CAT Data Sufficiency Solutions

Have given below solutions to the questions on data sufficiency

1. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(1,6)?

Stmt 1. m = 3. Not sufficient.
Stmt 2. n = 2.5 Not sufficient.

Either statement alone is not sufficient. Both put together, we can answer the questions. Ans B

2. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(5,6)?

Stmt 1. m = 5. Not sufficient. It tells us that all points lie on the line x = 5, but this is not enough
Stmt 2. n = 3 Not sufficient.

Both put together, this is enough.

In the above question, would n = 1 have been sufficient? Think about that.

3. A survey of 100 people tried to find the number of people who can write with both their left and right hands. What is the maximum number of people who could write left-handed and right-handed?

Stmt 1. 50 people can write only with their left hand. 40 people can write only with their right hand. Sufficient: Maximum of 10 people could write left-handed and right-handed.
Stmt 2. 50 people can write with their left hand. 40 people can write with their right hand. Sufficient: Maximum of 40 people could write left-handed and right-handed.

Answer Choice C

4. What is the slope of a line?

Stmt 1: The line makes 135 degrees with the negative direction of x - axis. Sufficient: The line makes 45 degrees with positive x axis. This should be enough.
Stmt 2: The line makes an isosceles right triangle with the coodinate axes and the product of the intercepts is negative. Sufficient: Either both intercepts are positive and equal or negative and equal. Slope = -1


Answer Choice C

CAT Data Sufficiency,

Following questions are courtesy Naveenan Ramachandran, head of 2IIM Mumbai.

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

1. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(1,6)?

Stmt 1. m = 3
Stmt 2. n = 2.5

2. A point P is identified as P(m,n). What is the ratio AP:BP given that the points A and B are identified as A(5,-4) and B(5,6)?

Stmt 1. m = 5
Stmt 2. n = 3

3. A survey of 100 people tried to find the number of people who can write with both their left and right hands. What is the maximum number of people who could write left-handed and right-handed?

Stmt 1. 50 people can write only with their left hand. 40 people can write only with their right hand.
Stmt 2. 50 people can write with their left hand. 40 people can write with their right hand.

4. What is the slope of a line?

Stmt 1: The line makes 135 degrees with the negative direction of x - axis.
Stmt 2: The line makes an isosceles right triangle with the coodinate axes and the product of the intercepts is negative.

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 Data Sufficiency : Mensuration

Here is another Data sufficiency question. The topic is Mensuration.

Question
A right circular cone has height H and radius R. A small cone is cut off at the top by a plane parallel to the base. At what height above the base the section has been made?

A. H = 20 cm
B. Volume of small cone: volume of large cone :: 1:15

Correct answer is Choice C. Both statements are required to answer the question

Explanatory Answer

From statement A, we know that the height of the initial cone is 20cm. However, nothing is said about the small cone. Hence, we cannot answer the question using statement A. So, we can eliminate choices (1) and (4).

We are down to choices (2), (3) or (5).

From Statement B, we know that the ratio of the volume of the small cone to that of the large cone is 1 : 15.

i.e. (1/3)*pi*r^2*h : (1/3)*pi*R^2*H is 1 : 15 (r is the base radius of the smaller cone and h is the height of the smaller cone)

or r^2 * h : R^2 * H is 1 : 15

From this information, we will not be able to find the answer to h. Hence, we can eliminate choice (2).

Combining the information in the two statements:

When a section is made the two cones are similar triangles. so h/H = r/R
R = rH/h

We know H = 20

h = (1/15) ( rH/h) ( rH/h)*H /r*r)

i.e., h^3 = (1/15)H^3. Substituting H = 20, we can get the value for h.
Choice (3) is therefore, the correct answer.

Data Sufficiency : Geometry

Data Sufficiency questions appear either in the Quantitative reasoning section or in the Data Interpretation and LR section of the CAT test.

Usually 5 to 10 questions appear in DS depending upon how many total questions are there in the section.

A data sufficiency question comes with the following set of instructions.

Directions to answer the DS question

1. Statement (A) ALONE is sufficient, but statement (B) alone is not sufficient to answer the question asked.
2. Statement (B) ALONE is sufficient, but statement (A) alone is not sufficient to answer the question asked.
3. BOTH statements (A) and (B) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
4. EACH statement ALONE is sufficient to answer the question asked.
5. Statements (A) and (B) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Here is a DS question from Geometry.

Question

Does the circle A with radius ‘a’ touch circle B with radius ‘b’ internally?

A. The two circles touch each other at only 1 point
B. The distance between centers of A and B is a-b

Correct Answer
Choice (2). Statement B alone is sufficient while statement A is not sufficient.

Explanatory Answer

Statement A: The two circles touch each other at only 1 point. If two circles touch each other at only one point, they could touch each other either externally or internally. Hence, this statement is not sufficient to answer the question.

Statement B: The distance between the centers of A and B is a-b. If two circles touch each other internally, then the distance between their centers is the difference between their radii.

As statement B clearly states that the distance between the centers is a - b, the two circles touch each other internally.

Choice (2) is the correct answer.

Some basics about Circles that Touch and Intersect
(i) If two circles touch each other externally, then the distance between their centers is a+ b. There will just be one point at which the circumference of the two circles touch each other.

(ii) If two touch each other internally, then the distance between their centers is a - b, where a > b. There will be only one point at which the circumference of the two circles touch each other.

(iii) If two circles intersect, then there will be two points at which the circumference of the two circles meet. We will not be able to determine the distance between their centers using the information about the radii of the two circles.

Geometry, Coordinate geometry and mensuration are favourite areas in which questions in DS typically appear. Practice with adequate number of questions from your CAT Preparation Material to get familiar with these questions.