Question for CAT Previous Year Questions: Mensuration
Try yourself:Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is
[2022]
Correct Answer : 9
Explanation
O is the centroid of the triangle ABC,that means,
BO : OD = 2 : 1
CO : OE = 2 : 1
Ar(BOC) : Ar(ODC) = 2 : 1
Ar(COB) : Ar(OEB) = 2 : 1
Since BD is the median, Ar(BDA) = Ar(BDC)
This means Ar(AEOD) = 2x
2x + x + x + 2x = 108
6x = 108
Ar(AEOD) = 2x = 36
Since ED is the line joining the midpoints of AB and AC, Ar(AED) = ¼ Ar(ABC)
Ar(AED) = ¼ Ar(108) = 27
Ar(EOD) = Ar(AEOD) - Ar(AED) = 36 - 27 = 9 sq cm
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Question for CAT Previous Year Questions: Mensuration
Try yourself:In a triangle ABC, AB = AC = 8cm. A circle drawn with BC as diameter passes through A . Another circle drawn with center at A passes through B and C. Then the area, in sq. cm , of the overlapping region between the two circles is
[2022]
Explanation
AB = AC = 8cm
Since BC is the diameter, angle A has to right-angled and angle B = angle C = 45 degrees.
Therefore, the radius of the circle centred at O is 4√ 2.
The radius of the circle centred at Aa is 8cm.
The common area between them = (half of the smaller circle) + (and the minor segment created by the chord BC in the bigger circle)
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Question for CAT Previous Year Questions: Mensuration
Try yourself:From a container filled with milk, 9 litres of milk are drawn and replaced with water. Next, from the same container, 9 litres are drawn and again replaced with water. If the volumes of milk and water in the container are now in the ratio of 16 : 9, then the capacity of the container, in litres, is
[2021]
Correct Answer : 45
Explanation
Let initial volume be V, final be F for milk.
The formula is given by : n is the number of times the milk is drawn and replaced.
so we get
here K =9
we get
we get
If considering 1 - 9/V = -4/5
V = 5, but this is not possible because 9 liters is drawn every time.
Hence : 1− 9/V = 4/5
V = 45 litres
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Question for CAT Previous Year Questions: Mensuration
Try yourself:If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is
[2021]
Explanation
All the sides of the rhombus are equal.
The area of a rhombus is 12 cm2.
Considering d1 to be the length of the longer diagonal, d2 to be the length of the shorter diagonal.
The area of a rhombus is
d1 x d2 = 24.
The length of the side of a rhombus is given by This is because the two diagonals and a side from a right-angled triangle with sides d1/2, d2/2 and the side length.
Hence,
or
In a rhombus the area of a Rhombus is given by :
The diagonals perpendicularly bisect each other. Considering the length of the diagonal to be 2a, 2b.
The area of a Rhombus is :
ab = 6.
The length of each side is :
2a is longer diagonal which is equal to
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Question for CAT Previous Year Questions: Mensuration
Try yourself:The sides AB and CD of a trapezium ABCD are parallel, with AB being the smaller side. P is the midpoint of CD and ABPD is a parallelogram. If the difference between the areas of the parallelogram ABPD and the triangle BPC is 10 sq cm, then the area, in sq cm, of the trapezium ABCD is
[2021]
Explanation
We are given that : Let DP =x
So AB =x
Now DP=CP
So CD = 2x
Now let height of trapezium be h
we can say A(Parallelogram ABPD ) = xh
And A (BPC) = 1/2xh
Now by condition xh - 1/2xh = 10
xh / 2 = 10
so xh =20
Now therefore area of trapezium ABCD = 1/2 (x + 2x)h = 3/2xh = 30
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Question for CAT Previous Year Questions: Mensuration
Try yourself:Let D and E be points on sides AB and AC, respectively, of a triangle ABC, such that AD : BD = 2 : 1 and AE : CE = 2 : 3. If the area of the triangle ADE is 8 sq cm, then the area of the triangle ABC, in sq cm, is
[2021]
Correct Answer : 30
Explanation
We have :
Now area of ADE = 1/2 x AD x AE x sin A
=
we get xy sinA =4Now Area of triangle ABC
=
We get,
we get Area of ABC = 30
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Question for CAT Previous Year Questions: Mensuration
Try yourself:Two pipes A and B are attached to an empty water tank. Pipe A fills the tank while pipe B drains it. If pipe A is opened at 2 pm and pipe B is opened at 3 pm, then the tank becomes full at 10 pm. Instead, if pipe A is opened at 2 pm and pipe B is opened at 4 pm, then the tank becomes full at 6 pm. If pipe B is not opened at all, then the time, in minutes, taken to fill the tank is
[2021]
Explanation
Let A fill the tank at x liters/hour and B drain it at y liters/hour
Now as per Condition 1 :
We get Volume filled till 10pm = 8x - 7y (1) .
Here A operates for 8 hours and B operates for 7 hours .
As per condition 2
We get Volume filled till 6pm = 4x - 2y (2)
Here A operates for 4 hours and B operates for 2 hours .
Now equating (1) and (2)
we get 8x - 7y =4x - 2y
so we get 4x = 5y
y = 4x/5
So volume of tank =
So time taken by A alone to fill the tank
= 144 minutes
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Question for CAT Previous Year Questions: Mensuration
Try yourself: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
[2021]
Explanation
Let us draw the rectangle.
Now, definitely, three sides should be fenced at Rs 100/ft, and one side should be fenced at Rs 200/ft.
In this question, we are going to assume that the L is greater than B.
Hence, the one side painted at Rs 200/ft should be B to minimise costs.
Hence, the total cost = 200B + 100B + 100L + 100L = 300B + 200L
Now, L x B = 60000
B = 60000/L
Hence, total cost = 300B + 200L = 18000000/L + 200L
To minimise this cost, we can use AM > = GM,
Hence, minimum cost = Rs 120000.
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Question for CAT Previous Year Questions: Mensuration
Try yourself:A park is shaped like a rhombus and has area 96 sq m. If 40 m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹125 per m, is
[2021]
Correct Answer : 3500
Explanation
We can say 40m is the perimeter of the park
so side of rhombus = 10
Now 1/2 x d1 x d2 = 96
So we get d1 x d2 = 192 (1) And as we know diagonals of a rhombus are perpendicular bisectors of each other : so, so we get Solving (1) and (2)
We get d1 = 12 and d2 = 16
Now the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹125 per m, is= (12 + 16)(125) = 3500
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Question for CAT Previous Year Questions: Mensuration
Try yourself:If a triangle ABC, ∠BCA=50o . D and E are points on AB and AC, respectively, such that AD=DE. If F is a point on BC such that BD=DF , then ∠FDE, in degrees, is equal to
[2021]
Explanation
We need to find out p.
Angle ADE = 180 - 2x
Angle BDF = 180 - 2y
Now, 180 - 2y + p + 180 - 2x = 180 [Straight line = 180 deg]
p = 2x + 2y - 180
Also, x + y + 50 = 180 [Sum of the angles of triangle = 180]
x + y = 130
p = 260 - 180 = 80 degrees.
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Question for CAT Previous Year Questions: Mensuration
Try yourself:Let ABCD be a parallelogram. The lengths of the side AD and the diogonal AC are 10cm and 20cm, respectively. If the angle ∠ADC is equal to 30o then the area of the parallelogram, in sq.cm is
[2021]
Explanation
Applying cosine rule in triangle ACD,
100 + X2 − 2 × 10 × Xcos30 = 400
Solving we get X =
Hence, area = 10Xsin 30 =
=
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Question for CAT Previous Year Questions: Mensuration
Try yourself: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
[2020]
Explanation
Let the length and the breadth of the rectangle be l and b respectively.
As the circle touches the two opposite sides, its diameter will be same as the breadth of the rectangle. Given, lb = 135 and
⇒
From this
∴ Required perimeter:
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Question for CAT Previous Year Questions: Mensuration
Try yourself:A circle is inscribed in a thombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is
[2020]
Explanation
Given the circle is inscribed in the rhombus of diagonals 12 and 16 . Let O be the point of intersection of the diagonals of the rhombus. Then, OE (radius) ⊥ DC.
Also
As area of ΔODC should be the same, we have, 1/2 x 6 x 8 = 6π/25
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Question for CAT Previous Year Questions: Mensuration
Try yourself: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
[2020]
Explanation
As the cone is cut at one-third height from the top (the vertex), the total volume is proportional to the cubes of the heights of the two parts.
Ratio of volumes two parts = = 1 : 27
Hence the bottom part will have volume of 27 - 1 i.e., 26 parts.
Given ( 26 - 1) i.e., 25 parts -225 cc.
Hence the required answer is 27 parts = 27 x 255 / 25 = 243 cc.
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Question for CAT Previous Year Questions: Mensuration
Try yourself:From an interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the three perpendiculars is s. Then the area of the triangle is