Course sections

1
The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is (1993)
2
Which is the least number that must be subtracted from 1856 so that the remainder, when divided by 7, 12, and 16, is 4? [1994]
3
The remainder obtained when a prime number greater than 6 is divided by 6 is [1995]
4
5^6-1 is divisible by a. 5 b.13 c. 31 d. None of these. breaking a larger exponent (1995)
5
Two positive integers differ by 4 and sum of their reciprocals is 10/21, then one of the numbers is a.  3 b.  1 c.  5 d. 21 [1995]
6
Three consecutive positive even numbers are such that thrice the first number exceeds double the third by 2, then the third number is   [1995].
7
If a number 774958A96B is to be divisible by 8 and 9 (1996)
8
If m and n are integers divisible by 5, which of the following is not necessarily true? a.  m – n is divisible by 5 [1997]
9
Consider a sequence, where the nth term is tn = n/n 2. The value of t3 * t4 * t5 * … t53 equals(2000)
10
A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour? ( 2001)
11
7^6n- 6^6n, where n is an integer >0, is divisible by (2002)
12
A child was asked to add first few natural numbers (that is, 1 2 3……) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong the child discovered he had missed one number in the sequence during addition. The number he missed was: (2002)
13
What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7 ? ( 2003)
14
What is the remainder when 4^96 is divided by 6 ? (2003)
15
Which among 2^1/2, 3^1/3, 4^1/4, 12^/12 is the largest? (2004)
16
The remainder, when 15^23 23^23 is divided by 19 is? a.4        b.15        c.0        d.18 (2004)
17
Let S be a set of positive integers such that every element n of S satisfies the conditions (2005)
18
When you reverse the digits of the number 13, the number increases by 18. How many other 2 digit numbers increase by 18 when their digits are reversed?  ( 2006)
19
a/b=1/3, b/c=2, c/d=1/2, d/e=3 then what is the value of abc/def? (2006)
20
Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many are such numbers perfect squares? (2007)
21
how-many-pairs-of-positive-integers-m-n-satisfy 1/m 4/n=1/12, where n is an odd number less than 60…. (2007)
22
What is the number of distinct terms in the expansion of (a b c)^20? (2008)
23
What is the remainder when 7^74 
24
In a tournament, there are n teams T1, T2 ….., Tn with n > 5. Each team consists of k players, k > 3… (2007)
25
a^2-ab b^2/a^2 ab b^2=1/3 then find a/b (CAT, 2009)
26
Let Sn denote the sum of squares of the first n odd natural numbers. If Sn = 533n, find the value of n. (2011)
27
The value of numbers  2^2004  and 5^2004 are written one after another. How many digits are there in all? (2011)
28
If p be a prime number, p>3 and let x be the product of positive number 1, 2, 3,…,(p-1), then consider the following statements (2012)
29
What is the sum of all two digit numbers which leave a remainder of 6 when divided by 8? (CAT, 2012)
30
Find the remainder of 21040 divided by 131. (CAT, 2012)
31
A three-digit number which on being subtracted from another three-digit number consisting of the same digits in reverse order gives 594. The minimum possible sum of all the three digits of this number is (2013)
32
If a1=1 and a(n 1)-3an 2=4n for every positive integer n, then a100 equals ? (2014)
33
Suppose n is an integer, such that the sum of the digits of n is 2, and 10^10<N<10^11
34
A sequence of 4 digits, when considered as a number in base 10 is four times the number it represents in base 6. What is the sum of the digits of the sequence? (2016)
35
What is the greatest power of 5 that divides 80! exactly (2016)
36
If 92x-1- 81x-1= 1944, then x is (2017)
37
The right most non-zero digit of 30^2720 is? (2005)
38
Each family in a locality has at most two adults, and no family has fewer than three children. .. (2004)
39
Let x and y be positive integers such that x is prime and y is composite. (2003)
40
If sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms? (2004)
41
The number of common terms in the two sequences 17, 21, 25, …, 417 and 16, 21, 26, …, 466 is (2008)
42
What is the remainder when 7 ^ 74 – 5 ^74 is divided by 4? (2008)
43
if a^2-ab b^2/a^2 ab b^2=1/3, then find (2009)
44
The largest number among the following that will perfectly… (2010)
45
How many integers, greater than 999 but not greater than 4000…(2010)
46
Find the smallest number which when increased by 5 is exactly divisible by 8, 11 and 24. (1994)
47
A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number (1997)
48
If n^3 is odd, which of the following statements are true? (1998)
49
What is the digit in unit’s place of 2^51? (1998)
50
If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? (1999)
51
The integers 34041 and 32506, when divided by a three digit integer n, leave the same remainder. What is the value of n? ( 2000)
52
Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product? (2001)

Section 1

if a^2-ab b^2/a^2 ab b^2=1/3, then find (2009)

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If  \frac { a^{ 2 }-ab+b^{ 2 } }{ a^{ 2 }+ab+b^{ 2 } } =\frac { 1 }{ 3 } , then find \frac { a }{ b } .

a.1       b.2          c.3        d. 4  (2009)

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Solution:

Algebraic Identities:

\frac { a^{ 2 }-ab+b^{ 2 } }{ a^{ 2 }+ab+b^{ 2 } } =\frac { 1 }{ 3 }

Firstly we will do cross multiplication.
3(a^{ 2 }-ab+b^{ 2 })=1(a^{ 2 }+ab+b^{ 2 })
3a^{ 2 }-3ab+3b^{ 2 }=a^{ 2 }+ab+b^{ 2 }
Note: When we multiply a number ‘a’ by 1, we again get that number because 1 is called the Multiplicative Identity in mathematics.

 

Ex: 2×1=2, we get 2 again,  ax1=a, we get the number ‘a’ again.

 

3a^{ 2 }-a^{ 2 }-3ab-ab+3b^{ 2 }+b^{ 2 }=0

 

On solving, we get:
 
2a^{ 2 }-4ab+2b^{ 2 }=0
 
Now take 2 as common and the above equation changes into the following form:
 
2(a^{ 2 }-2ab+b^{ 2 })=0

 

 
We know that { (a-b })^{ 2 }=a^{ 2 }-2ab+b^{ 2 }.
 
So we will put (a-b)2  = a2-2ab+b in the above equation to solve it.
2(a-b)2 =0 

 

On transposing 2 to the Right-hand side(RHS), it becomes:

 

{ (a-b) }^{ 2 }=\frac { 0 }{ 2 }    and the equation has now ‘0’ on its right hand side(RHS).

 

{ (a-b) }^{ 2 }=0 because 0 divided by 2 gives 0 again.

 

To know why division of zero by a number or by zero of a number gives 0 again, click here.

 

 Since { (a-b) }^{ 2 }= (a-b) x (a-b), we can write the above equation in the following form:

 

(a-b) x (a-b)=0

 

Now transpose (a-b) to the right hand side, the equation becomes: { (a-b) }=\frac { 0 }{ (a-b) }

 

 Now, the division by the expression (a-b) again gives ‘0’ on the right-hand side (RHS).

 

And we have (a-b) =0,

 

Now if we transpose b to the right-hand side (LHS), it becomes:

 

a=b

 

we are now left with 1xa=bx1 

 

 (The number 1 has appeared on both hand sides because they are now in multiplication. It is because the equation was 1x(a-b)=0 and we had no number on the right-hand side. When b was transposed there, the equation became 1xa=0+bx1,  since 0+n=n, we only have 1xa=bx1.)

 

So, b now can become a denominator of a by giving us 1 on the right-hand side.

 

\frac { a }{ b } =\frac { 1 }{ 1 }     or    \frac { a }{ b }=1

 

 Hence, the answer is 1 and the option (a) is correct.
 

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