G8_Tarekshwor KTM_8_2081

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  1. Sets \(P\) and \(Q\) are shown in a Venn diagram.
    1. Define subset.[1]
    2. Write the improper subset of set \(Q\).[1]
    3. If \(1, 3, 5\) are the only members of set \(P\), then what types of sets are \(P\) and \(Q\)? Write with reasons.[1]
    1. A set \(A\) is a subset of another set \(B\) if every element of \(A\) is also an element of \(B\). This is denoted as \(A \subseteq B\).
    2. From the Venn diagram, set \(Q = \{2, 4, 6, 8\}\).
      Thus,tThe improper subset of \(Q\) is the set \(Q\) itself
      \(\{2, 4, 6, 8\}\)
    3. If \(P = \{1, 3, 5\}\) then
      there are no common elements between \(P\) and \(Q\).
      Thus, \(P \cap Q = \emptyset\).
      Therefore, \(P\) and \(Q\) are disjoint sets.
  2. The highway distance between Kathmandu to Narayanghat is 140000 m.
    1. Write the distance in scientific notation.[1]
    2. Convert 1050 into the quinary number system.[1]
    3. Convert \(0.\overline{34}\) into a fraction.[2]
  3. Anish marked the price of a radio Rs. 3000. If he sold it allowing a discount of \(15\%\) and made a profit of Rs. 500.
    1. If marked price (MP) and discount percent (D) are represented by (MP) and (D) respectively, write the formula to find the selling price.[1]
    2. What is the selling price of the radio?[1]
    3. If the discount was not allowed, then what would be the profit?[2]
    1. Formula to find the selling price (SP) using marked price (MP) and discount percent (D):
      SP = MP – D% of MP
    2. Selling price of the radio:
      Marked Price (MP) = Rs. \(3,000\)
      Discount = \(15\%\)
      Thus
      SP = 85%of MP = \( \dfrac{85}{100} \times 3,000 = 2,550\)
      So, the selling price is Rs. 2,550.
    3. Profit if no discount was allowed.
      From part (ii), we got SP = Rs. 2,550, and Profit = Rs. 500
      Thus
      Cost Price (CP) = SP – Profit = 2,550 - 500 = 2,050
      Now
      CP=2050
      SP=3,000
      Then
      Profit= SP – CP = 3000-2050=950
      So, the new profit would be Rs. 950.
  4. Rajan deposited Rs. 60,000 at the rate of 10% p.a. in a savings account. After 5 years, he withdrew Rs. 40,000 and the total interest of 5 years.
    1. If interest (I), rate (R), and time (T) are given, write the formula to calculate the principal.[1]
    2. Find the interest of 5 years.[2]
    3. How long should he keep the remaining balance in the bank to get a total interest of Rs. 40,000 from the beginning?[2]
    1. Formula to calculate principal:
      Principal (P) = \( \dfrac{I \times 100}{T \times R} \)
    2. Interest of 5 years:
      Principal (P) = Rs. 60,000
      Time (T) = 5 years
      Rate (R) = 10%
      Thus,
      Interest (I) = \( \dfrac{60,000 \times 5 \times 10}{100} = 30,000 \)
      So, the interest for 5 years is Rs. 30,000.
    3. Time for remaining balance:
      Interest already earned in 5 years = Rs. 30,000
      Total target interest = Rs. 40,000
      Remaining interest needed (I) = 40,000 - 30,000 = Rs. 10,000
      Remaining Principal (P) after withdrawal = 60,000 - 40,000 = Rs. 20,000
      Rate (R) = 10%
      Thus,
      Additional Time (T) = \( \dfrac{I \times 100}{P \times R} = \dfrac{10,000 \times 100}{20,000 \times 10} = 5 \) years
      So, he should keep the remaining balance for an additional 5 years (total 10 years from the beginning).
  5. The length of a rectangular field is twice the breadth. A circular garden of radius 35 m is constructed in the field. The length of the field is 100 m.
    1. Write the formula to find the area of the rectangular field. [1]
    2. Calculate the area of the circular garden. [1]
    3. What is the area of the field excluding the garden? [2]
    4. Compare the perimeter of the field and the garden. [2]
    1. Area of rectangular field formula
      We know that,
      Area (\( A \)) = Length (\( l \)) \(\times\) Breadth (\( b \))
    2. Area of circular garden
      Given,
      Radius (\( r \)) = 35 m
      Using the formula,
      Area of garden (\( A_c \)) = \( \pi r^2 = \frac{22}{7} \times 35^2 = 3850 \) m²
    3. Area of the field excluding the garden
      First, find the breadth and area of the field,
      Length (\( l \)) = 100 m
      Breadth (\( b \)) = \( \frac{100}{2} = 50 \) m
      Area of field (\( A_r \)) = \( 100 \times 50 = 5000 \) m²
      Area excluding garden = \( 5000 - 3850 = 1150 \) m²
    4. Comparison of Perimeters
      Perimeter of field (\( P \)) = \( 2(l + b) = 2(100 + 50) = 300 \) m
      Circumference of garden (\( C \)) = \( 2\pi r = 2 \times \frac{22}{7} \times 35 = 220 \) m
      Comparison: The perimeter of the field is \( 300 - 220 = 80 \) m more than the garden's circumference.
    1. Write the combined form of \((m+n)(m-n)\).[1]
    2. Simplify: \(\left( \frac{m^a}{m^b} \right)^{a+b} \times \left( \frac{m^b}{m^c} \right)^{b+c} \times \left( \frac{m^c}{m^a} \right)^{c+a}\)[2]
  7. Two equations are given as: \(x + y = 4\) and \(x - y = 2\).
    1. What are the system of equations called?[1]
    2. Solve the above equations by using graph.[2]
  8. If two algebraic expressions are \(x^2 - 5x + 6\) and \(x^2 - 9\).
    1. Find the H.C.F. of the given algebraic expressions.[2]
    2. At what value of \(x\) is the value of the expression \(x^2 - 5x + 6\) equal to zero?[2]
  9. In the adjoining figure, line \(JI\) intersects straight lines \(AB\) and \(CD\) at points \(G\) and \(E\) respectively. \(\angle GHE = 45^\circ\).
    1. Write a pair of co-interior angles in the figure.[1]
    2. What type of triangle is \(\triangle GHE\) according to its angles?[2]
    3. At what value of \(\angle GED\) will the given line segments \(AB\) and \(CD\) be parallel?[1]
    1. Construct a parallelogram \(ABCD\) where \(AB = 7\) cm, \(AD = 6\) cm and \(\angle BAC = 75^\circ\).[3]
    2. In the given figure, \(\triangle ABC \sim \triangle AXY\). If \(AB = 4\) cm, \(AX = 6\) cm, and \(BC = 12\) cm, find the value of \(XY\).[2]
    1. Define regular tessellation.[1]
    2. In the given figure, the bearing of point \(B\) from point \(A\) is \(075^\circ\). What is the bearing of point \(A\) from point \(B\)?[2]
    3. \(A(-3, 2)\), \(B(-5, 4)\), and \(C(-2, 6)\) are the vertices of \(\triangle ABC\). Plot \(\triangle ABC\) on a graph and reflect it in the \(x\)-axis. Then, write the coordinates of the image points \(A'\), \(B'\), and \(C'\).[3]
  12. The monthly expenditure of Ramesh's family is given below:
    13. Baishakh Jestha Asar Shrawan
    Rs. 8,000 Rs. 12,500 Rs. 9,000 Rs. 4,500
    1. What is the monthly average expenditure of Ramesh's family?[1]
    2. Present Ramesh's family expenditure in a pie chart.[2]

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