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Mensuration

Q.No: 1
Test Name : CAT Paper 2004

A rectangular sheet of paper, when halved by folding it at the mid point of its longer side, results in a rectangle, whose longer and shorter sides are in the same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle is 2, what is the area of the smaller rectangle?

A
B
C
D
None of the above
Answer key/Solution
Solution:
Q.No: 2
Test Name : CAT Paper 2005

Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is

A
4
B
5
C
6
D
7
Answer key/Solution
Solution:
Q.No: 3
Test Name : CAT Paper 2005

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is

A
10
B
12
C
14
D
16
Answer key/Solution
Solution:
Q.No: 4
Test Name : CAT Paper 2007

Mark (1) if the questions can be answered using A alone but not using B alone.
Mark (2) if the question can be answered using B alone but not using A alone.
Mark (3) if the question can be answered using A and B together, but not using either A or B alone.
Mark (4) if the question cannot be answered even using A and B together.


ABC Corporation is required to maintain at least 400 Kilolitres of water at all times in its factory, in order to meet safety and regulatory requirements. ABC is considering the suitability of a spherical tank with uniform wall thickness for the purpose. The outer diameter of the tank is 10 meters. Is the tank capacity adequate to meet ABC's requirements?
A: The inner diameter of the tank is at least 8 meters.
B: The tank weights 30,000 kg when empty, and is made of a material with density of 3 gm/cc.

A
(1)
B
(2)
C
(3)
D
(4)
Answer key/Solution
Solution:
Q.No: 5
Test Name : CAT Paper 2008

Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest possible total surface area (in sq. cm) of the cylinder.

A
100∏ / 3
B
80∏ / 3
C
120∏ / 7
D
130∏ / 9
E
110∏ / 7
Answer key/Solution
Solution:
Q.No: 6
Test Name : CAT Paper 1991
Q 61 to 100 : Choose the best answer choice from those provided

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

A
2 : 3
B
3 : 4
C
1 : 4
D
1 : 2
E
NA
Answer key/Solution
Solution:
Q.No: 7
Test Name : CAT Paper 1991
Q 61 to 100 : Choose the best answer choice from those provided

There are 3 clubs A, B & C in a town with 40, 50 & 60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?

A
20
B
25
C
50
D
70
E
NA
Answer key/Solution
Solution:
Q.No: 8
Test Name : CAT Paper 1993
Q58 to 100 : Choose the appropriate answer choice.

Three identical cones with base radius r are placed on their bases so that each is touching the other two. The radius of the circle drawn through their vertices is

A
smaller than r
B
equal to r
C
larger than r
D
depends on the height of the cones.
Answer key/Solution
Solution:

It can be seen that, if we place the 3 cones in such a way that they touch each other, it will be similar to placing 3 circles touching, with vertices of the cone corresponding to the centers of the circles. The centers of the circle form an equilateral triangle with each side being 2r. A circle that passes through the centers will be the circumcircle to such a triangle. The radius of the circumcircle of an equilateral triangle is
Q.No: 9
Test Name : CAT Paper 1993
Q58 to 100 : Choose the appropriate answer choice.

A slab of ice 8 inches in length, 11 inches in breadth, and 2 inches thick was melted and resolidified into the form of a rod of 8 inches diameter. The length of such a rod, in inches, is nearest to

A
3
B
3.5
C
4
D
4.5
Answer key/Solution
Solution:
Q.No: 10
Test Name : CAT Paper 1994
Q51 – 90 : Choose the best alternative.

A right circular cone, a right circular cylinder and a hemisphere, all have the same radius, and the heights of the cone and cylinder are equal to their diameters. Then their volumes are proportional, respectively to

A
1 : 3 : 1
B
2 : 1 : 3
C
3 : 2 : 1
D
1 : 2 : 3
Answer key/Solution
Solution:
Q.No: 11
Test Name : CAT Paper 1994

A right circular cone of height h is cut by a plane parallel to the base and at a distance h/3 from the base, then the volumes of the resulting cone and the frustum are in the ratio

A
1 : 3
B
8 : 19
C
1 : 4
D
1 : 7
Answer key/Solution
Solution:
Q.No: 12
Test Name : CAT Paper 1994
Q51 – 90 : Choose the best alternative.

If a + b + c = 0, where then is equal to

A
zero
B
1
C
–1
D
abc
Answer key/Solution
Solution:
Q.No: 13
Test Name : CAT Paper 1995

Direction for questions 58 to 87: Answer the questions independently.

The sides of a triangle are 5, 12 and 13 units. A rectangle is constructed, which is equal in area to the triangle, and has a width of 10 units. Then the perimeter of the rectangle is

A
30 units
B
36 units
C
13 units
D
None of these
Answer key/Solution
Solution:
Q.No: 14
Test Name : CAT Paper 1996

A cube of side 12 cm is painted red on all the faces and then cut into smaller cubes, each of side 3 cm. What is the total number of smaller cubes having none of their faces painted?

A
16
B
8
C
12
D
24
Answer key/Solution
Solution:
Q.No: 15
Test Name : CAT Paper 1996

From a circular sheet of paper with a radius 20 cm, four circles of radius 5 cm each are cut out. What is the ratio of the uncut to the cut portion?

A
1 : 3
B
4 : 1
C
3 : 1
D
4 : 3
Answer key/Solution
Solution:
Q.No: 16
Test Name : CAT Paper 1996

A wooden box (open at the top) of thickness 0.5 cm, length 21 cm, width 11 cm and height 6 cm is painted on the inside. The expenses of painting are Rs. 70. What is the rate of painting per square centimetres?

A
Re. 0.7
B
Re. 0.5
C
Re. 0.1
D
Re. 0.2
Answer key/Solution
Solution:
Q.No: 17
Test Name : CAT Paper 2001
Directions for questions 1 to 37: Answer the questions independently.

Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles.

A
768 m2
B
534 m2
C
696.5 m2
D
684 m2
Answer key/Solution
Solution:
Q.No: 18
Test Name : CAT Paper 2001
Directions for questions 40 to 48: Answer the questions independently.

A rectangular pool 20 m wide and 60 m long is surrounded by a walkway of uniform width. If the total area of the walkway is 516 m2 , how wide, in metres, is the walkway?

A
43 m
B
4.3 m
C
3 m
D
3.5 m
Answer key/Solution
Solution:
Q.No: 19
Test Name : CAT Paper 2003 (L)
DIRECTIONS for Questions 126 to 150: Answer the questions independently of each other.

In the figure below, ABCDEF is a regular hexagon and ∠AOF = 90° . FO is parallel to ED. What is the ratio of the area of the triangle AOF to that of the hexagon ABCDEF?

A
B
C
D
Answer key/Solution
Solution:
It is very clear, that a regular hexagon can be divided
into six equilateral triangles. And triangle AOF is half
of an equilateral triangle.
Hence, the required ratio = 1 : 12.
Q.No: 20
Test Name : CAT Paper 2003 (R)
Directions for questions 57 to 59: Answer the questions on the basis of the information given below. Consider a cylinder of height h cm and radius as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at point B, gives a maximum of n turns (in other words, the string’s length is the minimum length required to wind n turns).

What is the vertical spacing between the two consecutive turns?

A
B
C
D
Cannot be determined
Answer key/Solution
Solution:
Q.No: 21
Test Name : CAT Paper 2003 (R)
Directions for questions 57 to 59: Answer the questions on the basis of the information given below. Consider a cylinder of height h cm and radius as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at point B, gives a maximum of n turns (in other words, the string’s length is the minimum length required to wind n turns).

The same string, when wound on the exterior four walls of a cube of side n cm, starting at point C and ending at point D, can give exactly one turn (see figure, not drawn to scale). The length of the string is

A
B
C
n cm
D
Answer key/Solution
Solution:
Q.No: 22
Test Name : CAT Paper 2003 (R)
Directions for questions 57 to 59: Answer the questions on the basis of the information given below. Consider a cylinder of height h cm and radius as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at point B, gives a maximum of n turns (in other words, the string’s length is the minimum length required to wind n turns).

In the set-up of the previous two questions, how is h related to n?

A
B
C
h = n
D
Answer key/Solution
Solution:
Q.No: 23
Test Name : CAT Paper 2003 (R)
Directions for questions 60 to 93: Answer the following questions independently.

Consider two different cloth-cutting processes. In the first one, n circular cloth pieces are cut from a square cloth piece of side a in the following steps: the original square of side a is divided into n smaller squares, not necessarily of the same size, then a circle of maximum possible area is cut from each of the smaller squares. In the second process, only one circle of maximum possible area is cut from the square of side a and the process ends there. The cloth pieces remaining after cutting the circles are scrapped in both the processes. The ratio of the total area of scrap cloth generated in the former to that in the latter is

A
1 : 1
B
C
D
Answer key/Solution
Solution:
Q.No: 24
Test Name : CAT Paper 2003 (R)
Directions for questions 60 to 93: Answer the following questions independently.

A piece of paper is in the shape of a right-angled triangle and is cut along a line that is parallel to the hypotenuse, leaving a smaller triangle. There was 35% reduction in the length of the hypotenuse of the triangle. If the area of the original triangle was 34 square inches before the cut, what is the area (in square inches) of the smaller triangle?

A
16.665
B
16.565
C
15.465
D
14.365
Answer key/Solution
Solution:
Q.No: 25
Test Name : CAT 2017 Actual Paper Slot 1

A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1:1:8: 27:27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to

A
10
B
50
C
60
D
20
Answer key/Solution
Solution:
Q.No: 26
Test Name : CAT 2017 Actual Paper Slot 1

A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is 9π cm3. Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is

Answer key/Solution
Solution:
Q.No: 27
Test Name : CAT 2017 Actual Paper Slot 2

If three sides of a rectangular park have a total length 400 ft, then the area of the park is maximum when the length (in ft) of its longer side is

Answer key/Solution
Solution:
Q.No: 28
Test Name : CAT 2018 Actual Paper Slot 1

A right circular cone, of height 12 ft, stands on its base which has diameter 8 ft. The tip of the cone is cut off with a plane which is parallel to the base and 9 ft from the base. With π = 22/7, the volume, in cubic ft, of the remaining part of the cone is

Answer key/Solution
Solution:
Q.No: 29
Test Name : CAT 2019 Actual Paper Slot 2

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

A
928π
B
1044(4 + π)
C
1026(1 + π)
D
8464π
Answer key/Solution
Solution:
Q.No: 30
Test Name : CAT 2019 Actual Paper Slot 2

The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is

A
B
C
12
D
Answer key/Solution
Solution:
Q.No: 31
Test Name : CAT Actual Paper 2020 Slot-1

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

A
B
C
D
Answer key/Solution
Solution:
Q.No: 32
Test Name : CAT Actual Paper 2020 Slot-1

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

A
264
B
256
C
232
D
243
Answer key/Solution
Solution:
Q.No: 33
Test Name : CAT Actual Paper 2021 Slot-3

The cost of fencing a rectangular plot is ₹ 200 per ft along one side, and ₹ 100 per ft along the three other sides. If the area of the rectangular plot is 60000 sq. ft, then the lowest possible cost of fencing all four sides, in INR, is

A
120000
B
100000
C
160000
D
90000
Answer key/Solution
Solution:
Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

It can be seen that, if we place the 3 cones in such a way that they touch each other, it will be similar to placing 3 circles touching, with vertices of the cone corresponding to the centers of the circles. The centers of the circle form an equilateral triangle with each side being 2r. A circle that passes through the centers will be the circumcircle to such a triangle. The radius of the circumcircle of an equilateral triangle is

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:
It is very clear, that a regular hexagon can be divided
into six equilateral triangles. And triangle AOF is half
of an equilateral triangle.
Hence, the required ratio = 1 : 12.

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

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