G8_Hetauda_8_2081

1. Two sets \(A\) and \(B\) are \(A = \{1, 3, 5, 7, 9\}\), \(B = \{2, 3, 5\}\).
1. Present the sets \(A\) and \(B\) in a Venn diagram.[1]
2. Illustrate all the proper subsets that can be made from set \(B\).[2]
1. The Venn diagram for sets \(A = \{1, 3, 5, 7, 9\}\) and \(B = \{2, 3, 5\}\) is shown below.
2. Set \(B = \{2, 3, 5\}\) has \(n = 3\) elements. So, its proper subsets are given below.

   Number of Elements	Proper Subsets	Count
   0	\(\emptyset\)	1
   1	\(\{2\}, \{3\}, \{5\}\)	3
   2	\(\{2,3\}, \{2,5\}, \{3,5\}\)	3
   Total Proper Subsets		7

2. Ram’s monthly income is Rs 40,000. The ratio of his saving to expenditure in a month is 2:3.
   1. In which month does he get an income of Rs 40,000? Find it.[1]
   2. How much amount does he save in a month? Find it.[2]
   3. By how much should Ram’s yearly expenditure be reduced to maintain a yearly expenditure of Rs 2,70,000 only?[2]
   4. How much simple interest will Ram get if he deposits his monthly saving amount in a bank at 10% interest rate for 2 years?[1]
1. Month of income:
   Since the question states "monthly income", he gets an income of Rs 40,000 in every month of the year.
2. Monthly Saving:
   Income = Rs. 40,000
   Ratio (Saving : Expenditure) = 2:3
   Sum of ratios = 2 + 3 = 5
   Thus,
   Monthly Saving = \( \dfrac{2}{5} \times 40,000 = 16,000 \)
   So, he saves Rs. 16,000 in a month.
3. Reduction in yearly expenditure:
   Monthly Expenditure = \( \dfrac{3}{5} \times 40,000 = 24,000 \)
   Current Yearly Expenditure = 24,000 × 12 = Rs. 2,88,000
   Target Yearly Expenditure = Rs. 2,70,000
   Thus,
   Reduction needed = 2,88,000 - 2,70,000 = 18,000
   So, his yearly expenditure should be reduced by Rs. 18,000.
4. Simple Interest:
   Principal (Monthly saving P) = Rs. 16,000
   Time (T) = 2 years
   Rate (R) = 10%
   Thus,
   Interest (I) = \( \dfrac{16,000 \times 2 \times 10}{100} = 3,200 \)
   So, Ram will get Rs. 3,200 as simple interest.
3. Ganesh bought a laptop for Rs 75000 and fixed its marked price \(20\%\) above the cost price. The laptop is sold to Ramesh after allowing a \(20\%\) discount.
1. Write the formula to find the discount percentage.[1]
2. Find the marked price of the laptop.[1]
3. What is Ganesh’s profit or loss percent in this transaction? Calculate it.[2]
1. Formula to find the discount percentage:
   Discount Percent = \( \dfrac{\text{Discount Amount}}{\text{Marked Price}} \times 100\% \)
   orDiscount Percent = \( \dfrac{MP - SP}{MP} \times 100\% \)
2. Marked price of the laptop.
   Cost Price (CP) = Rs. \(75,000\)
   Marked Price is \(20\%\) above CP.
   Thus
   MP = 120% of CP = \( \dfrac{120}{100} \times 75,000 = 90,000 \)
   So, the marked price is Rs. \(90,000\).
3. Profit or loss percent in the transaction.
   Marked Price (MP) = Rs. \(90,000\)
   Discount = \(20\%\)
   Thus
   Selling Price (SP) = 80% of MP = \( \dfrac{80}{100} \times 90,000 = 72,000 \)
   Given that
   Cost Price (CP) = Rs. \(75,000\)
   Since SP < CP, there is a loss. So
   Loss = \(75,000 - 72,000 = 3,000\)
   Now
   Loss Percent = \( \dfrac{3,000}{75,000} \times 100\% = 4\% \)
   So, Ganesh incurred a loss of \(4\%\).
4. The age of Sabin’s father is \(52\) years.
1. Define rational number.[1]
2. Convert the age of Sabin’s father into the quinary number system.[2]
3. How many maximum rational numbers can be made between \(1\) and \(2\)? Give a logical response.[1]
5. The lengths of the diagonals of a rhombus are 8 cm and 12 cm. The diameter of a circle is 7 cm.
   1. Write the formula to find the area of a circle. [1]
   2. Find the area of the rhombus. [1]
   3. How much more or less is the area of the rhombus than the area of the circle? Calculate it. [2]
   4. Are all the triangles formed by intersecting the two diagonals of a rhombus equal in area? Calculate if ‘yes’; give a reason if ‘no’. [1]
1. Area of circle formula
   We know that,
   Area (\( A \)) = \( \pi r^2 \) or \( \frac{\pi d^2}{4} \)
2. Area of the rhombus
   Diagonals (\( d_1 \)) = 8 cm and (\( d_2 \)) = 12 cm
   Area (\( A_r \)) = \( \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 8 \times 12 = 48 \) cm²
3. Comparison of areas
   Diameter of circle (\( d \)) = 7 cm, so Radius (\( r \)) = 3.5 cm.
   Area of circle (\( A_c \)) = \( \frac{22}{7} \times 3.5^2 = 38.5 \) cm².
   The area of the rhombus is more than the circle.
   Difference = \( 48 - 38.5 = 9.5 \) cm² more.
4. Relationship of triangles formed by diagonals
   Yes, all four triangles formed by the diagonals are equal in area.
   Reason: The diagonals of a rhombus bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles.
   Area of each triangle = \( \frac{48}{4} = 12 \) cm²
1. Write the factors of \(a^2 - b^2\).[1]
2. Simplify: \(\dfrac{x^2 - 9}{x + 3} \div \dfrac{x - 3}{x^2 - 5x + 6}\).[2]
7. The product of two consecutive natural numbers is \(12\).
1. Find the two numbers. What number should be added to one of these numbers so that the product becomes the square number \(16\)?[2]
2. Define quadratic equation.[1]
8. Two simultaneous equations are given: \(2x + y = 5\) and \(x - y = 1\).
1. Make a table showing the values of \(y\) for \(x = 0, 1, 2\) in both equations.[2]
2. Find the values of \(x\) and \(y\) by solving the equations graphically.[2]
9. In the adjoining figure, an isosceles triangle is shown with some measurements given.
1. What is the sum of the interior angles of the triangle?[1]
2. Find the values of \(x\) and \(y\).[2]
3. Experimentally verify that the base angles of an isosceles triangle are equal by constructing two isosceles triangles of different measurements.[3]
10. In the figure, the coordinates of vertices \(A(1,4)\) and \(B(1,2)\) of square \(ABCD\) with side \(2\,\text{cm}\) are given.
1. Find the length of diagonal \(BD\) of square \(ABCD\).[2]
2. Construct a square \(ABCD\) having side length \(4cm\) using a compass.[3]
3. Define regular tessellation.[1]
11. On a map, the scale is \(1\,\text{cm} = 2\,\text{km}\). The bearing of point \(Q\), which is \(8\,\text{cm}\) from point \(P\), is \(110^\circ\).
1. Find the actual distance from point \(P\) to point \(Q\).[1]
2. Compare the bearing of \(P\) from \(Q\) and the bearing of \(Q\) from \(P\).[2]
12. The ages (in years) of \(7\) students of class VIII are: \(11, 12, 11, 12, 13, 14, 12\).
1. What is the mode of the above data?[1]
2. Find the average age of the students.[1]
3. How much more or less is the median than the average? Compare.[1]