G8_Amarghadhi_8_2081

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  1. Two subsets of the universal set \(U = \{a, e, i, o, u\}\) are \(A = \{e, o, u\}\) and \(B = \{a, c, i\}\).
    1. What types of sets are \(A\) and \(B\)—overlapping or disjoint? Write it.[1]
    2. Write one proper and one improper subset of \(A\).[1]
    3. If \(e\) is eliminated from sets \(A\), then what types of sets are \(A\) and \(B\)? Write with a reason.[1]
    1. Sets \(A\) and \(B\) are disjoint sets.
      Because \(A = \{e, o, u\}\) and \(B = \{a, c, i\}\) have no common elements.
    2. One proper subset of \(A\): \(\{e, o\}\) and one improper subset of \(A\): \(\{e, o, u\}\) (which is \(A\) itself)
    3. Eliminating \(e\) from set \(A\) gives \(A = \{o, u\}\).
      Since \(A = \{o, u\}\) and \(B = \{a, c, i\}\) share no common elements, they remain disjoint sets (\(A \cap B = \emptyset\)).
  2. The binary and decimal number systems are illustrated in the table below, where the binary representation of \(15\) is blank.
    3. Binary number system \(0_2\) \(1_2\) \(10_2\) \(11_2\) \(100_2\) ?
    Decimal number system \(0\) \(1\) \(2\) \(3\) \(4\) \(15\)
    1. Write the digits used in the binary number system.[1]
    2. Which binary number should be placed in the blank?[2]
    3. Convert the distance of 240000 miles from the Earth to the Moon into scientific notation.[2]
  3. Ashok Mahato went to a stationery shop to buy a ball. He saw two balls \(A\) and \(B\). The ratio of their marked prices is \(3:2\). The marked price of ball \(A\) is Rs 3750, and that of ball \(B\) is Rs 2500. A discount of \(5\%\) is offered.
    4. ⚽️ 🏐
    1. Write the marked prices of balls \(A\) and \(B\) in proportion.[1]
    2. If Ashok Mahato decided to buy ball \(B\), what amount should he pay for it? Find it.[2]
    3. If the shopkeeper wanted to earn a \(10\%\) profit by selling ball \(B\) after the discount, at what price did he buy ball \(B\)?[2]
    1. Marked prices of balls \(A\) and \(B\) in proportion is
      Marked Price of A : Marked Price of B = 3:2
      or3750 : 1250 = 3:2
    2. Amount Ashok Mahato should pay for ball \(B\):
      Marked Price of ball \(B\) = Rs. \(2,500\)
      Discount = \(8\%\)
      Therefore
      SP= 92% of MP
      orSP= \(\dfrac{92}{100} \times 2500\)
      orSP= 2300
      So, Ashok should pay Rs. 2,300for ball B.
    3. Cost price of ball \(B\) for a \(15\%\) profit after discount:
      Selling Price (after \(8\%\) discount) = Rs. 2300
      Profit =\(15\%\)
      Therefore
      SP = \(115\%\) of CP
      or\(2300 =\dfrac{115}{100} \times \) CP
      orCP=\(\dfrac{2300 \times 100}{115} = 2000 \)
      So, the shopkeeper bought ball \(B\) for Rs. \(2000\).
  4. Bhargab deposited Rs 50,000 in a bank. After 2 years, he received simple interest of Rs 10,000.
    1. What is the amount after 2 years? Find it.[1]
    2. Find the rate of interest per annum.[2]
    3. If Bhargab had deposited the same amount for only 1 year, how much less interest would he have received?[1]
    1. Amount after 2 years:
      Principal (P) = Rs. 50,000
      Interest (I) = Rs. 10,000
      Thus,
      Amount (A) = P + I = 50,000 + 10,000 = 60,000
      So, the amount after 2 years is Rs. 60,000.
    2. Rate of interest per annum:
      Principal (P) = Rs. 50,000
      Time (T) = 2 years
      Interest (I) = Rs. 10,000
      We know,
      Rate (R) = \( \dfrac{I \times 100}{P \times T} = \dfrac{10,000 \times 100}{50,000 \times 2} = 10\% \)
      So, the rate of interest is 10% per annum.
    3. Difference in interest:
      Interest for 2 years = Rs. 10,000
      Interest for 1 year = \( \dfrac{50,000 \times 1 \times 10}{100} = 5,000 \)
      Thus,
      Less interest = 10,000 - 5,000 = 5,000
      So, he would have received Rs. 5,000 less interest.
  5. A wire of length 39.6 m is bent to form an equilateral triangle on a plane surface.
    1. Write the formula to find the area of an equilateral triangle. [1]
    2. What is the radius of the circle if the same wire is bent into a circle? Find it. [1]
    3. Find the area of the equilateral triangle. [2]
    4. How much more or less is the area of the circle compared to the area of the equilateral triangle when the same wire is used? Find it. [2]
    1. Area of an equilateral triangle formula
      We know that,
      Area (\( A \)) = \( \frac{\sqrt{3}}{4} a^2 \) (where \( a \) is the side length)
    2. Radius of the circle
      Given that the wire length is 39.6 m, which becomes the circumference (\( C \)) of the circle.
      We know that,
      \( 2\pi r = 39.6 \)
      or \( 2 \times \frac{22}{7} \times r = 39.6 \)
      or \( r = \frac{39.6 \times 7}{44} = 6.3 \) m
    3. Area of the equilateral triangle
      Here, the total length of the wire is the perimeter (\( P \)) of the triangle.
      Side (\( a \)) = \( \frac{39.6}{3} = 13.2 \) m
      Now calculating the area,
      Area (\( A_t \)) = \( \frac{\sqrt{3}}{4} \times (13.2)^2 \)
      or \( A_t = 0.433 \times 174.24 \approx 75.44 \) m²
    4. Comparison of areas
      First, calculating the area of the circle,
      Area (\( A_c \)) = \( \pi r^2 = \frac{22}{7} \times (6.3)^2 = 124.74 \) m²
      Finding the difference,
      Difference = \( A_c - A_t = 124.74 - 75.44 = 49.3 \) m²
      Thus, the area of the circle is 49.3 m² more than the area of the equilateral triangle.
    1. Find the value of: \((2a)^0\)[1]
    2. Simplify: \(x^{a-b} \times x^{b-c} \times x^{c-a}\)[2]
  7. Two algebraic expressions are \(x^{2} - 16\) and \(x^{2} - 9x + 20\).
    1. Find the Highest Common Factor (H.C.F.) of the given algebraic expressions.[2]
    2. What is the value of \(x\) when the expression \(x^{2} - 16\) is zero?[2]
  8. Two equations are given: \(x + 2y = 7\) and \(x + y = 5\).
    1. What type of equations are these?[1]
    2. Solve the above equations using the graphical method.[2]
  9. In the figure, straight lines \(AB\) and \(CD\) are cut by transversal \(EF\) at points \(G\) and \(H\) respectively. \(\angle BGH = (3y - 70)^{\circ}\) and \(\angle DHF = (2x - 9)^{\circ}\). Also, \(\angle AGH = 109^{\circ}\).
    1. What are \(\angle BGH\) and \(\angle DHF\) called in the figure?[1]
    2. Find the value of \(y\) from the figure.[2]
    3. At what degree value of \(x\) will the straight lines \(AB\) and \(CD\) be parallel? Write with reasons.[2]
    1. What is the sum of the interior angles of a regular polygon having \(n\) sides?[1]
    2. Construct a parallelogram with adjacent sides \(7cm\) and \(4cm\), and one diagonal \(8cm\), using compasses.[3]
    3. Measure the opposite sides of the parallelogram and write the conclusion based on your measurement.[1]
    1. What type of tessellation is shown in the given figure?[1]
    2. A man walks \(3\,\text{m}\) north and then turns east and walks \(4\,\text{m}\). What is the shortest distance between his starting point and final position? Calculate it.[1]
    3. \(A(2, 2)\), \(B(4, 6)\), and \(C(6, 3)\) are the vertices of \(\triangle ABC\). Draw \(\triangle ABC\) on graph paper and also plot its image obtained by reflecting it in the \(x\)-axis.[3]
  12. There are \(50\) students in Grade VIII of a school. The pie chart below shows their preferences for Mathematics and Science & Technology.
    1. Find the number of students who liked Science and Technology.[2]
    2. Write the name of the subject that represents the mode value.[1]

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