There were 1,350 examinees in an examination. Out of them, 700 passed in Account, 600 passed in economics, 350 passed in Statistics and 50 failed in all three subjects. Moreover, 200 passed in economics and statistics, 150 students passed in ...

Given \[ \begin{aligned} & a^2-4 a-1=0 \\ & a^2-4 a=1 \\ & a(a-4)=1 \\ & a-4=\frac{1}{a} \\ & a-\frac{1}{a}=4 \end{aligned} \] We have \(\mathrm{a}^2+3 \mathrm{a}+\frac{1}{\mathrm{a}^2}-\frac{3}{\mathrm{a}}\) \[ \begin{aligned} & \left(a^2+\frac{1}{a^2}\right)+3\left(a-\frac{Read more

Given

\[

\begin{aligned}

& a^2-4 a-1=0 \\

& a^2-4 a=1 \\

& a(a-4)=1 \\

& a-4=\frac{1}{a} \\

& a-\frac{1}{a}=4

\end{aligned}

\]

We have \(\mathrm{a}^2+3 \mathrm{a}+\frac{1}{\mathrm{a}^2}-\frac{3}{\mathrm{a}}\)

\[

\begin{aligned}

& \left(a^2+\frac{1}{a^2}\right)+3\left(a-\frac{1}{a}\right) \\

& \left(a-\frac{1}{a}\right)^2+3\left(a-\frac{1}{a}\right)+2 \\

& 4^2+3 \times 4+2=30

\end{aligned}

\]

Therefore, Correct option is (d).

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### Given Data: - Total examinees = 1350 - Passed in Accounts (\(A\)) = 700 - Passed in Economics (\(E\)) = 600 - Passed in Statistics (\(S\)) = 350 - Failed in all subjects = 50 - Passed in Economics and Statistics (\(E \cap S\)) = 200 - Passed in Accounts and Statistics (\(A \cap S\)) = 150 - PassRead more

### Given Data:

– Total examinees = 1350

– Passed in Accounts (\(A\)) = 700

– Passed in Economics (\(E\)) = 600

– Passed in Statistics (\(S\)) = 350

– Failed in all subjects = 50

– Passed in Economics and Statistics (\(E \cap S\)) = 200

– Passed in Accounts and Statistics (\(A \cap S\)) = 150

– Passed in Statistics only = 50

### Calculations:

1. **Total Passed in at least one subject**:

\[

\text{Total passed} = 1350 – 50 = 1300

\]

2. **Using Venn Diagram and Principle of Inclusion-Exclusion**:

\[

|A \cup E \cup S| = |A| + |E| + |S| – |A \cap E| – |E \cap S| – |A \cap S| + |A \cap E \cap S|

\]

Let’s denote \( x = |A \cap E \cap S| \).

3. **Substitute given values**:

\[

1300 = 700 + 600 + 350 – |A \cap E| – 200 – 150 + x

\]

Simplify the equation:

\[

1300 = 1650 – |A \cap E| – 350 + x

\]

\[

1300 = 1300 – |A \cap E| + x

\]

\[

0 = – |A \cap E| + x

\]

\[

x = |A \cap E|

\]

To find \( |A \cap E| \):

4. **Determine students passing only Economics**:

From the image:

\[

\text{Students passing Economics only} = 600 – 200 – x = 450

\]

5. **Determine students passing only Accounts**:

From the image:

\[

\text{Students passing Accounts only} = 700 – 150 – x = 550

\]

### Correct Results:

1. **Passed in at least one subject**:

\[

1300

\]

2. **Passed in all three subjects**:

\[

50

\]

3. **Passed in Economics only**:

\[

450

\]

4. **Failed in Accounts**:

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1350 – 700 = 650

\]