List of top Questions asked in CAT- 2012

\( A, B, C \) 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?

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 \( B \).

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

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 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?

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

\( p \) 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 certain number written in a certain base is 144. Which of the following is always true?

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:

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?

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 \( \left( \beta + \frac{\beta}{\alpha} \right) \).

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?

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

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

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?

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

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 \( OP \).

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.