G8_Dhanghadhi_8_2081

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  1. Answer the following questions from the given Venn diagram.
    1. Write the improper subset of the set \(P\).[1]
    2. Write the elements of set \(U\) by listing method.[1]
    3. How many subsets are formed from set \(Q\)?[1]
    1. From the Venn diagram, set \(P = \{5, 7, 2, 3\}\).
      The improper subset of \(P\) is the set itself.
      \(\{2, 3, 5, 7\}\)
    2. The universal set \(U\) includes all elements shown in the rectangle, which is
      \(U = \{1, 2, 3, 4, 5, 7\}\)
    3. From the diagram, set \(Q = \{1, 4, 2, 3\}\), so it has \(n = 4\) elements.
      Therefore, total number of subsets of \(Q\) with \(4\) elements \(2^4 = 16\).
      Number of Elements Subsets Count
      0 \(\emptyset\) 1
      1 \(\{1\}, \{2\}, \{3\}, \{4\}\) 4
      2 \(\{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}\) 6
      3 \(\{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}\) 4
      4 \(\{1, 2, 3, 4\}\) 1
      Total Subsets 16
  2. The marked price of a bag is Rs 5000.
    1. If marked price and discount percent are represented by \(MP\) and \(D\%\) respectively, write the formula to find the discount percent.[1]
    2. How much will be the discount amount of the bag if a discount of \(5\%\) on the marked price is given?[1]
    3. The shopkeeper gets \(25\%\) profit after selling the bag at \(5\%\) discount. What was the cost price of the bag?[2]
    1. Formula to find the discount percent:
      Discount Percent = \( \dfrac{\text{Discount Amount}}{MP} \times 100\% \)
      orDiscount Percent = \( \dfrac{MP - SP}{MP} \times 100\% \)
    2. Discount amount
      Marked Price (MP) = Rs. \(5,000\)
      Discount = \(5\%\)
      Thus
      Discount Amount = \( \dfrac{5}{100} \times 5,000 = 250 \)
      So, the discount amount is Rs. \(250\).
    3. Cost price
      Marked Price (MP) = Rs. \(5,000\)
      Discount = \(5\%\)
      Thus
      Selling Price (SP) = 95% of MP = \( \dfrac{95}{100} \times 5,000 = 4,750 \)
      Profit = \(25\%\)
      Thus
      SP= 125% of CP
      \(4,750 = \dfrac{125}{100} \times \text{CP}\)
      or\(\text{CP} = \dfrac{4,750 \times 100}{125} = \dfrac{475,000}{125} = 3,800\)
      So, the cost price of the bag was Rs. \(3,800\).
  3. Ram has deposited Rs 20,000 in an Agricultural Development Bank for 5 years at the rate of Rs 12 interest per annum for Rs 100.
    1. At what percent of interest rate per annum has Ram deposited the amount of money?[1]
    2. How much interest will Ram get in 5 years at the same rate of interest?[2]
    3. The ages of Ram’s two sons are 8 years and 12 years respectively. He divides Rs 20,000 between his sons in the ratio of their ages. How much does each get?[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. Interest for 5 years:
      Principal (P) = Rs 20,000
      Time (T) = 5 years
      Rate (R) = 12%
      We know,
      Interest (I) = \( \dfrac{P \times T \times R}{100} = \dfrac{20,000 \times 5 \times 12}{100} = 12,000 \)
      So, Ram will get Rs 12,000 interest.
    3. Division of money between sons:
      Ratio of their ages = 8 : 12 = 2 : 3
      Sum of ratios = 2 + 3 = 5
      Total amount = Rs 20,000
      Younger son's share (8 yrs) = \( \dfrac{2}{5} \times 20,000 = 8,000 \)
      Elder son's share (12 yrs) = \( \dfrac{3}{5} \times 20,000 = 12,000 \)
      So, the younger son gets Rs 8,000 and the elder son gets Rs 12,000.
  4. The total monthly income of five families is Rs 458000.
    1. Write the total income in scientific notation.[1]
    2. What is the monthly income of a family?[1]
    3. Convert \(0.\overline{25}\) into a fraction.[1]
    4. Convert \(27\) into the binary number system.[1]
  5. A circular playground has a radius of 100 m. A rectangular volleyball court of length 18 m and breadth 9 m is made inside it. (\(\pi = 3.14\))
    1. Write the formula to find the area of the playground. [1]
    2. What is the area of the volleyball court? [1]
    3. What is the area of the playground excluding the volleyball court? [2]
    4. How long should the wire be to fence the playground once? [1]
    1. Area of the playground formula
      Since the playground is circular,
      Area (\( A \)) = \( \pi r^2 \)
    2. Area of the volleyball court
      Given: Length (\( l \)) = 18 m, Breadth (\( b \)) = 9 m
      Area (\( A_v \)) = \( l \times b = 18 \times 9 = 162 \) m²
    3. Area of the playground excluding the court
      Total area of playground (\( A_m \)) = \( 3.14 \times (100)^2 = 31400 \) m²
      Remaining Area = \( 31400 - 162 = 31238 \) m²
    4. Length of wire for fencing
      The length of wire is equal to the circumference of the playground,
      Circumference (\( C \)) = \( 2\pi r = 2 \times 3.14 \times 100 = 628 \) m
    1. Express \(\dfrac{x^6}{x^3}\) as a power of \(x\).[1]
    2. Simplify: \(\dfrac{a}{a + b} + \dfrac{b}{a - b}\).[2]
    1. Define quadratic equation.[1]
    2. Solve by graphical method: \(x + y = 5\) and \(3x + y = 9\).[2]
    1. Find the H.C.F. of \(x^2 + 7x + 12\) and \(x^2 - 16\).[2]
    2. For what value of \(x\) does the expression \(x^2 - 8x + 15\) become zero?[2]
  9. In the figure, line \(PQ\) intersects straight lines \(AB\) and \(CD\) at points \(M\) and \(N\) respectively.
    1. Write a pair of alternate angles.[1]
    2. Find the value of \(x\).[2]
    3. At what value of \(\angle DNM\) will the lines \(AB\) and \(CD\) become parallel?[1]
    1. Construct a parallelogram \(ABCD\) with \(AB = 7cm\), \(BC = 5cm\), and \(\angle ABC = 60^\circ\).[3]
    2. In the adjoining figure, if \(\angle BAD = \angle ABC\) and \(AD = BC\), prove that \(\triangle ABC \cong \triangle ABD\).[2]
    1. What is the actual distance between two places represented by \(5.5\,\text{cm}\) on a map with scale \(1\,\text{cm} = 500\,\text{m}\)?[2]
    2. What type of quadrilateral is used to make a regular tessellation?[1]
    3. The vertices of \(\triangle ABC\) are \(A(2,1)\), \(B(4,1)\), and \(C(3,4)\). Find the coordinates of the image \(\triangle A'B'C'\) after reflecting \(\triangle ABC\) on the \(x\)-axis. Also draw the graph of the reflection.[3]
  12. The monthly expenditure of a family is given below:
    13. Title of expenditure Amount (Rs.)
    Food 20,000
    Education 10,000
    Health 6,000
    1. Calculate the average expenditure.[1]
    2. Represent the above data in a pie chart.[2]

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