CAT Question Paper with Solutions, Download PDF

Instructions

For the following questions answer them individually

Question 61

A test has 50 questions. A student scores 1 mark for a correct answer, -1/3 for a wrong answer, and -1/6 for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student cannot be less than

Video Solution

Solution

Let the number of questions answered correctly be x and the number of questions answered wrongly be y.

So, number of questions left unattempted = (50-x-y)

So, x - y/3 - (50-x-y)/6 = 32

=> 6x - 2y - 50 + x + y = 192 => 7x - y = 242 => y = 7x - 242

If x = 35, y = 3

If x = 36, y = 10

So, min. value of y is 3.

The number of wrongly answered questions cannot be less than 3.

Question 62

Twenty-seven persons attend a party. Which one of the following statements can never be true?

Video Solution

Solution

From the options a, c and d all can possibly occur. Hence option b. Besides, if all people have different number of acquaintances, then first person will have 26 acquaintance, second person will have 25 acquaintance, third person will have 24 and so on till 27 th person will have 0 acquaintance. 0 acquaintance is practically not possible.

Question 63

Let g(x) = max(5 - x, x + 2). The smallest possible value of g(x) is

Video Solution

Solution

Smallest possible value would be at 5-x = x+2 i.e. x= 1.5 as shown

Substituting we get smallest value as 3.5.

Question 64

The function f(x) = |x - 2| + |2.5 - x| + |3.6 - x|, where x is a real number, attains a minimum at

Video Solution

Solution

f(x) = |x - 2| + |2.5 - x| + |3.6 - x|

For x belonging to (-infinity to 2), f(x) = 2-x + 2.5-x + 3.6-x = 8.1-3x

This attains the minimum value at x=2. Value = 2.1

For x belonging to (2 to 2.5), f(x) = x-2 + 2.5-x + 3.6-x = 4.1-x

Attains the minimum value at x = 2.5. Value = 1.6

For x belonging to (2.5 to 3.6), f(x) = x-2 + x-2.5 + 3.6-x = x-0.9

Attains the minimum at x=2.5, value = 1.6

For x > 3.6, f(x) = x-2+x-2.5+x-3.6 = 3x - 8.1

Attains the minimum at x= 3.6, value = 2.7

So, min value of the function is 1.6 at x=2.5

Question 65

How many even integers n, where $$100 \leq n \leq 200$$ , are divisible neither by seven nor by nine?

Video Solution

Solution

Between 100 and 200 both included there are 51 even nos. There are 7 even nos which are divisible by 7 and 6 nos which are divisible by 9 and 1 no divisible by both. hence in total 51 - (7+6-1) = 39

There is one more method through which we can find the answer. Since we have to find even numbers, consider the numbers which are divisible by 14, 18 and 126 between 100 and 200. These are 7, 6 and 1 respectively.

Question 66

A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals

Video Solution

Solution

Since in all three cases the last digit is 1, the number should give remainder 1 when divided individually by 2,3,5 . So the no. may be 31 or 91 . Now 31 in base 2,3 and 5 give first digit as 1 in all the 3 cases while 91 gives exactly two out of the three cases the leading digit as 1. Hence option D.

Question 67

In a 4000 meter race around a circular stadium having a circumference of 1000 meters, the fastest runner and the slowest runner reach the same point at the end of the 5th minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race?

Video Solution

Solution

Let A , B and f,s be the distance traveled and speed of the fastest and the slowest person respectively. Also f=2s so in the given time A=2B. Since the ration of the speeds is 2:1, they will meet at 2-1 points = 1 pont.

Both meet each other for first time at starting point . let b travel distance equal to 1 circumference i.e. 1000m so A=2000m . Both meet after 5 min so speed of slowest is 1000/5=200m/min . So speed of the fastest is 400m/min. So time taken by A to complete race 4000/400 = 10 min

Question 68

Is $$a^{44} < b^{11}$$, given that a = 2 and b is an integer?

A. b is even

B. b is greater than 16

Video Solution

Solution

If b = 4, then the answer to the question is no. If b = 32, then the answer is yes. So, using statement A alone, we cannot answer the question.
Using statement B alone, we can conclude that $$a^{44} < b^{11}$$

Question 69

What are the unique values of b and c in the equation $$4x^2 + bx + c = 0$$ if one of the roots of the equation is (-1/2)?

A. The second root is 1/2.

B. The ratio of c and b is 1.

Video Solution

Solution

Using statement A, sum of roots = 0 and product of roots = 0, so b = c = 0.
Using statement B, sum of roots = x - 1/2 = -b/4 and product of roots = c/4 = b/4 = -x/2
So, we can calculate the values of b and c using either statement alone

Question 70

AB is a chord of a circle. AB = 5 cm. A tangent parallel to AB touches the minor arc AB at E. What is the radius of the circle?

A. AB is not a diameter of the circle.

B. The distance between AB and the tangent at E is 5 cm.

Video Solution

Solution

Let the radius be r. Using statement B alone, r, r-5 and 2.5 form a right-angled triangle. So, we can answer the question using statement B alone.

CAT Content

CAT 2003 Question Paper (Leaked)

A test has 50 questions. A student scores 1 mark for a correct answer, -1/3 for a wrong answer, and -1/6 for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student cannot be less than

Twenty-seven persons attend a party. Which one of the following statements can never be true?

Let g(x) = max(5 - x, x + 2). The smallest possible value of g(x) is

The function f(x) = |x - 2| + |2.5 - x| + |3.6 - x|, where x is a real number, attains a minimum at

How many even integers n, where $$100 \leq n \leq 200$$ , are divisible neither by seven nor by nine?

A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals

Is $$a^{44} < b^{11}$$, given that a = 2 and b is an integer?

A. b is even

B. b is greater than 16

What are the unique values of b and c in the equation $$4x^2 + bx + c = 0$$ if one of the roots of the equation is (-1/2)?

A. The second root is 1/2.

B. The ratio of c and b is 1.

AB is a chord of a circle. AB = 5 cm. A tangent parallel to AB touches the minor arc AB at E. What is the radius of the circle?

A. AB is not a diameter of the circle.

B. The distance between AB and the tangent at E is 5 cm.

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CAT Content

CAT 2003 Question Paper (Leaked)

A test has 50 questions. A student scores 1 mark for a correct answer, -1/3 for a wrong answer, and -1/6 for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student cannot be less than

Twenty-seven persons attend a party. Which one of the following statements can never be true?

Let g(x) = max(5 - x, x + 2). The smallest possible value of g(x) is

The function f(x) = |x - 2| + |2.5 - x| + |3.6 - x|, where x is a real number, attains a minimum at

How many even integers n, where $$100 \leq n \leq 200$$ , are divisible neither by seven nor by nine?

A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals

Is $$a^{44} < b^{11}$$, given that a = 2 and b is an integer? A. b is even B. b is greater than 16

What are the unique values of b and c in the equation $$4x^2 + bx + c = 0$$ if one of the roots of the equation is (-1/2)? A. The second root is 1/2. B. The ratio of c and b is 1.

AB is a chord of a circle. AB = 5 cm. A tangent parallel to AB touches the minor arc AB at E. What is the radius of the circle? A. AB is not a diameter of the circle. B. The distance between AB and the tangent at E is 5 cm.

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Also Read

Is $$a^{44} < b^{11}$$, given that a = 2 and b is an integer?

A. b is even

B. b is greater than 16

What are the unique values of b and c in the equation $$4x^2 + bx + c = 0$$ if one of the roots of the equation is (-1/2)?

A. The second root is 1/2.

B. The ratio of c and b is 1.

AB is a chord of a circle. AB = 5 cm. A tangent parallel to AB touches the minor arc AB at E. What is the radius of the circle?

A. AB is not a diameter of the circle.

B. The distance between AB and the tangent at E is 5 cm.