CAT 2018 QA Slot 2 Question Paper (Available) :Download Solutions with Answer Key PDF

Points A, P, Q and B lie on the same line such that P, Q and B are, respectively, 100 km, 200 km and 300 km away from A. Cars 1 and 2 leave A at the same time and move towards B. Simultaneously, Car 3 leaves B and moves towards A. Car 3 meets Car 1 at Q and Car 2 at P. If each car is moving in uniform speed, then the ratio of the speed of Car 2 to that of Car 1 is:

• (A) 1 : 4
• (B) 2 : 9
• (C) 1 : 2
• (D) 2 : 7

Let \( a_1, a_2, \ldots, a_{52} \) be positive integers such that \( a_1 < a_2 < \ldots < a_{52} \). Suppose, their arithmetic mean is one less than the arithmetic mean of \( a_2, a_3, \ldots, a_{52} \). If \( a_{52} = 100 \), then the largest possible value of \( a_1 \) is:

• (A) 48
• (B) 20
• (C) 45
• (D) 23

There are two drums, each containing a mixture of paints A and B.
In drum 1, A and B are in the ratio 18 : 7.
The mixtures from drums 1 and 2 are mixed in the ratio 3 : 4.
In this final mixture, A and B are in the ratio 13 : 7.
In drum 2, then A and B were in the ratio:

• (A) 251 : 163
• (B) 239 : 161
• (C) 220 : 149
• (D) 229 : 141

On triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively.
If the lengths of AB = 30 cm, AC = 25 cm, and CP = 20 cm, then the length of BQ (in cm) is: (TITA)

Let \( t_1, t_2, \ldots \) be real numbers such that \( t_1 + t_2 + \ldots + t_n = 2n^2 + 9n + 13 \), for every positive integer \( n \geq 2 \).

If \( t_k = 103 \), then \( k \) equals: (TITA)

From a rectangle \(ABCD\) of area \(768 cm^2\), a semicircular part with diameter \(AB\) and area \(72\pi cm^2\) is removed.

The perimeter of the leftover portion, in cm, is:

• (A) \(88 + 12\pi\)
• (B) \(80 + 16\pi\)
• (C) \(86 + 8\pi\)
• (D) \(82 + 24\pi\)

If \(N\) and \(x\) are positive integers such that \(NN = 2160\) and \(N^2 + 2N\) is an integral multiple of \(2x\), then the largest possible value of \(x\) is: (TITA)

A chord of length 5 cm subtends an angle of \(60^\circ\) at the centre of a circle.

The length (in cm) of a chord that subtends an angle of \(120^\circ\) at the centre of the same circle is:

• (A) \(2\pi\)
• (B) \(5\sqrt{3}\)
• (C) \(6\sqrt{2}\)
• (D) 8

If \( p^3 = q^4 = r^5 = s^6 \), then the value of \( \log_s(pqr) \) is equal to:

• (A) \( \frac{24}{5} \)
• (B) 1
• (C) \( \frac{47}{10} \)
• (D) \( \frac{16}{5} \)

In a tournament, there are 43 junior and 51 senior participants.

Each pair of juniors plays one match.

Each pair of seniors plays one match.

No junior-senior matches occur.

153 girl vs girl matches (junior)

276 boy vs boy matches (senior)

How many matches does a boy play against a girl? (TITA)

A 20% ethanol solution is mixed with another ethanol solution, say \( S \), of unknown concentration in the proportion 1:3 by volume.

This mixture is then mixed with an equal volume of 20% ethanol solution.

If the resultant mixture is a 31.25% ethanol solution, then the unknown concentration of \( S \) is:

• (A) 50%
• (B) 55%
• (C) 48%
• (D) 52%

The area of a rectangle and the square of its perimeter are in the ratio 1 : 25.

Then the lengths of the shorter and longer sides of the rectangle are in the ratio:

• (A) 3 : 8
• (B) 2 : 9
• (C) 1 : 4
• (D) 1 : 3

The smallest integer \( n \) for which \( 4n > 1719 \) holds is closest to:

The smallest integer \( n \) such that \( n^3 - 11n^2 + 32n - 28 > 0 \) is (TITA)

A parallelogram \(ABCD\) has area 48 sq cm. If length of \(CD = 8\) cm and that of \(AD = s\) cm,

which one of the following is necessarily true?

• (A) \( s \geq 6 \)
• (B) \( s \neq 6 \)
• (C) \( 5 \leq s \leq 7 \)
• (D) \( s \leq 6 \)

Find the value of the sum:
\[ 7 \times 11 + 11 \times 15 + 15 \times 19 + \ldots + 95 \times 99 \]

• (A) 80707
• (B) 80751
• (C) 80730
• (D) 80773

On a long stretch of east-west road, A and B are two points such that B is 350 km west of A.

One car starts from A and another from B at the same time.

- If they move towards each other, they meet in 1 hour.

- If both move towards the east, they meet in 7 hours.

Then the difference between their speeds (in km/hr) is: (TITA)

If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

• (A) 64
• (B) -32
• (C) 16
• (D) 48

A jar contains a mixture of 175 ml water and 700 ml alcohol.

Gopal removes 10% of the mixture and replaces it with water.

This process is repeated once more.

What is the final percentage of water in the mixture?

• (A) 25.4
• (B) 20.5
• (C) 30.3
• (D) 35.2

Points A and B are 150 km apart.

Cars 1 and 2 travel from A to B, but car 2 starts from A when car 1 is already 20 km away from A.

Each car travels at a speed of 100 kmph for the first 50 km, at 50 kmph for the next 50 km, and at 25 kmph for the last 50 km.

The distance, in km, between car 2 and B when car 1 reaches B is: (TITA)

A tank is emptied every day at a fixed time.

- Monday: A fills alone, completes at 8 pm

- Tuesday: B fills alone, completes at 6 pm

- Wednesday: A fills till 5 pm, B fills 5–7 pm


Find the time tank will be full on Thursday if both A and B work simultaneously all day.

• (A) 4:12 PM
• (B) 4:24 PM
• (C) 4:48 PM
• (D) 4:36 PM

Ramesh and Ganesh can together complete a work in 16 days.

After 7 days of working together, Ramesh got sick and his efficiency dropped by 30%.

The total work was completed in 17 days.

If Ganesh had worked alone after Ramesh got sick, how many days would he have taken to complete the remaining work?

• (A) 12
• (B) 14.5
• (C) 13.5
• (D) 11

If \( a \) and \( b \) are integers such that:
\[ 2x^2 - ax + 2 \geq 0 \quad and \quad x^2 - bx + 8 \geq 0 \quad for all real numbers x, \]
then the largest possible value of \( 2a - 6b \) is: (TITA)

The scores of Amal and Bimal in an examination are in the ratio 11:14. After an appeal, their scores increase by the same amount and their new scores are in the ratio 47:56. The ratio of Bimal’s new score to that of his original score is:

• (A) 3:2
• (B) 4:3
• (C) 5:4
• (D) 8:5

A triangle ABC has area 32 sq units and its side BC, of length 8 units, lies on the line \(x = 4\). Then the shortest possible distance between point A and the origin (0, 0) is:

• (A) \(4\sqrt{2}\) units
• (B) \(2\sqrt{2}\) units
• (C) 4 units
• (D) 8 units

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

• (A) 5
• (B) 8
• (C) 7
• (D) 6

A water tank has inlets of two types A and B. All inlets of type A, when open, bring in water at the same rate. All inlets of type B, when open, bring in water at the same rate.

The empty tank is completely filled in 30 minutes if 10 inlets of type A and 45 inlets of type B are open, and in 1 hour if 8 inlets of type A and 18 inlets of type B are open.

In how many minutes will the empty tank get completely filled if 7 inlets of type A and 27 inlets of type B are open? (TITA)

Gopal borrows Rs. \(X\) from Ankit at 8% annual interest. He then adds Rs. \(Y\) of his own money and lends Rs. \(X + Y\) to Ishan at 10% annual interest. At the end of the year, after returning Ankit’s dues, the net interest retained by Gopal is the same as that accrued to Ankit.

On the other hand, had Gopal lent Rs. \(X + 2Y\) to Ishan at 10%, then the net interest retained by him would have increased by Rs. 150.

If all interests are compounded annually, then find the value of \(X + Y\). \quad (TITA)

The arithmetic mean of \(x, y,\) and \(z\) is 80, and that of \(x, y, z, u,\) and \(v\) is 75, where:
\[ u = \frac{x + y}{2}, \quad v = \frac{y + z}{2} \]
Given that \(x \geq z\), find the minimum possible value of \(x\). (TITA)

Let \(f(x) = \max\{5x,\ 52 - 2x^2\}\), where \(x\) is any positive real number.

Then the minimum possible value of \(f(x)\) is: \quad (TITA)

For two sets A and B, let \( A \triangle B \) denote the set of elements which belong to A or B but not both.

If \(P = \{1, 2, 3, 4\}\), \(Q = \{2, 3, 5, 6\}\), \(R = \{1, 3, 7, 8, 9\}\), \(S = \{2, 4, 9, 10\}\), then the number of elements in \((P \triangle Q) \triangle (R \triangle S)\) is:

• (A) 7
• (B) 8
• (C) 9
• (D) 6

If \( A = \{6 \cdot 2^n - 35n - 1 : n = 1, 2, 3, \ldots\} \) and \( B = \{35(n - 1) : n = 1, 2, 3, \ldots\} \), then which of the following is true?

• (A) Neither every member of A is in B nor every member of B is in A
• (B) Every member of A is in B and at least one member of B is not in A
• (C) Every member of B is in A
• (D) At least one member of A is not in B

The strength of a salt solution is \(p%\) if 100 ml of the solution contains \(p\) grams of salt.

If three salt solutions A, B, C are mixed in the proportion \(1 : 2 : 3\), then the resulting solution has strength \(20%\).

If instead the proportion is \(3 : 2 : 1\), then the resulting solution has strength \(30%\).

A fourth solution, D, is produced by mixing B and C in the ratio \(2 : 7\).

Then the ratio of the strength of D to that of A is:

• (A) 3:10
• (B) 1:3
• (C) 2:5
• (D) 1:4