2 Marks Questions

Question 12 Marks

In what ratio must a grocer mix two varieties of tea worth ₹ 60 per kg and ₹ 65 per kg so that by selling the mixture at ₹ 68.20 per kg may gain 10%?

Answer

We have,
S.P. of mixture = ₹ 68.20 and Gain = 10%
$\therefore$ C.P. $=\frac{S P}{1+\frac{\text { Gain }}{100}}$
$\Rightarrow$ C.P. $=\left(\frac{68.20}{1+\frac{10}{100}}\right)=\left(\frac{682}{11}\right)=₹ 62$
The allegation grid is as given below:

By using the rule of allegation, we obtain
Tea of first kind: Tea of second kind = 3 : 2
Hence, the grocer must mix in the ratio 3 : 2

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Question 22 Marks

Find the values of x , if $\left|\begin{array}{ll}x+1 & x-1 \\ x-3 & x+2\end{array}\right|=\left|\begin{array}{cc}4 & -1 \\ 1 & 3\end{array}\right|$

Answer

We have $\left|\begin{array}{ll}x+1 & x-1 \\ x-3 & x+2\end{array}\right|=\left|\begin{array}{cc}4 & -1 \\ 1 & 3\end{array}\right|$
$\begin{array}{l}\Rightarrow( x +1)( x +2)-( x -1)( x -3)=4 \times 3-1 \times(-1) \\ \Rightarrow\left( x ^2+2 x + x +2\right)-\left( x ^2-3 x - x +3\right)=12+1 \\ \Rightarrow-2 x -1=13 \\ \Rightarrow-2 x =14 \\ \Rightarrow x =-7\end{array}$

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Question 32 Marks

If $A=\left[\begin{array}{rr}2 & -1 \\ 3 & 2\end{array}\right]$ and $B=\left[\begin{array}{rr}0 & 4 \\ -1 & 7\end{array}\right]$, find $3 A^2-2 B+I$

Answer

Given, $A =\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]$ and $B =\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]$
$3 A^2-2 B+I$
$\begin{array}{l}=3\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]-2\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]+\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ =3\left[\begin{array}{cc}4-3 & -2-2 \\ 6+6 & -3+4\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]+\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right] \\ =3\left[\begin{array}{cc}1 & -4 \\ 12 & 1\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]+\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ =\left[\begin{array}{cc}3 & -12 \\ 36 & 3\end{array}\right]-\left[\begin{array}{cc}0 & 8 \\ -2 & 14\end{array}\right]+\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ =\left[\begin{array}{cc}3-0+1 & -12-8+0 \\ 36+2+0 & 3-14+1\end{array}\right] \\ =\left[\begin{array}{cc}4 & -20 \\ 38 & -10\end{array}\right]\end{array}$

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Question 42 Marks

By using property of definite integrals, evaluate $\int_0^1|2 x-1| d x$

Answer

$\int_0^1|2 x-1| d x=\int_0^{\frac{1}{2}}|2 x-1| d x+\int_{\frac{1}{2}}^1|2 x-1| d x$
$=\int_0^{\frac{1}{2}}-(2 x-1) d x+\int_{\frac{1}{2}}^1(2 x-1) d x$
$=\left[-\left(x^2-x\right)\right]_0^{\frac{1}{2}}+\left[\left(x^2-x\right)\right]_{\frac{1}{2}}^1$
$=-\left(\frac{1}{4}-\frac{1}{2}\right)-0+0-\left(\frac{1}{4}-\frac{1}{2}\right)=\frac{1}{4}+\left(\frac{1}{4}\right)=\frac{1}{2}$

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Question 52 Marks

Jyoti buys a car for which she makes down payment of ₹3,50,000 and the balance is to be paid in 3 years by monthly installments of ₹34,000 each. If the financer charges interest at the rate of 12% per annum and uses flat rate method, find the actual price of the car.

Answer

Let the amount of loan be ₹P
EMI $=₹ 34000, i =\frac{12}{12 \times 100}=0.01, n =3 \times 12=36$
$EMI =\frac{ P + P n i}{n}$
$\Rightarrow 34000=\frac{ P (1+36 \times 0.01)}{36}=\frac{ P \times 1.36}{36}$
$\Rightarrow P =\frac{34000 \times 36}{1.36} \Rightarrow P =900,000$
$\therefore$ Price of the car $=$ down payment + loan
= ₹350000 + ₹900000 = ₹1250000

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Question 62 Marks

A simple interest of ₹ 1000 is paid on a certain sum of money at 10% p.a for 4 years. Find the sum.

Answer

Let $P$ be ₹ $x$, S.I. $=₹ 1000$, $r=10 \%$ p.a., $n=4$ years
$\begin{array}{l}\therefore \frac{x \times 10 \times 4}{100}=1000 \\ \Rightarrow x =2500 \\ \therefore \text { Sum }=₹ 2500\end{array}$

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Question 72 Marks

Calculate five yearly moving averages of the number of students who have studied in a school given below:

Year	1993	1994	1995	1996	1997	1998	1999	2000	2001	2002
No. of students	442	427	467	502	512	515	520	527	515	541

Answer

Calculation of 5-year moving averages:

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Applied Maths 2 Marks Questions

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