CMAT Geometry Questions

It is given, cylindrical box of radius 5 cm contains 10 solid spherical balls, each of radius 5 cm.

So, height of the cylinder = $$10\times\ 2r=10\times\ 2\times\ 5=100$$ cm.

So, volume of cylinder = $$\pi\ r^2h=\pi\ \times\ 5^2\times\ 100=2500\pi\ $$ $$cm^3$$

Now, volume of one sphere = $$\dfrac{4}{3}\pi\ r^3=\dfrac{4}{3}\times\pi\ \times\ 5^3=\dfrac{500}{3}\pi\ $$ $$cm^3$$

So, volume of 10 spheres = $$\dfrac{500}{3}\times\ 10\ \pi\ =\dfrac{5000}{3}\pi\ $$ $$cm^3$$

So, volume of empty space in the box = $$2500\pi\ -\dfrac{5000}{3}\pi=\dfrac{2500}{3}\ \pi$$ $$cm^3$$

So, the correct answer is option A.

Statement I: The ratio of the circumference of two circles is 3:4

So, ratio of their radius = 3:4

Now, ratio of their area will be equal to square of the ratio of their radius = $$3^2:4^2=9:16$$

So, statement I is true.

Statement II: The circumference and the area of a circle are numerically equal

or, $$2\pi\ r=\pi\ r^2$$

or, $$r=2$$ units

So, diameter = $$2r=2\times\ 2=4$$ units

So, statement II is true.

So, both statement I and statement II are true.

So, correct answer is option A.

We know the volume of a cylinder of radius $$r$$ and height $$h$$ is $$\pi r^2h$$

Given D = 35 cm, so $$r=17.5$$ cm

Also, $$h=32$$ cm

So, Volume = $$\pi r^2h=\pi\ \left(17.5\right)^2\times32\ cm^3$$

We know that the volume of a cone of radius $$r$$ and height $$h$$ is $$\dfrac{1}{3}\pi r^2h$$

Given the diameter of cone = 4 cm, so the radius = 2 cm

Height = 7 cm

So, the volume of the cone is $$\dfrac{1}{3}\pi r^2h=\dfrac{1}{3}\pi\left(2\right)^2\times7$$

To find the maximum number of cones that can be used, divide the volume of the cylinder by the volume of a cone

Max cones = $$\dfrac{\pi\left(17.5\right)^2\times32}{\dfrac{1}{3}\pi\left(2\right)^2\times7}=1050$$

Hence, the answer is 1050.

Statement I: If two chords XY and ZT of a circle intersects internally at point P, then PX - PY = PZ - PT

There is no such theorem since the length of XY and ZT can be different. Let's take any random example:

Here, we can clearly say PX - PY < PZ - PT

Hence, this statement is not true. 

Statement II: If two chords XY and ZT of a circle intersect internally at point P, then PXZ and PTY are similar triangles.

This statement is correct. When two chords intersect inside a circle, triangles formed by the intersecting chords (i.e., triangles PXZ and PTY) are indeed similar by the AA (Angle-Angle) similarity criterion. This happens because:

∠XPZ = ∠TPY (vertically opposite angles)

Now, if I join XT, the triangles formed will be XZT and XYT. Hence, the triangle formed on the same base has third angles equal. With this, we can say, Angle XZT = Angle XYT

Therefore, by the Angle-Angle (AA) similarity criterion, triangle PXZ is similar to triangle PTY.

So, the correct answer is: Statement I is false but Statement II is true.

Statement I: The perimeter of a triangle is greater than the sum of its three medians.

This is true. In a triangle:

The medians connect a vertex to the midpoint of the opposite side, dividing the triangle into two smaller triangles of equal area.

The length of each median is always less than the sum of the two sides it connects (triangle inequality).

Therefore, the sum of the three medians is always less than the perimeter of the triangle.

Statement II: In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.

This is true as well. By the triangle inequality:

For the triangle formed by

AD, AB, and BD, we have

AB+BD>AD.

Similarly, for the triangle formed by

AD, AC, and CD, we have

AC+CD>AD.

Adding these inequalities gives

AB+AC+BD+CD>2AD.

Since

BD+CD=BC, we conclude

AB+AC+BC>2AD.

Statement I relies on the fact that medians are always shorter than the sum of the sides they span.

Statement II is a direct result of the triangle inequality applied to segments involving AD, proving its validity.

Therefore, the correct answer is option A.

The volume of the sphere = $$\dfrac{4}{3}\times\ \pi\ \times\ \left(3\right)^3=36\pi\ $$

Now, the water level will increase equal to the volume of the sphere. 

Let the water level increasing by h cm.

Therefore, $$36\pi\ =\pi\ \times\ \left(4\right)^2\times h$$

$$h=\dfrac{9}{4}$$

Area of circle = $$\pi r^2$$

Area of rectangle = $$L\times B$$

Area of Square = $$S^2$$

(A) Circle with a diameter of 14 cm = $$\pi 7^2$$ = $$49\pi\ $$ = 49*22/7 = 154$$cm^2$$

(B) Rectangle with length 12 cm and breadth 10 cm = 12*10 = 120$$cm^2$$

(C) Rectangle with length 14 cm and breadth 11 cm = 14*11 = 154$$cm^2$$

(D) Square with side 12 cm = 12*12 = 144$$cm^2$$

Thus, A and C have the same area.

(A) Parallel if the system has no solution: True. If the two lines are parallel, they do not intersect, implying the system of equations has no solution. This occurs when the slopes of the lines are equal but their intercepts differ.

(D) Intersecting if the system has only one solution: True. If the lines intersect at exactly one point, the system has a unique solution. This happens when the lines have different slopes.

(B) Coincident if the system has a finite number of solutions: Incorrect. If the lines are coincident, the system has infinitely many solutions, not a finite number.

(C) Intersecting if the system has two solutions: Incorrect. Two linear equations can intersect at only one point, meaning the system cannot have two solutions.

 The nature of the solutions depends on how the lines relate geometrically:

Parallel lines → No solution (A is correct).

Intersecting lines → Unique solution (D is correct).

Lines cannot intersect at more than one point, ruling out C, and coincident lines lead to infinitely many solutions, ruling out B.

Therefore, the correct answer is option C.