GMAT GEOMETRY-2
Rectangular Solids, Cylinders and Spheres, Co-ordinate Geometry
Rectangular Solids, Cylinders and Spheres:
1. A rectangular box has the dimensions 12 inches x 10 inches x 8 inches. What is the
largest possible volume of a right cylinder that is placed inside the box?
A. 150π
B. 200π
C. 192π
D. 1440π
E. 360π
Explanation:
Volume of the cylinder=πr^2h
If the cylinder is placed on 8*10 face then it's maximum radius is 8/2=4 and volume =
π∗4^2∗12=196π;
If the cylinder is placed on 8*12 face then it's maximum radius is 8/2=4 and volume =
π∗4^2∗10=160π;
If the cylinder is placed on 10*12 face then it's maximum radius is 10/2=5 and volume =
π∗5^2∗8=200π;
So, the maximum volume is for 200π.
So the Answer is B.
2. A cylindrical tank has a height of 10 feet and a base with a radius of 6 feet. If the
thickness of the tank’s sides is negligible, what is the volume, in cubic feet, of the largest
rectangular solid that could be placed inside the tank?
A. 60
B. 360
C. 240√3
D. 360√2
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E. 720
Explanation:
The height is fixed, so if we have to maximize the volume, we need to maximize the base
area.
The area of a rectangular base is length * breadth , the area is maximum when Length =
breadth... so the square is a rectangle with maximum area...
Radius of the circular base = 6 ft, diameter = 12 ft.. If we have to inscribe a square within
this, it diagonal will be equal to the diameter of the circle...
Therefore diagonal of the base square = 12 ft, side = 6*sqrt(2)
Therefore volume = [6*sqrt(2)]^2 * 10
= 36 * 2 * 10
= 720.
So the answer is E.
3). A cylinder is placed inside of a sphere in such a way that the axis of cylinder
passes through the centre of the sphere. The volume of the sphere is 36π. If the radius of
the cylinder is 2/3rd the radius of the sphere, and if thecylinder completely fills the sphere,
what is the height of the cylinder?(Volume of the sphere =(4/3) 𝜋𝑟^3
A.
B.
C.
D.
E.
2
√5
2√2
2√5
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Explanation:
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Volume of sphere is 36∏ = (4/3 ∏ r^3)
So , 36*3/4=r^3
27=r^3
(Radius of sphere) r = 3
Radius of cylinder = 2/3 *r = 2/3 * 3 = 2
Now in figure OC = 2, OA = 3
So, my distance
AC= √5
So, length of cylinder = 2√5
So the Answer is D
Co-ordinate Geometry:
1. In the rectangular coordinate system, are the points (r, s) and (p, q) equidistant from the
origin?
I. r + s = 1
II. p= 1 – r and q= 1 – s
Explanation:
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The Question is about Is r^2+s^2 = p^2 + q^2 ?
Statement I is insufficient:
r + s = 1, nothing about p and q so this is insufficient.
Statement II is insufficient:
p = 1 - r and q = 1 - s, this makes r^2 + s^2 = 1 - 2 p + p^2 + 1 - 2 q + q^2 = p^2 + q^2, if and
only if p + q = 1; So this is Insufficient
Taken together,
If r + s = 1, then p + q = 1 - r + 1 - s = 2 - (r + s) = 2 - 1 = 1, and then,r^2 + s^2 = p^2 + q^2.
Sufficient
So the answer is C.
2. The line K has a positive slope m and it passes through the point (-4, 6). The area of the
triangle formed by K, the x-axis and y-axis is 54 square units. What is the possible value for
m?
A. ¼
B. ¾
C. 5/4
D. 7/4
E. cannot be determined.
Explanation:
Assume the equation of the line to be y = mx + c. Since this line goes thru (-4,6), the line
would satisfy this point. Putting x = -4 and y= 6, we get:
y = mx + 4m +6.
When x = 0, y = 4m +6.
When y = 0, x = [-(4m +6)/m]
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Therefore, area of triangle formed by the given line, x axis and y axis will be:
1/2 * (4m +6) * [(4m +6)/m] = 54
Solving further, we will get:
4m2 -15m + 9 = 0
Thus m= 3 or m=3/4.
So the answer is B.
3. A line has a slope of 1/6 and intersects the x-axis at (-24, 0). Where does this line
intersect the y-axis?
A. 4
B. -6
C. 6
D. -4
E. 0
Explanation:
Since the intersection with the y-axis will occur at (0, y).
Then the slope of 1/6 means y/24 = 1/6; thus(x, y) = (0, 4).
So the answer is A.
4. In the xy-plane, if line k has negative slope and passes through the point (s;-2), is the xintercept of line k positive?
I. s = 0
II. The y-intercept of line k is negative.
Explanation:
Line K has negative slope and passes through (s,-2),
We know that Line which has negative slope, x intercept and intercept are both positive or
both negative.
So the question is
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Is x intercept is positive?
Statement I is sufficient:
s = 0.
So y intercept is negative. Since it is a negative slope x intercept also negative.
So the answer to the question is always NO.
So sufficient.
Statement II is sufficient:
Given y intercept is negative, so x intercept also negative because of slope is negative.
So sufficient.
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So answer is D.
5. Point (x; y) lies in which quadrant of the rectangular coordinate system?
I. x + y < 0
II. y = 7
Explanation:
Statement I is sufficient:
If x= -3 and y = -2 then (x,y) lies in the III quadrant.
If x= -3 and y =2 then (x,y) lies in the II quadrant.
So not sufficient.
Statement II is insufficient:
Nothing about x.
So not sufficient.
Together it is sufficient.
y=7 and x+y <0 means x has to be less than -7.
So (x,y) lies in the II quadrant.
So the answer is C.
6. In the xy-plane, point P has coordinates (a; b) and point Q has coordinates (b;a). What is
the distance between P and Q?
I. a - b = 12
II. a + b = 35
Explanation:
To Find the distance, formula is = sqrt((x2-x1)2 +(y2-y1)2).
So distance between (a,b) and (b,a) is = sqrt((b-a)2+ (a-b)2)
= sqrt(2(a-b)2).
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Statement I is sufficient:
a-b = 12.
So sufficient.
Statement II is insufficient:
a+b = 35.
Which doesn’t say anything about a-b.
So not sufficient.
So the answer is A.
7. In the rectangular coordinate system below, the line y = x is the perpendicular bisector
of segment AB (not shown), and the y-axis is the perpendicular bisector of segment BC (not
shown). If the coordinates of point A are (3; 2), what are the coordinates of point C?
A. (-3;-2)
B. (-2; 3)
C. (-3; 2)
D. (2;-3)
E. (2; 3)
Explanation:
Let’s make this into points given:
1). y = x is the perpendicular bisector of segment AB
2). x = 0 (y-axis) is the perpendicular bisector of segment BC
3). If the coordinates of point A are (3, 2)
Combining 1 and 3, we can deduce that point B is the reflection of point A in the Axis
Bisector (y = x)
=> Coordinates of Point B would be reverse of A and that is (2, 3)
Note : here reflection is in X=Y , so x -coordinate and y - coordinate would exchange their
places.
So we can say :
4). If the coordinates of point B are (2, 3)
Combining 3 and 4, we can deduce that point C is the reflection of point B in the Y Axis (x =
8
0) line
=> Coordinates of Point C would be reverse of B and that is (-2, 3)
Note : here reflection is in Y axis , so sign of x -coordinate would change while sign of y
coordinate would remain same.
So the answer is B.
8. In the xy plane, is the slope of line k less than 0?
I. The x intercept of k is 0.
II. The y intercept of k is 0.
Explanation:
Statement I is insufficient:
The x- intercept of k is zero.
So the line k can pass through the origin, where x and y intercept is zero.
Slope can be less than zero or more than zero.
So not sufficient.
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Statement II is insufficient:
The same reasoning, slope can be negative or positive. So not sufficient.
Even together, still slope can be negative or positive. So not sufficient.
So the answer is E.
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