Progressions-12 (IIT-JEE 2009)

If the sum of first n terms of A.P is cn², then the sum of squares of these n terms is:

(a)n(4n²-1)c²/6          (b) n(4n²+1)c²/3        (c) n(4n²-1)c²/3        (d) n(4n²+1)c²/6

Solution follows here:

Progressions -11 (CAT-2006)

Consider the series S = {1,2,3,…1000}, how many arithmetic progressions can be formed from the elements of S that start with 1 and end with 1000 and have at least 3 elements?

(1) 3                (2) 4                (3) 6                (4) 7                (5) 8                           

Solution follows here:

Progressions-10 (CAT-2007)

Consider the set S = {2, 3, 4, ...., 2n + 1}, where n  is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X – Y?

                        (1) 0

                        (2) 1

                        (3) n/2

                        (4) n+1/2n

                        (5) 2008

Solution follows here:

Puzzle -26

		H
		G
A	B	O	C	D
		E
		F

A,B,C,D,E,F,G,H,O are distinct single digit non-zero positive integers placed as shown in the above diagram. Given that, sum of numbers placed north of ‘O’ = sum of numbers placed south of ‘O’ = sum of numbers placed east of ‘O’ = sum of numbers placed west of ‘O’. Then find all the possible values of ’O’?

(1)1,2,3                       (2)1,3,5                       (3)1,5,9                       (4)2,4,6                       (5)1,3,9

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CAT Analysis

            Being the last paper-based CAT exam, CAT-2008 has got its own significance. From 2009 onwards, CAT is being conducted online and CAT-Exam Q’s are not openly available. Considering CAT-2008 as a bench mark, it is thoroughly analysed here:

Total number of questions given in ‘Quant’ section is 25, each question contributing for 4 marks and there is a negative mark of 1 for each wrong answer. Out of all, 3 set problems each containing 2 questions are given. A set on number theory containing 2- sub questions is a harder and sitter type problem.  Geometry problem on circles is a harder one. I suggest leaving out such type of problems in the examination. Except these, remaining all come under easy to middle level problems. 

Geometry -18 (XAT-2011) (2 Marks)

A 25 ft long ladder is placed against the wall with its base 7 ft from the wall. The base of the ladder is drawn out so that the top comes down by half the distance that the base is drawn out. This distance is in the range:

(1)(2,7)                       (2)(5,8)                       (3)(9,10)         (4)(3,7)                       (5)None of these

Solution follows here:

P&C -12 (XAT-2011) (3 Marks)

In a bank the account numbers are all 8 digit numbers, and they all start with digit 2. So an account number can be represented as 2x₁x₂x₃x₄x₅x₆x₇. An account number is considered to be a ‘magic’ number if x₁x₂x₃ is exactly same as x₄x₅x₆ or x₅x₆x₇ or both. x_i can take values from 0 to 9, but 2 followed by seven zeroes is not a valid account number. What is the maximum possible number of customers having a ‘magic’ account number?

Solution follows here:

Puzzle-25

In the country of Twenty there are exactly twenty cities, and there is exactly one direct road between any two cities. No two direct roads have an overlapping road segment. After the election dates are announced, candidates from their respective cities start visiting the other cities. Following are the rules that the election commission has laid down for the candidates:

·        Each candidate must visit each of the other cities exactly once

·        Each candidate must use only the direct roots between two cities for going from one city to another

·        The candidate must return to his own city at the end of the campaign

·        No direct road between two cities would be used by more than one candidate

The maximum possible number of candidates is:

(1)5     (2)6     (3)7     (4)8     (5)9
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Numbers -17 (XAT-2011) (3 Marks)

The micromanometer in a certain factory can measure the pressure inside the gas chamber from 1 unit to 999999 units. Lately this instrument has not been working properly. The problem with the instrument is that it always skips the digit 5 and moves directly from digit 4 to 6. What is the actual pressure inside the gas chamber if the micromanometer displays 003016?

(1)2201          (2)2202          (3)2600          (4)2960          (5)None of these options

Solution follows here:

P&C - 11 (IIFT-2010)

How many subsets of {1,2,3,…11} contain at least one even integer?

Solution follows here:

Arithmetic-21 (IIFT-2009)

A petrol tank at a filling station has a capacity of 400 litres. The attendant sells 40 litres of petrol from the tank to one customer and then replenishes it with kerosene oil. This process is repeated with six customers. What quantity of pure petrol will the seventh customer get when he purchases 40 litres of petrol?
A. 20.50 litres            B. 21.25 litres            C. 24.75 litres            D. 22.40 litres

Solution follows here:

Algebra-25 (IIFT-2009)

The number of distinct terms in the expansion of (X+Y+Z+W)³⁰ is:

(1)4060                      (2)5456                      (3)27405                    (4)46376

Solution follows here:

Progressions-9 (FMS-2010)

Three non-zero numbers a,b,c form an arithmetic progression. Increasing a by 1 or increasing c by 2 results in a geometric progression. Then b equals:

(1)16               (2)14               (3)12               (4)10

Solution follows here:

Geometry-17 (FMS-2010)

Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions in to which they divide the plane is:

 (1)16              (2)20               (3)22               (4)24

Solution follows here:

Straight lines dividing a plane into maximum number of regions

To find maximum number of regions in to which ‘n’ straight lines can divide a plane:

“Maximum number of regions can be achieved when the lines are drawn such that no two are parallel and no three are concurrent”
Let us go case by case starting with one straight line.

One straight line can divide a plane into 2.

Two straight lines can divide a plane into 4.

Three straight lines can divide a plane into 7.

observe the sequence: 2, 4, 7,....

If we observe, it is a special sequence which can be written as

1+1, 3+1, 6+1,......
= (1)+1, (1+2)+1, (1+2+3)+1,.....

= ∑1 + 1, ∑2 + 1, ∑3 + 1,......

So, we can generalise like this:

Maximum number of regions that ‘n’ straight lines can divide a plane = ∑n + 1 = n(n+1)/2 + 1

Arithmetic-20 (FMS-2011)

The times between 7 and 8’O clock, correct to the nearest minute, when the hands of a clock will form an angle of 84 degrees are:

(1)7:23 and 7:53       (2)7:20 and 7:50       (3)7:22 and 7:53       
(4)7:23 and 7:52

Solution follows here:

Concept on Clocks

Speed of minutes hand = 360⁰ per hour = 6⁰ per minute

Speed of hours hand = 360⁰ per 12 hours = 0.5⁰ per minute

Relative speed of minutes hand over hours hand = 5.5⁰ per minute

If time is expressed as h:m, to find angle between minutes and hours hands:

At time h:0, angle between minutes and hours hands = 30h

After m minutes, ie., at h:m, minutes hand travels an angle of ‘5.5m’ relative to hours hand, and may or may not exceed the angle ‘30h’

=> the angle = 30h-5.5m (or) 5.5m-30h

If time is expressed as h:m, angle between minutes and hours hands 

= 30h-5.5m (or) 5.5m-30h

Puzzle-24 (CAT-2008)

Directions for Questions 1 and 2:

Choose (1) if Q can be answered from A alone but not from B alone.

Choose (2) if Q can be answered from B alone but not from A alone.   

Choose (3) if Q can be answered from A alone as well as from B alone.

Choose (4) if Q can be answered from A and B together but not from any of them alone. 

Choose (5) if Q cannot be answered even from A and B together.       

In a single elimination tournament, any player is eliminated with a single loss. The tournament is played in multiple rounds subject to the following rules:

(a) If the number of players, say n, in any round is even, then the players are grouped in to n/2 pairs.    The players in each pair play a match against each other and the winner moves on to the next round.

(b) If the number of players, say n, in any round is odd, then one of them is given a bye, that is, he automatically moves on to the next round. The remaining (n − 1) players are grouped into (n − 1)/2 pairs. The players in each pair play a match against each other and the winners move on to the next round. No player gets more than one bye in the entire tournament.

Puzzle-23 (CAT-2008)

Directions for Questions 1 and 2:

Five horses, Red, White, Grey, Black and Spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs. 3000, Rs. 2000 Rs. 1000 on Red, White and Black horses respectively and ends up with no profit and no loss.         

1. Which of the following cannot be true?

                        (1) At least two horses finished before Spotted

                        (2) Red finished last

                        (3) There were three horses between Black and Spotted

                        (4) There were three horses between White and Red

                        (5) Grey came in second      

2.  Suppose, in addition, it is known that Grey came in fourth. Then which of the following cannot be true?

                        (1) Spotted came in first

                        (2) Red finished last

                        (3) White came in second

                        (4) Black came in second 

(5) There was one horse between Black and White

Solution follows here:

Trigonometry-3 (CAT-2008)

Rahim plans to drive from city A to station C, at the speed of 70 km per hour, to catch a train arriving there from B. He must reach C at least 15 minutes before the arrival of the train. The train leaves C located 500 km south of A, at 8:00 am and travels at a speed of 50 km per hour. It is known that C is located between west and northwest of B, with BC at 60⁰ to AB. Also C is located between south and southwest of A with AC at 30⁰ to AB. The latest time by which Rahim must leave A and still catch the train is closest to
(1)6:15 am     (2) 6:30 am    (3) 6:45 am    (4) 7:00 am    (5) 7:15 am

Solution follows here:

Geometry-16 (CAT-2008)

Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest possible total surface area (in sq cm) of the cylinder.

(1)   100 π/3          (2) 80 π/3    (3) 120 π/7    (4) 130 π/9    (5) 110 π/7  

Solution follows here:

Profit & Loss

A and B invested capitals in the ratio of 3: 5 to start a business. A is the working partner and B is the sleeping partner. What percentage of total profit may be given as a salary to A, so that the ratio of the profit earned by them is 3: 2?

1)38%             2)24%             3)36%             4)40%             5) None of the above

Solution follows here:

Geometry Concepts -4

1. All triangles lying between two parallel lines and built on the same base have equal area.

If AB and CF are parallel lines,

∆ABC = ∆ABD = ∆ABE = ∆ABF

2. All parallelograms lying between two parallel lines and built on the same base have equal area.

If AB and CH are parallel lines,

oABEC = oABGD = oABHF

Geometry Concepts -3

When a circle is inscribed in a square, then

Diameter of the Circle = Side of the Square

(1) If the radius of the circle is given as ‘r’, then area/perimeter of the square can be determined as given below:

Side of square = Diameter of circle = 2r

=> Area of square = (2r)² = 4r² ,Where as Area of circle = πr²

Perimeter of square = 4(2r) = 8r, Where as Perimeter of circle = 2 π r

(2) If the side of the square is given as ‘x’, then area/perimeter of the circle can be determined as given below:

Diameter of circle = side of square = x => Radius of circle = x/2

=> Area of circle = π(x/2)² = πx²/4, Where as Area of square = x²

Perimeter of circle = 2 π (x/2) = πx, Where as Perimeter of square = 4x

Geometry Concepts -2

When a square is inscribed in a circle, then 

Diagonal of the Square = Diameter of the Circle

(1) If the radius of the circle is given as ‘r’, then area/perimeter of the square can be determined as given below:

Diagonal of square = Diameter of circle = 2r
=> side of square = (2r)/√2 = r√2

=> Area of square = (r√2)² = 2r²

Perimeter of square = 4(r√2) = 4√2r

(2) If the side of the square is given as ‘x’, then area/perimeter of the circle can be determined as given below:

Diameter of circle = Diagonal of square = x√2 
=> Radius of circle = x√2/2 = x/√2

=> Area of circle = π(x/√2)² = πx²/2

Perimeter of circle = 2 π (x/√2) = πx√2

Geometry Concepts -1

When a rectangle is inscribed in a circle, then

Diagonal of the Rectangle = Diameter of the Circle

(1) If the radius of the circle is given as ‘r’, then relation between area and perimeter of the rectangle can be determined as given below:

Diagonal of rectangle = Diameter of circle = 2r

=> √(a²+b²) = 2r => a²+b² = 4r² => (a+b)²-2ab = 4r²

=> (½ Perimeter of rectangle)² – 2(Area of rectangle) = 4r²

(2) If the length and breadth of the rectangle are given as ‘a’ and ‘b’, then area/perimeter of the circle can be determined as given below:

Diameter of circle = Diagonal of rectangle = √(a²+b²)

=> Radius of circle = ½ √(a²+b²)

=> Area of circle = π{½ √(a²+b²)}² = π(a²+b²)/4

Perimeter of circle = 2π (½ √(a²+b²)) = π √(a²+b²)

Numbers Concepts -2

Numbers Concepts -1

1. For any positive integer n, product of ‘n’ or more than ‘n’ consecutive positive integers is divisible by n!.

For example, 63*64*65*.......*91 is divisible by 29!
2. If (n-1)! is not divisible by ‘n’, then n is a prime number. 4 is a special case here, which obeys this rule but not a prime.
3. From 1 & 2 above, we can conclude that,  for any positive integer 'n-1', if the product of ‘n-1’  consecutive positive integers is not divisible by n, then n is a prime. But if divisible, we can't say that n is not a prime.
4. If a > b ≥ 3, then b^a > a^b where a,b Є Z+
5. Product of two successive integers always ends in 2,6, or 0
6. The general perception about even numbers is even numbers start with 0 and go on 2,4,6,8,... and the odd numbers are 1,3,5,.... But negative integers are also to be categorized in to even and odd sets.

Hence the even number set is: ....-4,-2,0,2,4,6,.....
and the odd number set is: ....-5,-3,-1,3,5,7,.....
7. Why '1' is not a prime number?
A prime number can be defined as a positive integer that has exactly two different positive divisors, 1 and itself.  As the number '1' has only one positive divisor (ie., itself), it is not a prime number. 

Probability-11 (GMAT)

Ellen invites 5 people to dinner, 2 women and 3 men. If Ellen's dining table is circular and the guests choose their seats at random around her table, what is the probability that no female is seated next to another female?

(A) 1/90          (B) 1/72          (C) 5/72          (D) 5/36          (E) 1/10

Solution follows here:

Algebra-24 (CAT-2008)

Puzzle-22

Directions for Questions 1 and 2:  The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

1. Neelam rides her bicycle from her house at A to her office at B, taking the shortest path. Then the     number of possible shortest paths that she can choose is

(1)  60              (2)   75               (3)     45               (4)   90              (5)   72

2. Neelam rides her bicycle from her house at A to her club at C, via B taking the shortest path. Then the number of possible shortest paths that she can choose is    
(1) 1170              (2)   630            (3)     792               (4)   1200              (5)   936

To enter your answers, click on “comments” below:

Geometry-15 (CAT-2008)

Consider a square ABCD with midpoints E, F, G, H of AB, BC, CD and DA respectively. Let L denote  the line passing through F and H. Consider points P and Q, on L and inside ABCD, such that the angles APD and BQC both equal 120°. What is the ratio of the area of ABQCDP to the remaining area inside ABCD?

(1) 4√2/3       (2) 2+√3       (3) (10-3√3)/9        (4) 1+(1/√3)         (5) 2√3-1

Solution follows here:

Quadratic Equations-3 (CAT-2008)

Directions for Questions 1 and 2: 

Let f(x) = ax² + bx + c, where a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

1. What is the other root of f(x) = 0?

(1)   −7        (2)   − 4         (3)     2         (4)   6        (5)   cannot be determined

2. What is the value of a + b + c?      

(1)  9          (2)   14          (3)     13          (4)   37     (5)   cannot be determined

Solution follows here:

Geometry-14 (CAT-2008)

Two circles, both of radii 1 cm, intersect such that the circumference of each one passes   through the centre of the circle of the other. What is the area (in sq cm) of the intersecting region?

(1) π/3 - √3/4            (2) 2π/3 + √3/2 (3) 4π/3 - √3/2 (4) 4π/3 + √3/2 (5) 2π/3 - √3/2

Solution follows here:

Numbers-16 (CAT-2008)

The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers say a and b, currently on the blackboard are erased and a new number a + b – 1 is written. What will be the number left on the board at the end?

(1)  820        (2) 821     (3) 781      (4) 819      (5) 780

Solution follows here:

Numbers-15 (CAT-2008)

Suppose the seed of any positive integer n is defined as follows:

seed(n) = n, if n < 10

= seed(s(n)), otherwise,

where s(n) indicates the sum of digits of n. For example, seed(7) = 7, seed(248) = seed(2+4+8) = seed(14) = seed(1+4) = seed(5) = 5 etc..

How many positive integers n, such that n < 500, will have seed(n) = 9?

 (1) 39             (2) 72                (3) 81              (4) 108                                   (5) 55

Solution follows here:

Probability-10 (IIT-2003)

A is targeting B, B and C targeting A. Probability of hitting the target by A,B and C are respectively 2/3,1/2 and 1/3. If A is hit then find the probability B but not C hits A.

Solution follows here:

Geometry-13

The lower part of the house (see the Figure) in the circle is a square, and the top is an equilateral triangle. Find a relation between the length of the side of the house and the radius of the circle:

Solution follows here:

Trigonometry-2 (CAT-2008)

In a triangle ABC, the lengths of the sides AB and AC equal 17.5cm and 9cm respectively. Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD=3cm, then what is the radius (in cm) of the circle circumscribing the triangle ABC?

(1)17.05   (2) 27.85   (3) 22.45   (4) 32.25   (5) 26.25

Solution follows here:

Puzzle-21 (IIT-2005)

A rectangle with sides (2m-1) cm X (2n-1) cm is divided in to squares by drawing parallel lines in such a way that distance between two consecutive parallel lines is 1 cm.

The number of rectangles that can be drawn on the figure such that both the length and breadth are in odd cm is:  

(a) 4^m+n+1          (b) mn(m+1)(n+1)        (c) m²n²              (d) (m+n+1)²

Solution follows here:

Puzzle-20 (IIT-1978)

Six X’s have to be placed in the squares of the following figure, such that each row contains at least one X. In how many different ways can this be done?

Solution follows here:

Probability-9 (IIT-1995)

The probability of India winning a test match against West Indies is ½. Assuming independence from match to match the probability that in a 5 match series India’s second win occurs at the third test is:  

(a) 1/8          (b) 1/4        (c) 1/2              (d) 2/3

Solution follows here:

Probability-8 (IIT-2004)

Out of first 100 natural numbers, three numbers are chosen without replacement. The probability that all these numbers are divisible both by 2 and 3 is:  

(a)4/11          (b) 4/55        (c) 4/33              (d) 4/1155

Solution follows here:

Probability-7 (IIT-2003)

For a student to qualify a competitive exam, he must pass at least two out of the three exams. The probability that he will pass the first exam is p. If he fails in one of the exams, then the probability of his passing in the next exam is p/2, otherwise it remains the same. Find the probability that the student will qualify the competitive exam.  

Solution follows here:

Probability-6 (FRM)

With a portfolio that consists of 20 well-diversified A-rated bonds and assuming a default probability of 1% per annum, what is the approximate probability of sustaining no losses on the portfolio over the next 12 months?

(a)0.82          (b) 0.8        (c) 0.85              (d) 0.99

Solution follows here:

Probability-5 (FRM)

A company has a constant 7% per year probability of default. What is the probability the company will be in default after three years?

(a)19.6%          (b) 22.5%        (c) 21%              (d) 7%

Solution follows here:

Probability-4 (FRM)

There are 10 students in a class. The probability of a student getting absent is 5%. The probability of any one student getting absent is completely independent of the presence or absence of any other student. What is the probability that exactly one student gets absent?

(a)5%          (b) 50%        (c) 32%              (d) 3%

Solution follows here:

Probability-3 (FRM)

Of all the employees at an office, 40% prefer coffee and 60% prefer tea. Of those who prefer coffee, 30% are females and of those who prefer tea, 40% are female. What is the probability that a randomly selected employee prefers coffee, given that the person selected is a female?

(a)1/3          (b) 2/3        (c) 1/9              (d) 2/9

Solution follows here:

Series-5 (CAT-2006)

Consider a sequence where nth term, t_n = n/(n+2), n=1,2,...

The value of t₃*t₄*t₅*....*t₅₃ equals:

(1) 2/495        (2) 2/477        (3) 12/55        (4) 1/1485                       (5) 1/2970

Solution follows here:

Numbers-14 (CAT-2006)

The number of solutions of the equation 2x+y=40 where both x and y are positive integers and x ≤ y is:

(1) 7                (2) 13                (3) 14              (4) 18                (5) 20

Solution follows here:

Probability-2 (IIT-JEE 2009)

A signal which can be green or red with probability 4/5 and 1/5 respectively, is received by station A, and then transmitted to station B. The probability of each station receiving the signal correctly is ¾. If the signal received at station B is green, then the probability that the original signal was green is
(a)3/5          (b) 6/7             (c)20/23                         (d)9/20
To enter your answer click on “comments” below:

P&C-10 (IIT-JEE 2009)

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1,2, and 3 only is

(a)55          (b) 66              (c)77                    (d)88

Solution follows here:

P&C-9 (Matching) (IIT-JEE 2008)

Consider all possible permutations of the letters of the word ENDEANOEL. Match the statements/expressions in Column-I with the statements/expressions in Column-II:

Column-I	Column-II
(a)   The number of permutations containing the word ENDEA is	(p) 5!
(b) The number of permutations in which the letter E occurs in the first and the last positions is	(q) 2*5!
(C) The number of permutations in which none of the letters D,L,N occurs in the last five positions is	(r) 7*5!
(d) The number of permutations in which the letters A,E,O occur only in odd positions is	(s) 21*5!

                                                           

Solution follows here: