G8_Mahalaxmi Lalitpur_8_2081

1. Two sets \(P\) and \(Q\) are shown in the adjoining Venn diagram.
1. Define subset.[1]
2. Write an improper subset of set \(P\).[1]
3. If two elements \(a\) and \(e\) are removed from set \(P\), then what type of sets are \(P\) and \(Q\)? Write with reason.[1]
1. Let A and B are two sets, then \(A\) is a subset of \(B\) if every element of \(A\) is also an element of \(B\). This is denoted as \(A \subseteq B\).
2. From the Venn diagram, \(P = \{b, c, d, a, e\}\).
   Now, the improper subset of \(P\) is the set \(P=\{a, b, c, d, e\}\) itself.
3. If elements \(a\) and \(e\) are removed from set \(P\), then
   New \(P = \{b, c, d\}\)
   New \(Q = \{a, e, i, o, u\}\) (unchanged)
   Now, \(P \cap Q = \emptyset\) because they share no common elements.
2. Sundar sold a piano of marked price Rs 20000 at \(15\%\) discount.
🎹🎹♪♫~
1. How much is the discount amount of the article whose marked price is \(MP\) and discount percentage is \(D\%\)?[1]
2. Find the selling price of the piano.[1]
3. If the piano is sold at \(20\%\) loss, find the cost price.[2]
4. If piano was sold at Rs. 20,825, then what would be the profit or loss? Find in percentage.[2]
1. Discount amount for an article with marked price \(MP\) and discount rate \(D\%\):
   Discount Amount = \( \dfrac{D}{100} \times MP \)
2. Selling price of the piano:
   Marked Price (MP) = Rs. \(20,000\)
   Discount = \(15\%\)
   Thus
   SP=85% of MP
   orSP= \( \dfrac{85}{100} \times 20,000 = 17,000\)
   So, the selling price is Rs. \(17,000\).
3. Cost price if sold at \(20\%\) loss.
   Selling Price (SP) = Rs. \(17,000\)
   Loss = \(20\%\)
   Thus
   SP=80% of CP
   or\(17,000 = \dfrac{80}{100} \times \text{CP}\)
   or\(\text{CP} = \dfrac{17,000 \times 100}{80} = \dfrac{1,700,000}{80} = 21,250\)
   So, the cost price was Rs. \(21,250\).
4. Profit or loss percent if SP was Rs. 20,825
   Cost Price = Rs. \(21,250\)
   Selling Price = Rs. \(20,825\)
   Since SP < CP, there is a loss.
   Loss = \(21,250 - 20,825= 425\)
   Now
   Loss Percent = \( \dfrac{425}{21,250} \times 100\% = 2\% \)
   So, Sundar would have incurred a loss of 2%.
3. The interest on Rs 5,040 in 5 years is Rs 2,520.
   1. Write the formula to find the simple interest if the amount (A) and principal (P) are given.[1]
   2. Find the rate of interest.[2]
   3. In how many years will the principal and interest be equal? Write with reason.[1]
1. Formula for Simple Interest:
   Simple Interest (I) = Amount (A) - Principal (P)
2. Rate of interest:
   Principal (P) = Rs. 5,040
   Time (T) = 5 years
   Interest (I) = Rs. 2,520
   We know,
   Rate (R) = \( \dfrac{I \times 100}{P \times T} = \dfrac{2,520 \times 100}{5,040 \times 5} = 10\% \)
   So, the rate of interest is 10% per annum.
3. Time for principal and interest to be equal:
   When Principal (P) and Interest (I) are equal, \( I = P \).
   We know, \( T = \dfrac{I \times 100}{P \times R} \).
   Since \( I = P \), then \( T = \dfrac{100}{R} = \dfrac{100}{10} = 10 \) years.
   Reason: Since the interest rate is 10% per year, it takes 10 years for the total interest to reach 100% of the principal (i.e., to become equal to the principal).
4. There are 2075 lemon plants in Numa's garden.
1. Write the number of plants in scientific notation.[1]
2. If the cost of one plant was \(\text{Rs.}\,70\), then find the total cost.[1]
3. Express the number of plants in the quinary number system.[2]
5. As shown in the figure, inside a circular ground having diameter 42 m, a rectangular pond having length 20 m and breadth 10 m is made.
   
   1. Write the formula to calculate the area of the pond. [1]
   2. What is the area of the circular ground? Find it. [1]
   3. Find the area of the remaining ground excluding the pond. [2]
   4. How much will it cost to plant dubo at the rate of Rs 25 per \(m^2\) in the remaining part of the ground? [1]
1. Area of the pond formula
   We know that,
   Since the pond is rectangular, Area (\( A \)) = \( l \times b \)
2. Area of the circular ground
   Given that,
   Diameter (\( d \)) = 42 m
   Radius (\( r \)) = \( \frac{42}{2} = 21 \) m
   Now using the formula, we get
   Area of ground (\( A_g \)) = \( \pi r^2 = \frac{22}{7} \times (21)^2 = 1386 \) m²
3. Area of the remaining ground
   First, calculating the area of the rectangular pond,
   Length (\( l \)) = 20 m and Breadth (\( b \)) = 10 m
   Area of pond (\( A_p \)) = \( 20 \times 10 = 200 \) m²
   Now to find the area excluding the pond,
   Remaining Area = \( A_g - A_p = 1386 - 200 = 1186 \) m²
4. Cost of planting grass (dubo)
   We know that,
   Remaining area = 1186 m²
   According to the question, the rate is Rs 25 per \(m^2\), so
   Total Cost = \( 1186 \times 25 = \) Rs 29,650
1. Write the expanded form of \(x^2 - y^2\).[1]
2. Simplify: \(\left(\dfrac{x^a}{x^b}\right)^{a+b} \times \left(\dfrac{x^b}{x^c}\right)^{b+c} \times \left(\dfrac{x^c}{x^a}\right)^{c+a}\)[2]
7. An equation \(x^2 - 7x + 12 = 0\) is given.
1. What is the degree of the given equation? Write it.[1]
2. For what values of \(x\) is the expression \(x^2 - 7x + 12 = 0\) equal to zero?[2]
1. Find the Highest Common Factor (H.C.F.) of \(2a^2 - 9a + 10\) and \(5(a - 2)\).[2]
2. Simplify: \(\dfrac{p - 2}{p + 2} - \dfrac{p - 2}{p^2 - 4}\)[2]
9. In the adjoining figure, \(AB\) intersects straight lines \(LM\) and \(PQ\) at points \(E\) and \(F\) respectively.
1. Write the name of a pair of co-interior angles.[1]
2. Find the value of \(x\).[2]
3. At what value of \(\angle MEG\) will the lines \(LM\) and \(PQ\) become parallel?[1]
1. Construct a square \(PQRS\) having side length \(6cm\).[3]
2. In the adjoining figure, \(ABCD\) is a parallelogram. Prove that \(\triangle ABD \cong \triangle CDB\).[2]
1. A tessellation is formed by regular octagons and squares. Write the type of tessellation.[1]
2. In \(\triangle ABC\), if the bearing of point \(B\) from point \(A\) is \(075^\circ\), find the bearing of point \(A\) from point \(B\).[2]
3. Reflect \(\triangle ABC\) (shown on graph paper) on the \(y\)-axis and write the coordinates of the vertices of the image triangle.[3]
12. The ages (in years) of \(10\) students of grade \(8\) are given below: \(13, 14, 12, 14, 13, 14, 13, 12, 14, 16\).
1. What is the mode of the above data? Write it.[1]
2. Find the average age (in years) of the students from the above data.[2]