा समूह A सँग खप्टिएको समूह बन्छ ?

[1]

2. The marked price of a computer set kept in a stall of Pokhara Festival is Rs 50,000 and the festival discount of 10% is allowed in the marked price of the computer set. २. पोखरा महोत्सवको स्टलमा राखिएको एउटा कम्प्युटरको अंकित मूल्य रु. ५०,००० छ । उक्त कम्प्युटरको मूल्यमा १०% महोत्सव छुट गरिएको रहेछ ।

Festival Offer

Pokhara Festival Computer Stall

Marked Price (M.P.) = Rs. 50,000

Discount Rate = 10%

a. If M.P. denotes the marked price and D denotes the discount amount, write the formula to calculate the selling price (S.P.).

क. यहाँ M.P. ले अंकित मूल्य र D ले छुट रकमलाई जनाउँछन् भने विक्रय मूल्य (S.P.) पत्ता लगाउने सूत्र लेख्नुहोस् ।

[1]

b. Find the discount amount that a customer gets while purchasing the computer set.

ख. उक्त कम्प्युटर सेट किन्दा ग्राहकले प्राप्त गर्ने छुट रकम पत्ता लगाउनुहोस् ।

[1]

c. If the cost price of the computer set for the shopkeeper is RS 36,000, what is the percentage profit to loss. Find it.

ग. सो कम्प्युटर सेटको क्रय मूल्य रु ३६,००० हो भने व्यापारीलाई कति प्रतिशत नाफा वा नोक्सान हुन्छ ? पत्ता लगाउनुहोस् ।

[2]

Solution (Strict Step-by-Step Vertical Method):

a. Formula for Selling Price:

$$\text{Selling Price (S.P.)} = \text{M.P.} – \text{D}$$

b. Given:

Marked Price ($\text{M.P.}$) = Rs. 50,000

Discount Rate ($\text{D}\%$) = 10%

Discount Amount ($\text{D}$) = $\text{D}\% \text{ of M.P.}$

$= \frac{10}{100} \times 50000$

$= \text{Rs. } 5,000$

c. Profit / Loss Percentage Calculation:

Method 1 (Step-Wise):

Selling Price ($\text{S.P.}$) = $\text{M.P.} – \text{D}$

$= 50000 – 5000$

$= \text{Rs. } 45,000$

Alternative Direct Formula:

$$\text{S.P.} = \frac{(100 – D\%) \times \text{M.P.}}{100}$$

$= \frac{(100 – 10) \times 50000}{100}$

$= \frac{90 \times 50000}{100} = \text{Rs. } 45,000$

Given Cost Price ($\text{C.P.}$) for shopkeeper = Rs. 36,000

Since $\text{S.P.} > \text{C.P.}$ ($45,000 > 36,000$), the shopkeeper makes a Profit.

Profit Amount = $\text{S.P.} – \text{C.P.}$

$= 45000 – 36000$

$= \text{Rs. } 9,000$

Profit Percentage = $\frac{\text{Profit Amount}}{\text{C.P.}} \times 100\%$

$= \frac{9000}{36000} \times 100\%$

$= \frac{1}{4} \times 100\% = 25\%$ Profit

3. Shiva has deposited Rs. 80,000 in a bank for 3 years at the rate of Rs. 10 simple interest per annum for Rs 100. ३. शिवले रु १०० को एक वर्षको १० रुपैयाँको साधारण व्याज दरले बैंकमा ३ वर्षका लागि रु ८०,००० जम्मा गरेका छन् ।

a. Write the formula to calculate the simple interest (I).

क. साधारण व्याज (I) पत्ता लगाउने सूत्र लेख्नुहोस् ।

[1]

b. How much interest does Shiva receive in 3 years? Find it.

ख. शिवले ३ वर्षमा कति व्याज प्राप्त गर्छन् ? पत्ता लगाउनुहोस् ।

[2]

c. Shiva has decided to distribute the principal amount Rs 80,000 to his two daughters and a son in the ratio of their ages. If the ages of his two daughters are 10 years and 4 years respectively and that of the son is 2 years, how much less amount does the son get than the eldest daughter?

ग. शिवले आफ्ना दुईवटा छोरीहरू र एउटा छोरालाई उनीहरूको उमेरको अनुपातमा साँवा रकम रु. ८०,००० लाई बाँड्ने निर्णय गरेछन् । यदि दुईटा छोरीहरूको उमेर क्रमशः १० वर्ष र ४ वर्ष रहेछ, साथै छोराको उमेर २ वर्ष रहेछ भने छोराले पाउने रकम जेठी छोरीको भन्दा कति कम हुन्छ ?

[2]

Solution (Exact Vertical Step-by-Step):

Derivation of Rate of Interest ($R$) from given condition:

According to the question, simple interest is Rs. 10 per year on Rs. 100.

$$\text{Principal } (P) = \text{Rs. } 100, \quad \text{Time } (T) = 1 \text{ year}, \quad \text{Interest } (I) = \text{Rs. } 10$$

Formula for Rate ($R$):

$$R = \frac{I \times 100}{P \times T} = \frac{10 \times 100}{100 \times 1} = 10\% \text{ p.a.}$$

a. Simple Interest Formula:

$$I = \frac{P \times T \times R}{100}$$

b. Given for 3 Years:

Principal ($P$) = Rs. 80,000

Time ($T$) = 3 years

Rate ($R$) = 10% per annum (derived above)

Simple Interest ($I$) = $\frac{P \times T \times R}{100}$

$= \frac{80000 \times 3 \times 10}{100}$

$= \text{Rs. } 24,000$

c. Ratio & Share Calculation:

Age of Eldest Daughter = 10 years

Age of Second Daughter = 4 years

Age of Son = 2 years

Ratio of ages = $10 : 4 : 2$

Let the respective shares be $10x$, $4x$, and $2x$.

Sum of ratios = $10x + 4x + 2x = 16x$

Total Amount to be distributed = Rs. 80,000

$16x = 80000$

$\implies x = \frac{80000}{16} = 5000$

Calculating Shares:

Share of Eldest Daughter ($10x$) = $10 \times 5000 = \text{Rs. } 50,000$

Share of Second Daughter ($4x$) = $4 \times 5000 = \text{Rs. } 20,000$

Share of Son ($2x$) = $2 \times 5000 = \text{Rs. } 10,000$

Difference in Amount:

$= \text{Eldest Daughter’s Share} – \text{Son’s Share}$

$= 50000 – 10000$

$= \text{Rs. } 40,000$

$\therefore$ The son gets Rs. 40,000 less than the eldest daughter.

4. The monthly salary of a teacher is Rs 45,975. ४. एउटा शिक्षकको मासिक तलब रु ४५,९९५ छ ।

a. Write the monthly salary of the teacher in scientific notation.

क. उक्त शिक्षकको मासिक तलबलाई वैज्ञानिक सङ्केतमा लेख्नुहोस् ।

[1]

b. Convert the aforementioned monthly salary of the teacher in quinary number system.

ख. माथि उल्लेखित शिक्षकको मासिक तलबलाई पञ्चआधार पद्धतिमा रूपान्तरण गर्नुहोस् ।

[1]

c. The teachers receive the additional amount of the salary of a month per year as a festival allowance. If the total expenses on making a meeting hall is Rs. 6,00,000, is it sufficient to pay the expenses on making the hall by the yearly income of the teacher? Write with the reason.

ग. शिक्षकहरूले प्रत्येक वर्ष चाडपर्व खर्च बापत एक महिनाको तलब बराबरको रकम थप प्राप्त गर्दछन् । यदि एउटा बैठक कक्ष निर्माण गर्दा रु. ६,००,००० खर्च लाग्छ भने सो शिक्षकको वर्षभरिको आम्दानीले बैठक कक्ष निर्माण खर्च तिर्न पुग्ला ? कारणसहित लेख्नुहोस् ।

[1]

d. Convert $0.\bar{23}$ into fraction.

घ. $0.\bar{23}$ (0.23) लाई भिन्नमा रूपान्तरण गर्नुहोस् ।

[2]

Solution (Strict Step-by-Step Vertical Method):

a. Scientific Notation:
Rs. 45,975 = $4.5975 \times 10^4$

b. Quinary (Base-5) System Conversion:

Step-by-Step Division Ladder (Handwritten Note):

5 45975 Remainder: 0

5 9195 Remainder: 0

5 1839 Remainder: 4

5 367 Remainder: 2

5 73 Remainder: 3

5 14 Remainder: 4

5 2 Remainder: 2

Read
Remainders
Bottom to Top

$\therefore 45,975 = (2432400)_5$

c. Annual Income vs Hall Construction Expenses:
Total Months Income = 12 months salary + 1 month festival allowance = 13 months salary
$= 13 \times 45,975 = \text{Rs. } 5,97,675$
Meeting Hall Cost = Rs. 6,00,000
Comparison: Rs. 5,97,675 < Rs. 6,00,000.
No, it is NOT sufficient. The total annual income (Rs. 5,97,675) is short by Rs. 2,325 to cover the hall construction expense (Rs. 6,00,000).

d. Converting $0.\bar{23}$ into Fraction (Step-by-Step):

Let $x = 0.\bar{23}$

$\implies x = 0.232323…$ — (i)

Multiplying equation (i) by $100$:

$100 \times x = 100 \times 0.232323…$

$\implies 100x = 23.232323…$

$\implies 100x = 23.\bar{23}$ — (ii)

Work Space & Concept Explanation:

Decimal Shift Calculation:
When multiplying $0.232323… \times 100$, since $100$ has two zeros, the decimal point shifts 2 places to the right $\rightarrow 23.232323…$, which is abbreviated back with the recurring bar as $23.\bar{23}$.

Common Student Error to Avoid:

❌ Incorrect: Writing $100x = 23.23$ (omitting the recurring bar). If you write $23.23$ as a terminating decimal, subtracting $x = 0.232323…$ yields $22.997677…$ instead of an exact integer $23$!

✔️ Correct: Always retain the recurring bar ($23.\bar{23}$ or $23.232323…$) so that the infinite decimal digits cancel out perfectly!

Subtracting equation (i) from equation (ii) vertically:

100x = 23.232323…

– x = 0.232323…

99x = 23.000000…

Solving for $x$:

$99x = 23$

$\implies x = \frac{23}{99}$

$\therefore 0.\bar{23}$ in fraction form is $\frac{23}{99}$.

5. A circular swimming pool of radius 14 m is constructed within the rectangular garden having length 70 m and breadth 56 m as shown in the figure. ५. चित्रमा देखाए अनुसार लम्बाइ 70 m र चौडाइ 56 m भएको आयताकार चौरको भित्र 14 m अर्धव्यास भएको वृत्ताकार पोखरी निर्माण गरिएको छ ।

a. Write the formula to find the area covered by the circular swimming pool in the ground.

क. वृत्ताकार पोखरीले चौरमा ओगटेको क्षेत्रफल निकाल्ने सूत्र लेख्नुहोस् ।

[1]

b. What is the area covered by the swimming pool in the ground? Find it.

ख. पोखरीले चौरमा ओगटेको क्षेत्रफल कति हुन्छ ? पत्ता लगाउनुहोस् ।

[1]

c. Find the area of the remaining portion of the ground except the swimming pool.

ग. चौरमा पोखरी बाहेक बाँकी रहेको भागको क्षेत्रफल निकाल्नुहोस् ।

[2]

d. If the cost of fencing the circumference of the pond is Rs 2,000 per metre, is it sufficient to cover the expenses of making the fence by the total amount of Rs 6,00,000? Write with the reason.

घ. यदि सो पोखरीको वरिपरि घेरामा रु २,००० प्रति मिटरका दरले बार लगाउने हो भने पोखरीमा बार बनाउदा लाग्ने जम्मा खर्च रु. ६,००,००० ले पर्याप्त होला ? कारण सहित लेख्नुहोस् ।

[1]

6. a. What is the value of $x^0$?

६. क. $x^0$ को मान कति हुन्छ ?

[1]

b. Simplify:

ख. सरल गर्नुहोस् :

$$\frac{3}{x-a} + \frac{4}{x+a}$$

[2]

7. a. Solve graphically the given system of linear equations:

७. क. दिइएका रेखीय समीकरणहरू लेखाचित्र विधिबाट समाधान गर्नुहोस् ।

$2x + y = 8$ and $x + y = 5$

[2]

b. What is the equation $ax^2 + bx + c = 0, a \neq 0$ called?

ख. $ax^2 + bx + c = 0$ समीकरणमा $a \neq 0$ छ भने, उक्त समीकरणको नाम लेख्नुहोस् ।

[1]

Solution (Graphical Method Step-by-Step):

1. For Equation (i): $2x + y = 8 \implies y = 8 – 2x$

x	1	3	4
y = 8 – 2x	6	2	0

Ordered pairs: $(1, 6)$, $(3, 2)$, $(4, 0)$

2. For Equation (ii): $x + y = 5 \implies y = 5 – x$

x	0	3	5
y = 5 – x	5	2	0

Ordered pairs: $(0, 5)$, $(3, 2)$, $(5, 0)$

Graph Plotting Visualization (Intersection at P(3, 2))

Point of Intersection = $(3, 2) \implies x = 3, y = 2$

b. The standard form equation $ax^2 + bx + c = 0$ (where $a \neq 0$) is called a Quadratic Equation.

8. a. Find the L.C.M. of:

८. क. ल. स. निकाल्नुहोस् :

$a^2 + a – 6$ and $a^2 – 9$

[2]

b. What are the possible values of $x$ that can satisfy the equation $x^2 – 3x – 4 = 0$?

ख. समीकरण $x^2 – 3x – 4 = 0$ मान्य हुनका लागि $x$ का सम्भाव्य मानहरू के के होलान् ? पत्ता लगाउनुहोस् ।

[2]

9. Observe the given figure and answer the following questions. ९. दिइएको चित्र अवलोकन गरी तलका प्रश्नहरूको उत्तर दिनुहोस् ।

a. Write a pair of corresponding angles.

क. एक जोडा सङ्गत कोणहरू उल्लेख गर्नुहोस् ।

[1]

b. Find the measurement of $\angle ECD$.

ख. $\angle ECD$ को नाप पत्ता लगाउनुहोस् ।

[2]

c. For what value of $x$, the lines AB and CD become parallel?

ग. $x$ को मान कति भएमा रेखाहरू AB र CD समानान्तर हुन्छन् ?

[1]

10. a. Construct a parallelogram ABCD with $AB = 8\text{ cm}, BC = 6\text{ cm}$ and $\angle ABC = 135^\circ$.

१०. क. $AB = 8\text{ cm}, BC = 6\text{ cm}$ र $\angle ABC = 135^\circ$ भएको एउटा समानान्तर चतुर्भुज ABCD रचना गर्नुहोस् ।

[3]

b. Make the triangles ADC and ABC from the parallelogram ABCD. Prove that $\Delta ADC$ and $\Delta ABC$ are congruent.

ख. समानान्तर चतुर्भुज ABCD बाट दुईवटा त्रिभुज ADC र ABC बनाउनुहोस् । सोही चित्रमा बनेका $\Delta ADC$ र $\Delta ABC$ अनुरुप हुन्छन् भनी प्रमाणित गर्नुहोस् ।

[2]

Steps of Construction for Parallelogram ABCD:

Step 1: Draw a line segment $AB = 8\text{ cm}$ using a ruler.
Step 2: At point $B$, construct an angle $\angle ABC = 135^\circ$ using a compass ($90^\circ + 45^\circ$).
Step 3: From point $B$, cut an arc of length $6\text{ cm}$ on the ray of $135^\circ$ to mark vertex $C$ (since $BC = 6\text{ cm}$).
Step 4: With $C$ as center and radius $8\text{ cm}$ (opposite side $CD = AB$), draw an arc.
Step 5: With $A$ as center and radius $6\text{ cm}$ (opposite side $AD = BC$), draw another arc intersecting the previous arc at point $D$.
Step 6: Join $AD$ and $CD$ to complete the required parallelogram $ABCD$.

Proof (Step-by-Step Table in English):

To Prove: $\Delta ADC \cong \Delta ABC$

Parallelogram $ABCD$ with Diagonal $AC$ (S.S.S. Markings)

― AB = CD (Single Tick) ⸗ AD = BC (Double Tick) — AC = AC (Common Side)

Statements	Reasons
1. $AD = BC$ (Side)	1. Opposite sides of a parallelogram are equal.
2. $DC = AB$ (Side)	2. Opposite sides of a parallelogram are equal.
3. $AC = AC$ (Side)	3. Common side in both triangles.
4. $\Delta ADC \cong \Delta ABC$	4. By S.S.S. Axiom. (Proved)

11. a. Write the name of the tessellation that can be formed by using equilateral triangles.

११. क. समबाहु त्रिभुजहरू प्रयोग गरेर बनाउन सकिने टेसेलसनको नाम लेख्नुहोस् ।

[1]

b. In the given figure, the bearing of the point B from the point C is $250^\circ$. Find the bearing of the point C from the point B.

ख. दिइएको चित्रमा बिन्दु B को दिशास्थिति बिन्दु C बाट $250^\circ$ छ भने बिन्दु B बाट बिन्दु C को दिशास्थिति पत्ता लगाउनुहोस् ।

Bearing Diagram (दिशास्थिति चित्र)

[2]

c. $A(-3, 2), B(-5, 4)$ and $C(-2, 6)$ are the vertices of $\Delta ABC$. Present $\Delta ABC$ on graph and rotate $\Delta ABC$ about the origin through positive $90^\circ$. Write the coordinates of the images $A’, B’, C’$.

ग. $\Delta ABC$ का शीर्षबिन्दुहरू $A(-3, 2), B(-5, 4)$ र $C(-2, 6)$ छन् । $\Delta ABC$ लाई रेखाचित्रमा प्रस्तुत गरी उद्गम बिन्दु O को वरिपरि धनात्मक दिशामा $90^\circ$ को कोणले परिक्रमण गर्नुहोस् र परिक्रमण पछि बनेको प्रतिबिम्बका शीर्षबिन्दुहरू $A’, B’, C’$ का निर्देशकअङ्क लेख्नुहोस् ।

[3]

Interactive Graph Simulator: $+90^\circ$ Rotation about Origin $(0,0)$

12. The expenditure of five months of a family is given in the table below. १२. एउटा परिवारको पाँच महिनाको खर्च तलको तालिकामा दिइएको छ ।

Shrawn (साउन)	Bhadra (भदौ)	Aswin (असोज)	Kartik (कार्तिक)	Mangsir (मंसिर)
Rs. 30,000	Rs. 25,000	Rs. 50,000	Rs. 45,000	Rs. 3,000

a. What is the average monthly expenditure of the given family?

क. माथि उल्लेखित परिवारको मासिक औसत खर्च कति रहेछ ?

[1]

b. Present the expenditure of the five months of the family in a pie chart.

ख. उक्त परिवारको पाँच महिनाको खर्चलाई वृत्तचित्रमा प्रस्तुत गर्नुहोस् ।

[2]

Expenditure Pie Chart (५ महिनाको खर्चको पाई-चार्ट)

THE END

Pokhara Metropolitan City • Basic Level Examination (Class 8) Model Question Paper

हाम्रा आधिकारिक सामाजिक सञ्जालहरू

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