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All questions of Trigonometry and Triangles for Grade 10 Exam

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 If the length of a shadow cast by a pole is  √3  times the length of the pole, then the angle of elevation of the sun is​
  • a)
    45°
  • b)
    60°
  • c)
    30°
  • d)
    90°
Correct answer is option 'C'. Can you explain this answer?

Anjana Khatri answered

Consider the height of tower be h

∴ height of shadow =√3h .

In a triangle ABC, 

tan ∠ACB = h / √3h

tan ∠ACB = 1 / √3

∠ACB = 30degree.

Therefore, angle of elevation is 30degree .

 If the angle of elevation of a cloud from a point 60 metres above a lake is 30o and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is​
  • a)
    30 m
  • b)
    120 m
  • c)
    200 m
  • d)
    500 m
Correct answer is option 'B'. Can you explain this answer?

Neha Patel answered
Let AB be the surface of the lake and P be the point of observation such that AP = 60 m. Let C be the position of the cloud and C be its reflection in the lake.
Then CB = 
Draw PM⊥CB
Let CM = h
∴ CB = h + 60 m

 The angle of elevation of the sun, when the length of the shadow of a tree is equal to the height of the tree, is:
  • a)
    45°
  • b)
    60°
  • c)
    30°
  • d)
    None of these
Correct answer is option 'C'. Can you explain this answer?

Ananya Das answered

Consider the diagram shown above where QR represents the tree and PQ represents its shadow
We have, QR = PQ
Let ∠QPR = θ
tan θ = QR/PQ = 1 (since QR = PQ)
⇒ θ = 45°
i,e., required angle of elevation = 45°

A tower stands vertically on the ground. From a point on the ground which is 25 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 45o. Then the height (in meters) of the tower is​
  • a)
    25
  • b)
    25√3
  • c)
    12.5
  • d)
    25√2
Correct answer is option 'A'. Can you explain this answer?

Vikram Kapoor answered

A point on the ground which is 25 m away from the foot of the tower i. BC= 25 m
Let the height of the tower be x
The angle of elevation of the tower is found to be 45 degree.i.e.∠ACB=45°
In ΔABC
Using trigonometric ratios

Hence the height of the tower is 25 m.

The ratio of the length of rod and its shadow is 1: , then the angle of elevation of the sun is:​
  • a)
    600
  • b)
    300
  • c)
    450
  • d)
    900
Correct answer is option 'B'. Can you explain this answer?

Nisha Choudhury answered
Here, AB = Length of rod
BC = Length of shadow.

So 1 : ✓3 = AB/BC
1/✓3 = AB/BC

We know that AB/BC = tan theta

As, tan 30 deg = 1/✓3
So, angle of elevation of the sun is theta  = 30 degree

A tree casts a shadow 4 m long on the ground, when the angle of elevation of the sun is 45o. The height of the tree is:​
  • a)
    5.2 m
  • b)
    4 m
  • c)
    3 m
  • d)
    4.5 m
Correct answer is option 'B'. Can you explain this answer?

Nisha Choudhury answered
In a triangle with one angle being 90 degrees (which the tree makes with the ground) and the other being 45 degrees (the angle of elevation), the 3rd angle is bound to be 45 degrees (180 - 90 - 45 = 45). 

We also know that sides opposite to equal angles are equal.

Hence, the height of the tree will also be 4m.

A 1.2m tall boy stands at a distance of 2.4m from a lamp post and casts a shadow of 3.6m on the ground. The height of the lamp post is
  • a)
    2m
  • b)
    3m
  • c)
    4m
  • d)
    6m
Correct answer is option 'A'. Can you explain this answer?

Ashwini chauhan answered
Solution:

Given, the height of the boy = 1.2 m
Distance of the boy from the lamp post = 2.4 m
Length of the shadow cast by the boy = 3.6 m

Let the height of the lamp post be h meters.

Using the concept of similar triangles, we can say that:

Height of the lamp post / Height of the boy = Distance of the lamp post from the boy / Length of the shadow cast by the boy

h/1.2 = (2.4+distance of the lamp post from the boy)/3.6

h = (1.2 x 2.4)/3.6

h = 0.8 m

Therefore, the height of the lamp post is 2 meters (1.2 + 0.8).

Hence, the correct option is A.

The angle of elevation from a point 30 metre from the base of tree as level ground to the top of the tree is 60°. The height of the tree is : 
  • a)
    60√3 m
  • b)
    30√3 m
  • c)
    30 m
  • d)
    30/√3 m
Correct answer is option 'B'. Can you explain this answer?

Krishna Iyer answered
Angle of elevation is 60
Base = 30m
Height of the tree = Perpendicular
So in the right triangle
Where base is given and we have to find perpendicular we have only tan θ
So, Tan θ = P/B
Tan 60 = P/30
√30 = P/30
P = 30√30

The angles of elevation of the top of a tower from two points on the ground at distances 8m and 18m from the base of the tower and in the same straight line with it are complementary. The height of the tower is
  • a)
    8m
  • b)
    16m
  • c)
    12m
  • d)
    18m
Correct answer is option 'C'. Can you explain this answer?

Mysterio Man answered
Let AB= height of the tower=?,
and angle of elevations be ,
angle ACB =@ and angle ADB=90-@( as both the angles are complementary angles),
in right triangle ABC tan@=AB/BC,
tan@=AB/8---(1).,
in right triangle ABD tan(90-@)=AB/18,
we know that tan (90-@)=cot@,
so, cot@=AB/18,--(2),
Cot@=1/tan@,
from( 1)&(2),
AB/8=1/AB/18,
AB/8=18/AB,
AB²=18×8,
AB=√18×8=√3×3×2×2×2×2=3×2×2=12m

Tree is broken by the wind the top struck the ground at 30° at a distance of 30m. away from the root. Find the height of the tree.
  • a)
    45.9
  • b)
    88.60
  • c)
    54.63
  • d)
     51.96
Correct answer is option 'D'. Can you explain this answer?

Neha Patel answered
let , 
the height of standing part of the tree be = h 
the height of  fallen part (forms hypotenuse) be = x
then the total height of the tree will be = h + x 
now, 
tan 30 = h/30 m
     1/√3 = h/30 m 
    30/√3 = h
    ⇒ h= 30/√3 m           .... 1
similarly, 
cos 30 = 30 m/ x 
   √3/2 = 30 / x
√3x = (30)2 
 √3x = 60 m 
⇒ x = 60 / √3 m           ....2
       ( we now have both value of h and x )
   on adding equation1 & 2 :
 ⇒ h + x = 30 /√3 +60 /√3
         =90 /√3 m 
         = 60√3 m
so , the total height of the tree is 60√3 m .

A pole of height 60m has a shadow of length 20√3m at a particular instant of time. The angle of elevation of the sun at this point of time
  • a)
    30°
  • b)
    45°
  • c)
    60°
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Kuldeep Raj answered
Given,

[•] Height of the pole (AB) = 60m.

[•] Pole having shadow of length (BC) = 20√3m.

[•] Angle of elevation of the Sun (AC) = ?




Solution:-

[•] Let the angle of elevation be taken as "θ".

[•] Let the height of the pole (AB) be taken as "Perpendicular" or "Opposite side".

[•] Let the shadow of pole (BC) be taken as "Base" or "Adjacent".



We can find the Angle of elevation of the Sun by using Tan θ. So, Let us see..

Tan θ = Opposite side / Adjacent = AB/BC.

Tan θ = 60/20√3.

Tan θ = 3/√3.



Rationalize the denominator term i.e. √3.

=> 3/√3 × √3×√3.

=> 3√3/(√3)².

Square root and Square gets cancel for (√3)². Then, we get 3 in denominator.

=> 3√3/3.

Now the 3 and 3 gets cancel in both numerator and denominator. Then we get √3 as answer.

=> √3.

Now, what is the trignometric ratio of √3 in Tan θ? It is 60 degree.




Hence, Option (c) 60 degree is the correct answer..

If the length of the shadow of a tower is √3 times that of its height, then the angle of elevation of the sun is
  • a)
    30°
  • b)
    45°
  • c)
    60°
  • d)
    75°
Correct answer is option 'A'. Can you explain this answer?

Ananya Das answered
Let AB be the tree and AP be the shadow.
Let AB = x meters. Then AP = x√3 meters
Also ∠APB = θ
In right angled triangle ABP 

Therfore the angle of elevation of the Sun is 30°.

A ladder 12m long just reaches the top of a vertical wall. If the ladder makes an angle of 45° with the wall, then the height of the wall is
  • a)
    12m
  • b)
    6m
  • c)
    12√2m
  • d)
    6√2m
Correct answer is option 'D'. Can you explain this answer?

Aarav murthy answered
° with the ground, what is the distance between the wall and the base of the ladder?

Let's call the distance between the wall and the base of the ladder "x".

We can use trigonometry to solve for x.

We know that the ladder is 12m long and makes an angle of 45° with the ground. This means that the opposite side (the distance between the wall and the ladder) and the adjacent side (the distance between the base of the ladder and the wall) are equal.

Using the trigonometric function for tangent (tan), we can set up the following equation:

tan(45°) = opposite/adjacent

tan(45°) = x/12

We can solve for x by multiplying both sides by 12:

x = 12 * tan(45°)

Using a calculator, we can find that tan(45°) is equal to 1.

So, x = 12 * 1 = 12m.

Therefore, the distance between the wall and the base of the ladder is 12m.

From a point P on the level ground, the angle of elevation of the top of a tower is 30°. If the tower is 100m high, the distance between P and the foot of the tower is
  • a)
    100√3m
  • b)
    200√3m
  • c)
    300√3m
  • d)
    150√3m
Correct answer is option 'A'. Can you explain this answer?

Keerthana Singh answered
Let QR be the height of the tower, then QR = 100 mQ And angle of elevation of the top of the tower be ∠PPR = 30° 
Therefore, the distance between P and the foot of the tower is 100√3 metres.

A tree is broken by wind and its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 45° with the ground. The entire length of the tree is​
  • a)
    20 m
  • b)
    10(1+√2)m
  • c)
    10 m
  • d)
    10√2 m
Correct answer is option 'B'. Can you explain this answer?

Drishti Kumari answered
Base = 10m
Angle of elevation = 45 degree
tan 45 ^ = P / b ( Let p supoose h )
1= h / 10
h = 10 m
height of half tree = 10 m
Now the length of broken part i.e, hypotenuse
H^2 = P ^2 + B ^2
H ^2 = 10^2 + 10 ^2
H ^2 = 100 +100
H = root under 100
H = 10 root 2

Hence , the length of entire tree = 10 + 10 root2
10 ( 1 + root 2 )
That's why B is the correct optipn .

If the length of the shadow of a tower is equal to its height, then the angle of elevation of the sun is
  • a)
    30°
  • b)
    45°
  • c)
    60°
  • d)
    65°
Correct answer is option 'B'. Can you explain this answer?

Om Khanna answered
Let AB be the tower and AC be its shadow 

And AB = AC= x m
Let the angle of elevation of the sun be θ.
Then ∠ACB = θ
In right angled triangle ABC 

Therefore, the angle of elevation of the Sun is 45°.

Chapter doubts & questions for Trigonometry and Triangles - Mathematics for Grade 10 2024 is part of Grade 10 exam preparation. The chapters have been prepared according to the Grade 10 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Grade 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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