CAT 2010 QA Slot 1 Question Paper(Available) :Download Solutions with Answer Key

If \( r, s, t \) are consecutive odd integers with \( r < s < t \), which of the following must be true?

• (A) \( r + t = 2s \)
• (B) \( r + t = 2s + 2 \)
• (C) \( r + s = t - 2 \)
• (D) \( r + t = 2s + 5 \)

Let \( S \) be the set of rational numbers with the following properties:


[I.] \( \frac{1}{2} \in S \)
[II.] If \( x \in S \), then both \( \frac{1}{x+1} \in S \) and \( \frac{x}{x+1} \in S \)


Which of the following is true?

• (A) \( S \) contains all rational numbers in the interval \( 0 < x < 1 \)
• (B) \( S \) contains all rational numbers in the interval \( -1 < x < 1 \)
• (C) \( S \) contains all rational numbers in the interval \( -1 < x < 0 \)
• (D) \( S \) contains all rational numbers in the interval \( 1 < x < \infty \)

\( P, Q, R \) are three consecutive odd numbers in ascending order. If the value of three times \( P \) is three less than two times \( R \), find the value of \( R \).

• (A) 5
• (B) 7
• (C) 9
• (D) 11

Consider the following statements: When two straight lines intersect, then:


[I.] Adjacent angles are complementary
[II.] Adjacent angles are supplementary
[III.] Opposite angles are equal
[IV.] Opposite angles are supplementary


Which of these statements are correct?

• (A) I and III are correct
• (B) II and III are correct
• (C) I and IV are correct
• (D) II and IV are correct

A pole has to be erected on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates \( A \) and \( B \) on the boundary is 7 metres. The distance of the pole from one of the gates is:

• (A) 8 metres
• (B) 8.25 metres
• (C) 5 metres
• (D) None of these

From a square piece of cardboard measuring \( 2a \) on each side, a box with no top is to be formed by cutting out from each corner a square with sides \( b \) and bending up the flaps. The value of \( b \) for which the box has the greatest volume is:

• (A) \( \frac{a}{5} \)
• (B) \( \frac{a}{4} \)
• (C) \( \frac{a}{6} \)
• (D) \( \frac{2a}{3} \)

The sum of the areas of two circles which touch each other externally is \( 153\pi \). If the sum of their radii is 15, find the ratio of the larger to the smaller radius.

• (A) 4
• (B) 2
• (C) 3
• (D) None of these

Consider the following statements:


[I.] If \( a^x = b^x = c^x = abc \), then \( xyz = 1 \)
[II.] If \( a^p = b^q = c^r \) and \( a^q b^r c^p = 1 \), then \( xyz = 1 \)
[III.] If \( x^a = y^b = z^c \) and \( ab + bc + ca = 0 \), then \( xyz = 1 \)

• (A) I and II are correct
• (B) II and III are correct
• (C) Only I is correct
• (D) All I, II and III are correct

If \( a, b, c \) are three real numbers, then which of the following is not true?

• (A) \( |a + b| \leq |a| + |b| \)
• (B) \( |a - b| \leq |a| + |b| \)
• (C) \( |a - b| \leq |a| - |b| \)
• (D) \( |a - c| \leq |a - b| + |b - c| \)

Let \( S \) denote the infinite sum:
\[ S = 2 + 5x + 9x^2 + 14x^3 + 20x^4 + \ldots \quad where |x| < 1 \]
and the coefficient of \( x^n \) is \( \frac{1}{2}n(n+3) \). Then \( S \) equals:

• (A) \( \frac{2 - x}{(1 + x)^3} \)
• (B) \( \frac{2 - x}{(1 - x)^3} \)
• (C) \( \frac{2x}{(1 - x)^3} \)
• (D) \( \frac{2 + x}{(1 + x)^3} \)

ABCD is a rectangle. Points \( P \) and \( Q \) lie on \( AD \) and \( AB \) respectively. If triangles \( PAQ, QBC, \) and \( PCD \) all have the same areas and \( BQ = 2 \), then \( AQ = \) ?

• (A) \( 1 + \sqrt{5} \)
• (B) \( 1 - \sqrt{5} \)
• (C) \( \sqrt{7} \)
• (D) \( 2\sqrt{7} \)

For what value of \( k \) does the following pair of equations yield a unique solution for \( x \), such that the solution is positive?
\[ x^2 - 3y^2 = 0
x^2 - 6y^2 + k = 0 \]

• (A) 2
• (B) 0
• (C) \( \sqrt{2} \)
• (D) \( -\sqrt{2} \)

In an examination, the average marks obtained by students who passed was \( x% \), while the average of those who failed was \( y% \). The average marks of all students taking the exam was \( z% \). Find in terms of \( x, y, z \), the percentage of students taking the exam who failed.

• (A) \( \frac{x - z}{x - y} \)
• (B) \( \frac{z - y}{x - z} \)
• (C) \( \frac{z - y}{x - y} \)
• (D) \( \frac{y - z}{y - x} \)

If \( a = \log 2, b = \log 3, c = \log 4 \), then the value of \( \log(abcd) \) would be:

• (A) \( \log_{10} 24 \)
• (B) \( \log_2 24 \)
• (C) \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{\log_5 4} \)
• (D) \( \frac{1}{a} - \frac{1}{b} - \frac{1}{c} + \frac{1}{\log_4 5} \)

If three positive real numbers \( a, b, c \) (with \( c > a \)) are in Harmonic Progression, then \( \log(a + c) + \log(a - 2b + c) \) is equal to:

• (A) \( 2\log b \)
• (B) \( 2\log (a - c) \)
• (C) \( 2\log (c - a) \)
• (D) \( \log a + \log b + \log c \)

Let \( f \) be an injective map with domain \{x, y, z\ and range \{1, 2, 3\ such that exactly one of the following statements is correct and the remaining are false.


[I.] \( f(x) = 1 \Rightarrow f(y) = 1, f(z) = 2 \)
[II.] \( f(x) = 2 \Rightarrow f(y) = 1, f(z) = 1 \)
[III.] \( f(x) = 1, f(y) = 1, f(z) = 2 \)


Then the value of \( f(1) \) is:

• (A) \( x \)
• (B) \( y \)
• (C) \( z \)
• (D) None of the above

For constructing the working class consumer price index number of a particular town, the following weights corresponding to different groups of items were assigned:

Food = 55, Fuel = 15, Clothing = 10, Rent = 10, Miscellaneous = 10


It is known that the rise in food prices is double that of fuel, and the rise in miscellaneous group prices is double of that in rent.

In October 2006, the increased D.A. by a factor of 1.82 (i.e., by 82%) fully compensated for the rise in prices of food and rent but did not compensate for anything else.

Another factory of the same locality increased D.A. by 46.5%, which compensated for the rise in fuel and miscellaneous groups.


Which is the correct combination of the rise in prices of food, fuel, rent, and miscellaneous groups?

• (A) 320.14, 159.57, 95.64, 164.28
• (B) 311.14, 159.57, 90.64, 198.28
• (C) 321.14, 162.57, 84.46, 175.38
• (D) 317.14, 158.57, 94.64, 189.28

In a factory making radioactive substances, it was considered that three cubes of uranium together are hazardous. So the company authorities decided to have the stack of uranium interspersed with lead cubes. But there is a new worker in the company who does not know the rule. So he arranges the uranium stack the way he wanted. What is the number of hazardous combinations of uranium in a stack of 5?

• (A) 3
• (B) 7
• (C) 8
• (D) 10

A line graph on a graph sheet shows the revenue for each year from 1990 through 1998 by points and joins the successive points by straight line segments. The point for revenue of 1990 is labeled A, that for 1991 is B, and that for 1992 is C. What is the ratio of growth in revenue between 1991–92 and 1990–91?

\medskip
Statement I:
The angle between AB and X-axis when measured with a protractor is \( 40^\circ \), and the angle between CB and x-axis is \( 80^\circ \).


Statement II:
The scale of y-axis is \( 1 cm = 1000 \).

Geetanjali Express, which is 250 m longer when moving from Howrah to Tatanagar, crosses Subarnarekha bridge in 30 seconds. What is the speed of Geetanjali Express?

\medskip
Statement I: Bombay Mail, which runs at 60 km/h, crosses the Subarnarekha bridge in 30 seconds.


Statement II: Bombay Mail, when running at 90 km/h, crosses a lamp post in 10 seconds.