G8_Rampur_Palpa_8_2081

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  1. Study the given Venn diagram and answer the following questions.
    1. List the elements of sets \(A\) and \(B\) in listing method.[1]
    2. Write the improper subset of set \(B\).[1]
    3. If set \(A\) contains only the element ‘\(r\)’, then what type of set relation exists between \(A\) and \(B\)? Give a reason.[1]
    1. From the Venn diagram:
      \( A = \{p, q, r\} \)
      \( B = \{r, s, t\} \)
    2. The improper subset of set \(B\) is the set \(B\) itself:
      \( \{r, s, t\} \)
    3. If set \(A\) contains only the element \(r\), then \(A = \{r\}\) and \(B = \{r, s, t\}\).
      In this case, \(A\) is a proper subset of \(B\) because every element of \(A\) is in \(B\)
  2. Anisha bought a scooter for Rs 300000 and sold it to Divya after \(2\) years, deducting \(20\%\) of the price. Divya sold it to Gauri at a profit of Rs 6000.
    1. Find the selling price of Anisha.[1]
    2. Compare the price of Anisha and Gauri of the scooter in the ratio form.[2]
    1. Selling price of Anisha:
      Cost Price for Anisha = Rs. \(300,000\)
      She sold it after deducting \(20\%\) of the price (i.e., a \(20\%\) depreciation or loss):
      Thus
      Selling Price = 80% of 300,000= \(\dfrac{80}{100} \times 300,000 = 240,000\)
      So, Anisha sold the scooter to Divya for Rs. 240,000.
    2. Price comparison between Anisha (original buyer) and Gauri (final buyer):
      Anisha’s purchase price = Rs. \(300,000\)
      Divya’s purchase price = Rs. \(240,000\)
      Next
      Divya sold it to Gauri at a profit of Rs. 6,000, so
      Gauri’s purchase price = \(240,000 + 6,000 = 246,000\)
      Now, ratio of Anisha’s price to Gauri’s price is
      \(300,000 : 246,000\)
      or\(50 : 41\)
      So, the ratio is \(50 : 41\).
  3. Ganesh borrowed a sum of Rs 27,000 from his friend Bishnu. He paid an interest of Rs 5,400 to Bishnu at the end of 2 years.
    1. Write the formula to find the rate of interest.[1]
    2. Find the rate of interest at which Ganesh borrowed the sum.[1]
    3. At the same rate of interest, calculate the interest for 3 years.[2]
    4. If Ganesh had not paid any interest till the end of 3 years, how much amount would be needed to clear the loan?[2]
    1. Formula to find the rate of interest:
      Rate (R) = \( \dfrac{I \times 100}{P \times T} \)
    2. Rate of interest:
      Principal (P) = Rs. 27,000
      Time (T) = 2 years
      Interest (I) = Rs. 5,400
      Thus,
      Rate (R) = \( \dfrac{5,400 \times 100}{27,000 \times 2} = 10\% \)
      So, the rate of interest is 10% per annum.
    3. Interest for 3 years:
      Time (T) = 3 years
      Rate (R) = 10%
      Thus,
      Interest (I) = \( \dfrac{27,000 \times 3 \times 10}{100} = 8,100 \)
      So, the interest for 3 years is Rs. 8,100.
    4. Amount to clear loan after 3 years:
      Principal (P) = Rs. 27,000
      Interest for 3 years (I) = Rs. 8,100
      Thus,
      Amount (A) = P + I = 27,000 + 8,100 = 35,100
      So, Rs. 35,100 would be needed to clear the loan.
  4. The road between Rampur to Gaidakot is \(95\) kilometers.
    1. Write the distance in scientific notation in meters.[1]
    2. If Utsab drives his bike at the rate of \(19\,\text{km/hr}\), calculate the time taken to travel from Rampur to Gaidakot.[1]
    3. Write \(35\) in the quinary number system.[1]
    4. The monthly salary of Prakash is \(\text{Rs.}\,43{,}689\). Find his expenditure and saving amount if their ratio is \(2:1\).[2]
  5. In the adjoining figure, LOVE is a square park. Inside that park, there is a circular pond. The side of the square is 25 m and the diameter of the pond is 14 m.
    1. Write the formula to find the area of a circle. [1]
    2. Find the area of the circular pond. [1]
    3. Find the area of the park including the pond. [1]
    4. How much does it cost to fence the park at the rate of Rs 650 per meter? [1]
    1. Area of circle formula
      We know that,
      Area (\( A \)) = \( \pi r^2 \)
    2. Area of the circular pond
      Given that,
      Diameter of pond (\( d \)) = 14 m
      Therefore,
      Radius (\( r \)) = \( \frac{14}{2} = 7 \) m
      Now using the formula, we get
      Area of pond (\( A_p \)) = \( \pi r^2 = \frac{22}{7} \times 7^2 = 154 \) m²
    3. Area of the park including the pond
      Given that the park is square,
      Side (\( l \)) = 25 m
      Therefore,
      Area of park (\( A \)) = \( l^2 = 25^2 = 625 \) m²
      (Note: Since the pond is inside the park, the total area including the pond is 625 m².)
    4. Cost of fencing the park
      We know that,
      Perimeter of square (\( P \)) = \( 4l = 4 \times 25 = 100 \) m
      According to the question, the rate is Rs 650 per meter, so
      Total Cost = \( 100 \times 650 = \) Rs 65,000
    1. What is the value of \(\left(\dfrac{3x}{y}\right)^0\)?[1]
    2. Simplify: \(\dfrac{x^2}{x - 7} + \dfrac{49}{7 - x}\)[2]
  7. Two equations are given below: \(x + 3y = 8\) and \(2x + y = 6\).
    1. What is this system of equations called?[1]
    2. Solve the above equations using a graph.[2]
    1. Find the H.C.F. of \(x^2 - 5x + 6\) and \(x^2 - 4\).[2]
    2. At what value of \(x\) does the expression \(x^2 - 7x - 18\) become zero?[2]
  9. In the figure, \(PQ \parallel RS\). \(TR\) and \(TU\) are two transversals. If \(\angle PTR = 60^\circ\) and \(\angle QTU = 70^\circ\),
    1. Find the values of \(x^\circ\) and \(y^\circ\).[2]
    2. Experimentally verify that the angles of an equilateral triangle are equal. (Two different triangles are necessary.)[3]
    1. By which axiom are the given triangles \(\triangle ABC\) and \(\triangle PQR\) congruent?[1]
    2. Construct a parallelogram \(ABCD\) in which \(AB = 6cm\), \(AD = 5cm\), and \(\angle BAC = 60^\circ\).[3]
    3. How can the area of the parallelogram be found? Write your argument.[1]
  11. The vertices of \(\triangle ABC\) are \(A(3,2)\), \(B(5,6)\), and \(C(7,2)\).
    1. How many triangular surfaces are there in a tetrahedron?[1]
    2. In \(\triangle ABC\), if the bearing of point \(C\) from \(A\) is \(090^\circ\), find the bearing of \(A\) from point \(C\). Also, find the distance between points \(A\) and \(B\).[2]
    3. Plot \(\triangle ABC\) on a graph and find the coordinates of the image of \(\triangle ABC\) after reflection in the \(x\)-axis. Also, plot its image on the same graph.[2]
  12. The numbers of students in classes \(1\) to \(5\) in a school are given below. Study the table and answer the following questions:
    13. Class 1 2 3 4 5
    No. of students \(12\) \(18\) \(15\) \(25\) \(20\)
    1. What is the average number of students per class?[1]
    2. Present the above information in a pie chart.[2]

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