G8_Jaljala_8_2081

1. Given Venn diagram:
1. Define overlapping set.[1]
2. Make a proper subset of set \(M\).[1]
3. If the element \(b\) is removed from the above Venn diagram, which type of set relation will exist between \(L\) and \(M\)? Write with reason.[1]
1. Let A and B are two sets, then they are overlapping sets if they have at least one common element. In other words, if \(A \cap N \neq \emptyset\), then \(A\) and \(B\) are overlapping sets.
   
2. From the Venn diagram, \(M = \{b, c, d\}\).
   One proper subset of \(M\) is: \(\{c, d\}\)

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

3. If element \(b\) is removed, then
   \(L = \{a, e\}\)
   \(M = \{c, d\}\)
   Since they have no common elements, \(L \cap M = \emptyset\), \(L\) and \(M\) become disjoint sets.
2. Suman went to the furniture shop to buy a set of a table and chairs. The marked price of a set of a table and 4 chairs was Rs 12000.
1. If the price of a table is Rs 6000, then how much is the cost for 4 chairs?[1]
2. Suman gets \(5\%\) discount on the set of table and chairs. Find the price after discount.[2]
3. If the shopkeeper earned \(14\%\) profit even after offering \(5\%\) discount, at what price did the shopkeeper purchase the set?[2]
1. Cost for 4 chairs:
   Marked price of the set (table + 4 chairs) = Rs. \(12,000\)
   Price of the table = Rs. \(6,000\)
   Therefore,
   Cost of 4 chairs = \(12,000 - 6,000 = 6,000\)
   So, the cost for 4 chairs is Rs. \(6,000\).
2. Price after \(5\%\) discount:
   Marked Price (MP) = Rs. \(12,000\)
   Discount = \(5\%\)
   Thus
   SP= 95% of MP
   orSP= \( \dfrac{95}{100} \times 12,000 = 11,400\)
   So, Suman pays Rs. \(11,400\) after the discount.
3. Cost price of the set for the shopkeeper:
   Selling Price after discount = Rs. \(11,400\)
   Profit \(14\%\)
   Thus
   SP = \(114\%\) of CP
   or\(11,400 = \dfrac{114}{100} \times \text{CP}\)
   or\(\text{CP} = \dfrac{11,400 \times 100}{114} = \dfrac{1,140,000}{114} =10000\)
   So, the shopkeeper purchased the set for Rs. \(10000\).
3. A hotel of Mardi Himal has deposited Rs 5,00,000 in bank A and Rs 3,00,000 in bank B at the rate of 6% per annum.
   1. Write the formula to find simple interest.[1]
   2. How much interest does the hotel earn in 4 years from bank A?[1]
   3. Find the ratio of interest received from bank A and bank B in 4 years.[2]
1. Formula to find simple interest:
   Simple Interest (I) = \( \dfrac{P \times T \times R}{100} \)
2. Interest from Bank A:
   Principal (P) = Rs. 5,00,000
   Time (T) = 4 years
   Rate (R) = 6%
   Thus,
   Interest (I) = \( \dfrac{5,00,000 \times 4 \times 6}{100} = 1,20,000 \)
   So, the hotel earns Rs. 1,20,000 from Bank A.
3. Ratio of interest from Bank A and Bank B:
   Interest from Bank A (\(I_A\)) = Rs. 1,20,000
   Interest from Bank B (\(I_B\)) = \( \dfrac{3,00,000 \times 4 \times 6}{100} = 72,000 \)
   Thus,
   Ratio = \( \dfrac{1,20,000}{72,000} = \dfrac{5}{3} \)
   So, the ratio of interest received from Bank A and Bank B is 5:3.
4. There are two tanks. The first tank contains \(2.64 \times 10^3\) liters and the second contains \(3.56 \times 10^2\) liters of water.
1. How much water is there altogether from both tanks? Express in scientific notation.[1]
2. How much water should be added to the second tank to make it equal to the first tank?[1]
3. Convert \(0.57\) into a fraction.[1]
4. Prove that \(2081 = 3131_5\).[2]
5. There is a football ground with 90 m length and 60 m width inside a circular stadium with diameter 140m.
   
   1. Write the formula to find the radius when diameter is given. [1]
   2. Find the area of the football ground. [1]
   3. Find the area of the circular stadium excluding the football ground. [2]
   4. Find the cost of planting grass on the football ground at the rate of Rs 250 per \(m^2\). [1]
1. Radius formula
   We know that,
   Radius (\( r \)) = \( \frac{\text{Diameter (d)}}{2} \)
2. Area of the football ground
   Given that,
   Length (\( l \)) = 90 m
   Width (\( b \)) = 60 m
   Now using the formula, we get
   Area (\( A_g \)) = \( l \times b = 90 \times 60 = 5400 \) m²
3. Area of stadium excluding football ground
   For the circular stadium,
   Diameter (\( d \)) = 140 m
   Radius (\( r \)) = \( \frac{140}{2} = 70 \) m
   Calculating total area of stadium,
   Total Area (\( A_s \)) = \( \pi r^2 = \frac{22}{7} \times 70^2 = 15400 \) m²
   Now to find area excluding ground,
   Required Area = \( A_s - A_g = 15400 - 5400 = 10000 \) m²
4. Cost of planting grass
   We know that,
   Area of football ground (\( A_g \)) = 5400 m²
   According to the question, the rate is Rs 250 per \(m^2\), so
   Total Cost = \( 5400 \times 250 = \) Rs 13,50,000
1. Express \(\dfrac{x^a}{x^{-b}}\) as a power of \(x\).[1]
2. Simplify: \(\dfrac{x^2 + 5x + 6}{x^2 - 9} \times \dfrac{x - 3}{x + 3}\)[2]
7. Two equations are given below: \(x + y = 6\) and \(2x + y = 9\).
1. What is this system of equations called?[1]
2. Solve the above equations using a graph.[2]
1. Find the Highest Common Factor (H.C.F.) of \(x^2 - 7x + 10\) and \(x^2 - 4\).[2]
2. At what value of \(m\) does the expression \(m^2 - 5m + 6\) become zero?[2]
9. In the figure, \(AB \parallel CD\) and \(QR\) is a transversal.
1. Write the alternate angle of \(\angle BIJ\).[1]
2. What type of triangle is \(\triangle IKJ\) according to its angles?[2]
3. At what value of \(\angle AIK\) will the lines \(AB\) and \(CD\) be parallel?[1]
1. In the given figure, \(AB = CD\) and \(AB \parallel CD\). Prove that \(\triangle AOB \cong \triangle COD\).[2]
2. Construct a rectangle \(ABCD\) in which \(AB = 7cm\) and \(AD = 4cm\).[3]
11. Answer the following questions:
1. What type of quadrilateral is used to make a regular tessellation?[1]
2. From the given figure, write the bearing of point \(B\) from point \(A\) and the bearing of point \(A\) from point \(B\).[2]
3. Find the vertices of the image square \(A'B'C'D'\) formed by reflecting square \(ABCD\) with vertices \(A(2,3)\), \(B(6,3)\), \(C(6,7)\), and \(D(2,7)\) in the \(x\)-axis. Also draw the graph of the reflection.[3]
1. In a class test of mathematics, 10 students obtained the following marks: \(10, 15, 12, 15, 14, 12, 15, 14, 15, 10\). Find its mode.[1]
2. A man with a monthly income of \(\text{Rs.}\,30{,}000\) plans his budget as follows:

   Items	Food	Education	Miscellaneous
   Expenditure (Rs.)	15000	6000	9000

   Present the above data in a pie chart.[2]