CAT Coordinate Geometry Question and Solution

Question

What is the area enclosed by the region defined by y = |x -1| + 2, the line x = 1; X-axis and Y-axis?

Explanation

Solution is available on the video given below. 

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 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?

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.