G8_Lalitpur_8_2081

1. If set \(A = \{\text{even numbers up to }15\}\) and set \(B = \{\text{prime numbers up to }15\}\),
1. Define overlapping sets.[1]
2. Make any two proper subsets from set \(B\).[1]
3. What change in the outcome of set \(B\) makes the two sets \(A\) and \(B\) disjoint?[1]
1. Two sets are called overlapping sets if they have at least one element in common.
   In other words, sets \(A\) and \(B\) are overlapping if \(A \cap B \neq \emptyset\).
2. Given that,
   \(B = \{\text{prime numbers up to }15\}= \{2, 3, 5, 7, 11, 13\}\)
   Two proper subsets of \(B\) are
   \(\{2, 3\}\) and \(\{5, 7, 11\}\)
3. Given that,
   \(A = \{\text{even numbers up to }15\}= \{2, 4, 6, 8, 10, 12, 14\}\)
   To make \(A\) and \(B\) disjoint, the common element \(2\) must be removed from set \(B\).
2. The ratio of boys and girls of class eight of a school is \(5:7\) respectively. If the total number of students is \(60\), then
1. How many girls are there? Find it.[1]
2. Change the total number of students into the binary number system.[1]
3. Write 350000 in scientific notation.[1]
4. Convert \(1.\overline{24}\) into a fraction.[1]
3. Raju Lama visited a computer store to get \(5\) printers and a laptop. A set of \(5\) printers and a laptop is available for Rs 465000.
1. If \(15\%\) discount is allowed on those machineries, find the discount amount.[2]
2. If the shopkeeper earned \(20\%\) profit even after allowing \(15\%\) discount, at what price did the shopkeeper purchase such machineries?[2]
3. By how much is the marked price of a laptop less or more than Rs 95000, if the price of a printer is Rs 75000?[1]
1. Discount amount when \(15\%\) discount is allowed.
   Marked Price (MP) of the set = Rs. \(465,000\)
   Discount = \(15\%\)
   Thus
   Discount Amount = \( \dfrac{15}{100} \times 465,000 = 69,750 \)
   So, the discount amount is Rs. \(69,750\).
2. Cost price of the machineries
   Marked Price (MP) = Rs. \(465,000\)
   Discount = \(15\%\)
   Thus
   Selling Price (SP) = 85% of MP = \( \dfrac{85}{100} \times 465,000 = 395,250 \)
   Profit = \(20\%\)
   Thus
   SP = 120% of CP
   or\(395,250 = \dfrac{120}{100} \times \text{CP}\)
   or\(\text{CP} = \dfrac{395,250 \times 100}{120} = \dfrac{39,525,000}{120} = 329,375\)
   So, the shopkeeper purchased the machineries for Rs. \(329,375\).
3. Difference between the laptop’s marked price .
   Price of one printer = Rs. \(75,000\)
   Price of 5 printers = \(5 \times 75,000 = 375,000\)
   Total marked price of set = Rs. \(465,000\)
   Thus
   Marked Price of Laptop = \(465,000 - 375,000 = 90,000\)
   Thus
   \(95,000 - 90,000 = 5,000\)
   Marked price of the laptop is Rs. 5,000 less than Rs. 95,000.
4. Dilliram deposited Rs 120,000 in a bank at the rate of 12% per annum.
   1. Write a formula to calculate simple interest.[1]
   2. How much interest does Dilliram get after 6 years?[1]
   3. If he has to pay 5% of interest as income tax, how much amount will he receive after 6 years?[2]
1. Formula for Simple Interest:
   Simple Interest (I) = \( \dfrac{P \times T \times R}{100} \)
2. Interest after 6 years:
   Principal (P) = Rs. 120,000
   Time (T) = 6 years
   Rate (R) = 12%
   Thus,
   Interest (I) = \( \dfrac{120,000 \times 6 \times 12}{100} = 86,400 \)
   So, he gets Rs. 86,400 as interest.
3. Total amount after tax deduction:
   Total Interest = Rs. 86,400
   Income Tax (5%) = \( \dfrac{5}{100} \times 86,400 = 4,320 \)
   Net Interest = 86,400 - 4,320 = Rs. 82,080
   Total Amount Received (A) = Principal + Net Interest = 120,000 + 82,080 = 202,080
   So, he will receive a total of Rs. 202,080.
5. Mohan constructed a rectangular garden and a circular fish pond in the garden of his house with equal areas.
   
   1. Find the area of a triangle having base b cm and height h cm. [1]
   2. Find the area of the circular fish pond. [1]
   3. Find the perimeter of the rectangular garden. [2]
   4. Which of the garden or fish pond needs more cost to fence at the same rate of cost? [1]
1. Area of triangle formula
   We know that,
   Area (\( A \)) = \( \frac{1}{2} \times b \times h \)
2. Area of the fish pond
   Given: Radius (\( r \)) = 14 m
   Area (\( A_p \)) = \( \pi r^2 = \frac{22}{7} \times 14^2 = 616 \) m²
3. Perimeter of the rectangular garden
   Area of garden = Area of pond = 616 m²
   Length (\( l \)) = 56 m
   Breadth (\( b \)) = \( \frac{\text{Area}}{\text{Length}} = \frac{616}{56} = 11 \) m
   Perimeter (\( P_r \)) = \( 2(l + b) = 2(56 + 11) = 134 \) m
4. Comparison of fencing cost
   Perimeter of pond (Circumference) = \( 2\pi r = 2 \times \frac{22}{7} \times 14 = 88 \) m
   Perimeter of garden = 134 m
   Since the perimeter of the garden is greater than that of the pond,
   The rectangular garden needs more cost to fence.
1. What exponent of \(y\) will be equal to \(1\)?[1]
2. Prove that: \((x^{a-b})^{a+b} \cdot (x^{b-c})^{b+c} \cdot (x^{c-a})^{c+a} = 1\).[2]
7. An algebraic fraction is given: \(\dfrac{x}{x^2 + 3x + 2}\) \(\div\) \(\dfrac{2}{x^2 - 1}\).
1. Find the L.C.M. of the denominators.[1]
2. Simplify the given fraction and reduce it to the lowest term.[2]
8. There are two numbers \(x\) and \(y\) such that their sum is \(8\) and difference is \(4\).
1. Construct the simultaneous equations based on the given statements.[1]
2. What are the numbers? Calculate by using graphical method.[2]
9. Study the given figure and answer the following questions.
1. Find the value of \(x\) and \(y\).[2]
2. Compare the angles \(x\) and \(y\).[1]
3. Draw a bearing angle of \(030^\circ\).[1]
10. Study the given figure and answer the following questions.
1. Define congruent figures.[1]
2. If \(\triangle ADE \sim \triangle ABC\), find the length of \(DE\).[2]
3. In a regular polyhedron, the number of vertices is \(8\) and the number of edges is \(12\). Calculate the number of faces by using Euler’s formula.[2]
1. Construct a parallelogram having adjacent sides \(7cm\) and \(4cm\) and the angle between them is \(60^\circ\).[3]
2. The vertices of \(\triangle ABC\) are \(A(-3,2)\), \(B(5,3)\), and \(C(1,6)\). Sketch it on graph paper and reflect it in the \(x\)-axis. Write down the coordinates of the image.[3]
12. The total expenditure of a family is \(\text{Rs.}\,36{,}000\) for four months. Expenditure of each month is shown in the adjoining pie chart.
1. Find the expenditure in each month.[2]
2. How much is the average expenditure of such a family in one month? Calculate it.[1]