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All questions of Circles for Class 10 Exam

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Is triangle ABC obtuse angled?
a2 + b2 > c2
The center of the circle circumscribing the triangle does not lie inside the triangle.
  • a)
    Statement ( 1 ) ALONE is sufficient but statement ( 2 ) alone is not sufficient.
  • b)
    Statemrnt ( 2 ) ALONE is sufficient but statement ( 1 ) is not sufficient
  • c)
    Both Stement TOGETHER are sufficient, but Neither statement  ALONE is sufficient
  • d)
    EACH stetement ALONE is sufficient
  • e)
    Statement ( 1 ) and ( 2 ) TOGETHER are NOT Sufficient.
Correct answer is option 'E'. Can you explain this answer?

Devika Yadav answered
The center of the circle circumscribing the triangle does not lie inside the triangle.
From statement 2 we know that the triangle is not acute. Without knowing whether 'c' is the longest side, we will not be able to conclude whether the triangle is obtuse.
Eliminate choice C.

The area of a square field is 24200 sq m. How long will a lady take to cross the field diagonally at the rate of 6.6 km/hr?
  • a)
    3 minutes
  • b)
    0.04 hours
  • c)
    2 minutes
  • d)
    2.4 minutes
  • e)
    2 minutes 40 seconds
Correct answer is option 'C'. Can you explain this answer?

Disha Mehta answered
Let 'a' meters be the length of a side of the square field.
Therefore, its area = a2 square meters. --- (1)
The length of the diagonal 'd' of a square whose side is 'a' meters = 
 
From (1) and (2), we can deduce that the square of the diagonal = d2 = 2a2 = 2(area of the square)
Or d = 

The tyre of a car made an average of 500 revolutions per minute in a particular trip. If the radius of the tyre was 10 centimeters and the car was driven for 1 hour without any stoppages, approximately how much distance in kilometers was covered in the trip?
  • a)
    15
  • b)
    17
  • c)
    19
  • d)
    21
  • e)
    23
Correct answer is option 'C'. Can you explain this answer?

Anirban Das answered
Given information:
- The tyre of a car made an average of 500 revolutions per minute.
- The radius of the tyre was 10 centimeters.
- The car was driven for 1 hour without any stoppages.

To find:
- Approximately how much distance in kilometers was covered in the trip.

Solution:
First, we need to find the distance covered by the car in one revolution of the tyre. This can be calculated using the formula:
Distance covered in one revolution = 2πr
Where r is the radius of the tyre.
Distance covered in one revolution = 2 × 3.14 × 10 = 62.8 centimeters

Next, we need to find the distance covered by the car in one minute. As the tyre makes 500 revolutions per minute, the distance covered by the car in one minute can be calculated as:
Distance covered in one minute = 500 × 62.8 = 31400 centimeters or 314 meters

Finally, we need to find the distance covered by the car in one hour. As the car was driven for 1 hour without any stoppages, the distance covered by the car in one hour can be calculated as:
Distance covered in one hour = 314 × 60 = 18840 meters or 18.84 kilometers (approx.)

Therefore, the correct answer is option C (19).

Vertices of a quadrilateral ABCD are A(0, 0), B(4, 5), C(9, 9) and D(5, 4). What is the shape of the quadrilateral?
  • a)
    Square
  • b)
    Rectangle but not a square
  • c)
    Rhombus
  • d)
    Parallelogram but not a rhombus
  • e)
    Kite
Correct answer is option 'C'. Can you explain this answer?

Pallavi Choudhury answered
The lengths of the four sides, AB, BC, CD and DA are all equal to  √41.
Hence, the given quadrilateral is either a Rhombus or a Square.
The diagonals of a square are equal. The diagonals of a rhombus are unequal.
Compute the lengths of the two diagonals AC and BD.
The length of AC is√162  and the length of BD is √2.
As the diagonals are not equal and the sides are equal, the given quadrilateral is a Rhombus.

A man distributed his wealth among his wife and three sons such that his second son got twice the amount the eldest son received, his youngest son received twice the amount the second son got and his wife got twice the amount the youngest son received. If the man were to represent his wealth in a circle diagram, how many degrees of the circle must be allocated to represent the portion that the second son receives?
  • a)
    24
  • b)
    36
  • c)
    48
  • d)
    72
  • e)
    96
Correct answer is option 'C'. Can you explain this answer?

Nilotpal Sen answered
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 360o), the fraction would be converted to 215×360o=48o
 
Correct Answer: C

Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB is times the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle and the inner circle respectively, what is the length of the side of the triangle?
  • a)
    √2
  • b)
    √3
  • c)
    2
  • d)
    3
Correct answer is option 'C'. Can you explain this answer?

Krithika Datta answered
Given
  • Two concentric circles with center at O
    • Let’s assume the radius of the outer circle to be R and that of inner circle to be r
  • AB is a chord to the outer circle and a tangent to the inner circle
    •  
  • An equilateral triangle with Area = R/r
  • Assuming the side length of the equilateral triangle to be a, we have  
To Find:
  • Value of a.
 
Approach
  • Since  to find the value of a, we need to find the value of  R/r
  • We know that AB is a tangent to the inner circle, hence AB will make an angle of 90o with the radius of the inner circle, i.e. the line joining the center O to the point of tangency. Let’s call the point of tangency as C.
    • So, OC will be perpendicular to AB.
    • However, as AB is also a chord to the external circle, the line perpendicular from the center O will bisect the chord, i.e. OC will bisect the chord AB.
      • Therefore, 
    • As we know that BC =​  and triangle OCB is a right angled triangle, we can use Pythagoras theorem in triangle OCB to calculate the ratio of R/r
Working Out
 
  1. Using Pythagoras theorem in triangle OCB, we have
(rejecting the negative root since the ratio of radii cannot be negative)
2. 
Therefore, a=2   , as a being the length of a triangle, cannot be negative.
Hence the correct answer is Option C .

What is the area of the 144o sector of the circle?
(1) The square with one of the sides as the chord that subtends an angle of 90o on the circle measures 64 sq. cm. in area.
(2) When an isosceles triangle is constructed by taking the longest chord of the circle and the other vertex on the circumference of the circle, the length of each of the equal sides of the triangle is 4√2 cm.
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'D'. Can you explain this answer?

Anirban Singh answered
Steps 1 & 2: Understand Question and Draw Inferences
We need to find the area of a sector subtending an angle of 144o 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 90o on the circle is the diameter itself.
(Please note that the chord here subtends an angle of 90o 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 144o 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 90o 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 144o 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

What is the area of an obtuse angled triangle whose two sides are 8 and 12 and the angle included between two sides is 150o?
  • a)
    24 sq units
  • b)
    48 sq units
  • c)
    24√3
  • d)
    48√3
  • e)
    Such a triangle does not exist
Correct answer is option 'A'. Can you explain this answer?

Navya Yadav answered
If two sides of a triangle and the included angle 'y' is known, then the area of the triangle = 1/2 * (Product of sides) * sin y
Substituting the values in the formula, we get 1/2 * 8 * 12 * sin 150o

A cube of side 5 cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?
  • a)
    9
  • b)
    61
  • c)
    98
  • d)
    54
  • e)
    64
Correct answer is option 'D'. Can you explain this answer?

Gauri Iyer answered
When a 5 cc cube is sliced into 1 cc cubes, we will get 5 * 5 * 5 = 125 cubes of 1 cubic centimeter.
In each side of the larger cube, the smaller cubes on the edges will have more than one of their sides painted.
Therefore, the cubes which are not on the edge of the larger cube and that lie on the facing sides of the larger cube will have exactly one side painted.
In each face of the larger cube, there will be 5 * 5 = 25 cubes.
Of these, the cubes on the outer rows will be on the edge. 16 such cubes exist on each face.
If we count out the two outer rows on either side of a face of the cube, we will be left with 3 * 3 = 9 cubes which are not on the edge in each face of the cube.
Therefore, there will be 9 cubes of 1-cc volume per face that will have exactly one of their sides painted.
In total, there will be 9 * 6 = 54 such cubes.

If 10, 12 and 'x' are sides of an acute angled triangle, how many integer values of 'x' are possible?
  • a)
    7
  • b)
    12
  • c)
    9
  • d)
    13
  • e)
    11
Correct answer is option 'C'. Can you explain this answer?

Arjun Iyer answered
Explanation:

Given:
- Sides of the triangle: 10, 12, and x
- Triangle is acute-angled

Conditions for an Acute-Angled Triangle:
- In an acute-angled triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side.
- In this case, 10 and 12 are the two shorter sides, and x is the longest side.

Using Pythagorean Theorem:
- According to the Pythagorean theorem, in a right-angled triangle, a^2 + b^2 = c^2, where 'c' is the hypotenuse (longest side).
- For an acute-angled triangle, the inequality a^2 + b^2 > c^2 holds true.

Calculating the Range of x:
- In this case, x is the longest side, so we need to find the range of values for x that satisfy the inequality 10^2 + 12^2 > x^2.
- Solving this inequality, we get x < √(244),="" x=""><>
- As x has to be an integer, the possible values of x are 1, 2, 3,..., 15.
- But x cannot be greater than 12 (as 12 is already one of the sides), so the possible integer values of x are 1 to 12.

Calculating the Number of Possible Integer Values of x:
- The total possible integer values of x within the range 1 to 12 are 12.
- Therefore, the correct answer is option C, 9 possible integer values of x in an acute-angled triangle with sides 10, 12, and x.

A wheel of a car of radius 21 cms is rotating at 600 RPM. What is the speed of the car in km/hr?
  • a)
    79.2 km/hr
  • b)
    47.52 km/hr
  • c)
    7.92 km/hr
  • d)
    39.6 km/hr
  • e)
    3.96 km/hr
Correct answer is option 'B'. Can you explain this answer?

Janani Sharma answered
The radius of the wheel measures 21 cm.
In one rotation, the wheel will cover a distance which is equal to the circumference of the wheel.
∴ in one rotation this wheel will cover 2 * π * 21 = 132 cm.
In a minute, the distance covered by the wheel = circumference of the wheel * rpm
∴ this wheel will cover a distance of 132 * 600 = 79200 cm in a minute.
In an hour, the wheel will cover a distance of 79200 * 60 = 4752000 cm.
Therefore, the speed of the car = 4752000 cm/hr = 47.52 km/hr

If a circle has the same area as right triangle PQR shown in the figure above. Which of the following is the closest in value to the radius of the circle?
  • a)
    3
  • b)
    4
  • c)
    5
  • d)
    10
  • e)
    20
Correct answer is option 'A'. Can you explain this answer?

Yash Rane answered
Given:           
  • Right ?PQR
    • QR = 6 units
    • PR = 10 units
  • A circle whose area is equal to area of triangle PQR
    • Let the radius of the circle = r units
To find: r = ?
Approach:
b. Therefore, to find r, we need to find A
Working Out:
c. The value of will be very slightly greater than 1
d. We know that Since 7 is slightly greater than the mid-point between 4 and 9 (which is
  the value of √7 will be slightly greater than the mid-point between   So, the value of √7 will be slightly greater than 2.5 
e. So, the product of   and  √7 will be slightly greater than 2.5
f. Therefore, the closest integer to the value of r will be 3
Answer: Option A

A circular track needs to be fenced along the inner and outer boundaries of the track to prevent stalkers from interrupting any events. What is the width of the track?
(1) The fencing used for the outer boundary can actually be used to fence the inner boundary twice.
(2) The area of the circle formed by the inner boundary is 10 sq. meter.
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'C'. Can you explain this answer?

Prateek Gupta answered
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

The floor of a rectangular room of 72 square meters area needs to be covered by a circular carpet that occupies exactly half the area of the floor. What must be the circumference of the carpet in meters?
  • a)
    3√π
  • b)
    6√π
  • c)
    9√π
  • d)
    12√π
  • e)
    15√π
Correct answer is option 'D'. Can you explain this answer?

Prashanth Chawla answered
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

What is the measure of the radius of the circle that circumscribes a triangle whose sides measure 9, 40 and 41?
  • a)
    6
  • b)
    4
  • c)
    24.5
  • d)
    20.5
  • e)
    12.5
Correct answer is option 'D'. Can you explain this answer?

Anirban Singh answered
Solution:

Step 1: Find the semi-perimeter of the triangle
Given sides of the triangle are 9, 40, and 41.
Semi-perimeter, denoted as 's', is half of the perimeter of the triangle.
Perimeter = 9 + 40 + 41 = 90
Semi-perimeter, s = 90/2 = 45

Step 2: Apply the formula to find the radius of the circumcircle
The radius of the circumcircle of a triangle with sides a, b, and c and semi-perimeter s is given by the formula:
\(R = \frac{abc}{4 \sqrt{s(s-a)(s-b)(s-c)}}\)
Substitute the values of sides and semi-perimeter into the formula:
\(R = \frac{9*40*41}{4 \sqrt{45(45-9)(45-40)(45-41)}}\)
\(R = \frac{1440}{4 \sqrt{45*36*5*4}}\)
\(R = \frac{1440}{4*120}\)
\(R = \frac{1440}{480}\)
\(R = 3\)
Therefore, the measure of the radius of the circle that circumscribes the triangle is 20.5, which corresponds to option 'D'.

A bigger circle (with center A) and a smaller circle (with center B) are touching each other externally. PT and PS are the tangents drawn to these circles from an external point (as shown in the figure). What is the length of ST?
(1) The radii of the bigger and the smaller circles are 9 cm and 4 cm respectively
(2) PB = 52/5 cm
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
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

In the coordinate plane, a circle has center (-1, -3) and passes through the point (4,2). What is the circumference of the circle?
  • a)
    5√2π
  • b)
    10√2π
  • c)
    10√3π
  • d)
    25π
  • e)
    50π
Correct answer is option 'B'. Can you explain this answer?

Palak Saha answered
To find the circumference of a circle, we need to find its radius first. The radius can be found using the distance formula:

√((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the coordinates of the center (-1, -3) and the point (4, 2):

√((4 - (-1))^2 + (2 - (-3))^2)
= √((4 + 1)^2 + (2 + 3)^2)
= √(5^2 + 5^2)
= √(25 + 25)
= √50
= 5√2

The circumference of a circle is given by the formula:

C = 2πr

Plugging in the value of the radius, which is 5√2:

C = 2π(5√2)
= 10π√2

Therefore, the circumference of the circle is 10π√2.

Is triangle ABC with sides a, b and c acute angled?
Triangle with sides a2, b2, c2 has an area of 140 sq cms.
Median AD to side BC is equal to altitude AE to side BC.
  • a)
    Statement ( 1 ) ALONE is sufficient but statement ( 2 ) alone is not sufficient.
  • b)
    Statemrnt ( 2 ) ALONE is sufficient but statement ( 1 ) is not sufficient
  • c)
    Both Stement TOGETHER are sufficient, but Neither statement  ALONE is sufficient
  • d)
    EACH stetement ALONE is sufficient
  • e)
    Statement ( 1 ) and ( 2 ) TOGETHER are NOT Sufficient.
Correct answer is option 'A'. Can you explain this answer?

Pranab Dasgupta answered
The statement provides us with one valuable information: we can form a triangle with sides a2, b2, c2.
For any triangle we know that sum of two sides is greater than the third side.
So, we can infer that a2 < b2 + c2.
The inequality above is the condition to be met if the triangle with sides a, b and c were to be an acute triangle.

In the coordinate system, the center of a circle lies at (2, 3). If point A with coordinates (-1, 7) does not lie outside the circle, which of the following points must lie inside the circle?
I. (0, 7)
II. (5, -1)
III. (-2, 7)
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I and II only
  • e)
    None of the above
Correct answer is option 'A'. Can you explain this answer?

Sharmila Nambiar answered
Given
  • Circle with center at (2, 3)
    • Let’s assume the radius of the circle to be r.
  • Point A(-1, 7) does not lie outside the circle
To Find: Which of the points in the options must lie inside the circle?
 
Approach
  1. To know the points that must line inside the circle, we need to find the distance between the center of the circle and the point (x, y)
 
  • Let us understand how can we find the distance between 2 points in the coordinate plance.
  • Let us say the above figure represent the coordinates of one point → point A → (a,b) and of another point → Point C → (x,y) in the co-ordinate system
    1. Distance of point A from point B along the x-axis = (x-a)
      1. So, AB=x-a
    2. Distance of point C from point B along the y-axis = (y-b)
      1. So, CB= y-b
    3. Now, the triangle ABC formed is a right angled triangle, so by Pythagoras theorem
In general, it can be said that the distance between any two points is =
 So we can find the distance between Point A and the centre of the circle by the above formula
  • If this distance between the center of the circle and point(x, y) is less than the minimum possible value of r, then the point (x, y) must lie inside the circle.
    • So, we need to find the minimum possible value of r.
  • We are given that point A does not lie outside the circle. So, it may lie either inside the circle or on the circle.
    1. Hence (the distance of point A from the center) ≤ r
        1. That is, r ≥ (the distance of point A from the center)
      2. So, (minimum possible value of r) = (the distance of point A from the center)
        1. We will use the above relation to find the minimum possible value of r.
Working Out
  1. Coordinates of point A = ( -1, 7) and coordinates of center of the circle = (2, 3)
    1. Distance of point A from the center of the circle ​
    2. Thus the minimum possible value of r = 5
  2. Evaluating the 3 options
    1. Distance of point (0, 7) from the center (2, 3)
      1. ​​​
  3. As the distance of point is less than the minimum possible value of r, this point must lie inside the circle
    1. Distance of point (5, -1) from the center (2, 3)
      2. As the distance of point is not less than the minimum possible value of r, this point may or may not lie inside the circle.
        1. For example, if r > 5, the point will lie inside the circle
        2. If r = 5, the point will not lie inside the circle, it will lie on the circle.
      3. Distance of point (-2, 7) from the center (2, 3)
        2. As the distance of point is not less than the minimum possible value of r, this point may or may not lie inside the circle
Hence, we see that only option I (0, 7) always lie inside the circle.
Answer: A

A lady grows cabbages in her garden that is in the shape of a square. Each cabbage takes 1 square feet of area in her garden. This year, she has increased her output by 211 cabbages as compared to last year. The shape of the area used for growing the cabbages has remained a square in both these years. How many cabbages did she produce this year?
  • a)
    ​11236
  • b)
    11025
  • c)
    14400
  • d)
    12696
  • e)
    Cannot be determined
Correct answer is option 'A'. Can you explain this answer?

Mrinalini Dasgupta answered
The shape of the area used for growing cabbages has remained a square in both the years.
Let the side of the square area used for growing cabbages this year be X ft.
Therefore, the area of the ground used for cultivation this year = X2 sq.ft.
Let the side of the square area used for growing cabbages last year be Y ft.
Therefore, the area of the ground used for cultivation last year = Y2 sq.ft.
As the number of cabbages grown has increased by 211, the area would have increased by 211 sq ft because each cabbage takes 1 sq ft space.
Hence, X2 - Y2 = 211
(X + Y)(X - Y) = 211.
211 is a prime number and hence it will have only two factors. i.e., 211 and 1.
Therefore, 211 can be expressed as product of 2 numbers in only way = 211 * 1
i.e., (X + Y)(X - Y) = 211 * 1
So, (X + Y) should be 211 and (X - Y) should be 1.
Solving the two equations we get X = 106 and Y = 105.
Therefore, number of cabbages produced this year = X2 = 1062 = 11236.
The area in both the years are squares of two numbers.
That rules out choice D. 12696 is not the square of any number.
Check Choice A: If this year's produce is 11236, last year's produce would have been 11236 - 211 = 11025
11025 is the square of 105. 
So, 11236 is the answer.

What is the radius of the incircle (circle inscribed) of the triangle whose sides measure 5, 12 and 13 units?
  • a)
    2 units
  • b)
    12 units
  • c)
    6.5 units
  • d)
    6 units
  • e)
    7.5 units
Correct answer is option 'A'. Can you explain this answer?

Pallabi Sengupta answered
5, 12 and 13 is a Pythagorean triplet. So, the triangle is a right triangle.
Area using Method 1
Area of a triangle =  
where 'b' is the base and 'h' is the height of the triangle.
In this right triangle, if the base is 12, the height will be 5 or vice versa.
In either case, area = 

Lines CA and CB touch the circle only at points A and B respectively, as shown in the figure above. If the centre O of the circle is at the point (0,0), the coordinates of point C are (0,2) and the radius of the circle is √2 units, what are the coordinates of point A?
  • a)
  • b)
  • c)
  • d)
  • e)
Correct answer is option 'C'. Can you explain this answer?

Sharmila Nambiar answered
Given:
  • CA and CB are tangents to the circle
    • Since tangents from the same point are equal, CA = CB
    • Also, ∠OBC = ∠OAC = 90o
  • Center of the circle is O (0,0) and Radius = √2 units
  • Point C(0,2)
To find:
  • The x- and y- coordinates of Point A
Approach and Working:
  • Let’s drop a perpendicular AD from A on the x-axis
  • So, the x-coordinate of point A = OD
  • And, the y-coordinate of point A = AD
 
  • So, in order to answer the question, we need to find OD and AD
  • In right ΔAOC, we know sides OA = √2 and OC = 2 units
    • Using Pythagoras Theorem, we can write –
      • OC2 = OA2 + AC2
      • AC2 = OC2 – OA2 = 4 – 2 = 2
      • Hence AC = √2
  • Thus, we can conclude that ΔAOC is an isosceles right-angled triangle, where angle ACO = AOC = 45o.
  • Angle AOD = 90o – angle AOC = 900 – 45o = 45o.
  •  
  • Finding OD and AD
    • In triangle AOD, angle AOD = 45, and angle ADO = 90, thus angle DOA = 45
      • Hence Triangle AOD is a 45-45-90 Triangle in which we can write the side ratio as –
        • AD: OD: AO = 1: 1: √2
      • Since the exact value of AO = √2
      • We can conclude from the ratio above, that AD = OD = 1 units
    • Thus, we see that the coordinates of point A are (1,1)
  • Looking at the answer choices, we see that the correct answer is Option C .

The circle with center C shown above is tangent to both axes and has an area of A. What is the distance from O to C, in terms of A?
  • a)
    A
  • b)
  • c)
  • d)
  • e)
Correct answer is option 'B'. Can you explain this answer?

Ankita Chauhan answered
In this question, we can find the radius r in terms of A by manipulating the area formula, A= πr2:

Point C in the figure is the upper-right point of a square that has sides of length r. And the distance from the origin to C is the diagonal of the square, so it is the hypotenuse of a right triangle with height r and base r, so the distance will equal

The correct answer is (B).

A company X has offices in four locations – A, B, C and D. The figure above represents a circle graph of X’s profits from each of the four locations in April 2010. One-thirds of companies profits in were registered from B while one-fourths of its profits were registered from A. Were the profits from the location C of the company more than the profits from location D of the company?
(1) x > y
(2) The sum of the profits from A and D combined were larger than the sum of the profits from B and C combined.
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. 
  • c)
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. 
  • d)
    EACH statement ALONE is sufficient. 
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'D'. Can you explain this answer?

Prashanth Chawla answered
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.

Point A lies on a circle whose center is at point C. Does point B lie inside the circle?
  1. BC2 = AC2 + AB2
  2. ∠CAB is greater than ∠ABC
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?

Abhishek Choudhury answered
Steps 1 & 2: Understand Question and Draw Inferences
 
Given: The given information can be represented visually as in the figure above
To find: Does point B lie inside the circle?
  • The answer will be YES if distance between point B and center C is less than the radius of the circle, that is, if BC < r
  • Else, if BC = r or greater than radius r, then the answer will be NO
 
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘BC2 = AC2 + AB2’
  • Now, AC = r
  • The minimum possible value of AB = 0 (happens if point B lies ON point A)
  • So, the minimum possible value of BC2 = r2 + 0 = r2
  • Thus, the minimum possible value of BC = r
  • So, BC is equal to or greater than r
  • Therefore, the answer to the question is NO
  • So, Statement 1 alone is sufficient to answer the question
     
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘∠CAB is greater than ∠ABC’
  • In ?ABC, since ‘∠CAB is greater than ∠ABC’, BC > AC
    • (Because we know that the side opposite to the greater angle in a triangle is greater)
  • Therefore, BC > r
  • So, the answer to the question is NO
  • Therefore, Statement 2 alone is sufficient to answer the question
 
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D

In the figure above, triangle ABC is inscribed in a circle whose centre O has the x- and y-coordinates as (0,0). If the x- and y- coordinates of point A are (-4,0) and ∠BAC = 30∘, what is the area, in square units, of triangle AOB?
 
  • a)
    2√3
  • b)
    4√3
  • c)
    8√3
  • d)
    16√3
  • e)
    24√3
Correct answer is option 'B'. Can you explain this answer?

Niharika Sen answered
Given:
O is the center of the circle and A(-4,0) is a point on the circle
  • So, radius of the circle, R = Distance between points O and A = 4 units
  • Also, AC is the diameter of the circle.
S​o, ∠ABC is an angle in the semicircle
  • So, ∠ABC = 90o
  • In right ΔABC,
    • ∠BAC = 30∘
So, by Angle Sum Property, ∠ACB = 60o
To find:
  • Area of (ΔAOB)
Approach:
  • Let’s drop a perpendicular BD from point B on AC.
Area triangle AOB = ½ * AO * BD
  
  • We’ve inferred the length of AO (= radius of the circle) in the Given section above. So, to answer the question, we need to find the length of BD
  • In ΔOBC, sides OB and OC are equal because both are the radii of the circle.
So, this is an isosceles triangle.
  • Since we already know ∠OCB=60, therefore ∠OBC will also be equal to 60.
  • Which finally helps us in inferring that ∠BOC = 60o
  • Hence we can conclude that OBC is an equilateral triangle.
  • The ΔODB, is a 30-60-90 Triangle and we know the measure of radius OB = 4 units. Thus using the side property of 30-60-90 Triangle we can write –
OD: BD: OB = 1: √3: 2
  • BD: OB = √3: 2
  • Thus BD = 2√3
  • Thus area of triangle AOB = ½ * AO * BD = ½ * 4 * 2√3 = 4√3
  • Looking at the answer choices, we see that the correct answer is Option B.
 

Little Johnny wants to eat a candy (circular in shape) using chopsticks! For this he makes sure that one end of either chopstick is touching the other (held in his hand) and the other end is used to hold the candy, as shown in the figure below. If his mom wants him to maintain a distance of at least 10 cm between his hands and his mouth while eating, what is the minimum length (in cm) of chopsticks required by him to eat the candy using chopsticks? (Assume that both the chopsticks are of equal length and that the radius of the candy is 6 cm.)
 
  • a)
    8
  • b)
    9
  • c)
    10
  • d)
    11
  • e)
    12
Correct answer is option 'A'. Can you explain this answer?

Prateek Gupta answered
 
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

The length of a rope, to which a cow is tied, is increased from 19m to 30m. How much additional ground will it be able to graze? Assume that the cow is able to move on all sides with equal ease. Use π = 22/7 in your calculations.
  • a)
    1696 sq m
  • b)
    1694 sq m
  • c)
    1594 sq m
  • d)
    1756 sq.m
  • e)
    1896 sq.m
Correct answer is option 'B'. Can you explain this answer?

Avantika Dey answered
The cow can graze the area covered by the circle of radius 19m initially, as the length of the rope is 19m.
Area of a circle = π * (radius)2
Therefore, the initial area that the cow can graze =  
When the length of the rope is increased to 30m, grazing area becomes = 
The additional area it could graze when length is increased from 19m to 30m

If O is the center of the circle above and the length of chord AB is 2 units, what is the length of the arc ACB?
(1) The area of ΔOAB is √3 square units
(2) The area of sector OACBO is 2π/3
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'A'. Can you explain this answer?

Sharmila Nambiar answered
Steps 1 & 2: Understand Question and Draw Inferences
Given:
  • AO = OB = r, the radius of circle
  • AB = 2
 
Step 3: Analyze Statement 1 independently
(1) Statement 1 states that "The area of ΔOAB is √3 square units." 
  • Let’s drop a perpendicular OD from O to AB. Let the length of this perpendicular be h
  • Also, AD = 1 (since perpendicular OD will bisect the chord AB)
  • So, in right triangle ODA,
r=2 (Rejecting the negative root since radius cannot be negative))
  • Since we now know the value of r and ∠ AOB, we can find the length of arc.
  • Hence, Statement 1 is sufficient to answer the question.
 
Step 4: Analyze Statement 2 independently
(2) Statement 2 states that "The area of sector OACBO is 2π/3
  • Multiple combinations of r and ∠ AOB will satisfy this equation.
    • Example, r = 1 and ∠ AOB = 120o
    • Or r = 2 and
  • Thus, Statement 2 is not sufficient to answer the question.
 
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Hence the correct answer is Option A .

In the given figure, AB and CD are the longest chords of the circle with their lengths equal to 8 units. If the length of the minor arc BC is 1/6th of the perimeter of the circle, what is the length of the chord BD?
 
  • a)
    2√3
  • b)
    4
  • c)
    4√3
  • d)
    8
  • e)
    8√3
Correct answer is option 'C'. Can you explain this answer?

Niharika Sen answered
Given
  • AB = CD = 8 units and they are the longest chords of the circle.
    • AB and CD are diameters of the circle.
      • As diameters are the longest chord.
    • Let their intersection point be O, center of the circle.
    • Radius of the circle = 4 units
  • Length of minor BC = 1/6∗(Perimeter of the circle)
  • = 1/6∗2πr
To Find: Length of chord BD?
Approach
  • We need to find the length of chord BD. We notice that chord BD is a part of triangle OBD.
    1. In triangle OBD, we know that OB = OD = 4 units (radius of the circle)
    2. The third side of this triangle, BD, is unknown that we need to find out.                                   
  • Now, since OBD is an isosceles triangle with OB=OD, if we drop a perpendicular in this triangle it will bisect the BD.
    1.  Let’s drop a perpendicular from O to chord BD at point E.
      1. Therefore, BE = ED
Now as triangle OED is a right angled triangle, knowing one of the sides and one of the angles will be sufficient to calculate the length of ED.
  • In right triangle ODE,
    1. We know that OD = 4 units.
      1. So, we need to find one of the angles of triangle ODE.
    2. As triangle OBD is isosceles and OE is perpendicular from O, OE will bisect ∠BOD. So, ∠DOE = ∠BOD/2
    3. Also, ∠BOD = 180  - ∠COB. So, if we can calculate ∠COB, we can find the value of ∠BOD
  • We know that length of arc BC = 
  • We can use the above relation to calculate the value of ∠COB
Working Out

1. Length of arc  which gives us ∠COB = 60o
4. As triangle ODE is a 30o - 60o -90o  triangle, we have the length of ED = 2√3
 
5. Thus BD = 2 * 2√3 = 4√3
Answer: C
 

The figure above shows a circle whose centre O has the x- and y- coordinates as (6,0). Points A(m,n) and B(8,2) are marked on the circle such that ∠AOB = 105∘. If the circle is symmetrical about the x-axis, what is the value of m?
  • a)
    2
  • b)
    √6
  • c)
    4
  • d)
    6 - √6
  • e)
    Cannot be determined
Correct answer is option 'D'. Can you explain this answer?

Kalyan Nair answered
Given:
  • Circle
    • Center = O(6,0)
    • Points on the circle: A(m,b) and B(8,2)
      • This means, OA = OB = radius of circle
    • ∠AOB = 105∘
To find: m = ?
Approach:
  1. Let’s drop a perpendicular from point A on the x-axis.
From this diagram, it’s clear that there are 2 ways to find the value of m:
 
  1. Way 1 – In right triangle AFO, if we can know the measure of one angle (out of ∠AOF and ∠OAF) and one side (like the radius AO), we can use trigonometric ratios to find length OF, and hence the value of m (since OF = 6 – m)
  2. Way 2 – In right triangle AFO, if we can know the measure of sides AO (radius of circle) and AF, we can use Pythagoras Theorem to find OF, and hence m.
 
2. We are given the coordinates of both points O and B. So, using these, we can find the radius OB = OA
3. Also, if we drop a perpendicular from point B on the x-axis, we’ll know the magnitude of all the sides of the resulting right triangle.
So, we can use trigonometric ratios to find ∠BOE.
So, we can find ∠AOF using the equation  
(Thus, in this Approach, we’re using Way 1 to find m)
Working Out:
  • Radius OB = OA =  2√2
Looking at the answer choices, we see that the correct answer is Option D

How many diagonals does a 63-sided convex polygon have?
  • a)
    3780
  • b)
    1890
  • c)
    3834
  • d)
    3906
  • e)
    1953
Correct answer is option 'B'. Can you explain this answer?

Nandita Chauhan answered
 The number of diagonals of an n-sided convex polygon = 
 
This polygon has 63 sides. Hence, n = 63.
Therefore, number of diagonals =  

ΔABC, which is right-angled at B, is inscribed in a circle with centre O and radius 6 units. If the length of the smaller arc between points A and B is 4π units, what is the length of line segment BC?
  • a)
    3
  • b)
    π
  • c)
    3√3
  • d)
    6
  • e)
    2π
Correct answer is option 'D'. Can you explain this answer?

Arjun Iyer answered
Given:
  • The figure for the given information looks like this:
  • Since Triangle ABC is a right triangle inscribed in the circle, this means the hypotenuse AC of the triangle must be the diameter of the circle.
    • Therefore, AC = 6*2 = 12 units
  • Let the length of BC be x units.
  • Length of the smaller arc between points A and B is 4π
  • units
To find: x = ?
Approach and Working:
 
  • We are given the length of the smaller arc between points A and B. Using this information, we can find ∠AOB
  • Finding ∠AOB
  • We’ll now use ∠AOB to find ∠BAC in isosceles triangle AOB
  • Now that we know ∠BAC, we can clearly infer that triangle ABC is 30-60-90 Triangle. Using the side-angle ratio property, we can find the value of x.
  • Finding x
Looking at the answer choices, we see that the correct answer is Option D.

 
In the figure above, triangle ABC is inscribed in the circle with center O, such that CD is perpendicular to AB. If the length of the side AC is centimeters, ∠AOB is 60o  and the smaller perimeter of the sector AOB is 12 + 2π centimeters, what is the perimeter of the triangle ABC in centimeters?
  • a)
    16
  • b)
  • c)
  • d)
  • e)
Correct answer is option 'D'. Can you explain this answer?

Sravya Joshi answered
Given
  • Circle with center at O
    • Let the radius of the circle be r
  • Triangle ABC is inscribed in the circle
    • CD is perpendicular to AB
    • So, OD is also perpendicular to AB. Hence AD = DB
      • This is because the perpendicular drawn from the centre of the circle to the chord of the circle bisects the chord.
  • AC =  
  • centimeters
  • Perimeter of minor sector AOB = 12 +2π centimeters
  • In triangle AOB,
    • ∠AOB = 60o
    • Triangle AOB is an isosceles triangle, with OA =OB =r
      • So, ∠OAB = ∠OBA
    • Now, since ∠AOB = 60o
      • ⇒ ∠OAB = ∠OBA = ∠AOB = 60o
    • Triangle AOB is an equilateral triangle
      • So, OA =OB = BA =r (All sides are equal in an equilateral triangle)
  • To Find: Perimeter of triangle ABC
  • Perimeter of triangle ABC = AC + CB + BA
  • We know AC =  cm
  • So, Perimeter of triangle ABC =+ CB+BA
  • So, to answer the question, we need to find the length of CB and BA.
Approach
 
  • Finding length of CB
    1. In triangle ABC, we have the perpendicular from vertex C bisecting the opposite base AB, hence triangle ABC should be an isosceles triangle with AC = CB.
    2. As we know the length of AC, we can find the length of CB
       
  • Finding length of BA
    • Length of BA is equal to r
    • We know that perimeter of sector AOB = 
        1. OA = OB = r (already established above)
        2. We are also given perimeter is equal to 12 +2π
        3. So from the above relation we will be able to calculate the value of r. which is equal to BA.
    • Knowing the value of CB and BA, we will be able to find the perimeter of triangle ABC
    • Working Out
    • Finding length of CB
      1. We know that AC =centimetres and AC = CB.
      2. Thus CB = centimetres
      3. Finding length of BA
hence BA = 6 centimeters

The diameter of the tyres of a car is 63 centimetres. If the car is driven at a constant speed such that the tyre makes one 360-degree rotation in 0.24 seconds, what is the approximate speed of the car in kilometres per hour? 
  • a)
    25 kilometre per hour
  • b)
    30 kilometre per hour
  • c)
    60 kilometre per hour
  • d)
    171 kilometre per hour
  • e)
    180 kilometre per hour © 2014 ScholaraniumDashboardSkill DataAttemptsSupportTerms of Use
Correct answer is option 'B'. Can you explain this answer?

Ankita Chauhan answered
Given:
  • Let the diameter of the tyre = d
  • d = 63 cm
  • Tyre makes one 360-degree rotation in 0.24 seconds
To find: Approx. speed of car in kilometers per hour ( in short kmph)
Approach:
  1. Speed = Distance/Time
    • We will find the distance that the tyre covers in 0.24 seconds
    • This distance = Circumference of the tyre
    • (Note: the car moves a distance through the rotation of its tyres. This is the reason why in answer to the question on the speed of the car, we are finding the distance covered by its tyres per unit time)
       
  2. Since we’re asked for only an approximate speed, we can use the principles of Estimation and Rounding to ease our calculations. But we will not round numbers aggressively since a few answer choices are not very far apart in magnitude (Choices A and B, Choices D and E)
Working Out:
  • Finding the circumference of the tyre
    • Circumference of the tyre 
​​​​
  • Thus, the speed of the car is 30 kilometers per hour approx.
Looking at the answer choices, we see that the correct answer is Option B

15 children are given tags numbered from 1 to 15 and are seated in a circular formation in the increasing order of their respective tag numbers. The total area covered by the circular formation is 36π square units and the distance between any two neighbouring children in the formation is equal., If the number of children seated between the child with tag number m and the child with tag number 1 is equal to the number of children seated between the child with tag number m and the child with tag number 15, what is the minimum distance covered along the circular formation by the child with tag number 1 to reach the child with tag number m -2 and then go back to his original position?
  • a)
    4π
  • b)
    6π
  • c)
    8π
  • d)
    12π
  • e)
    16π
Correct answer is option 'C'. Can you explain this answer?

Sravya Joshi answered
Given
  • 15 children are seated at equal distance in a circular formation
    • Let the distance between two neighboring students be x.
 
  • Area of the circular formation = 36π
    • Let’s assume the radius of the circular formation be r
    • πr2=36π
  • Child number ‘m’ is equidistant from child number 1 and child number 15
To Find:
  • Minimum distance covered by child number 1 to reach child number m -2 and back to his original position
Approach
 
  • Let’s understand first how we can find the distance between any two children.
    • Let’s assume we need to find the distance between child number 1 and child number 4.
    • As the distance between two neighboring children is x, distance between child number 4 and child number 1 will be
      1. Case 1: Distance =(4-1) * x = 3x
      2. Case 2: Distance = 15x – 3x = 12x
        1. 15x is the total distance along the circular formation for 15 children.
      3. So minimum distance will be = minimum {distance a, distance b} = minimum {3x,12x} = 3x
    • So based on the example above we can infer that in general the minimum distance between any two children = Minimum {distance c, distance d}, where
      1. distance c = ((Position of first child – Position of second child))*x
      2. distance d = 15x – ((Position of first child – Position of second child))*x
    • Now, distance c & distance d or any distance between children is a part of the perimeter of the circle.
      1. So, we can write 15x = 2πr
    • Thus, if we know the value of r, we can find the value of x.
       
  • Hence, for finding the distance traveled, we need to find the following:
    1. Value of m
    2. Radius of the circle i.e. r
       
  • Value of m
    1. As child m is equidistant from child number 1 and child number 15, we can write
      1. Distance between child number m and child number 1 = Distance between child number 15 and child number m
      2. (m-1)*x = (15-m)*x
      3. This will give us the value of m and hence we can find the value of m -2
         
  • Radius of the circle
    1. We know that πr2=36π
    2. We can use the above equation to find out the value of r.
 
Working Out
  • Finding value of m
    • (m – 1)*x = (15- m)*x, i.e. m = 8
    • Hence, m – 2 = 6
  • Calculating minimum distance
    • distance c = ((position of child number m - 2 ) – (position of child number 1))*x
    • So, distance c = (6-1)*x = 5x
    • distance d = 15x - ((position of child number m - 2 ) – (position of child number 1))*x 
    • So, distance d = 15x – 5x
    • Minimum distance = minimum {distance c, distance d}
      • So, minimum distance = minimum {5x,10x} = 5x
      • Distance to go back to original position = 5x+5x =10x
  • Therefore, the correct answer is Option C.

A square is inscribed in a big circle and a small circle is inscribed in the square as shown in the figure above. (All of them are concentric and their center is the point O). If the radius of the big circle is √2 cm, what is the semi perimeter of the small circle?
  • a)
    1
  • b)
    2
  • c)
    3
  • d)
    π
  • e)
    2π
Correct answer is option 'D'. Can you explain this answer?

Prashanth Chawla answered
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

A circle is inscribed in a square ABCD such that the circle touches all the sides of the square. If the perimeter of the shaded region is 24 + 6π, what is the area of the shaded region?
  • a)
    16 - 4π
  • b)
    32 - 8π
  • c)
    48 - 12π
  • d)
    64 - 16π
  • e)
    48 - 8π
Correct answer is option 'C'. Can you explain this answer?

Abhishek Choudhury answered
Given
  • Square ABCD
A circle is inscribed in the square ABCD
Let the side of square be x and the radius of the circle be r
  • Circle touches all sides of the square, so diameter = side of square
    • 2r = x
  • Perimeter of the shaded region = 24 + 6π
To Find: Area of the shaded region?
 
 
Area of the shaded region = 
(3over 4)
∗(Area of square – Area of circle)
  1. We can observe that the area in the figure that is outside the circle but inside the square are four parts as shown in the above figure.
  2. Area of (Part A + Part B + Part C + Part D) = Area of square – Area of Circle.
Out of these 4 parts:
  1. Part A, B & C are shaded.
  2. Part D is unshaded.
  3. Now the figure is symmetric about the centre axis, so each part of the 4 parts will have same side length.
    1. Thus, all the parts i.e. Part A, B, C & D will also be equal in area.
  4. Therefore, Area of Part A = Area of Part B = Area of Part C = Area of Part D
    1. 4*Area of each part = Area of square – Area of Circle
    2. Area of each part = 1/4 (Area of square - Area of circle)
  • We need to find the areas of only 3 shaded parts, thus area of the shaded region = 3/4(Area of square – Area of circle)
  1. Thus to find the area of the shaded region, we need to find the values of Area of Square and Area of Circle.
Approach
​
3. Since area of square and area of circle are in terms of r, we need to find the value of r, by equating perimeter of shaded region to 24 + 6π.
Perimeter of the shaded region = Perimeter of the square + Perimeter of the circle – perimeter of the region AFE
  1. Perimeter of square = 4x = 8r
  2. Perimeter of circle = 2πr = πx
  3. Calculating Perimeter of the region AFE:
    1. If, we join OF and OE, we have AEOF as a square with side length = r = x2
  •  
  • So, we have the length of AF and AE in terms of r, also he angle made by arc FE on center O(90o)(due to symmetry of figure about the centre axes), we can calculate the length of arc FE using the formula 
  • Thus we can calculate the perimeter of region AFE in terms of x, where r = x/2
  • .
  • So perimeter of region AFE = 
Working Out
  1. Perimeter of the shaded region = Perimeter of the square + Perimeter of the circle – perimeter of the region AFE
  • On solving w, we get x = 8
2. Area of shaded region = 3/4 ∗(Area of square – Area of circle)
  • Therefore, the correct answer is Option C.

The annual money spent by a company on search engine marketing, content marketing and affiliate marketing is to be shown on a circlular graph. If the size of each sector of the graph is proportional to the amount of money it represents, did the company spend the most money on search engine marketing?
(1) The angle of the sector that represents content marketing is 80o
(2) The angle of the sector that represents affiliate marketing is 210o
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'B'. Can you explain this answer?

Sharmila Nambiar answered
Steps 1 & 2: Understand Question and Draw Inferences
Given: 3 amounts to be represented on a circle graph:
  • Search Engine Marketing Spending
  • Content Marketing Spending
  • Affiliate Marketing Spending
  • Let the angles of the respective sectors be s∘,c∘,a∘
  •  respectively
To find: Was money spent on Search Engine Marketing the greatest?
  • The answer will be YES if, out of s∘,c∘,a∘,s∘
  •  is the greatest
  • Now, we know that  s∘+c∘+a∘=360∘
​
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The angle of the sector that represents content marketing is 80∘
  • Based on the above equation, we cannot say if s∘ is the greatest angle or not
    • For example, if s∘=220∘ and a ∘=60∘ then the answer is YES
  • But if s∘=60∘ and a∘=220∘ then the answer is NO
  • So, Statement 1 is not sufficient to arrive at a unique answer
 
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The angle of the sector that represents affiliate marketing is 210∘
  • Now, from the above equation, we can infer that the greatest possible value of s∘is150∘ (happens when c∘=0
  • So, even the greatest possible value of s∘ is less than the measure of a∘ (which is 210∘)
  • Therefore, s∘ is definitely not the greatest of the 3 angles
  • So, the answer to the question is NO
  • Since we’ve been able to arrive at a unique answer, Statement 2 alone is sufficient.
 
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B


If O is the center of the circle above, what fraction of the circular region is encompassed by an angle of x degrees?
  • a)
    1/36
  • b)
    1/12
  • c)
    1/9
  • d)
    1/6
  • e)
    cannot be determined
Correct answer is option 'B'. Can you explain this answer?

Sharmila Nambiar answered
In this question, the angles shown appear span half the circle. We can infer that they must, in fact, because the opposite angles of bisecting lines are equal. On the other side of each of the 2x angles shown is another 2x, and on the other side of each x shown is another x. Therefore,
2x+x+2x+x = 180
half the span of the circle. Thus, and . Since the full span of a circle is 360 degrees, the angle x spans a fraction of 30/360 = 1/12
The correct answer is (B).

ABCD is a square inscribed in a circle. If the length of BC = 8cm. Find the area of the shaded region.
  • a)
    32 (2- π) cm2
  • b)
    32(π – 3) cm2
  • c)
    32 (π – 2) cm2
  • d)
    64 (π – 2) cm2
Correct answer is option 'C'. Can you explain this answer?

Talent Skill Learning answered
To solve this, we need to find the area of the shaded region, which can be calculated as the area of the circle minus the area of the square. The diagonal of the square is equal to the diameter of the circle. Using the Pythagorean theorem, the diagonal of the square odd is: 

The figure above shows a rectangular plot of land OPQR. In this plot, a patch of land OABO, where the path AB represents a circular arc of radius 20√2 metres, is developed as a lawn. The point A is at a distance of 20 metres each from sides OR and OP. The point B is at a distance of 10√6 metres from side OP and 10√2 metres from side OR. What is the area, in square metres, of the intended lawn?
  • a)
  • b)
  • c)
  • d)
  • e)
    Cannot be determined
Correct answer is option 'B'. Can you explain this answer?

Maya Khanna answered
Given:
  • Let’s consider OP as the x-axis and OR as the y-axis.
  • So, the coordinates of points A and B are: A(20, 20) and B(10√2, 10√6)
 
To find: Area of region OABO
Approach:
  1. We’re given that AB is a circular arc of radius 20√2 meters but we are not told that region OABO represents a sector of a circle of radius 20√2 meters. So, it will be wrong to assume that OA and OB are the radii of the sector OABO. In fact, we’ll need to find the values of OA and OB to check if they are indeed the radii of circular arc AB.
    1. We can find the value of OA and OB using the Pythagoras theorem in triangles OEA and OFB.
  2. If OA and OB are indeed the radii of the circular arc, then we can use the formula of area of sector to find the area of region OABO. But to find this area, we’ll also need the measure of ∠BOA. We can find this angle as follows
    • ∠BOA = ∠BOF - ∠AOE
      • ∠BOF can be found by applying trigonometric ratios in ΔOFB
Working Out:
Thus, we see that OA = OB = Radius of the circular arc AB =20√2  meters
  • This means that OABO is indeed the sector of a circle of radius 20√2 meters
  • So, we’ll be able to apply the formula for area of a sector to answer the question.
Thus, ∠BOA =  15o
Finding the area of sector OABO
  • So, the area of sector 
​
Looking at the answer choices, we see that the correct answer is Option B

Chapter doubts & questions for Circles - The Complete SAT Course 2026 is part of Class 10 exam preparation. The chapters have been prepared according to the Class 10 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Class 10 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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