G8_Raghuganga_magdi_8_2081

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  1. The elements of sets \(A\) and \(B\) are shown in the adjoining Venn diagram.
    1. State whether the sets \(A\) and \(B\) are overlapping or disjoint. Give a reason.[1]
    2. Write two proper subsets of set \(A\).[1]
    3. What needs to be done to make the given sets disjoint?[1]
    1. Sets \(A\) and \(B\) are overlapping sets.
      Because they share the common element \(3\), i.e., \(A \cap B = \{3\} \neq \emptyset\).
    2. From the Venn diagram, \(A = \{2, 3\}\).
      Two proper subsets of \(A\) are: \(\{2\}\) and \(\{3\}\).
      Number of Elements Proper Subsets Count
      0 \(\emptyset\) 1
      1 \(\{2\}, \{3\}\) 2
      Total Proper Subsets 3
    3. To make sets \(A\) and \(B\) disjoint, the common element \(3\) must be removed from either set \(A\) or set \(B\) (or both).
      For example
      Remove \(3\) from \(A\): Then \(A = \{2\}\), \(B = \{3, 4\}\) → disjoint
      Remove \(3\) from \(B\): Then \(A = \{2, 3\}\), \(B = \{4\}\) → disjoint
  2. Grishma has a book of 150 pages.
    1. Write the number of pages in scientific notation.[1]
    2. Write the number of pages in the quinary number system.[1]
    3. Write the number of pages in the binary number system.[1]
    4. If she has read 15 pages, find the ratio of the completed pages to the remaining pages.[1]
  3. Ganesh buys a mobile at Rs 36000. He fixed the marked price of the mobile \(30\%\) above the cost price. If he sells the mobile at a \(25\%\) discount, then
    1. What is the marked price of the mobile?[1]
    2. What is the discount amount?[1]
    3. What is the selling price of the mobile?[1]
    4. What is his profit or loss percent from that mobile? Find it.[1]
    1. Marked price of the mobile:
      Cost Price (CP) = Rs. \(36,000\)
      Marked Price (MP) is \(30\%\) above CP:
      MP = \( \dfrac{130}{100} \times 36,000 = 1.3 \times 36,000 = 46,800 \)
      So, the marked price is Rs. \(46,800\).
    2. Discount amount:
      Discount = \(25\%\) of MP = \( \dfrac{25}{100} \times 46,800 = 11,700 \)
      So, the discount amount is Rs. \(11,700\).
    3. Selling price of the mobile:
      SP = MP – Discount = \(46,800 - 11,700 = 35,100\)
      So, the selling price is Rs. \(35,100\).
    4. Profit or loss percent:
      CP = Rs. \(36,000\)
      SP = Rs. \(35,100\)
      Since SP < CP, there is a loss.
      Loss = \(36,000 - 35,100 = 900\)
      Thus
      Loss Percent = \( \dfrac{900}{36,000} \times 100\% = 2.5\% \)
      So, Ganesh incurred a loss of \(2.5\%\).
  4. Santosh borrowed a sum of Rs 7,000 from his friend Roopal. He paid an interest of Rs 1,400 to Roopal at the end of 2 years.
    1. Write the formula to find the rate of interest.[1]
    2. At which rate of interest did Santosh borrow the sum?[1]
    3. At the same rate of interest, calculate the interest for 5 years.[1]
    4. If Santosh had not paid any interest till the end of 5 years, how much amount would he need to clear the debt?[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. 7,000
      Time (T) = 2 years
      Interest (I) = Rs. 1,400
      Thus,
      Rate (R) = \( \dfrac{1,400 \times 100}{7,000 \times 2} = 10\% \)
      So, the rate of interest is 10% per annum.
    3. Interest for 5 years:
      Time (T) = 5 years
      Rate (R) = 10%
      Thus,
      Interest (I) = \( \dfrac{7,000 \times 5 \times 10}{100} = 3,500 \)
      So, the interest for 5 years is Rs. 3,500.
    4. Amount to clear debt after 5 years:
      Principal (P) = Rs. 7,000
      Interest for 5 years (I) = Rs. 3,500
      Thus,
      Amount (A) = P + I = 7,000 + 3,500 = 10,500
      So, he would need Rs. 10,500 to clear the debt.
  5. The perimeter of a square land is 440 ft.
    1. Write the formula for the perimeter of a square. [1]
    2. Find the length of the land. [1]
    3. Find the cost of levelling the land at the rate of Rs 50 per \( \text{ft}^2 \). [2]
    4. How much does it cost to fence the land at the rate of Rs 70 per foot? [1]
    1. Perimeter of a square formula
      We know that,
      Perimeter (\( P \)) = \( 4l \) (where \( l \) is the length of a side)
    2. Length of the land
      Given that,
      Perimeter (\( P \)) = 440 ft
      Now using the formula, we get
      \( 4l = 440 \)
      or \( l = \frac{440}{4} = \) 110 ft
    3. Cost of levelling the land
      We know that,
      Area of land (\( A \)) = \( l^2 = 110^2 = \) 12100 sq. ft
      According to the question, the rate is Rs 50 per sq. ft, so
      Total Cost = \( 12100 \times 50 = \) Rs 6,05,000
    4. Cost of fencing the land
      Given that,
      Perimeter (\( P \)) = 440 ft
      According to the question, the rate is Rs 70 per foot, so
      Total Cost = \( 440 \times 70 = \) Rs 30,800
    1. For what power of \(4\) does the value become \(\dfrac{1}{64}\)?[1]
    2. If \(a = 1\), \(b = 2\), and \(c = 3\), find the value of \(a^{b} \times b^{c} \times c^{a}\).[2]
    1. For what value of \(a\), \(\dfrac{3}{a - 5}\) is undefined?[1]
    2. Simplify: \(\dfrac{4}{x^{2} + 3x + 2} \div \left( \dfrac{2}{x^{2} - 1} \right)\)[2]
  8. If two algebraic expressions are \(a^{2} - 3a + 2\) and \(a^{2} - 9a + 10\),
    1. Find the Highest Common Factor (H.C.F.).[2]
    2. Solve the equation \(x^{2} - 6x + 8 = 0\) by the factorization method.[2]
  9. From the adjoining figure,
    1. Write the name of a pair of alternate angles.[1]
    2. What is the value of \(x\)?[2]
    3. Calculate the value of \(y\).[1]
    1. Construct a square \(ABCD\) in which side \(AB = 6cm\).[3]
    2. In the adjoining figure, \(\angle RPQ = \angle PQS\) and \(QS = PR\). Show that \(\triangle PQR \cong \triangle QPS\).[2]
    1. Write the formula to calculate the distance between two points.[1]
    2. Find an interior angle of a regular pentagon.[2]
    3. \(A(2,1)\), \(B(-4,5)\), and \(C(2,3)\) are the vertices of \(\triangle ABC\). Find the vertices of the image \(\triangle A'B'C'\) after reflecting \(\triangle ABC\) on the \(x\)-axis using a graph.[3]
  12. Study the following table and solve the given problems:
    13. Class V VI VII VIII Total
    No. of students \(53\) \(43\) \(46\) \(50\) \(192\)
    1. Present the above information in a pie chart.[2]
    2. What is the average number of students per class?[1]

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