(i) a^{m} * a^{n} = a^{m} + ^{n}
(ii) a^{m}/a^{n} = a^{m} - ^{n}
(iii) (a^{m})^{n} = a^{m} * ^{n}
(v) a^{-n} = 1/a^{n}
(vi) ^{n}$\sqrt{a}$^{m} = a^{m}/^{n}
(vii) (ab)^{m}= a^{m} ∙ b^{m}.
(viii) (a/b)^{m} = a^{m}/b^{m}
Divisibility rules:
Co prime or relatively prime: Two numbers are said to be relatively prime if they have only 1 as the common factor. E.g: (4,5); (8,9). If a number is divisible by p,q where both are co primes then it is also divisible by p*q;
The sequence of numbers like 1,2,3,4... are said to be in arithmetic progression with common difference d = 1; Generalizing this the arithmetic series is of the form: a, a+d, a+2d, a+3d... a+(n-1)d.
Common difference d is T_{n} - T_{(n-1) }i.e. next term minus previous term. This is uniform throughout.
_{ }
Properties:
Geometric Progression
The series is in geometric progression if the numbers increase or decrease by a common ratio. So the series is a, ar, ar^{2}, ar^{3}, ar^{4}.... ar^{n-1}
^{ }
If r > 1 then Sum = a ( r^{n} - 1) / ( r - 1)
If r < 1 then Sum = a ( 1 - r^{n}) / ( 1 - r )^{ }
^{ }
If an infinite geometric progression series is Sum = a / ( 1 - r )^{ }
^{ }
Calculate the Sum of factors of a number:
Step 1: Get prime factors of a number say 240
240 = 2^{4} * 3^{1} * 5^{1}
Step 2: Sum of factors formula is
240 = (2^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{4}) * (3^{0} + 3^{1}) * (5^{0} + 5^{1})
Step 3: 31*4*6 = 744
Calculate the Number of factors of a number:
Step 1: Get the prime factors of a number
240 = 2^{4} * 3^{1} * 5^{1}
Step 2: Number of factors of a number.
Number of factors = ( 4 + 1 ) * ( 1 + 1) * ( 1 + 1) = 5 * 2 * 2 = 20
Thus the powers of the numbers are increased by one and multiplied.
Calculate the sum and number of even factors of a number:
Step 1: Get the prime factors of a number
240 = 2^{4} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of even factors
240 = 2^{4} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of odd factors
30 = 2^{1} * 3^{1} * 5^{1}
^{ }
Step 2: Sum of factors formula is
30 = (2^{0} + 2^{1} ) * (3^{0} + 3^{1}) * (5^{0} + 5^{1})
New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.
Q. What is the number of projects in which Gyani alone is involved?
Uniquely equal to zero.
Uniquely equal to 1.
Uniquely equal to 4
Cannot be determined uniquely.
Ans . D
Putting the value of M in either equation, we get G + B = 17
Hence neither of two can be uniquely determined.
Q. What is the number of projects in which Medha alone is involved?
Uniquely equal to zero.
Uniquely equal to 1.
Uniquely equal to 4
Cannot be determined uniquely.
Ans . B
G + B = M + 16 Also, M + B + G + 19 = (2 × 19) – 1
i.e. (G + B) = 18 – M Thus, M + 16 = 18 – M
i.e. M = 1
There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.
Q.How many schools had none of the three viz., laboratory, library or play – ground?
Ans.d
Q.What was the ratio of schools having laboratory to those having library?
Ans.b
Q.There are 3 clubs A, B & C in a town with 40, 50 & 60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?
Ans.b
Q.Amar, Akbar, and Anthony came from the same public school in the Himalayas. Every boy in that school either fishes for trout or plays frisbee. All fishermen like snow while no frisbee player likes rain. Amar dislikes whatever Akbar likes and likes whatever Akbar dislikes. Akbar likes rain and snow. Anthony likes whatever the other two like. Who is a fisherman but not a frisbee player?
Ans.b
Q.Let x, y and z be distinct positive integers satisfying x < y < z and x + y + z = k. What is the smallest value of k that does not determine x, y, z uniquely?
Ans.d
Q.The product of all integers from 1 to 100 will have the following numbers of zeros at the end.
Ans.b
Q.A five digit number is formed using digits 1, 3, 5, 7 and 9 without repeating any one of them. What is the sum of all such possible numbers?
Ans.a
Q.Out of 100 families in the neighbourhood, 45 own radios, 75 have TVs, 25 have VCRs. Only 10 families have all three and each VCR owner also has a TV. If 25 families have radio only, how many have only TV?
Ans.c
Ghoshbabu is staying at Ghosh Housing Society, Aghosh Colony, Dighospur , Calcutta. In Ghosh Housing Society 6 persons read daily Ganashakti and 4 read Anand Bazar Patrika; in his colony there is no person who reads both. Total number of persons who read these two newspapers in Aghosh Colony and Dighospur is 52 and 200 respectively. Number of persons who read Ganashakti in Aghosh Colony and Dighospur is 33 and 121 respectively; while the persons who read Anand Bazar Patrika in Aghosh Colony and Dighospur are 32 and 117 respectively.
Q.Number of persons in Dighospur who read only Ganashakti is
Ans.b
Q.Number of persons in Aghosh Colony who read both of these newspapers is
Ans.a
Q.Number of persons in Aghosh Colony who read only one paper
Ans.c
Q.If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio
Ans.c
Q.Fourth term of an arithmetic progression is 8. What is the sum of the first 7 terms of the arithmetic progression?
Ans.c
Q. The table below shows the agewise distribution of the population of Reposia. The number of people aged below 35 years is 400 million.If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females in that age group?
82.8 million
90.8 million
80 million
90 million
Ans . B
Score more than 80% marks and move ahead else stay back and read again!