IIM CAT Preparation Tips

IIM CAT Preparation Tips: June 2011

Jun 11, 2011

Geometry Concept testers - Solutions

Given below is the answer key to the set of True of False questions

1. Any two triangles that are congruent to each other will also be similar to each other - TRUE

2. If in a triangle with sides a, b and c, if a^2 + b^2 > c^2 the triangle has to be acute-angled - FALSE. This is true only if c is the largest side. Thinking about it differently, for any triangle, in sum of the squares of the two larger sides will be greater than the square of the smallest side. So, this would imply that any triangle has to be acute angled :)

3. Any parallelogram inscribed inside a circle has to be a rectangle - TRUE
Opposite angles of a parallelogram are equal
Opposite angles of a cyclic quadrilateral are supplementary

4. If in a triangle the orthocenter, incenter and circumcenter are collinear the triangle has to be isosceles - TRUE
In any triangle, orthocenter, circumcenter and centroid will be collinear. So, if orthocenter, incenter and circumcenter are collinear, then all 4 points lie on the same straight line, which implies the triangle has to be isosceles

5. There will be a unique circle passing through any three points - FALSE
There is a unique circle passing through any three non-collinear points

6. If two circles with centers A and B and radii r and R intersect, then AB > R - r - TRUE
If two circles intersect, then AB, R and r form sides of a triangle. => r + AB > R => AB > R - r

7. Circumradius of a triangle cannot be greater than the three sides of the triangle - FALSE
Think about this differently. Draw a large circle, draw a small triangle inside this circle in a small segment of this circle. The circle is the circumcircle for this triangle

8. For any obtuse-angled triangle, the orthocenter and circumcenter will lie outside the triangle - TRUE

9. In a scalene triangle, the sum of the three medians will be greater than the sum of the three altitudes - TRUE
Shortest distance from a point to a line is the perpendicular distance. => Altitude is always less than or equal to median. In a scalene triangle, altitude will be less than the median

10. In circumcircle and incircle are concentric, the triangle has to be equilateral - TRUE

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Jun 7, 2011

Geometry - Concept testers

For each of the statements given below, say true or false

1. Any two triangles that are congruent to each other will also be similar to each other
2. If in a triangle with sides a, b and c, if a^2 + b^2 > c^2 the triangle has to be acute-angled
3. Any parallelogram inscribed inside a circle has to be a rectangle
4. If in a triangle the orthocenter, incenter and circumcenter are collinear the triangle has to be isosceles.
5. There will be a unique circle passing through any three points
6. If two circles with centers A and B and radii r and R intersect, then AB > R - r
7. Circumradius of a triangle cannot be greater than the three sides of the triangle
8. For any obtuse-angled triangle, the orthocenter and circumcenter will lie outside the triangle
9. In a scalene triangle, the sum of the three medians will be greater than the sum of the three altitudes.
10. In circumcircle and incircle are concentric, the triangle has to be equilateral

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Jun 4, 2011

Solutions to Number Theory and Counting questions

Given below are the solutions to these Number theory questions.

1. From the digits 2,3,4,5,6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?

Any multiple of 12 should be a multiple of 4 and 3. First, let us look at the constraint for a number being a multiple of 3. Sum of the digits should be a multiple of 3. Sum of all numbers from 2 to 7 is 27. So, if we have to drop a digit and still retain a multiple of 3, we should drop either 3 or 6.

So, the possible 5 digits are 2, 4, 5, 6, 7 or 2, 3, 4, 5, 7.

When the digits are 2, 4, 5, 6, 7. the last two digits possible for the number to be a multiple of 4 are 24, 64, 52, 72, 56, 76. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 36 different numbers

When the digits are 2, 3, 4, 5, 7. the last two digits possible for the number to be a multiple of 4 are 32, 52, 72, 24. For each of these combinations, there are 6 different numbers possible. So, with this set of 5 digits we can have 24 different numbers

Overall, there are 60 different 5-digit numbers possible

2. All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

If a number written in base 6 ends with a zero, it should be a multiple of 6. In other words, the question wants us to find all numbers from 1 to 200 that are not multiples of 6 or 7. There are 33 multiples of 6 less than 201. There are 28 multiples of 7 less than 201. There are 4 multiples of 6 & 7 (or multiple of 42) from 1 to 200.

So, total multiples of 6 or 7 less than 201 = 33 + 28 - 4 = 57. Number of numbers with non-zero units digit = 200-57 = 143.

3. All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will not contain any zero?

Any multiple of 6 will end in a zero. There are 25 such numbers. Beyond this, we can have zero as the middle digit of a 3-digit number. This will be the case for numbers from 37-41, 73-77, 109-113 and 145-149. There are 20 such numbers. Overall, there are 45 numbers that have a zero in them.

4. How many factors of 1080 are perfect squares?

1080 = 2^3 * 3^3 * 5. For any perfect square, all the powers of the primes have to be even numbers. So, if the factor is of the form 2^a * 3^b * 5^c. The values 'a' can take are 0 and 2, b can take are 0 and 2, and c can take the value 0. Totally there are 4 possibilities. 1, 4, 9, and 36.

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Jun 3, 2011

Few Questions on Number Theory and Counting (combined)

CAT has been consistently asking questions combining basic number theory and counting. So, it is probably good practice to have a go at these.

1. From the digits 2,3,4,5,6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?

2. All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

3. All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will not contain any zero?

4. How many factors of 1080 are perfect squares?

Solutions to these questions can be found here.