CDC Model

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SEE - 2080

CDC MATHS MODEL SET

Time: 3 hrs Full Marks: 75

Attempt all the questions. All the working must be shown.

Question 1

In a survey conducted to determine suitable educational trips for students studying in Class 10 at Shree Sharada Secondary School, it was found that 40 students considered Pokhara to be suitable, 30 students considered Lumbini to be suitable, and 45 students considered Ilam to be suitable. When 15 students from the class said that all three are suitable destinations, 5 students did not provide any opinion.

[1K] (a) If P, L and I denote the set of students who prefer Pokhara, Lumbini and Ilam respectively, write the cardinality notation of students for whom all places are suitable.

Show Solution

\( n(P \cap L \cap I ) = 15 \)

[1U] (b) Show the above information in a Venn diagram.

Show Solution

Venn-Diagram

[3A] (c) How many students are studying in class 10 in Sarada Secondary School? Calculate.

Show Solution

From venn diagram,

\begin{align*} n(U) & = 25+15+15+30+5 \\ & = 90 \end{align*}

[1HA] (d) In a survey, suppose out of those five students who didn't express their opinion indicate Lumbini as a best place then what will be the ratio of students who consider only Pokhara to the students who consider only Lumbini as the suitable destination?

Show Solution

\begin{align*}\text{Ratio} & = \frac{n_o(P)}{n_o(L) } \\ & = \frac{25}{15+5} \\ & = \frac{25}{20}\\ &=\frac{5}{4} \\ & = 5:4 \end{align*}

Question 2

Bipin wants to deposit Rs 1,00,000 in a bank for 2 years. The bank has given Bipin three options (annual compound interest, half-yearly compound interest and quarterly compound interest) at the rate of 10% compound interest per year.

[1K] (a) From the above mentioned option which option is beneficial for Bipin to gain more interest?

Show Solution

Bipin will get more interest by selecting quarterly compound interest.

[2U] (b) How much compound amount can he get at the end of 2 years at the rate of half yearly compound interest?

Show Solution

\begin{align*} P&=100000\\ R&=10\\ T& = 2 \text{ years}\\ \text{Semi-annual C.A. } & = P\left( 1+ \frac{R}{200} \right)^{2T}\\ & = 100000\left( 1+ \frac{10}{200} \right)^{2\times 2}\\ & = 100000( 1.05)^4 \\ & = \text{Rs. } 121550.63\\ \end{align*}

[2A] (c) At the end of 1 year, if Bipin withdraws the amount received according to semi-annual compound interest and then deposits it for the rest of the period to receive quarterly interest, how much amount will he get at the end?

Show Solution

\begin{align*} \text{Semi-annual C.A. for 1 year } & = 100000\left( 1 + \frac{10}{200} \right)^{2\times 1} = \text{Rs. } 110250 \\ \text{Case II: } P = 110250, T = 1, R = 10\% \\ \text{Quarterly C.A. } & = 110250\left( 1+ \frac{10}{400}\right)^{4\times 1} \\ & = 110250(1.025)^4 \\ & = \text{Rs. } 121695.37 \end{align*}

Question 3

Sajan has Rs 1,00,00,000 with him. Sajan purchased a car for Rs. 30,00,000 and land for Rs. 70,00,000. For 2 years, the price of the car has been decreasing at a compound rate of 5% per annum, while the price of land has been increasing at a certain compound rate.

[1K] (a) Write the formula to find out the compound depreciation.

Show Solution

\[ P_T = P\left( 1- \frac{R}{100}\right)^{T} \]

[1U] (b) What will be the price of a car after two years? Calculate.

Show Solution

\begin{align*} P_T & = 3000000\left( 1 - \frac{5}{100} \right)^2 \\ & = 3000000(0.95)^2 \\ & = \text{Rs. } 2707500 \end{align*}

[2HA] (c) After 2 years the total value of car and land is Rs 1,05,72,700, what is the increase compound rate in the price of land? Calculate.

Show Solution

\begin{align*} \text{Car} + \text{Land} & = 10572700 \\ \text{or, Land} & = 10572700 - \text{Car} \\ & = 10572700 - 2707500 \\ & = \text{Rs. } 7865200 \\ \\ \text{For land: } \\ P_T & = P\left( 1 + \frac{R}{100} \right)^T \\ \text{or, } 7865200 & = 7000000\left( 1 + \frac{R}{100} \right)^2 \\ \text{or, } \frac{7865200}{7000000} & = \left( 1 + \frac{R}{100} \right)^2 \\ \text{or, } 1.1236 & = \left( 1 + \frac{R}{100} \right)^2 \\ \text{or, } \sqrt{1.1236} & = 1 + \frac{R}{100} \\ \text{or, } 1.06 - 1 & = \frac{R}{100} \\ \text{or, } 0.06 & = \frac{R}{100} \\ \text{or, } R & = 100 \times 0.06 \\ \therefore R & = 6\% \end{align*}

Question 4

Pravin went to the bank to exchange US dollars. Buying rate: $1 = Rs. 131.05, Selling rate: $1 = Rs. 131.65.

[2A] (a) How many dollars will he get in exchange of Rs 1,57,980?

Show Solution

\begin{align*} Rs. \ 157980 & = \$x \text{ (say)} \\ \$1 & = Rs. \ 131.65 \\ \text{Using chain rule, } \\ x \times 131.65 & = 157980 \\ \text{or, } x & = \frac{157980}{131.65} \\ \text{or, } x & = 1200 \\ \therefore Rs. \ 157980 & = \$ 1200 \end{align*}

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