Test: Circles- 1 - Class 10 MCQ

Let, Radius of smaller circle = r
and, Radius of Bigger circle = R

Approach

Steps 1 & 2: Understand Question and Draw Inferences

The figure shows the sector AOB that subtends an angle of 90^o at the center of the circle.

Therefore ACB is the major arc that subtends an angle of 270^o at the center.

We need to find the area of the circle. For this we need to find the radius of the circle.

Step 3: Analyze Statement 1

Given that length of major arc ACB is 9π cm.

Note that the ratio of arc lengths is equal to the ratio of angle subtended by the arcs at the center of the circle.

Let the circumference of the circle be C. (Note that the circle is essentially an arc subtending 360^o at the center.)

Therefore 

⇒c=12π

Let us assume the radius as r cm.

Therefore, 2πr = 12π

⇒  r = 6 cm.

We can calculate the area by using the formula A = πr²

We get a unique answer.

So statement 1 is sufficient to arrive at a unique answer.

Step 4: Analyze Statement 2

ΔOAB is right angled with OA and OB as the perpendicular sides. (Note that both OA and OB are radii of the circle).

We know that in a right angled triangle, the area of the triangle is equal to ½*(product of the perpendicular sides).

We can calculate the area by using the formula A = πr²

We get a unique answer.

So statement 2 is sufficient to arrive at a unique answer.

Step 5: Analyze Both Statements Together (if needed)

We arrived at a unique answer in both Step 3 and Step 4 above. So this step is not required.

Correct Answer: D

Steps 1 & 2: Understand Question and Draw Inferences

Assume that the inner radius, outer radius and width of the circular track are r, R and w.

Therefore R = r + w

We need to find w.

w = R - r

So we need the values of r and R to find W. 

Step 3: Analyze Statement 1

This statement says that the fencing used for outer boundary (perimeter of the outer boundary) is twice the size of the fencing for inner boundary (perimeter of inner boundary).

Perimeter of outer boundary = 2πR

Perimeter of inner boundary = 2πr

Therefore,

2πR = 2*2πr

R = 2r.

However, this information is not enough to find w.

Therefore statement 1 is not sufficient to arrive at a unique answer.

Step 4: Analyze Statement 2

The area of the inner circle is 10 sq. meters.

But we still don’t know R and so we cannot calculate the value of w.

Step 5: Analyze Both Statements Together (if needed)

Combining the information we got from statements (1) and (2);

Therefore statement (1) and statement (2) together are sufficient to arrive at a unique answer.

Correct Answer: C

Steps 1 & 2: Understand Question and Draw Inferences

Since one-thirds of the total profits > one-fourths of the total profits,

the area of region B > area of region A

The question is:

Is the area of region C > area of region D?

Step 3: Analyze Statement 1

Since O is the centre of the circle and x > y,

area of region D > area of region C. (For two sectors in a circle, the sector with the greater angle at the centre has the greater area)

Therefore, the profits from company C were NOT greater than the profits from company
D.

SUFFICIENT.

Step 4: Analyze Statement 2

According to this statement

area of region A + area of region D > area of region C + area of region B

Since area of region A is smaller than area of region B, for this inequality to hold true, the area of region D must be larger than area of region C.

Therefore, the profits from company C were NOT greater than the profits from company D.

SUFFICIENT.

Step 5: Analyze Both Statements Together (if needed)

Since we have obtained an answer, there is no need to combine the statements.

Given that the area of the rectangular floor = 72 square meters

And that the carpet occupies exactly half the area.

Area of carpet = 36 square meters.

Let the radius of carpet = R meters

Therefore πR² = 36

Therefore the circumference  = 

 meters

From the given information we can see that the chopsticks act as tangents to the circular candy (As shown in the above figure.)

ΔOTP is right angled with OT and PT as the perpendicular sides. Therefore applying Pythagoras theorem on ΔOTP.

10² = 6² + PT²

⇒PT² = 100 – 36 = 64

⇒PT = 8

Therefore the chopsticks must be of at least 8 cm length.

Correct Answer: A

Let the amounts of wealth received by the eldest son, second son, youngest son and the wife be E, S, Y and W respectively.

Given that

S = 2E

Y = 2S

W = 2Y

So if we assume that Eldest son got “x”, we have

E = x; S = 2x; Y = 4x; W = 8x;

and the total wealth = x + 2x + 4x + 8x = 15x.

So the fraction of wealth received by the second son is 2/15

So if we represent it on a circle (over 360^o), the fraction would be converted to 215×360^o=48^o

Correct Answer: C

Let us assume that

the radius of big circle = R

side of the square = S

radius of the small circle = r

Observe that the diameter of the big circle = diagonal of the square.

• 2R = √2 * S
• S = √2 * R

Also the diameter of the small circle = side of the square

• 2r = S
• 2r = √2 * R

​   

Semi perimeter of the small circle = πr = π(R/√2)

Putting R = √2 in the above equation, we get:

Semi perimeter of the small circle = π

Correct Answer: D

Steps 1 & 2: Understand Question and Draw Inferences

Given that PT is a tangent to the small circle and PS is a tangent to the big circle.

ΔPTB and ΔPSA are right angled at T and S respectively.

We need to find the length of ST. 

Step 3: Analyze Statement 1

(1) The radii of the bigger and the smaller circles are 9 cm and 4 cm respectively.

If we drop a perpendicular BD on side AS, we get:

BDST is a rectangle (since all angles of this quadrilateral are right angles).

Therefore, since opposite sides of a rectangle are equal, SD = BT = 4 cm

This means, AD = 9 – 4 = 5cm

In right triangle ADB, by applying Pythagoras Theorem, we get:

BD² + AD² = AB²

That is, BD² = (9+4)² – (5)²

BD² = (13+5)(13-5)

BD² = 18*8 = 2⁴*3²

Therefore, BD = 2²*3 = 12

Since opposite sides of a rectangle are equal, ST = BD = 12 cm

Since we have been able to determine a unique length of ST, Statement (1) is sufficient.

Step 4: Analyze Statement 2

(2) PB =  52/5 cm

In right triangle BTP, we know the length of only one side: BP.

In order to find the lengths of the other sides of this triangle, we need one more piece of information – either one of the two unknown angles, or one of the two unknown sides.

Since we don’t have this information, we will not be able to find the lengths of the unknown sides of triangle BTP.

Due to a similar reasoning, we will not be able to find the length of the sides of triangle ASP.

So, we will not have enough information to find the length of ST.

Therefore statement 2 is not sufficient to arrive at a unique answer.

Step 5: Analyze Both Statements Together (if needed)

We have arrived at a unique answer in step 3 above. Hence this step is not required.

Correct Answer: A

Steps 1 & 2: Understand Question and Draw Inferences

We need to find the area of a sector subtending an angle of 144^o at the center of the circle.

For this we need either the radius of the circle or the area of a known portion of the circle.

Step 3: Analyze Statement 1

The chord that subtends an angle of 90^o on the circle is the diameter itself.

(Please note that the chord here subtends an angle of 90^o on the circle, and not on the center of the circle)

Area of the square with the side as the diameter is 64 sq. cm.

Therefore the length of the diameter = √(64) = 8 cm.

• Radius of the circle = 4 cm.

We can find the area of the 144^o sector by using the formula 

Therefore statement 1 is sufficient to arrive at a unique answer.

Step 4: Analyze Statement 2

The longest chord of the circle is the diameter itself.

We know that the diameter subtends an angle of 90^o on the circle.

Therefore the isosceles triangle constructed would be an isosceles right angled triangle with the diameter as the hypotenuse.

Given that the length of the equal sides is 4√2 cm.

Let the diameter = D

Therefore, D² = (4√2)² + (4√2)² (Applying Pythagoras)

⇒  D² = 64

⇒  D = 8 cm.

⇒  Radius = 4 cm.

We can find the area of the 144^o sector by using the formula 

Therefore statement 2 is sufficient to arrive at a unique answer.

Step 5: Analyze Both Statements Together (if needed)

We arrived at a unique answer in both Step 3 and Step 4 above. So this step is not required.

Correct Answer: D

Step 1: Question statement and Inferences

Since each part has equal area measuring 8π square units,

1/8 * π * r² = 8π

r = 8

The circle is divided into eight equal arcs, i.e.

AB = BC = CD = DE = EF = FG = GH = HA

length of each arc = 1/8 * 2 * π * r

Substituting r = 8, we have

length of each arc = 2π

Step 2:Finding required values

length of boundary AOBCODEOFGOHA = radius AO + radius OB + arc BC + radius CO + radius OD + arc DE + radius EO + radius OF +  arc FG + radius GO + radius OH + arc HA

length of boundary AOBCODEOFGOHA = 8 * radius + 4 * length of arc

Step 3: Calculating the final answer

Substituting the values of radius and length of arc obtained above,

length of boundary AOBCODEOFGOHA = 8 * 8 + 4 * 2 π  = 64 + 8 π

Correct Answer: E