CBSE-CLASS-X-BOARD-MATH-IMPORTANT-QUESTIONS-CHAPTER-7-to-9

BOARD IMPORTANT QUESTIONS FROM CHAPTER 7 TO 9 AND HINTS

                               CHAPTER 7

Question 1: 

Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).

Question 2: Find a point on the y-axis that is equidistant from the points A(6, 5) and B(– 4, 3).

Question 3: Find the distance between the points  (a, b), (– a, – b)

Question 4: Determine if the points (1, 5), (2, 3), and (– 2, – 11) are collinear.

Question 5: Find the point on the x-axis that is equidistant from (2, –5) and (–2, 9).

Question 6: . If Q(0, 1) is equidistant from P(5, –3) and R(x, 6), find the values of x. Also find the distances QR and PR.

Question 7: In what ratio does the point (– 4, 6) divide the line segment joining the points A(– 6, 10) and B(3, – 8)?

Question 8: Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment joining the points A(2, – 2) and B(– 7, 4).

Question 9: Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).

Question 10: If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = 3 / 7 AB and P lies on the line segment AB.

Question 11: Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are (0, –1), (2, 1), and (0, 3). Find the ratio of this area to the area of the given triangle.

 

CHAPTER 8

Learn all the trigonometric ratio values in your book and the trigonometric relation.

 

Question 1

Consider ΔACB, right-angled at C, in which AB = 29 units, BC = 21 units and ∠ABC = θ (see Fig. 8.10). Determine the values of:

1. cos²θ + sin²θ
2. cos²θ − sin²θ

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Question 2

In ΔACB, if sin A = ³⁄₄, calculate:

• cos A
• tan A

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Question 3

Given 15 cot A = 8, find:

• sin A
• sec A

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Question 4

Given sec θ = ¹³⁄₁₂, calculate all other trigonometric ratios.

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Question 5

If ∠A and ∠B are acute angles such that cos A = cos B, show that:

∠A = ∠B

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Question 6

If cot θ = ⁷⁄₈, evaluate:

1. ^{(1 + sin θ)(1 − sin θ)}⁄ _{(1 + cos θ)(1 − cos θ)}
2. cot²θ

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Question 7

If 3 cot A = 4, check whether:

^{1 − tan2A}⁄ _{1 + tan²A} = cos²A − sin²A

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Question 8

In ΔABC, right-angled at B, if tan A = ¹⁄₃, find:

1. sin A cos C + cos A sin C
2. cos A cos C − sin A sin C

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Question 9

Evaluate the following:

1. sin 60° cos 30° + sin 30° cos 60°
2. 2 tan²45° + cos²30° − sin²60°
3. ^{cos 45°}⁄_{sec 30°} + ^{cosec 30°}⁄_{cos 45°}
4. sin 30° + tan 45° − cosec 60°
5. sec 30° + cos 60° + cot 45°

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Question 10

If tan(A + B) = 3 and tan(A − B) = ¹⁄₃, where 0° < A + B ≤ 90° and A > B, find A and B.

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Question 11

Prove that:

^{cot A − cos A cosec A − 1}⁄ _{cot A + cos A cosec A + 1}

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Question 12

Write all the other trigonometric ratios of ∠A in terms of sec A.

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Question 13

Choose the correct option. Justify your answer.

(i) 9 sec²A − 9 tan²A =

• (A) 1
• (B) 9
• (C) 8
• (D) 0

(ii) ^{tan θ + cot θ}⁄ _{sec θ + cosec θ} = ¹⁄ _{cot θ − tan θ}

(Hint: Write the expression in terms of sin θ and cos θ)

(iii) ^{2 + sec A sin A}⁄ _{sec A + 1 − cos A}

(iv) cos A − sin A + 1 = cosec A + cot A

(Use the identity cosec²A = 1 + cot²A)

(v) ^{1 + sin A}⁄ _{1 − sin A} = sec A + tan A

(vi) ^{sin θ − 2 sin θ}⁄ _{2 cos θ − cos θ} = tan θ

(vii) (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A

                                       CHAPTER 9

1.  An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3 m below the top of the pole to undertake the repair work (see Fig. 9.5). What should be the length of the ladder that she should use, which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take square root of 3 = 1.73.)

2. From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building, and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P.

3. The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multistoreyed building and the distance between the two buildings.

4. From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

5. A tree breaks due to a storm, and the broken part bends so that the top of the tree touches the ground, making an angle of 30° with it. The distance between the foot of the tree and the point where the top touches the ground is 8 m. Find the height of the tree.

6. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

7. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

8.  The angle of elevation of the top of a building from the foot of the tower is 30°, and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

9. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.

10.  As observed from the top of a 75 m high lighthouse from the sea level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

11. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.