G8_Pokhara_8_2081

1. If the sets \(A = \{3, 4, 5, 7\}\) and \(B = \{2, 3, 4, 9\}\),
1. Which of the sets \(A\) and \(B\) are overlapping or disjoint? Write it.[1]
2. Form any two proper subsets of set \(A\).[1]
3. If set \(C = \{2, 3, 4, 9\}\), then the set \(C\) is an proper subset of set \(B\). Give a reason.[1]
1. Sets \(A\) and \(B\) are overlapping sets.
   Because they share common elements \(3\) and \(4\), i.e., \(A \cap B = \{3, 4\} \neq \emptyset\).
2. Set \(A = \{3, 4, 5, 7\}\).
   Two proper subsets of \(A\) are
   \(\{3, 4\}\) and \(\{5, 7\}\)

   Number of Elements	Proper Subsets	Count
   0	\(\emptyset\)	1
   1	\(\{3\}, \{4\}, \{5\}, \{7\}\)	4
   2	\(\{3,4\}, \{3,5\}, \{3,7\}, \{4,5\}, \{4,7\}, \{5,7\}\)	6
   3	\(\{3,4,5\}, \{3,4,7\}, \{3,5,7\}, \{4,5,7\}\)	4
   Total Proper Subsets		15

3. Given \(B = \{2, 3, 4, 9\}\) and \(C = \{2, 3, 4, 9\}\), we see that \(C = B\).
   Since \(C\) contains exactly the same elements as \(B\), we say \(C\) is a improper subset of \(B\).
2. The marks obtained by \(12\) students of Grade Eight in the Annual Examination in Mathematics are given below: \(22, 30, 25, 26, 24, 29, 27, 27, 28, 31, 33, 23\).
1. Sita said, “Mean divides the given data into two equal parts.” But Gita said, “Median divides the given data into two equal parts.” Whose statement is true?[1]
2. Find the mean from the above data.[2]
3. The monthly salary of Hari Rana is \(\text{Rs.}\,36{,}000\).
1. Write the amount in scientific notation.[1]
2. Express the number in the quinary number system.[2]
3. If Hari’s salary is twice as much as Suraj’s salary and Suraj’s salary is thrice as much as Shyam’s salary, find the annual salary of Shyam.[2]
4. Ram has a stationery shop at his house. He sells a calculator with a marked price Rs 450 at a discount of \(12\%\), and he sells a ball at Rs 850 after giving \(15\%\) discount.
1. If marked price (\(M\)) and discount (\(D\)) are given, then write the formula to find the price after discount.[1]
2. How much is the discount amount of the calculator? Find it.[1]
3. Find the marked price of the ball.[1]
4. By how much is the discount amount on the calculator more or less than the discount amount on the ball?[1]
1. Formula to find the price after discount:
   Price after discount = \( M - D \)
   or, if discount is given as a percentage \(D\%\):
   Price after discount = \( (100-D)\% \times M \)
2. Discount amount of the calculator.
   Marked Price (\(M\)) = Rs. \(450\)
   Discount = \(12\%\)
   Thus
   Discount Amount = \( \dfrac{12}{100} \times 450 = 54 \)
   So, the discount amount on the calculator is Rs. \(54\).
3. Marked price of the ball.
   Selling Price (SP) = Rs. \(850\)
   Discount = \(15\%\)
   Thus
   SP = 85% of Marked Price (MP)
   or\(850 = \dfrac{85}{100} \times \text{MP}\)
   or\(\text{MP} = \dfrac{850 \times 100}{85} = 1,000\)
   So, the marked price of the ball is Rs. \(1,000\).
4. Difference between discount amounts.
   Discount on calculator = Rs. \(54\) (from part ii)
   Discount on ball = \(15\%\) of\(1,000 = \dfrac{15}{100} \times 1,000 = 150\)
   Thus
   Difference = \(150 - 54 = 96\)
   So, the discount amount on the calculator is Rs. \(96\) less than that on the ball.
5. Rina took a loan of Rs 50,000 from Pawan at 18% per annum simple interest for 3 years.
   1. How much simple interest is paid by Rina in 3 years? Find it.[1]
   2. Find the total amount of principal and interest to be paid by Rina.[1]
   3. If Rina took the loan at 12% per annum, then how much less interest would she have to pay?[2]
1. Interest paid in 3 years:
   Principal (P) = Rs. 50,000
   Time (T) = 3 years
   Rate (R) = 18%
   We know,
   Interest (I) = \( \dfrac{P \times T \times R}{100} = \dfrac{50,000 \times 3 \times 18}{100} = 27,000 \)
   So, Rina has to pay Rs. 27,000 as interest.
2. Total amount to be paid:
   Amount (A) = Principal (P) + Interest (I)
   A = 50,000 + 27,000 = 77,000
   So, the total amount to be paid is Rs. 77,000.
3. Difference in interest if rate was 12%:
   Original Rate = 18%, New Rate = 12%
   Difference in Rate = 18% - 12% = 6%
   Thus,
   Interest saved = \( \dfrac{50,000 \times 3 \times 6}{100} = 9,000 \)
   So, she would have to pay Rs. 9,000 less interest.
6. Bindu decided to exchange her square-shaped land of 42 m side with a rectangular land of equal area.
   1. What was the area of her square-shaped land? [1]
   2. If the length of the rectangular land to be exchanged is 72 m, find the breadth of the land. [1]
   3. What will be the length of wire required to fence the rectangular land three times? [2]
   4. To fence the above square and rectangular land one round each, which land needs more wire and by how much? [1]
1. Area of square land
   Side (\( l \)) = 42 m
   Area (\( A \)) = \( l^2 = 42 \times 42 = 1764 \) m²
2. Breadth of the rectangular land
   Area of rectangle = Area of square = 1764 m²
   Length (\( L \)) = 72 m
   Breadth (\( B \)) = \( \frac{\text{Area}}{\text{Length}} = \frac{1764}{72} = 24.5 \) m
3. Wire required for rectangular land
   Perimeter (\( P_r \)) = \( 2(L + B) = 2(72 + 24.5) = 193 \) m
   Wire for 3 rounds = \( 193 \times 3 = 579 \) m
4. Comparison of wire required
   Perimeter of square land (\( P_s \)) = \( 4 \times 42 = 168 \) m
   Perimeter of rectangular land (\( P_r \)) = 193 m
   The rectangular land needs more wire.
   Difference = \( 193 - 168 = 25 \) m more wire.
1. Express \(x^m \times x^n\) as a power of \(x\).[1]
2. Simplify: \(\dfrac{a}{(a - b)(a - c)} + \dfrac{b}{(a - c)(b - a)}\).[2]
8. Two equations are given below: \(2x - y = 5\) and \(x - y = 1\).
1. Write the degree of the given equations.[1]
2. Solve the above equations by using graphical method.[2]
1. Find the H.C.F. of: \(3x^3 - 15x^2\) and \(2x^3 - 50x\).[2]
2. Write a quadratic equation having roots \(4\) and \(5\) of \(x\).[2]
10. In the adjoining figure, \(TU\) intersects straight lines \(PQ\) and \(RS\) at points \(V\) and \(W\) respectively.
1. Write a pair of alternate angles from the figure.[1]
2. What type of triangle is \(\triangle VWX\) according to the angles of the triangle?[2]
3. At what value of \(\angle QVX\) will the given line segments \(PQ\) and \(RS\) be parallel?[1]
1. Construct a parallelogram \(ABCD\) with \(AB = 6cm\), \(BC = 4.5cm\), and \(\angle ABC = 45^\circ\).[3]
2. In the given figure, if \(\triangle ABC \sim \triangle DEF\), find the measurement of \(DF\).[2]
1. Write down the bearing of point \(A\) from point \(O\).[1]
2. \(L(0,-2)\), \(M(5,-4)\), and \(N(2,5)\) are the vertices of \(\triangle LMN\). Find the coordinates of the vertices of its image when it is reflected about the \(x\)-axis. Also, draw the graph of the reflection.[3]
3. If the centre of a circle is \(A(4,4)\) and \(B(7,4)\) is any point on its circumference, find the area of the circle.[2]