CAT 2009 QA Question Paper(Available) :Download Solutions with Answer Key

If \( mx^n - nx^m = 0 \), then what is the value of \( \frac{1}{x^m + x^n} + \frac{1}{x^m - x^n} \) in terms of \( x \)?

• (a) \( \frac{2mn}{x^n(m^2 - n^2)} \)
• (b) \( \frac{2mn}{x^n(n^2 - m^2)} \)
• (c) \( \frac{2mn}{x^m(m^2 - n^2)} \)
• (d) \( \frac{2mn}{x^m(n^2 - m^2)} \)

If \( \log (0.57) = -0.244 \), then the value of \( \log 57 + \log (0.57) + \log \sqrt{0.57} \) is:

• (a) 0.902
• (b) 2.146
• (c) 1.902
• (d) 1.146

In a certain zoo, there are 42 animals in one sector, 34 in the second sector and 20 in the third sector. Out of this, 24 graze in sector one and also in sector two. 10 graze in sector two and sector three, 12 graze in sector one and sector three. These figures also include four animals grazing in all the three sectors are now transported to another zoo. Find the total number of animals.

• (a) 38
• (b) 36
• (c) 54
• (d) None of the above

The ratio of the roots of \( bx^2 + nx + n = 0 \) is \( p:q \), then

• (a) \( \frac{q \sqrt{p}}{\sqrt{q}} + \frac{p \sqrt{q}}{\sqrt{p}} = 0 \)
• (b) \( \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} = 0 \)
• (c) \( \frac{q}{\sqrt{p}} + \frac{p}{\sqrt{q}} = 0 \)
• (d) \( \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} \neq 0 \)

The average age of a couple is 25 years. The average age of the family just after the birth of the first child was 18 years. The average age of the family just after the second child was born was 15 years. The average age of the family after the third and the fourth children (who are twins) were born was 12 years. If the present average age of the family of six persons is 16 years, how old is the oldest child?

• (a) 6 years
• (b) 7 years
• (c) 8 years
• (d) 9 years

10% of the voters did not cast their vote in an election between two candidates. 10% of the votes polled were found invalid. The successful candidate got 54% of the valid votes and won by a majority of 1620 votes. The number of voters enrolled on the voters list was:

• (A) 25000
• (B) 33000
• (C) 35000
• (D) 40000

The resistance of a wire is proportional to its length and inversely proportional to the square of its radius. Two wires of the same material have the same resistance and their radii are in the ratio \(9:8\). If the length of the first wire is 162 cm, find the length of the other.

• (A) 64 cm
• (B) 120 cm
• (C) 128 cm
• (D) 132 cm

A 20 litre vessel is filled with alcohol. Some of the alcohol is poured out into another vessel of equal capacity, which is then completely filled by adding water. The mixture thus obtained is then poured into the first vessel to capacity. Then \(6\frac{2}{3}\) litres is poured from the first vessel into the second. Both vessels now contain an equal amount of alcohol. How much alcohol was originally poured from the first vessel into the second?

• (A) 9 litres
• (B) 10 litres
• (C) 12 litres
• (D) 12.5 litres

The shaded portion of the figure shows the graph of which of the following?

• (A) \( x (y - 2x) \geq 0 \)
• (B) \( x (y - 2x) \leq 0 \)
• (C) \( x \left( y + \frac{x}{2} \right) \geq 0 \)
• (D) \( x \left( y - \frac{x}{2} \right) \leq 0 \)

If \( f\left(x + \frac{y}{8}, y - \frac{x}{8}\right) = xy \), then \( f(m,n) + f(n,m) = 0 \) is true:

• (A) only when \( m = n \)
• (B) only when \( m \neq n \)
• (C) only when \( m = -n \)
• (D) for all \( m \) and \( n \)

A person closes his account in an investment scheme by withdrawing Rs. 10,000. One year ago he had withdrawn Rs. 6000. Two years ago he had withdrawn Rs. 5000. Three years ago he had not withdrawn any money. How much had he deposited approximately at the time of opening the account 4 years ago, if the annual simple interest rate is 10%?

• (A) Rs. 15600
• (B) Rs. 16500
• (C) Rs. 17280
• (D) None of these

It takes 6 technicians a total of 10 hours to build a new server from direct computer, with each working at the same rate. If six technicians start to build the server at 11:00 am, and one technician per hour is added beginning at 5:00 pm, at what time will the server be completed?

• (A) 6:40 pm
• (B) 7:00 pm
• (C) 7:20 pm
• (D) 8:00 pm

A ship 55 km from the shore springs a leak which admits 2 tonnes of water in 6 min; 80 tonnes would sink her, but the pumps can throw out 12 tonnes an hour. Find the average rate of sailing so that she may just reach the shore as she begins to sink.

• (A) 5.5 km/h
• (B) 6.5 km/h
• (C) 7.2 km/h
• (D) 8.5 km/h

In a 400 metre race around a circular stadium having a circumference of 1000 metres, 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. 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 mins
• (B) 15 mins
• (C) 10 mins
• (D) 5 mins

A train crosses a platform 100 metres long in 60 seconds at a speed of 45 km/hr. The time taken by the train to cross an electric pole is:

• (A) 8 seconds
• (B) 1 minute
• (C) 52 seconds
• (D) Data inadequate

If \( x = 1 + 2a + 3a^2 + 4a^3 + \ldots \) (\(|a| < 1\)) and \( y = 1 + 3b + 6b^2 + \ldots\) (\(|b| < 1\)), then find \( 1 + ab + (ab)^2 + (ab)^3 + \ldots \) in terms of \(x\) and \(y\).

• (A) \( \frac{x^{1/2}y^{1/3}}{x^{1/2} + y^{1/3} - 1} \)
• (B) \( \frac{xy}{x + y - 1} \)
• (C) \( \frac{x^{1/3}y^{2/3}}{x^{1/3}y^{1/2} - 1} \)
• (D) None of these

Two vertical lamp-posts of equal height stand on either side of a road 50 m wide. At a point \(P\) on the road between them, the elevation of the tops of the lamp-posts are \(60^\circ\) and \(30^\circ\). Find the distance of \(P\) from the lamp-post which makes angle of \(60^\circ\).

• (A) 25 m
• (B) 12.5 m
• (C) 16.5 m
• (D) 20.5 m

\(M\) is the centre of the circle. \(\ell(QS) = 10\sqrt{2}\), \(\ell(PR) = \ell(RS)\) and \(PR \perp QS\). Find the area of the shaded region. (\(\pi = 3\))

• (A) 100 sq. units
• (B) 114.4 sq. units
• (C) 50 sq. units
• (D) 200 sq. units

There are three coplanar parallel lines; if any \(p\) points are taken on each of the lines, then find the maximum number of triangles with the vertices of these points.

• (A) \(p^2(4p - 3)\)
• (B) \(p^3 - (4p - 3)\)
• (C) \(p(4p - 3)\)
• (D) \(p^3\)

A and B throw with one die for a stake of Rs. 11 which is to be won by the player who first throws a 6. If A has the first throw, what are their respective expectations?

• (A) Rs. 7, Rs. 4
• (B) Rs. 6, Rs. 5
• (C) Rs. 4, Rs. 7
• (D) Rs. 5, Rs. 6