Unit 5: Trigonometry, Seating and Coded Inequalities - Subjective Questions

PEA308 — Advanced Analytical Skills-Ii • Practice Questions with Detailed Answers

20 questions

A tower stands vertically on level ground. From a point $30\text{ m}$ away from its base, the angle of elevation of its top is $45^\circ$ . Find the height of the tower.

Define the terms angle of elevation and angle of depression. Explain their use in height and distance problems with a suitable diagram description.

From the top of a $50\text{ m}$ high building, the angle of depression of a car on the ground is $30^\circ$ . Find the horizontal distance of the car from the building.

A man observes the top of a tower at an angle of elevation of $30^\circ$ . After walking $40\text{ m}$ towards the tower, the angle becomes $60^\circ$ . Find the height of the tower.

A kite is flying at a height of $60\text{ m}$ above the ground. The string attached to it makes an angle of $60^\circ$ with the ground. Find the length of the string, assuming it is taut.

A train moves away from a pole at a constant speed. The angle of elevation of the top of the pole changes from $60^\circ$ to $30^\circ$ in $10$ seconds. If the height of the pole is $20\sqrt{3}\text{ m}$ , find the speed of the train.

Explain the method for solving moving-object problems involving angles of elevation. State the key assumptions generally used in such questions.

Eight persons $A, B, C, D, E, F, G,$ and $H$ are sitting in a row facing north. $B$ sits third to the right of $A$ . $C$ sits second to the left of $B$ . $D$ sits immediately to the right of $C$ . If $A$ sits at the extreme left end, determine the positions of $A, B, C,$ and $D$ .

Explain the difference between a linear seating arrangement and a circular seating arrangement. How does the direction in which persons face affect the interpretation of left and right?

Six persons $P, Q, R, S, T,$ and $U$ sit in a row facing south. $Q$ is second to the left of $P$ . $R$ is immediately to the right of $Q$ . $S$ sits at the extreme right end. If $P$ is third from the left end, determine the positions of $P, Q, R,$ and $S$ from the observer's left.

Eight people $A, B, C, D, E, F, G,$ and $H$ sit around a circular table facing the centre. $B$ sits second to the right of $A$ . $C$ sits immediately to the left of $B$ . $D$ sits opposite $A$ . Determine the relative positions of $B, C,$ and $D$ with respect to $A$ .

Describe a systematic approach for solving circular seating arrangement problems involving persons facing the centre and persons facing outside the centre.

Define an inequality. Distinguish between the symbols $>$ , $<$ , $\geq$ , $\leq$ , and $\neq$ with examples.

Solve the inequality $3x - 7 > 11$ and represent the solution on a number line.

Solve the compound inequality $-2 \leq 3x + 4 < 10$ .

Explain the rule for multiplying or dividing an inequality by a negative number. Solve $-4x + 5 \geq 17$ .

In coded inequalities, the symbols have the following meanings: $P @ Q$ means $P \geq Q$ , $P \# Q$ means $P < Q$ , $P \$ Q$ means $P = Q$ , $P \% Q$ means $P > Q$ , and $P \& Q$ means $P \leq Q$ . Determine the relation between $M$ and $R$ if $M @ N \# O \$ P \% Q \& R$ .

What is meant by comparison of roots of equations? Explain how the nature and relative position of roots of a quadratic equation $ax^2 + bx + c = 0$ can be studied using the discriminant.

Compare the roots of the equations $x^2 - 7x + 12 = 0$ and $x^2 - 9x + 20 = 0$ . State the relation between their larger roots and smaller roots.

Let $\alpha$ and $\beta$ be the roots of $2x^2 - 9x + 4 = 0$ . Find $\alpha + \beta$ , $\alpha\beta$ , and determine whether both roots are positive.

Question 1

A tower stands vertically on level ground. From a point $30\text{ m}$ away from its base, the angle of elevation of its top is $45^\circ$ . Find the height of the tower.

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Answer 1

Given:

Distance from the observation point to the base $= 30\text{ m}$
Angle of elevation $= 45^\circ$

Using the trigonometric relation:

$\tan \theta = \frac{\text{Perpendicular}}{\text{Base}}$

$\tan 45^\circ = \frac{h}{30}$

Since $\tan 45^\circ = 1$ :

$1 = \frac{h}{30}$

$h = 30\text{ m}$

Therefore, the height of the tower is $30\text{ m}$ .

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Question 2

Define the terms angle of elevation and angle of depression. Explain their use in height and distance problems with a suitable diagram description.

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Answer 2

Angle of Elevation:

The angle between the horizontal line through the observer's eye and the line of sight when the object is above the observer's level is called the angle of elevation.

Angle of Depression:

The angle between the horizontal line through the observer's eye and the line of sight when the object is below the observer's level is called the angle of depression.

Use in problems:

Use the angle of elevation when observing the top of a tower, building, tree, or hill from ground level.
Use the angle of depression when observing an object located below an elevated point, such as a boat viewed from a lighthouse.
The horizontal lines at the observer and object levels are parallel. Therefore, the angle of depression is equal to the corresponding angle of elevation.

Diagram description:

Draw a horizontal line through the observer.
Draw the observer's line of sight towards the object.
The angle formed above the horizontal line is the angle of elevation.
The angle formed below the horizontal line is the angle of depression.

These angles help form a right-angled triangle, allowing the use of $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ .

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Question 7

Explain the method for solving moving-object problems involving angles of elevation. State the key assumptions generally used in such questions.

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Answer 7

Method of solving moving-object problems:

Draw the object, observer, ground line, and vertical height clearly.
Mark the initial and final positions of the moving object.
Form separate right-angled triangles for both positions.
Use the appropriate trigonometric ratio, generally:

$\tan \theta = \frac{\text{height}}{\text{horizontal distance}}$

Find the distances at the two positions.
Calculate the distance travelled using the difference of distances when the object moves in a straight line towards or away from the fixed object.
If time is given, calculate speed using:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Common assumptions:

The ground is level.
The tower, pole, lighthouse, or building is vertical.
The moving object travels in a straight line.
The observer's eye level is taken as ground level unless otherwise stated.
The angle is measured from the horizontal line.

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Question 8

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Answer 8

Since all persons face north, left and right are interpreted normally.

Step 1: $A$ sits at the extreme left end.

$A \quad \_ \quad \_ \quad \_ \quad \_ \quad \_ \quad \_ \quad \_$

Step 2: $B$ sits third to the right of $A$ .

$A \quad \_ \quad \_ \quad B \quad \_ \quad \_ \quad \_ \quad \_$

Step 3: $C$ sits second to the left of $B$ .

Thus, $C$ occupies the first position.

$C \quad \_ \quad \_ \quad B \quad \_ \quad \_ \quad \_ \quad \_$

But this conflicts with the condition that $A$ is at the extreme left end.

Conclusion: The given conditions are inconsistent. If $A$ is in the first position and $B$ is third to its right, then the second position to the left of $B$ is also the first position, already occupied by $A$ . Therefore, no valid seating arrangement is possible under these conditions.

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Question 9

Explain the difference between a linear seating arrangement and a circular seating arrangement. How does the direction in which persons face affect the interpretation of left and right?

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Answer 9

Linear seating arrangement:

Persons sit in a straight line.
There are fixed end positions, such as extreme left and extreme right.
Positions can be numbered from left to right.
The number of persons to the left or right can be determined directly.

Circular seating arrangement:

Persons sit around a circular table or in a circle.
There are no fixed end positions.
One person's position is often fixed first to remove rotational ambiguity.
The seating order is considered clockwise or anticlockwise.

Effect of facing direction:

If a person faces north in a linear arrangement, their left is the observer's left and their right is the observer's right.
If a person faces south, their left and right are reversed from the observer's perspective.
In a circle, persons facing the centre have their left in the clockwise direction and right in the anticlockwise direction.
Persons facing outside have their left in the anticlockwise direction and right in the clockwise direction.

Correctly identifying the facing direction is essential before placing any person.

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Question 10

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Answer 10

All persons face south. Therefore, a person's left is towards the observer's right, and a person's right is towards the observer's left.

Step 1: $P$ is third from the observer's left.

$\_ \quad \_ \quad P \quad \_ \quad \_ \quad \_$

Step 2: $Q$ is second to the left of $P$ .

Since $P$ faces south, $P$ 's left is towards the observer's right. Therefore, $Q$ is second to the observer's right of $P$ .

$\_ \quad \_ \quad P \quad \_ \quad Q \quad \_$

Step 3: $R$ is immediately to the right of $Q$ .

Since $Q$ faces south, $Q$ 's right is towards the observer's left. Hence, $R$ sits immediately to the observer's left of $Q$ .

$\_ \quad \_ \quad P \quad R \quad Q \quad \_$

Step 4: $S$ sits at the extreme right end.

$\_ \quad \_ \quad P \quad R \quad Q \quad S$

Therefore:

$P$ is in position $3$ .
$Q$ is in position $5$ .
$R$ is in position $4$ .
$S$ is in position $6$ .

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Question 11

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Answer 11

Since all persons face the centre:

Right means moving in the anticlockwise direction.
Left means moving in the clockwise direction.

Fix $A$ at any position because a circular arrangement can be rotated without changing relative positions.

Step 1: $B$ sits second to the right of $A$ .

Thus, starting from $A$ , move two places anticlockwise to place $B$ .

Step 2: $C$ sits immediately to the left of $B$ .

Since everyone faces the centre, move one place clockwise from $B$ to place $C$ .

Therefore, $C$ is first to the right of $A$ .

Step 3: $D$ sits opposite $A$ .

With eight persons, the opposite position is four places away from $A$ .

Relative positions:

$B$ is second to the right of $A$ .
$C$ is first to the right of $A$ .
$D$ is opposite $A$ .

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Question 12

Describe a systematic approach for solving circular seating arrangement problems involving persons facing the centre and persons facing outside the centre.

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Answer 12

Systematic approach:

Read the facing direction carefully.
- Facing centre: left is clockwise and right is anticlockwise.
- Facing outside: left is anticlockwise and right is clockwise.
Fix one person at a reference position.
- In a circular arrangement, placing one person anywhere removes rotational repetition.
Place direct relations first.
- Conditions such as "immediately left", "second to the right", and "opposite" should be placed before indirect conditions.
Use a clockwise reference.
- Mark seats consecutively around the circle to avoid confusion.
Handle opposite positions correctly.
- For an even number of persons, the opposite seat is halfway around the circle.
- For $n$ persons, the opposite position is $\frac{n}{2}$ places away when $n$ is even.
Check every condition after completing the arrangement.
- Verify directions based on each person's facing orientation.

Important note: Never assume that the observer's left or right is the same as the seated person's left or right without considering the facing direction.

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Question 13

Define an inequality. Distinguish between the symbols $>$ , $<$ , $\geq$ , $\leq$ , and $\neq$ with examples.

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Answer 13

An inequality is a mathematical statement that compares two quantities or expressions that may not be equal.

Common inequality symbols:

$a > b$ : $a$ is greater than $b$ .
- Example: $8 > 5$ .
$a < b$ : $a$ is less than $b$ .
- Example: $3 < 9$ .
$a \geq b$ : $a$ is greater than or equal to $b$ .
- Example: $x \geq 4$ means $x$ can be $4$ or any value greater than $4$ .
$a \leq b$ : $a$ is less than or equal to $b$ .
- Example: $y \leq 10$ means $y$ can be $10$ or any value less than $10$ .
$a \neq b$ : $a$ is not equal to $b$ .
- Example: $6 \neq 7$ .

Key point: Unlike an equation, which shows equality, an inequality expresses an order or comparison between values.

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Question 18

What is meant by comparison of roots of equations? Explain how the nature and relative position of roots of a quadratic equation $ax^2 + bx + c = 0$ can be studied using the discriminant.

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Answer 18

For a quadratic equation:

$ax^2 + bx + c = 0, \quad a \neq 0$

The discriminant is:

$D = b^2 - 4ac$

The discriminant helps determine the nature of roots.

Cases:

If $D > 0$ , the equation has two distinct real roots.
If $D = 0$ , the equation has two equal real roots.
If $D < 0$ , the equation has no real roots; the roots are complex.

The roots are given by:

$x = \frac{-b \pm \sqrt{D}}{2a}$

To compare roots of two equations:

Find their roots directly, if possible.
Compare their numerical values.
If direct calculation is difficult, analyse signs and evaluate the quadratic expression at selected values.
Use the sum and product of roots:

$\alpha + \beta = -\frac{b}{a}$

$\alpha\beta = \frac{c}{a}$

These relations help identify whether roots are positive, negative, equal, or located in a particular interval.

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Unit4 Unit6

Unit 5: Trigonometry, Seating and Coded Inequalities - Subjective Questions

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