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Test: Geometry- 2 - CAT MCQ


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15 Questions MCQ Test - Test: Geometry- 2

Test: Geometry- 2 for CAT 2024 is part of CAT preparation. The Test: Geometry- 2 questions and answers have been prepared according to the CAT exam syllabus.The Test: Geometry- 2 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Geometry- 2 below.
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Test: Geometry- 2 - Question 1
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Test: Geometry- 2 - Question 2
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A square is inscribed in a semi circle of radius 10 cm. What is the area of the inscribed square? (Given that the side of the square is along the diameter of the semicircle.)

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Test: Geometry- 2 - Question 3
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Two circles of an equal radii are drawn, without any overlap, in a semicircle of radius 2 cm. If these are the largest possible circles that the semicircle can accommodate, what is the radius (in cm) of each of the circles?

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Test: Geometry- 2 - Question 4
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PQRS is a Trapezzium, in which PQ is Parralel to RS, and PQ = 3 (RS) . The diagnol of the Trapezzium intersect each other at X, then the ratio of,  ar ( ∆ PXQ)  : ar ( ∆ RXS)  is?
Detailed Solution for Test: Geometry- 2 - Question 4

In ∆ PXQ and ∆ RXS

=> angle P = angle R

    angle Q  =  angle S 

:- ∆ PXQ  ~ ∆ RXS  ( AA similarity rule)  

ar ( ∆ PXQ) /  ar ( ∆ RXS) = ( PQ / RS) ^ 2 

 =  ( 3 / 1 ) ^ 2

=     9 / 1 

Therefore,  ar ( ∆ PXQ) :  ar          ( ∆ RXS) 

=   9:1

Test: Geometry- 2 - Question 5
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Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?
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Test: Geometry- 2 - Question 6
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A pond 100 m in diameter is surrounded by a circular grass walk-way 2 m wide. How many square metres of grass is the on the walk-way?
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Test: Geometry- 2 - Question 7
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The dimensions of a rectangular box are in the ratio of 1:2:4 and the difference between the costs of covering it with the cloth and a sheet at the rate of Rs 20 and Rs 20.5 per sq m respectively is Rs 126. Find the dimensions of the box.
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Test: Geometry- 2 - Question 8
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The ratio of the area of a square inscribed in a semicircle to that of the area of a square inscribed in the circle of the same radius is
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Test: Geometry- 2 - Question 9
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The ratio of the area of a square to that of the square drawn on the its the diagonal is
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Test: Geometry- 2 - Question 10
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What is the area of the triangle in which two of its medians 9 cm and 12 cm long intersect at the right angles?
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Test: Geometry- 2 - Question 11
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Four horses are tethered at four comers of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m2 at the centre. Find the ungrazed area.
Detailed Solution for Test: Geometry- 2 - Question 11

Total area of plot = 14 * 14 = 196m2
Horses can graze in quarter circle of radius = 7m
Grazed area = 4 * (pie r2)/4 = 154 m2
Area of plot when horses cannot reach = (196 - 154) = 42m2
Ungrazed area = 42 - 20 = 22m2

Test: Geometry- 2 - Question 12
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Two sides of a triangle are 4 and 5. Then, for the area of the triangle, which one of the following bounds is the sharpest?
Detailed Solution for Test: Geometry- 2 - Question 12

Let AB = 4 and BC = 5 and AB is perpendicular to BC

then Area = 1/2 AB . AC = 1/2 . 4.5 = 10 

Test: Geometry- 2 - Question 13
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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?
Detailed Solution for Test: Geometry- 2 - Question 13


Let the longer side of the rectangle is 2a and the shorter side is b as shown in the figure.

Using pythagoras theorem, we get
a2 + b2 = 22 = 4   ...(1)

Also, AM ≥ GM

⇒ a2 + b2 ≥ 2ab

⇒ 4 ≥ 2ab

⇒ 2 ≥ ab

∴ Highest value of ab = 2, when a = b.

⇒ Ratio of longer side to shorter side = 2a : b = 2 : 1.

Hence, option (a).

Test: Geometry- 2 - Question 14
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The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is
Detailed Solution for Test: Geometry- 2 - Question 14

In a quadrilateral, the longest side is less than the sum of other three sides and greater than the least of the difference of any 2 of the other three sides.

Let the fourthe sides of the quadrilateral be x.

⇒ 2 - 1 < x < 1 + 2 + 4
⇒ 1 < x < 7
∴ x can be 2, 3, 4, 5 or 6

Hence, x can take 5 integral values.

Hence, option (a).

Test: Geometry- 2 - Question 15
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Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. If T is the mid point of CD, then the length of AT, in cm, is
Detailed Solution for Test: Geometry- 2 - Question 15

Side of the regular hexagon = 2 cm.
Consider the figure below.

Consider the isosceles ∆ATF

TU is the altitude from T to AF. 
We know, in a regular hexagon the distance between any two parallel sides = √3 × side.
∴ TU = √3 × 2 = 2√3 cm.

Since ATF is an isosceles triangle, U will be the mid-point of AF
∴ AU = 1 cm.

In ∆ATU
AT2 = TU2 + AU2
⇒ AT2 = (2√3)2 + 12
⇒ AT2 = 13
⇒ AT = √13

Hence, option (c).

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