Solutions to CAT Geometry Data Sufficiency Questions

Have given below the solutions to the questions on geometry DS. The solutions are courtesy Vimal Gopinath (person incharge of 2IIM Bengaluru)

Qn 1: Is triangle ABC obtuse angled?
I) a^2 + b^2 > c^2 - Not enough. We don’t have info about b^2 or a^2
II) The circumcenter of the triangle does not lie inside the triangle - Not enough. The triangle could be right-angled as well.
Combination is also not enough, it is valid for both right angled and obtuse angled triangles.
D

Qn 2: Do the two circles with centers A and B and radii R and r intersect each other
I) AB > R - r – Not enough. May intersect, may be “parallel” or "disjoint"
II) AB > R + r – Sufficient. Circles cant intersect. Have to be separate.
A

Qn 3: Trapezium ABCD is such that AB is parallel to CD. Is this trapezium anisosceles trapezium?
I) Angle B and D are supplementary Sufficient. If two base angles are equal, then the trapezium has to be isosceles.
II) The quadrilateral is inscribed inside a circle. Sufficient. All trapeziums inscribed in circles have to be isosceles. (Think about the proof for this)
C

Qn 4: Circle C has center O, and a chord AB such that angle AOB = 80 degrees.Does point E lie inside the circle

I) Angle AEB > 50 degrees Insufficient.E could lie on the minor segment ADB or slightly outside or slightly inside the circle.
II) Angle AEB < 30 degrees. Sufficient. All the angles inside the circle will be in the range from 40 – 140 degrees. Anything less than 40 will have to be outside the circle.)
A

CAT Geometry Data Sufficiency questions

For the following questions,

Mark A) If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone
Mark B) If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together
Mark C) If the question can be answered with either statement alone
Mark D) If the question cannot be answered with the information provided

Qn 1: Is triangle ABC obtuse angled
I) a^2 + b^2 > c^2
II) The circumcenter of the triangle does not lie inside the triangle

Qn 2: Do the two circles with centers A and B and radii R and r intersect each other
I) AB > R - r
II) AB > R + r

Qn 3: Trapezium ABCD is such that AB is parallel to CD. Is this trapezium an isosceles trapezium?
I) Angle B and D are supplementary
II) The quadrilateral is inscribed inside a circle

Qn 4: Circle C has center O, and a chord AB such that angle AOB = 80 degrees. Does point E lie inside the circle
I) Angle AEB > 50 degrees
II) Angle AEB < 30 degrees

CAT Geometry Solutions

Have given below the solutions to the questions on basic geometry

1. Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?

This is just a counting question, with the caveat that sum of two sides should be greater than the third. Let us assume a < b < c

a = 1, Possible triangle 1, 7, 7
a = 2, possible triangle 2, 6, 7
a = 3, possible triangles 3, 6, 6 and 3, 5, 7
a = 4, possible triangles 4, 4, 7 and 4, 5, 6

Again, from comments,
a = 5, possible triangle is 5,5,5,

There are totally 7 triangles possible

2. Triangle ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

xz = 12

x,z can be 1, 12 or 2, 6 or 3, 4

Possible triangles
1-12-12
2-6-5; 2-6-6; 2-6-7
3-4-2; 3-4-3; 3-4-5; 3-4-6.

As pointed out in the comments section, I have missed the triangle 3-4-4.

There are totally 9 triangles.

3. Triangle has sides a^2, b^2 and c^2. Then the triangle with sides a, b, c has to be - a) Right angled b) Acute-angled c) Obtuse angled d) can be any of these three

Assuming a < b < c, we have a^2 + b^2 > c^2. This implies the triangle with sides a, b, c has to be acute-angled.

P.S: Big thanks to 'maniac' for pointing out the errors

CAT Geometry

Have given below a few questions on basic geometry

1. Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?

2. Triangle ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

3. Triangle has sides a^2, b^2 and c^2. Then the triangle with sides a, b, c has to be - a) Right angled b) Acute-angled c) Obtuse angled d) can be any of these three

Solutions to questions on Coordinate Geometry

Have given the solutions to questions on Cogeo

1. Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?

Any point on the line x/3 + y/5 =1 will have the shortest overall distance. However, we need to have integral coordinates. So, we need to find points with integral coordinates as close as possible to the line 5x + 3y =15.

Substitute x =1, we get y = 2 or 3
Substitute x = 2, we get y = 1 or 2


Sum of distances for ( 1, 2) = sqrt(8) + sqrt (10)
Sum of distances for ( 1, 3) = sqrt(13) + sqrt (5)
Sum of distances for ( 2, 1) = sqrt(2) + sqrt (20)
Sum of distances for ( 2, 2) = sqrt(5) + sqrt (13)

sqrt(5) + sqrt (13) is the shortest distance. Sum of abscissa + ordinate = 4

2. Region R is defined as the region in the first quadrant satisfying the condition 3x + 4y < 12. Given that a point P with coordinates (r, s) lies within the region R, what is the probability that r > 2?

Line 3x + 4y =12 cuts the x-axis at (4, 0) and y axis at (0, 3)

The region in the first quadrant satisfying the condition 3x + 4y < 12 forms aright triangle with sides 3, 4 and 5. Area of this triangle = 6 sq units.

The lines x = 2 and 3x + 4y = 12 intersect at (2, 1.5). So, the region r > 2, 3x + 4y < 12 also forms a right triangle. This right triangle has base sides 2, 1.5. Area of this triangle = 1.5

Probability of the point lying in said region = 1.5/6 = 1/4

3. Region Q is defined by the equation 2x + y < 40. How many points (r, s) exist such that r is a natural number and s is a multiple of r?

When r = 1, s can take 37 values [37/1]
When r = 2, s can take 17 values [35/2]
When r = 3, s can take 11 values [33/3]
When r = 4, s can take 7 values [31/4]
When r = 5, s can take 5 values [29/5]
When r = 6, s can take 4 values [27/6]
When r = 7, s can take 3 values [25/7]
When r = 8, s can take 2 values [23/8]
When r = 9, s can take 2 values [21/9]
When r = 10, s can take 1 values [19/10]
When r = 11, 12, 13 s can take 1 values one value each

Totally, there are 92 values possible

CAT Coordinate geometry

Few questions on coordinate geometry. These are slightly unconventional questions. CAT has stayed away from raw, brute-force cogeo questions for a while now. The questions that we see in CAT have not been algebraic. They have either been of the visual type or of the type where one has to do trial and error. So, the cogeo basics are important, but no point learning all kinds of fancy formula stuff.

1. Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?

2. Region R is defined as the region in the first quadrant satisfying the condition 3x + 4y < 12. Given that a point P with coordinates (r, s) lies within the region R, what is the probability that r > 2?

3. Region Q is defined by the equation 2x + y < 40. How many points (r, s) exist such that r is a natural number and s is a multiple of r?

Solutions to questions on permutations and combinations

Have given below solutions to the questions on permutations and combinations

1. a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)

Exactly one of ab, bc and ca is odd => Two are odd and one is even

abc is a multiple of 4 => the even number is a multiple of 4

The arithmetic mean of a and b is an integer => a and b are odd

and so is the arithmetic mean of a, b and c. => a+ b + c is a multiple of 3

c can be 4 or 8.
c = 4; a, b can be 3, 5 or 5, 9
c = 8; a, b can be 3, 7 or 7, 9

Four triplets are possible

2. A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?

Number should be a multiple of 3 and 4. So, the sum of the digits should be a multiple of 3. WE can either have all seven digits as 3, or have three 2's and four 3's, or six 2's and a 3. (The number of 2's should be a multiple of 3).

For the number to be a multiple of 4, the last 2 digits should be 32. Now, let us combine these two.

All seven 3's - No possibility
Three 2's and four 3's - The first 5 digits should have two 2's and three 3's in some order. No of possibilities = 5!/3!*2! = 10

Six 2's and one 3 - The first 5 digits should all be 2's. So, there is only one number 2222232

3. How many factors of 2^5 * 3^6 * 5^2 are perfect squares?

Any factor of this number should be of the form 2^a * 3^b * 5^c. For the factor to be a perfect square a,b,c have to be even. a can take values 0, 2, 4. b can take values 0,2, 4, 6 and c can take values 0,2. Total number of perfect squares = 3 * 4 * 2 = 24


4. How many factors of 2^4 * 5^3 * 7^4 are odd numbers?

Any factor of this number should be of the form 2^a * 3^b * 5^c. For the factor to be an odd number, a should be 0. b can take values 0,1, 2,3, and c can take values 0, 1, 2,3, 4. Total number of odd factors = 4 * 5 = 20

5. A number when divided by 18 leaves a remainder 7. The same number when divided by 12 leaves a remainder n. How many values can n take?

Number can be 7, 25, 43, 61, 79.
Remainders when divided by 12 are 7 and 1.

n can take exactly 2 values

Permutation and Combination (with a touch of Number Theory)

Have given below a few questions on Counting that also include a bit of Number Theory. These types of questions are commonly tested.

1. a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)

2. A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?

3. How many factors of 2^5 * 3^6 * 5^2 are perfect squares?

4. How many factors of 2^4 * 5^3 * 7^4 are odd numbers?

5. A number when divided by 18 leaves a remainder 7. The same number when divided by 12 leaves a remainder n. How many values can n take?