G8_Biratnagar_8_2081

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  1. Study the given Venn diagram and answer the following questions.
    1. Define disjoint sets.[1]
    2. Write any one proper subset of set \(P\).[1]
    3. If the common element \(5\) is removed from the Venn diagram, then what will be the relation between sets \(P\) and \(Q\)? Write it.[1]
    1. Two sets are called disjoint sets if they have no elements in common.
      In other words, sets \(A\) and \(B\) are disjoint if \(A \cap B = \emptyset\).
    2. From the Venn diagram:
      Set \(P = \{1, 2, 3, 4, 5\}\)
      One proper subset of \(P\) is \(\{1, 2, 3\}\)
      Elements Subsets Count
      0 \(\emptyset\) 1
      1 \(\{1\}, \{2\}, \{3\}, \{4\}, \{5\}\) 5
      2 \(\{1,2\}, \{1,3\}, \{1,4\}, \{1,5\}, \{2,3\}\)
      \(\{2,4\}, \{2,5\}, \{3,4\}, \{3,5\}, \{4,5\}\)      		 10
      3 \(\{1,2,3\}, \{1,2,4\}, \{1,2,5\}, \{1,3,4\}, \{1,3,5\}\)
      \( \{1,4,5\}, \{2,3,4\}, \{2,3,5\}, \{2,4,5\}, \{3,4,5\}\)      		 10
      4 \(\{1,2,3,4\}, \{1,2,3,5\}, \{1,2,4,5\}, \{1,3,4,5\}, \{2,3,4,5\}\) 5
      Total Proper Subsets 31
    3. If element \(5\) is removed from the diagram (i.e., from both \(P\) and \(Q\)), then
      New \(P = \{1, 2, 3, 4\}\)
      New \(Q = \{6, 7, 8, 9\}\)
      Now, \(P \cap Q = \emptyset\), therefore, sets \(P\) and \(Q\) become disjoint sets.
  2. The marked price of a television is Rs24000. If the shopkeeper got Rs2400 profit after selling it with \(15\%\) discount, then
    1. If marked price and discount amount are represented by \(MP\) and \(D\) respectively, write the formula to find the selling price after discount.[1]
    2. How much discount had been given by the shopkeeper to sell the television? Find it.[2]
    3. Find the cost price of the television.[2]
    1. Formula to find the selling price after discount:
      Selling Price (SP) = MP – D
      or, if discount is given as a percentage \(D\%\):
      SP = \((100 -D)\% \times MP\)
    2. Discount amount
      Marked Price (MP) = Rs. \(24,000\)
      Discount = \(15\%\)
      Thus
      Discount = \( \dfrac{15}{100} \times 24,000 = 3,600 \)
      So, the discount given was Rs. \(3,600\).
    3. Cost price of the television.
      Marked Price (MP) = Rs. \(24,000\)
      Discount = \(15\%\)
      Thus
      Selling Price (SP) = 85% of MP = \( \dfrac{85}{100} \times 24,000 = 20,400 \)
      Profit = Rs. \(2,400\)
      We know that
      CP = SP – Profit=\(20,400 - 2,400 = 18,000\)
      So, the cost price of the television was Rs. \(18,000\).
  3. Sunil has deposited RS 300,000 in Rastriya Banijya Bank for 3 years at the rate of Rs 12 simple interest per annum for every Rs 100.
    1. At what percent of interest rate per annum had Sunil deposited the amount?[1]
    2. After 3 years, how much total money does Sunil get with principal and interest? Calculate it.[1]
    3. If Sunil decides to distribute RS 300,000 to his brothers Chandan and Ram in the ratio 2:3, then compare the amount received by Chandan and Ram.[2]
    1. Interest rate in percentage:
      Since the interest for Rs 100 in 1 year is Rs 12, the rate is 12%.
      So, Rate (R) = 12% p.a.
    2. Total money after 3 years (Amount):
      Principal (P) = Rs 300,000
      Time (T) = 3 years
      Rate (R) = 12%
      Interest (I) = \( \dfrac{P \times T \times R}{100} = \dfrac{300,000 \times 3 \times 12}{100} = 108,000 \)
      Total Amount (A) = P + I = 300,000 + 108,000 = 408,000
      So, Sunil gets a total of Rs 408,000.
    3. Comparison of amount received by Chandan and Ram:
      Total Amount = Rs 300,000
      Ratio (Chandan : Ram) = 2 : 3
      Sum of ratios = 2 + 3 = 5
      Chandan's share = \( \dfrac{2}{5} \times 300,000 = 120,000 \)
      Ram's share = \( \dfrac{3}{5} \times 300,000 = 180,000 \)
      Comparison: Ram receives Rs 60,000 (180,000 - 120,000) more than Chandan.
  4. Raju takes a bus to Dharan from Biratnagar. The wheel of the bus rotates \(35750\) times in an hour.
    1. Write \(35750\) in scientific notation.[1]
    2. How many times will the wheel rotate in \(90\) minutes?[1]
    3. Find the value of \(\sqrt{48} + \sqrt{75} - \sqrt{3}\).[2]
    4. Convert \(0.\overline{24}\) into a fraction.[1]
  5. In the given figure, ABCD is a square and a circle is drawn inside it.
    1. Write the formula to find the area of a circle. [1]
    2. How much is the radius of the circle? [1]
    3. Find the area of the shaded region. [2]
    4. Compare the circumference of the circle and the perimeter of the square. [1]
    1. Area of circle formula
      We know that,
      Area (\( A \)) = \( \pi r^2 \)
    2. Radius of the circle
      In the figure, the side of the square is equal to the diameter of the circle.
      Diameter (\( d \)) = 28 cm
      Radius (\( r \)) = \( \frac{d}{2} = 14 \) cm
    3. Area of the shaded region
      Area of square (\( A_s \)) = \( 28 \times 28 = 784 \) cm²
      Area of circle (\( A_c \)) = \( \frac{22}{7} \times 14^2 = 616 \) cm²
      Shaded Area = \( 784 - 616 = 168 \) cm²
    4. Comparison of boundary lengths
      Circumference of circle (\( C \)) = \( 2\pi r = 2 \times \frac{22}{7} \times 14 = 88 \) cm
      Perimeter of square (\( P \)) = \( 4 \times 28 = 112 \) cm
      Comparison: The perimeter of the square is \( 112 - 88 = 24 \) cm more than the circumference.
    1. Express \(x^m \times x^{-1}\) as a power of \(x\).[1]
    2. Simplify: \(\dfrac{a}{(a - b)^2} + \dfrac{b}{(a - b)^2}\).[2]
  7. Two equations are given: \(x + y = 6\) and \(x - y = 2\).
    1. What is meant by simultaneous equations?[1]
    2. Solve the given equations by using a graph.[2]
    1. Find the L.C.M. of the algebraic expressions: \(x^2 - 7x + 12\) and \(3x^2 - 27\).[2]
    2. Find the quadratic equation in which the values of \(x\) are \(2\) and \(3\).[2]
  9. In the adjoining figure, when \(XY\) and \(XZ\) meet the line segments \(PQ\) and \(RS\), a \(\triangle XYZ\) is formed.
    1. Write the relation between \(\angle XYZ\) and \(\angle XZY\).[1]
    2. Find the value of \(x\).[2]
    3. At which value of \(\angle PXY\) will the line segments \(PQ\) and \(RS\) be parallel?[1]
    1. Construct a rectangle \(ABCD\) in which \(AB = 7cm\) and \(BC = 5cm\).[3]
    2. In rectangle \(ABCD\), prove that \(\triangle ABC \cong \triangle ACD\) by drawing diagonal \(AC\).[2]
    1. What is meant by regular tessellation?[1]
    2. In the adjoining figure, if the bearing of point \(S\) from point \(R\) is \(060^\circ\), find the bearing of point \(R\) from point \(S\).[2]
    3. Find the coordinates of the images \(M'\), \(N'\), and \(O'\) of \(\triangle MNO\) with vertices \(M(2,1)\), \(N(4,3)\), and \(O(-1,2)\) after reflection on the \(x\)-axis.[3]
  12. The monthly expenses of Shital's meals are given in the table below:
    13. Month Ashoj Kartik Mangsir Poush Magh
    Expenditure (Rs.) 4000 2500 2000 1700 1800
    1. What is the monthly average expenditure of Shital on her meals?[1]
    2. Present Shital’s expenditure in a pie chart.[2]

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