G8_DMUN_8_2018

1. Study the Venn diagram alongside and answer the following questions.
1. Write the sets \(M\) and \(N\) by description method and tabulation method.[1]
2. Write the universal set \(U\).[1]
3. Are \(M\) and \(N\) disjoint or overlapping sets? Why?[1]
1. From the Venn diagram
   The sets \(M\) and \(N\) by tabulation method are as follows.
   \(M = \{3, 6, 9, 12\}\)
   \(N = \{6, 12, 18, 24\}\)
   The sets \(M\) and \(N\) by description method are as follows.
   \(M = \{x \mid x \text{ is a multiple of } 3 \text{ and } x \leq 12\}\)
   \(N = \{x \mid x \text{ is a multiple of } 6 \text{ and } 6 \leq x \leq 24\}\)
2. The universal set \(U\) is
   \(U = \{3, 6, 9, 12, 18, 21, 24, 30\}\)
3. Sets \(M\) and \(N\) are overlapping sets.
   Because they share common elements \(6\) and \(12\)
   \(M \cap N = \{6, 12\} \neq \emptyset\).
1. Which two digits are used for binary number system?[1]
2. Convert \(1234_5\) into decimal number.[2]
3. Convert \(0.\overline{24}\) into a fraction.[1]
4. Divide Rs 500 in the ratio \(7:5\).[2]
3. A mobile set of marked price Rs 22000 is sold at \(10\%\) discount.
1. Write the formula to find the discount amount.[1]
2. Find the discount amount of the mobile set.[1]
3. Find the selling price of the mobile.[1]
1. Formula to find the discount amount:
   Discount Amount = \( \dfrac{D}{100} \times \text{MP} \)
2. Discount amount of the mobile set.
   Marked Price (MP) = Rs. \(22,000\)
   Discount = \(10\%\)
   Thus
   Discount Amount = \( \dfrac{10}{100} \times 22,000 = 2,200 \)
   So, the discount amount is Rs. \(2,200\).
3. Selling price of the mobile.
   Marked Price (MP) = Rs. \(22,000\)
   Discount = \(10\%\)
   Thus
   SP = 90% of MP = \( \dfrac{90}{100} \times 22,000 = 19,800 \)
   So, the selling price is Rs. \(19,800\).
4. The simple interest on Rs 4,000 for 5 years is Rs 2,400.
   1. If principal (P), time (T), and rate of interest (R) are given, write the formula to find interest (I).[1]
   2. Find the rate of interest.[2]
   3. At the same interest rate, how much will be the interest on Rs 2,000 for 4 years?[2]
1. Formula to find interest:
   Simple Interest (I) = \( \dfrac{P \times T \times R}{100} \)
2. Rate of interest:
   Principal (P) = Rs. 4,000
   Time (T) = 5 years
   Interest (I) = Rs. 2,400
   We know,
   Rate (R) = \( \dfrac{I \times 100}{P \times T} = \dfrac{2,400 \times 100}{4,000 \times 5} = 12\% \)
   So, the rate of interest is 12% per annum.
3. Calculation of new interest:
   New Principal (P) = Rs. 2,000
   New Time (T) = 4 years
   Rate (R) = 12%
   Thus,
   Interest (I) = \( \dfrac{P \times T \times R}{100} = \dfrac{2,000 \times 4 \times 12}{100} = 960 \)
   So, the interest on Rs. 2,000 for 4 years will be Rs. 960.
1. Find the value of \((5x)^0\).[1]
2. Simplify: \(x^{a-b} \times x^{b-c} \times x^{c-a}\).[2]
1. Find the H.C.F. of \(x^2 - 16\) and \(x^2 - 8x + 16\).[2]
2. Solve: \(x^2 - 5x + 6 = 0\).[2]
1. Solve by graphical method: \(x + y = 6\), \(x - y = 2\).[2]
2. Simplify: \(\dfrac{a}{a - b} - \dfrac{b}{a + b}\).[2]
8. In the adjoining figure, ABCD is a square and point O is the centre of the circle.
   
   1. Find the radius of the circle. [1]
   2. Find the area of the square ABCD. [2]
   3. Find the area of the circle. [2]
   4. Find the area of the shaded portion. [2]
1. Radius of the circle
   In the figure, the diagonal AC of the square is the diameter of the circle.
   Diameter (\( d \)) = 14 cm
   Radius (\( r \)) = \( \frac{d}{2} = \frac{14}{2} = 7 \) cm
2. Area of the square ABCD
   When diagonal (\( d \)) is given, the area of a square is:
   Area (\( A_s \)) = \( \frac{1}{2} \times d^2 = \frac{1}{2} \times 14^2 = 98 \) cm²
3. Area of the circle
   We know that,
   Area (\( A_c \)) = \( \pi r^2 = \frac{22}{7} \times 7^2 = 154 \) cm²
4. Area of the shaded portion
   Shaded Area = Area of circle - Area of square
   or Shaded Area = \( 154 - 98 = 56 \) cm²
9. Solve the given questions from the adjoining figure.
1. Write the relation between \(\angle AGH\) and \(\angle GHD\).[1]
2. If \(AB \parallel CD\), find the value of \(x\).[2]
3. Find each interior angle of a regular octagon.[1]
1. Construct parallelogram \(PQRS\) having \(PQ = 6cm\), \(QR = 5cm\) and \(\angle PQR = 45^\circ\).[3]
2. In the adjoining figure, if \(AO = OC\) and \(BO = OD\), prove that \(\triangle ABO \cong \triangle ODC\).[2]
1. Write the definition of regular tessellation.[1]
2. If the bearing of place \(D\) from place \(C\) is \(060^\circ\), find the bearing of place \(C\) from place \(D\).[2]
3. Plot \(\triangle KLM\) with vertices \(K(6,6)\), \(L(4,5)\), and \(M(6,2)\) on graph paper. Then plot the image \(\triangle K'L'M'\) after rotation about the origin through \(+90^\circ\) on the same graph paper and write the coordinates of the vertices of \(\triangle K'L'M'\).[3]
12. Ten students of class \(8\) have obtained the following marks in mathematics: \(50, 60, 60, 70, 80, 40, 60, 90, 50, 80\).
1. Find the mean.[2]
2. Find the mode.[1]