CAT 2003 QA - Leaked questions and solutions

Previous year paper questions for CAT 2003 QA - Leaked

1. CAT 2003 QA - Leaked | Algebra - Simple Equations

How much does R owe to S in Thai Baht?

A.
428
B.
416
C.
334
D.
324

2. CAT 2003 QA - Leaked | Algebra - Simple Equations

How much does M owe to S in US Dollars?

A.
3
B.
4
C.
5
D.
6

3. CAT 2003 QA - Leaked | Venn Diagram

What is the number of projects in which Gyani alone is involved?

A.
Uniquely equal to zero
B.
Uniquely equal to 1
C.
Uniquely equal to 4
D.
Cannot be determined uniquely

4. CAT 2003 QA - Leaked | Venn Diagram

What is the number of projects in which Medha alone is involved?

A.
Uniquely equal to zero
B.
Uniquely equal to 1
C.
Uniquely equal to 4
D.
Cannot be determined uniquely

5. CAT 2003 QA - Leaked | Algebra - Functions & Graphs

The number of non-negative real roots of 2^x – x – 1 = 0 equals

A.
0
B.
1
C.
2
D.
3

6. CAT 2003 QA - Leaked | Algebra - Functions & Graphs

When the curves, y = log₁₀ x and y = x⁻¹are drawn in the X-Y plane, how many times do they intersect for values of x ≥ 1?

A.
Never
B.
Once
C.
Twice
D.
More than twice

7. CAT 2003 QA - Leaked | Geometry - Mensuration

Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be

A.
85.5
B.
92.5
C.
90.5
D.
87.5

8. CAT 2003 QA - Leaked | Algebra - Simple Equations

Which one of the following conditions must p, q and r satisfy so that the following system of linear simultaneous equations has at least one solution, such that p + q + r ≠ 0?

x + 2y − 3z = p

2x + 6y − 11z = q

x − 2y + 7z = r

A.
5p − 2q − r = 0
B.
5p + 2q + r = 0
C.
5p + 2q − r = 0
D.
5p − 2q + r = 0

Answer: Option A

Explanation :

On substituting the values of p, q and r in the options we see that the values of p, q and r satisfy only the equation 5p − 2q − r = 0.

Hence, option (a).

Alternatively,

In order to understand this solution we need to have an understanding of a few concepts of higher mathematics like “Rank of a matrix”.

A number p is said to be the rank of a matrix A if

(i) A possesses at least one p-rowed determinant whose value is not zero

(ii) A does not possess any non zero (p + 1) rowed determinant.

In other words, the number of non-zero rows in the row-reduced form of a matrix is called the rank of a matrix.

For example,

Consider the matrix,

$A = (\begin{matrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 3 & 4 & 5 \end{matrix})$

Matrix A has only one 3 rowed determinant, namely

$|\begin{matrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 3 & 4 & 5 \end{matrix}|$

The value of this determinant is zero. Hence, the rank of A will be less than 3.

Now we will try to find a 2 rowed non zero determinant.

$|\begin{matrix} 4 & 5 \\ 1 & 2 \end{matrix}|$ = 42 - 5.1 = 3 ≠ 0

The fact that every 3 rowed determinant of A is zero and there is at least one 2 rowed determinant of A which is not zero, is generally expressed by saying that the rank of A is 2.

Consider the system of equations,

x + 2y − 3z = p

2x + 6y − 11z = q

x − 2y + 7z = r

The coefficient matrix (whose elements are coefficients of the three unknowns x, y and z) of the system is

$(\begin{matrix} 1 & 2 & - 3 \\ 2 & 6 & - 11 \\ 1 & - 2 & 7 \end{matrix})$

And the Augmented matrix is

$(\begin{matrix} 1 & 2 & - 3 & p \\ 2 & 6 & - 11 & à \\ 1 & - 2 & 7 & r \end{matrix})$

For the system of equations to possess solutions it is necessary that the rank of the augmented matrix and the rank of the coefficient matrix should be equal.

Let us first find the rank of the coefficient matrix.

We will first find the value of the only 3 rowed determinant of A,

$|\begin{matrix} 1 & 2 & - 3 \\ 2 & 6 & - 11 \\ 1 & - 2 & 7 \end{matrix}|$ = 1(6 × 7 - 11 × (-2)) - 2(2 × 7 - (-11)1) - 3(2 × (-2) - 6 × 1) = 0

Hence, the rank of the coefficient matrix will be less than 3.

The following is a two-rowed non-zero determinant in the coefficient matrix

$|\begin{matrix} 1 & 2 \\ 2 & 6 \end{matrix}|$ = 6 - 4 = 2 ≠ 0

Hence, the rank of the coefficient matrix is 2.

For the system of equations to possess solutions the rank of the augmented matrix should also be 2.

We will first try to simplify the augmented matrix using row operations.

$(\begin{matrix} 1 & 2 & - 3 & p \\ 2 & 6 & - 11 & q \\ 1 & - 2 & 7 & r \end{matrix})$

By the row operation R₃ → R₃ - R₁

we reduce the augmented matrix to the equivalent matrix,

$~ (\begin{matrix} 1 & 2 & 3 & p \\ 2 & 6 & - 11 & q \\ 0 & - 4 & 10 & r - p \end{matrix})$

$~ (\begin{matrix} 1 & 2 & - 3 & p \\ 0 & 2 & - 5 & q - 2 p \\ 0 & - 4 & 10 & r - p \end{matrix})$ (By the row operation R₂ → R₂ - 2R₁)

$~ (\begin{matrix} 1 & 2 & - 3 & p \\ 0 & 2 & - 5 & q - 2 p \\ 0 & 0 & 0 & - 5 p + 2 q + r \end{matrix})$ (By the row operation R₃ → R₃ - 2R₂)

For the augmented matrix to have rank 2 every 3 rowed determinant should be zero.

Hence,

5p − 2q − r = 0

Hence, option (a).

9. CAT 2003 QA - Leaked | Arithmetic - Profit & Loss

A leather factory produces two kinds of bags, standard and deluxe. The profit margin is Rs. 20 on a standard bag and Rs. 30 on a deluxe bag. Every bag must be processed on machine A and on machine B. The processing times per bag on the two machines are as follows:

The total time available on machine A is 700 hours and on machine B is 1250 hours. Among the following production plans, which one meets the machine availability constraints and maximizes the profit?

A.
Standard 75 bags, Deluxe 80 bags
B.
Standard 100 bags, Deluxe 60 bags
C.
Standard 50 bags, Deluxe 100 bags
D.
Standard 60 bags, Deluxe 90 bags

10. CAT 2003 QA - Leaked | Algebra - Progressions

The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the same progression. Then which element of the series should necessarily be equal to zero?

A.
1st
B.
9th
C.
12th
D.
None of the above

11. CAT 2003 QA - Leaked | Geometry - Circles

The ratio of the sum of the lengths of all chord roads to the length of the outer ring road is

A.
$\sqrt{5} : 2$
B.
$\sqrt{5} : 2 π$
C.
$\sqrt{5} : π$
D.
None of the above

Answer: Option C

Explanation :

Consider the diagram below as per the conditions given in the question,

Speed on outer ring road = 30π km/hr

Speed on inner ring road = 20π km/hr

And speed on chord roads = $15 \sqrt{5}$ km/hr

Let R and r be radii of the Outer Ring road and the Inner Ring road respectively.

∵ Outer Ring road is twice as long as Inner Ring road, we have,

∴ 2πR = 2 × (2πr)

∴ R = 2r

∴ Length of inner ring road = 2πr km and outer ring road = 4πr km

In the figure above, in ∆OW2N1,

W₂N₁ = $\sqrt{{(2 r)}^{2} + r^{2}} = \sqrt{5} r$ km

∴ Length of the chord road = $\sqrt{5} r$ km

∴ Ratio of the length of all chord roads to length of the outer ring road = $\frac{4 \sqrt{5} r}{4 π r} = \frac{\sqrt{5}}{π}$

Hence, option (c).

12. CAT 2003 QA - Leaked | Geometry - Circles

Amit wants to reach N₂ from S₁. It would take him 90 minutes if he goes on minor arc S₁ - E₁ on OR, and then on the chord road E₁ - N₂. What is the radius of the outer ring road in km?

A.
60
B.
40
C.
30
D.
20

Answer: Option C

Explanation :

Total distance to be travelled =

S₁–E₁ arc of the Outer Ring road + Chord E₁– N₂

Given the speeds,

Total time required to travel from S₁ to N₁ = $\frac{π r}{30 π} + \frac{\sqrt{5} r}{15 \sqrt{5}} = \frac{90}{60}$

∴ r = 15 km

∴ Radius of the Outer Ring road R = 2r = 30 km

Hence, option (c).

13. CAT 2003 QA - Leaked | Geometry - Circles

Amit wants to reach E2 from N1 using first the chord N1 – W2 and then the inner ring road. What will be his travel time in minutes on the basis of information given in the above question?

A.
60
B.
45
C.
90
D.
105

Answer: Option D

Explanation :

Total distance to be travelled, chord N₁– W₂ + W₂– E₂ arc of the Inner Ring road

Given the speeds,

Total time required to travel from N₁ to E₂ = $\frac{\sqrt{5} r}{15 \sqrt{5}} + \frac{π r}{20 π}$

$= \frac{r}{15} + \frac{r}{20}$

$= \frac{35 r}{300} = \frac{35 \times 15}{300} = \frac{7}{4} hrs$

= 105 minutes

Hence, option (d).

14. CAT 2003 QA - Leaked | Algebra - Simple Equations

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

A.
6
B.
12
C.
3
D.
9

15. CAT 2003 QA - Leaked | Modern Math - Permutation & Combination

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

A.
There is a person in the party who is acquainted with all the twenty-six others.
B.
Each person in the party has a different number of acquaintances.
C.
There is a person in the party who has an odd number of acquaintances.
D.
In the party, there is no set of three mutual acquaintances.

16. CAT 2003 QA - Leaked | Algebra - Functions & Graphs

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

A.
4.0
B.
4.5
C.
1.5
D.
None of the above

17. CAT 2003 QA - Leaked | Algebra - Functions & Graphs

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

A.
x = 2.3
B.
x = 2.5
C.
x = 2.7
D.
None of the above

18. CAT 2003 QA - Leaked | Algebra - Number Theory

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

A.
40
B.
37
C.
39
D.
38

19. CAT 2003 QA - Leaked | Algebra - Number System

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

A.
31
B.
63
C.
75
D.
91

20. CAT 2003 QA - Leaked | Arithmetic - Time, Speed & Distance

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?

A.
20 min
B.
15 min
C.
10 min
D.
5 min

21. CAT 2003 QA - Leaked | Algebra - Number Theory | Data Sufficiency

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Is a⁴⁴ < b¹¹, given that a = 2 and b is an integer?

b is even
b is greater than 16

A.
1
B.
2
C.
3
D.
4

22. CAT 2003 QA - Leaked | Algebra - Quadratic Equations | Data Sufficiency

What are the unique values of b and c in the equation 4x² + 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

A.
1
B.
2
C.
3
D.
4

23. CAT 2003 QA - Leaked | Geometry - Circles | Data Sufficiency

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

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

A.
1
B.
2
C.
3
D.
4

24. CAT 2003 QA - Leaked | Algebra - Progressions | Data Sufficiency

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Is $(\frac{1}{a^{2}} + \frac{1}{a^{4}} + \frac{1}{a^{6}} + . . .) > (\frac{1}{a} + \frac{1}{a^{3}} + \frac{1}{a^{5}} + . . .) ?$

A. −3 ≤ a ≤ 3
B. One of the roots of the equation 4x2 − 4x + 1 = 0 is a

A.
1
B.
2
C.
3
D.
4

Answer: Option A

Explanation :

R.H.S. and L.H.S. are infinite GP's with common ratio = $\frac{1}{a^{2}}$ .

∴ We need to find whether

$\frac{\frac{1}{a^{2}}}{1 - \frac{1}{a^{2}}} < \frac{\frac{1}{a}}{1 - \frac{1}{a^{2}}}$

$∴ \frac{1}{a^{2}} > \frac{1}{a}$ ...(i)

This statement (i) is always true for a < 1.

Using statement A alone:

It does not give any conclusion regarding value of a i.e. whether it is less than 1 or not.

So, statement A alone is not sufficient to answer the question.

Using statement B alone:

4a²− 4a + 1 = 0 …(ii)

∴ a = 1/2

So, statement B alone is sufficient to answer the question.

Hence, option (a).

25. CAT 2003 QA - Leaked | Geometry - Triangles | Data Sufficiency

D, E, F are the mid-points of the sides AB, BC and CA of triangle ABC respectively. What is the area of DEF in square centimetres?

A. AD = 1 cm, DF = 1 cm and perimeter of DEF = 3 cm
B. Perimeter of ABC = 6 cm, AB = 2 cm, and AC = 2 cm

A.
1
B.
2
C.
3
D.
4

26. CAT 2003 QA - Leaked | Arithmetic - Percentage | Data Sufficiency

At the end of year 1998, Shepard bought nine dozen goats. Henceforth, every year he added p% of the goats at the beginning of the year and sold q% of the goats at the end of the year where p > 0 and q > 0. If Shepard had nine dozen goats at the end of year 2002, after making the sales for that year, which of the following is true?

A.
p = q
B.
p < q
C.
p > q
D.
p = q/2

27. CAT 2003 QA - Leaked | Geometry - Quadrilaterals & Polygons

Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the number of convex corners in such a polygon is 25, the number of concave corners must be

A.
20
B.
0
C.
21
D.
22

28. CAT 2003 QA - Leaked | Algebra - Progressions

The 288th term of the series a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, f, f... is

A.
u
B.
v
C.
w
D.
x

29. CAT 2003 QA - Leaked | Algebra - Quadratic Equations

Let p and q be the roots of the quadratic equation x²− (α − 2)x − α − 1 = 0. What is the minimum possible value of p² + q²?

A.
0
B.
3
C.
4
D.
5

30. CAT 2003 QA - Leaked | Geometry - Circles

There are two concentric circles such that the area of the outer circle is four times the area of the inner circle. Let A, B and C be three distinct points on the perimeter of the outer circle such that AB and AC are tangents to the inner circle. If the area of the outer circle is 12 square centimetres then the area (in square centimetres) of the triangle ABC would be

A.
$π \sqrt{12}$
B.
$\frac{9}{π}$
C.
$\frac{9 \sqrt{3}}{π}$
D.
$\frac{6 \sqrt{3}}{π}$

Answer: Option C

Explanation :

Consider the diagram below, as per the given conditions.

Let r and R be the radii of the inner circle and outer circle respectively.

As area of the outer circle is 4 times the area of the inner circle, we have, R = 2r

In ∆OAM, sin θ = $\frac{O M}{O A} = \frac{r}{2 r} = \frac{1}{2}$

∴ ∠OAM = θ = 30°

Similarly,

∠OAM = ∠OBM = ∠OAC = ∠OCA = 30°

∠OBC = ∠OCB = 30°

∴ ∠BAC = ∠ACB = ∠CBA

∴ ∆ABC is an equilateral triangle.

AB = 2 × $\sqrt{(O A^{2} - O M^{2})} = 2 \sqrt{3} r$

Area of ∆ABC = $\frac{\sqrt{3}}{4}$ × (AB)² = $\frac{\sqrt{3}}{4}$ × 4 × 3 × r² = $r \sqrt{3} r^{2}$ ...(i)

But, area of the outer circle = π(2r)² = 4πr² = 12

∴ r² = $\frac{3}{π}$

∴ Area of ∆ABC = $3 \sqrt{3} \times \frac{3}{π} = \frac{9 \sqrt{3}}{π}$

Hence, option (c).

31. CAT 2003 QA - Leaked | Algebra - Inequalities & Modulus

Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?

A.
The minimum possible value of a² + b² + c² + d² is 4m² − 2m + 1
B.
The minimum possible value of a² + b² + c² + d² is 4m² + 2m + 1
C.
The maximum possible value of a² + b² + c² + d² is 4m² − 2m + 1
D.
The maximum possible value of a² + b² + c² + d² is 4m² + 2m + 1

32. CAT 2003 QA - Leaked | Geometry - Circles

Three horses are grazing within a semi-circular field. In the diagram given below, AB is the diameter of the semi-circular field with centre at O. Horses are tied up at P, R and S such that PO and RO are the radii of semi-circles with centres at P and R respectively, and S is the centre of the circle touching the two semi-circles with diameters AO and OB. The horses tied at P and R can graze within the respective semi-circles and the horse tied at S can graze within the circle centred at S. The percentage of the area of the semi-circle with diameter AB that cannot be grazed by the horses is nearest to

A.
20
B.
28
C.
36
D.
40

Answer: Option B

Explanation :

Consider the diagram below.

Let r be the radius of the smaller semi-circles and s be the radius of the smaller circle.

OS = 2r − s

PS = r + s

PO = r

But, ∆PSO is a right-angled triangle.

∴ PS² = PO² + SO²

(r + s)² = r² + (2r − s)²

∴ r² + s² + 2rs = r² + 4r² + s² − 4rs

$∴ s = \frac{2 r}{3}$

∴ Total area not grazed = $\frac{1}{2}$ π(2r)² - $[2 \times \frac{1}{2} π r^{2} + π {(\frac{2 r}{3})}^{2}]$

= 2πr² - πr² - $\frac{4}{9}$ πr²

= $\frac{5}{9}$ πr²

∴ Required percentage = $\frac{\frac{5}{9} π r^{2}}{2 π r^{2}}$ × 100

= $\frac{5}{9 \times 2}$ × 100

≈ 28%

Hence, option (b).

33. CAT 2003 QA - Leaked | Geometry - Quadrilaterals & Polygons

In the figure below, ABCDEF is a regular hexagon and ∠AOF = 90°. FO is parallel to ED. What is the ratio of the area of the triangle AOF to that of the hexagon ABCDEF?

A.
$\frac{1}{12}$
B.
$\frac{1}{6}$
C.
$\frac{1}{24}$
D.
$\frac{1}{18}$

Answer: Option A

Explanation :

We can see that triangles AMF, AMB, BMC, CMD, DME and EMF are all equilateral triangles. Also, AE, XY and BD bisect MF, AB, ED and CM.

∴ All the small right triangles in the figure are congruent.

∴ A(∆AOF) = $\frac{1}{12}$ × the area of the hexagon ABCDEF

Hence, option (a).

34. CAT 2003 QA - Leaked | Modern Math - Permutation & Combination

How many three digit positive integers, with digits x, y and z in the hundred's, ten's and unit's place respectively, exist such that x < y, z < y and x ≠ 0?

A.
245
B.
285
C.
240
D.
320

35. CAT 2003 QA - Leaked | Geometry - Trigonometry

A vertical tower OP stands at the centre O of a square ABCD. Let h and b denote the lengths OP and AB respectively. Suppose ∠APB = 60°, then the relationship between h and b can be expressed as

A.
2b² = h²
B.
2h² = b²
C.
3b² = 2h²
D.
3h² = 2b²

Answer: Option B

Explanation :

OP = h and AB = b

Now, OA = $\frac{A C}{2} = \frac{b \sqrt{2}}{2} = \frac{b}{\sqrt{2}}$

OP is a perpendicular tower at the centre O of the square.

In ∆PAB, PA = PB

∴ ∠PAB = ∠PBA = ∠APB = 60°

∴ ∆PAB is an equilateral triangle.

∴ AP = b

In the right-angled ∆AOP, we have,

AP² = OP² + OA²

∴ b² = h² + $\frac{b^{2}}{2}$

∴ 2h² = b²

Hence, option (b).

36. CAT 2003 QA - Leaked | Geometry - Triangles

In a triangle ABC, AB = 6, BC = 8 and AC = 10. A perpendicular dropped from B, meets the side AC at D. A circle of radius BD (with centre B) is drawn. If the circle cuts AB and BC at P and Q respectively, then AP : QC is equal to

A.
1 : 1
B.
3 : 2
C.
4 : 1
D.
3 : 8

37. CAT 2003 QA - Leaked | Geometry - Triangles

In the diagram given below, ∠ABD = ∠CDB = ∠PQD = 90°. If AB : CD = 3 : 1, the ratio of CD : PQ is

A.
1 : 0.69
B.
1 : 0.75
C.
1 : 0.72
D.
None of the above

Answer: Option B

Explanation :

Consider the figure below,

As AB || CD and ∠ABD = ∠CDB = ∠PQD = 90°

∴ ∠BAP = ∠CDP and ∠ABP = ∠DCP

∆CPD ~ ∆BPA …(AAA test)

$∴ \frac{C P}{P B} = \frac{x}{3 x} = \frac{1}{3}$

If CP = y,

PB = 3y

Now, ∆CBD ~ ∆PBQ …(AA test)

$\frac{C D}{P Q} = \frac{C B}{P B} = \frac{y + 3 y}{3 y} = \frac{4}{3}$ = 1 : 0.75

Hence, option (b).

38. CAT 2003 QA - Leaked | Geometry - Mensuration

There are 8436 steel balls, each with a radius of 1 centimetre, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. The number of horizontal layers in the pile is

A.
34
B.
38
C.
36
D.
32

Answer: Option C

Explanation :

The first layer has 1 ball.

Second layer has 1 + 2 = 3 balls

Third layer has 1 + 2 + 3 = 6 balls

∴ The nth layer of the stack would have $[\frac{n (n + 1)}{2}]$ balls

∴ Total balls in all the layers = $\sum_{}^{} [\frac{n (n + 1)}{2}]$ = 8436

∴ $\frac{1}{2} \times [\frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2}]$ = 8436

Only n = 36 satisfies the above equation.

Hence, option (c).

39. CAT 2003 QA - Leaked | Algebra - Progressions

If the product of n positive real numbers is unity, then their sum is necessarily

A.
a multiple of n
B.
equal to n + 1/n
C.
never less than n
D.
a positive integer

40. CAT 2003 QA - Leaked | Algebra - Progressions | Algebra - Logarithms

If log₃ 2, log₃ (2^x − 5), log₃ (2^x − 7/2) are in arithmetic progression, then the value of x is equal to

A.
5
B.
4
C.
2
D.
3

41. CAT 2003 QA - Leaked | Geometry - Circles

In the figure given below, AB is the chord of a circle with centre O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ∠ACD = y° and ∠AOD = x° such that x = ky, then the value of k is

A.
3
B.
2
C.
1
D.
None of the above

42. CAT 2003 QA - Leaked | Geometry - Circles

In the figure below, the rectangle at the corner measures 10 cm × 20 cm. The corner A of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm?

A.
10 cm
B.
40 cm
C.
50 cm
D.
None of the above

43. CAT 2003 QA - Leaked | Algebra - Inequalities & Modulus

Given that −1 ≤ v ≤ 1, −2 ≤ u ≤ −0.5 and −2 ≤ z ≤ −0.5 and w = vz/u, then which of the following is necessarily true?

A.
−0.5 ≤ w ≤ 2
B.
−4 ≤ w ≤ 4
C.
−4 ≤ w ≤ 2
D.
−2 ≤ w ≤ −0.5

44. CAT 2003 QA - Leaked | Modern Math - Permutation & Combination

There are 6 boxes numbered 1, 2,..., 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is

A.
5
B.
21
C.
33
D.
60

45. CAT 2003 QA - Leaked | Algebra - Functions & Graphs

Consider the following two curves in the X-Y plane

y = x³ + x² + 5

y = x² + x + 5

Which of the following statements is true for −2 ≤ x ≤ 2?

A.
The two curves intersect once
B.
The two curves intersect twice
C.
The two curves do not intersect
D.
The two curves intersect thrice

46. CAT 2003 QA - Leaked | Algebra - Progressions

In a certain examination paper, there are n questions. For j = 1, 2, ..., n, there are 2^{(n − j)}students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is

A.
12
B.
11
C.
10
D.
9

47. CAT 2003 QA - Leaked | Algebra - Inequalities & Modulus

If x, y, z are distinct positive real numbers, then $\frac{x^{2} (y + z) + y^{2} (x + z) + z^{2} (x + y)}{x y z}$ would be

A.
greater than 4
B.
greater than 5
C.
greater than 6
D.
None of these

Answer: Option C

Explanation :

The given expression may be represented as

$\frac{x}{y} + \frac{x}{z} + \frac{y}{x} + \frac{y}{z} + \frac{z}{x} + \frac{z}{y}$

We know that, A.M. ≥ G.M.

$∴ \frac{(\frac{x}{y} + \frac{x}{z} + \frac{y}{x} + \frac{y}{z} + \frac{z}{x} + \frac{z}{y})}{6} \geq \sqrt[6]{\frac{x}{y} \times \frac{x}{z} \times \frac{y}{x} \times \frac{y}{z} \times \frac{z}{x} \times \frac{z}{y}}$

$\frac{x}{y} + \frac{x}{z} + \frac{y}{x} + \frac{y}{z} + \frac{z}{x} + \frac{z}{y} \geq 6$

∴ The given expression will have a minimum value of 6.

But x, y and z are distinct, so the value will always be greater than 6.

Hence, option (c).

48. CAT 2003 QA - Leaked | Modern Math - Permutation & Combination

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 other point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition

A.
11 ≤ e ≤ 66
B.
10 ≤ e ≤ 66
C.
11 ≤ e ≤ 65
D.
0 ≤ e ≤ 11

49. CAT 2003 QA - Leaked | Algebra - Number Theory

The number of positive integers n in the range 12 ≤ n ≤ 40 such that the product (n − 1)(n − 2)...3.2.1 is not divisible by n is

A.
5
B.
7
C.
13
D.
14

50. CAT 2003 QA - Leaked | Algebra - Number Theory | Modern Math - Sets

Let T be the set of integers {3, 11, 19, 27, ..., 451, 459, 467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is

A.
32
B.
28
C.
29
D.
30

51. CAT 2003 QA - Leaked | Miscellaneous | Data Sufficiency

In each question there are two statements: A and B.

Choose 1 if the question can be answered by one of the statements alone but not by the other.
Choose 2 if the question can be answered by using either statement alone.
Choose 3 if the question can be answered by using both the statements together but cannot be answered using either statement alone.
Choose 4 if the question cannot be answered even by using both the statements A and B.

A game consists of tossing a coin successively. There is an entry fee of Rs. 10 and an additional fee of Re. 1 for each toss of the coin. The game is considered to have ended normally when the coin turns heads on two consecutive throws. In this case the player is paid Rs. 100. Alternatively, the player can choose to terminate the game prematurely after any of the tosses. Ram has incurred a loss of Rs. 50 by playing this game. How many times did he toss the coin?

A. The game ended normally.
B. The total number of tails obtained in the game was 138.

A.
1
B.
2
C.
3
D.
4

52. CAT 2003 QA - Leaked | Miscellaneous | Data Sufficiency

In each question there are two statements: A and B.

Choose 1 if the question can be answered by one of the statements alone but not by the other.
Choose 2 if the question can be answered by using either statement alone.
Choose 3 if the question can be answered by using both the statements together but cannot be answered using either statement alone.
Choose 4 if the question cannot be answered even by using both the statements A and B.

Each packet of SOAP costs Rs. 10. Inside each packet is a gift coupon labelled with one of the letters S, O, A, and P. If a customer submits four such coupons that make up the word SOAP, the customer gets a free SOAP packet. Ms. X kept buying packet after packet of SOAP till she could get one set of coupons that formed the word SOAP. How many coupons with label P did she get in the above process?

A. The last label obtained by her was S and the total amount spent was Rs. 210.
B. The total number of vowels obtained was 18.

A.
1
B.
2
C.
3
D.
4

Answer: Option C

Explanation :

From Statement A,

Number of soaps purchased by Ms. X = $\frac{210}{10}$ = 21

Also the last label obtained by her is S.

But this is not sufficient to get the number of P's.

∴ Statement A alone is not sufficient.

From Statement B,

The number of O's and A's is 18.

But this is also not individually sufficient to arrive at the required answer.

∴ Statement B alone is not sufficient.

After combining both the statements A and B, we can conclude that 18 out of 21 coupons are O's and A's and that the 21st is an S.

∴This means that the remaining two are P's.

Hence, option (c).

53. CAT 2003 QA - Leaked | Arithmetic - Time, Speed & Distance | Data Sufficiency

In each question there are two statements: A and B.

Choose 1 if the question can be answered by one of the statements alone but not by the other.
Choose 2 if the question can be answered by using either statement alone.
Choose 3 if the question can be answered by using both the statements together but cannot be answered using either statement alone.
Choose 4 if the question cannot be answered even by using both the statements A and B.

If A and B run a race, then A wins by 60 seconds. If B and C run the same race, then B wins by 30 seconds. Assuming that C maintains a uniform speed, what is the time taken by C to finish the race?

A. A and C run the same race and A wins by 375 metres.
B. The length of the race is 1 km.

A.
1
B.
2
C.
3
D.
4

Answer: Option C

Explanation :

From the question, if A and C participate in a race, A will win by 90 seconds.

From statement A,

∵ A beats C by 375 m or 90 sec.

∴ C's speed = $\frac{375 m}{90 s}$

But the length of the track is not known.

∴ Statement A is not sufficient to find the time taken by C to complete the race.

From Statement B we cannot find the speed of C.

∴ Statement B alone is not sufficient.

After combining both the statements A and B, the time taken by C to complete the race

$= \frac{Length of the track}{C' s speed} = \frac{1000}{(\frac{375}{90})}$ = 240 sec

Hence, option (c).

CAT 2003 QA - Leaked | Previous Year Questions

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