2 Marks Questions

Question 12 Marks

Evaluate: $-8 \bmod 5$

Question 22 Marks

If the matrix $\left[\begin{array}{ccc}x+4 & x & x \\ x & x+4 & x \\ x & x & x+4\end{array}\right]$ is singular, find x.

Answer

$
\left|\begin{array}{ccc}
x+4 & x & x \\
x & x+4 & x \\
x & x & x+4
\end{array}\right|=0
$
(Operate $C_1 \rightarrow C_1+C_2+C_3$ and take $(3 x+4)$ out from new $C_1$
$\Rightarrow(3 x +4)\left|\begin{array}{ccc}1 & x & x \\ 1 & x+4 & x \\ 1 & x & x+4\end{array}\right|=0\left(\right.$ Operate $\left.C _2 \rightarrow C _2- xC _1, C _3 \rightarrow C _3- xC _1\right)$
$\Rightarrow(3 x+4)\left|\begin{array}{lll}1 & 0 & 0 \\ 1 & 4 & 0 \\ 1 & 0 & 4\end{array}\right|=0 \Rightarrow(3 x+4) \cdot 4 \cdot 4=0$
$\begin{array}{l}\Rightarrow 16(3 x+4)=0 \\ \Rightarrow 3 x+4=0 \Rightarrow x=-\frac{4}{3}\end{array}$

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Question 32 Marks

Using matrix method, solve the following system of equations:
$
\begin{array}{l}
x-2 y+3 z=6 \\
x+4 y+z=12 \\
x-3 y+2 z=1
\end{array}
$

Answer

The given system of equations can be written as $AX = B$
$
\text { where } A=\left[\begin{array}{rrr}
1 & -2 & 3 \\
1 & 4 & 1 \\
1 & -3 & 2
\end{array}\right], X=\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right] \text { and } B=\left[\begin{array}{r}
6 \\
12 \\
1
\end{array}\right].
$
Now, $| A |=1(8+3)+2(2-1)+3(-3-4)=-8 \neq 0$
$
\Rightarrow A^{-1} \text { exists }
$
$\Rightarrow$ the given system of equations has a unique solution $X=A^{-1} B$
$
\begin{array}{l}
A_{11}=11, A_{12}=-1, A_{13}=-7, \\
A_{21}=-5, A_{22}=-1, A_{23}=1, \\
A_{31}=-14, A_{32}=2, A_{33}=6.
\end{array}
$
$\begin{array}{l}\text { So, } A^{-1}=\frac{1}{-8}\left[\begin{array}{rrr}11 & -5 & -14 \\ -1 & -1 & 2 \\ -7 & 1 & 6\end{array}\right] \\ \therefore\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\frac{1}{-8}\left[\begin{array}{rrr}11 & -5 & -14 \\ -1 & -1 & 2 \\ -7 & 1 & 6\end{array}\right]\left[\begin{array}{r}6 \\ 12 \\ 1\end{array}\right] \\ =\frac{1}{-8}\left[\begin{array}{r}66-60-14 \\ -6-12+2 \\ -42+12+6\end{array}\right]=\frac{1}{-8}\left[\begin{array}{r}-8 \\ -16 \\ -24\end{array}\right] \\ \Rightarrow x=1, y=2, z =3\end{array}$

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Question 42 Marks

Evaluate the definite integral:
$
\int_2^4 \frac{x}{x^2+1} d x
$

Answer

Put $x ^2+1= t \Rightarrow 2 xdx = dt \Rightarrow xdx =\frac{1}{2} d t$
When $x =2, t =2^2+1=5$ and when $x =4, t =4^2+1=17$
$
\begin{array}{l}
\therefore \int_2^4 \frac{x}{x^2+1} d x=\frac{1}{2} \int_5^{17} \frac{d t}{t}=\frac{1}{2}[\log |t|]_5^{17} \\
=\frac{1}{2}(\log 17-\log 5)=\frac{1}{2} \log \frac{17}{5}
\end{array}
$

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Question 52 Marks

Rahul purchased an old scooter for ₹ 16000. If the cost of the scooter after 2 years depreciates to ₹14440, find the rate of depreciation.

Answer

The current cost of the scooter, $C _0=16000$
Cost after two years, $C =14440$
Let the rate of depreciation be R , then
$
\begin{array}{l}
C=C_0\left(1-\frac{R}{100}\right)^{T} \\
\Rightarrow 14400=16000\left(1-\frac{R}{100}\right)^2 \\
\Rightarrow \frac{14400}{16000}=\left(1-\frac{R}{100}\right)^2 \\
\Rightarrow\left(\frac{38}{40}\right)^2=\left(1-\frac{R}{100}\right)^2 \\
\Rightarrow \frac{38}{40}=1-\frac{R}{100} \\
\Rightarrow \frac{R}{100}=1-\frac{38}{40} \\
\Rightarrow R=\frac{2 \times 100}{40} \\
\Rightarrow R=5 \%
\end{array}
$

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Question 62 Marks

Mr. X took a loan of ₹2,000 for 6 months. Lender deducts ₹200 as interest while lending. Find the effective rate of interest charged by lender.

Answer

Since the money Lender deducts ₹200 as interest while lending a loan of ₹2000 for 6 months, therefore ₹200 may be treated as interest on ₹1800 for 6 months. Consequently, interest rate per six months is
$
i=\frac{200}{1800}=\frac{1}{9}
$
Thus, the equivalent effective rate of interest, is given by
$
\begin{array}{l}
\text { Now, } r_{\text {eff }}=(1+i)^2-1 \\
=\left(1+\frac{1}{9}\right)^2-1=0.23456 \\
=23.45 \%
\end{array}
$

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Question 72 Marks

Construct 3-yearly moving averages from the following data:

Year:	2010	2011	2012	2013	2014	2015	2016
Imported cotton consumption in India (in '000 bales):	129	131	106	91	95	84	93

Answer

Construction of 3-yearly moving average

Year	Imported cotton consumption in India (in '000 bales)	3-yearly moving totals	3-yearly moving averages
2010	129	-	-
2011	131	366	122.00
2012	106	328	109.33
2013	91	292	97.33
2014	95	270	90.00
2015	84	272	90.66
2016	93	-	-

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Applied Maths 2 Marks Questions

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