CAT Circles Practice Questions With Solutions

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CAT Quant Circles Practice Questions

VARC DILR

Question 1.

Let C be the circle $x^{2} + y^{2} + 4x - 6y - 3 = 0$ and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to $60^{\circ}$ . Then, the point at which L touches the line $x$ = 6 is

(6, 6)

(6, 3)

(6, 8)

(6, 4)

Question 2.

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals

2 : 1

1 : 2

8 : 5

5 : 8

Question 3.

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a : b. If the radius of the circle is r, then the area of the triangle is

$\frac{abr^{2}}{2(a^{2}+b^{2})}$

$\frac{2abr^{2}}{a^{2}+b^{2}}$

$\frac{4abr^{2}}{a^{2}+b^{2}}$

$\frac{abr^{2}}{a^{2}+b^{2}}$

Question 4.

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

2 : 1

1 : 1

$\sqrt{5} : 1$

$\sqrt{2} : 1$

Question 5.

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

$\frac{P^2}{16} - R^2$

$\frac{P^2}{8} - 2R^2$

$\frac{P^2}{2} - 2PR$

$\frac{P^2}{8} - \frac{R^2}{2}$

Question 6.

In a triangle ABC, AB = AC = 8 cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through Band C. Then the area, in sq. cm, of the overlapping region between the two circles is

$16\pi$

$16(\pi - 1)$

$32(\pi - 1)$

$32\pi$

Question 7.

A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

243

232

256

264

Question 8.

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

$3\sqrt{\pi}(5+\frac{12}{\pi})$

$4\sqrt{\pi}(3+\frac{9}{\pi})$

$3\sqrt{\pi}(\frac{5}{2}+\frac{6}{\pi})$

$5\sqrt{\pi}(3+\frac{9}{\pi})$

Question 9.

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

$\frac{6\pi}{25}$

$\frac{5\pi}{18}$

$\frac{3\pi}{25}$

$\frac{2\pi}{15}$

Question 10.

Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

8.8

7.8

6.6

7.2

Great Job! continue working on more practice questions?

Question 1.

Two circular tracks T1 and T2 of radii 100 m and 20 m, respectively touch at a point A. Starting from A at the same time, Ram and Rahim are walking on track T1 and track T2 at speeds 15 km/hr and 5 km/hr respectively. The number of full rounds that Ram will make before he meets Rahim again for the first time is

Question 2.

The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is

$\frac{14\pi}{3}$

$\frac{123\pi}{7}$

$\frac{12\pi}{5}$

$\frac{205\pi}{9}$

Question 3.

AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to

9.3

7.8

9.1

8.5

Question 4.

In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is

2.5

1.5

3.5

0.5

Question 5.

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

$\frac{1}{\surd2}$

$\frac{\pi}{3}$

$\surd2$

Question 6.

A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

$1026(1 + \pi)$

$8464\pi$

$928\pi$

$1044(4 + \pi)$

Question 7.

In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is

$\sqrt{13}$

$\sqrt{14}$

$\sqrt{11}$

$\sqrt{12}$

Question 8.

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

24, 10

25, 9

25, 10

24, 12

Question 9.

In a circle with center O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

$(\frac{\pi}{4\sqrt{3}})^{\frac{1}{2}}$

$(\frac{\pi}{6})^{\frac{1}{2}}$

$(\frac{\pi}{3\sqrt{3}})^{\frac{1}{2}}$

$(\frac{\pi}{4})^{\frac{1}{2}}$

Question 10.

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

$5\sqrt{3}$

$2\pi$

$8$

$6\sqrt{2}$

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CAT Circles Practice Questions With Solutions

CAT Quant Circles Practice Questions

Question 1.

Let C be the circle x2+y2+4x−6y−3=0x^{2} + y^{2} + 4x - 6y - 3 = 0x2+y2+4x−6y−3=0 and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to 60∘60^{\circ}60∘. Then, the point at which L touches the line xxx = 6 is

Question 2.

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals

Question 3.

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a : b. If the radius of the circle is r, then the area of the triangle is

Question 4.

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

Question 5.

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

Question 6.

In a triangle ABC, AB = AC = 8 cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through Band C. Then the area, in sq. cm, of the overlapping region between the two circles is

Question 7.

A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

Question 8.

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

Question 9.

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

Question 10.

Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

Question 1.

Question 2.

The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is

Question 3.

AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to

Question 4.

In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is

Question 5.

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

Question 6.

Question 7.

In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is

Question 8.

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

Question 9.

Question 10.

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

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Let C be the circle $x^{2} + y^{2} + 4x - 6y - 3 = 0$ and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to $60^{\circ}$ . Then, the point at which L touches the line $x$ = 6 is