CAT 2012 QA Slot 1 Question Paper(Available) :Download Solutions with Answer Key

Consider a sequence \( S \) whose \( n \)th term \( T_n \) is defined as \( T_n = 1 + \frac{3}{n} \), where \( n = 1, 2, \ldots \). Find the product of all the consecutive terms of \( S \) starting from the 4th term to the 60th term.

• (A) 1980.55
• (B) 1985.55
• (C) 1990.55
• (D) 1975.55

Let \( P = \{2, 3, 4, \ldots, 100\} \) and \( Q = \{101, 102, 103, \ldots, 200\} \). How many elements of \( Q \) are there such that they do not have any element of \( P \) as a factor?

• (A) 20
• (B) 24
• (C) 21
• (D) 23

What is the sum of all the 2-digit numbers which leave a remainder of 6 when divided by 8?

• (A) 612
• (B) 594
• (C) 324
• (D) 872

Which of the terms \( 2^{1/3}, 3^{1/4}, 4^{1/6}, 6^{1/8}, 10^{1/12} \) is the largest?

• (A) \( 2^{1/3} \)
• (B) \( 3^{1/4} \)
• (C) \( 4^{1/6} \)
• (D) \( 10^{1/12} \)

If the roots of the equation \[ (a^2 + b^2)x^2 + 2(b^2 + c^2)x + (b^2 + c^2) = 0 \]
are real, which of the following must hold true?

• (A) \( c^2 \ge a^2 \)
• (B) \( c^4 \ge a^2(b^2 + c^2) \)
• (C) \( b^2 \ge a^2 \)
• (D) \( a^4 \le b^2(a^2 + c^2) \)

Find the remainder when \( 2^{1040} \) is divided by 131.

• (A) 1
• (B) 3
• (C) 5
• (D) 7

In the figure below, \( \angle MON = \angle MPO = \angle NQO = 90^\circ \), \( OQ \) is the bisector of \( \angle MON \), and \( QN = 10 \), \( OR = 40/\sqrt{7} \). Find \( OP \).
   (A) 4.8

• (B) 4.5
• (C) 4
• (D) 5
   

If \( (a^2 + b^2), (b^2 + c^2) \) and \( (a^2 + c^2) \) are in geometric progression, which of the following holds true?

• (A) \( b^2 - c^2 = \dfrac{a^2 - c^2}{b^2 + a^2} \)
• (B) \( b^2 - a^2 = \dfrac{a^2 - c^2}{b^2 + c^2} \)
• (C) \( b^2 - c^2 = \dfrac{a^2 - c^2}{b^2 + a^2} \)
• (D) \( a^2 - b^2 = \dfrac{b^2 + c^2}{b^2 + a^2} \)

\( p \) is a prime and \( m \) is a positive integer. How many solutions exist for the equation \[ p^5 - p = (m^2 + m + 6)(p - 1)? \]

• (A) 0
• (B) 1
• (C) 2
• (D) Infinite

A certain number written in a certain base is 144. Which of the following is always true?

• (A) I. Square root of the number written in the same base is 12
• (B) II. If base is increased by 2, the number becomes 100
• (C) Neither I nor II
• (D) Both I and II

A rectangle is drawn such that none of its sides has length greater than ‘\( a \)’. All lengths less than ‘\( a \)’ are equally likely. The chance that the rectangle has its diagonal greater than ‘\( a \)’ (in terms of %) is:

• (A) 29.3%
• (B) 21.5%
• (C) 66.66%
• (D) 33.33%

If \( x \) is a real number and \( \lfloor x \rfloor \) is the greatest integer \( \leq x \), then \[ 3\lfloor x \rfloor + 2 - \lfloor x \rfloor = 0 \]
Will the above equation have any real root?

• (A) Yes
• (B) No
• (C) Will have real roots for \( x < 0 \)
• (D) Will have real roots for \( x > 0 \)

If \[ x = \frac{x}{y+z}, \quad y = \frac{y}{z+x}, \quad z = \frac{z}{x+y} \]
then which of the following statements is/are true?

• (A) I and II
• (B) I and III
• (C) II and III
• (D) None of these

If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \[ x^2 - 10x + 15 = 0, \]
then find the quadratic equation whose roots are \( \left( \alpha + \frac{\alpha}{\beta} \right) \) and \( \left( \beta + \frac{\beta}{\alpha} \right) \).

• (A) \( 15x^2 + 71x + 210 = 0 \)
• (B) \( 5x^2 - 22x + 56 = 0 \)
• (C) \( 3x^2 - 44x + 78 = 0 \)
• (D) Cannot be determined 

A vessel has a milk solution in which milk and water are in the ratio 4:1. By addition of water to it, milk solution with milk and water in the ratio 4:3 was formed. On replacing 14 L of this solution with pure milk, the ratio of milk and water changed to 5:3. What is the volume of the water added?

• (A) 12 L
• (B) 32 L
• (C) 24 L
• (D) 14 L

A car \( A \) starts from a point \( P \) towards another point \( Q \). Another car \( B \) starts (also from \( P \)) 1 hour after the first car \( A \), and overtakes it after covering 30% of the distance \( PQ \). After that, the cars continue. On reaching \( Q \), car \( B \) reverses and meets car \( A \), after covering \( 2\frac{1}{3} \) of the distance \( QP \). Find the time taken by car \( B \) to cover the distance \( PQ \) (in hours).

• (A) 3
• (B) 4
• (C) 5
• (D) \( 3\frac{1}{3} \)

\( A, B, C \) can independently do a work in 15, 20, and 30 days respectively. They work together for some time after which \( C \) leaves. A total of ₹18000 is paid for the work and \( B \) gets ₹6000 more than \( C \). For how many days did \( A \) work?

• (A) 6
• (B) 4
• (C) 8
• (D) 2

In the figure, \( OABC \) is a parallelogram. The area of the parallelogram is 21. Coordinates are: \( O = (0,0) \), \( A = (7,0) \), and point \( C \) lies on line \( x = 3 \). Find coordinates of \( B \).
   (A) (3, 10)

• (B) (10, 3)
• (C) (10, 10)
• (D) (8, 3)

Find the complete set of values that satisfy the relations \[ |x - 3| < 2 \quad and \quad |x| - 2| < 3 \]

• (A) \( (-5, 5) \)
• (B) \( (-5, -1) \cup (1, 5) \)
• (C) \( (1, 5) \)
• (D) \( (-1, 1) \)

If \( ax^2 + bx + c = 0 \) and \( 2a, b, 2c \) are in arithmetic progression, then which of the following are the roots of the equation?

• (A) \( \frac{a}{c} \)
• (B) \( \frac{a - c}{b} \)
• (C) \( \frac{a}{2}, \frac{c}{2} \)
• (D) \( \frac{c - a}{b - a} \)

A solid sphere of radius 12 inches is melted and cast into a right circular cone whose base diameter is \( \sqrt{2} \) times its slant height. If the radius of the sphere and the cone are the same, how many such cones can be made and how much material is left out?

• (A) 4 and 1 cubic inch
• (B) 3 and 12 cubic inches
• (C) 4 and 0 cubic inch
• (D) 3 and 6 cubic inches

If \( \log_x(a - b) - \log_x(a + b) = \log_x\left(\frac{b}{a}\right) \), find \( \frac{a^2 + b^2}{b^2 + a^2} \)

• (A) \( \frac{b^2}{a^2} \)
• (B) 2
• (C) 3
• (D) 6

Letters of the word “ATTRACT” are written on cards and kept on a table. Manish lifts three cards at a time, writes all possible combinations of the three letters on a piece of paper, and then replaces the cards. He is to strike out all the words which look the same when seen in a mirror. How many words is he left with?

• (A) 40
• (B) 20
• (C) 30
• (D) None of these

A set \( S = \{1, 2, 3, \ldots, n\} \) is partitioned into \( n \) disjoint subsets \( A_1, A_2, \ldots, A_n \), each containing four elements. It is given that in each subset, one element is the arithmetic mean of the other three. Which of the following statements is true?

• (A) \( n \neq 1 \) and \( n \neq 2 \)
• (B) \( n \neq 1 \) but can be equal to 2
• (C) \( n \neq 2 \) but can be equal to 1
• (D) It is possible to satisfy for \( n = 1 \) as well as for \( n = 2 \)

When asked for his taxi number, the driver replied,
“If you divide the number of my taxi by 2, 3, 4, 5, 6 each time you will find a remainder of one. But if you divide it by 11, the remainder is zero.”
What is the taxi number?

• (A) 121
• (B) 1001
• (C) 1881
• (D) 781

A student is asked to form numbers between 3000 and 9000 with digits 2, 3, 5, 7 and 9. If no digit is to be repeated, in how many ways can the student do so?

• (A) 24
• (B) 120
• (C) 60
• (D) 72
  (C) 60

The side of an equilateral triangle is 10 cm long. By drawing parallels to all its sides, the distance between any two parallel lines being the same, the triangle is divided into smaller equilateral triangles, each of which has sides of length 1 cm. How many such small triangles are formed?

• (A) 60
• (B) 90
• (C) 120
• (D) None of these