CAT Model Paper 11 Questions and Answers with Explanation Part 3
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16. If , n being a natural number, then which of the following is equal to ?
(A)
(B)
(C)
(D) None of these
Answer:
Solution:
Option: a
Approach 1: using the options
exists in all options.
We will calculate it first.
Approach 2: Assumption
Since the question is of the form ‘variable to variable’ . Assume any value for “n” and “x” .
17. In a certain examination paper, there are n questions. For , there are students who answered j or more questions wrongly. If the total number of wrong answer is 4095, then the value of n is
(A) 12
(B) 11
(C) 10
(D) 9
Answer: A
Solution:
Option a- Let us say there are only 3 questions. Thus there are students who have answered 1 or more questions wrongly, students who have answered 2 or more questions wrongly and students who must have answered all 3 wrongly. Thutal numbers of wrong answers .
In our question, the total number of wrong answers
Thus .
18. Two circles centered at points A and B, respectively, intersect at points E and F as shown. These circles intersect segment AB at points C and D. If AC = 1 and CD = DB = 2, determine EF.
19. What are the last two digits of ?
(A) 54
(B) 34
(C) 44
(D) 64
Answer: D
Solution:
Option d:
(since always ends in 24)
The last two digits of the above product
20. A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition.
(A)
(B)
(C)
(D)
Answer: A
Solution:
Option (a) the question is basically asking us to find out the minimum number and maximum number of lines that can be drawn using 1 2 points.
We can draw a minimum of 11 lines. We can visualize this as one central point and all lines connecting to this point. i.e.. one point has a degree of 11 and the other 11 have a degree of 1.
There can be a maximum of lines that can be drawn
Group Questions
23. Is ?
(1)
(2)
(A) a
(B) b
(C) c
(D) d
Answer: c
Solution:
Option (c)
The question is “Does n lie between ?”
(1) INSUFFICIENT: If we add n to both sides of the inequality, we can rewrite it as the following:
We cannot decide the answer based on this
If and then
However, if and , n is less than .
(2) INSUFFICIENT: can be rewritten as . However, this statement contains no information about n. Hence, answer cannot be determined based on this statement alone as well.
(1) and (2) SUFFICIENT: If we combine the two statements by plugging the value for x into the first statement, we get .
The only values for n that satisfy this inequality are those greater than 1.
The connect answer is (c) .
24.
Person A and Person B are both at point A (above) . Starting at the same time, Person A drives to point B while person B drives to point C. Who arrives at this destination first? (1) person A՚s average speed is that of person B՚s. (2) person B՚s average speed is 20 kilometers per hour greater than person A՚s.
(A) a
(B) b
(C) c
(D) d
Answer: a
Solution:
Option (a)
Since, , triangle ABC is a triangle. Such triangle have fixed side ratios as follows:
Thus, we can call person A՚s distance (AB) x, while person B՚s distance (AC) is Person B has a greater distance to travel.
let՚s first analysis statement (1) alone: Person A՚s average speed is that of person B՚s.
This indicates that person B is travelling 1.5 times faster than person A. If person A՚s rate is r, than person B՚s rate is 1.5 r. However, recall that person B also has a greater distance to travel.
To determine who will arrive first, we use the distance formula: Whoever has a shorter TIME will arrive first.
Since, person B is travelling for less time, he will arrive first.
Statement (1) alone is sufficient.
25. Last year, the five employees of company X took an average of 16 vacations days each. What was the average number of vacation days taken by the same employees this year?
(1) Three employees had a 50 % increase in their number of vacation days, and two employees had a 50 % decrease.
(2) Three employees had 10 more vacation days each, and two employees had 5 fewer vacation days each.
(A) a
(B) b
(C) c
(D) d
Answer: b
Solution:
The average number of vacation days taken this year can be calculated by dividing the total number of vacation days by the number of employees. Since we know the total number of employees, we can rephrase the question as: How many total vacation days did the employees of company X take this year?
(1) INSUFFICIENT: Since we don՚t know the specific details of how many vacation days each employee took the year before, we cannot determine the actual numbers that a 50 % increase or a 50 % decrease represent. For example, a 50 % increase for someone who look 40 vacation days last year is going to affect the overall average more than the same percentage increase for someone who took only 4 days of vacation last year.
(2) SUFFICIENT: If three employees took 10 more vacation days each, and two employees took 5 fewer vacation days each, then we can calculate how the number of vacation days taken this year differs from the number taken last year.
(10 more days/employees) (3 employees) (5 fewer days/employee) (2 employees)
26. What is the average (arithmetic mean) height of the n people of a certain group?
(1) The average height of the tallest people in the group is 6 feet 2.5 inches, and the average height of the rest of the people in the group is 5 feet 10 inches.
(2) The sum of the heights of the n people is 178 feet 9 inches.
(A) a
(B) b
(C) c
(D) d
Answer: A
Explanation:
Statement 1: sufficient
there՚s no need to simplify further, because ‘n’ is gone: you get one number. Therefore, this statement is sufficient.
Statement 2: Insufficient: Looking at statement (2) first, we see that it is not sufficient, because the average (arithmetic mean) of a group of numbers is defined as
With statement (2) , we only have the numerator of this expression (the of people in the group is unknown) , so we can՚t figure out the average.