Thursday, January 27, 2011

IMPORTANT FACTS AND FORMULAE

Square Root: If x² = y, we say that the square root of y is x and we write, √y = x.

Thus, √4 = 2, √9 = 3, √196 = 14.

Cube Root: The cube root of a given number x is the number whose cube is x. We denote the cube root of x by ³√x.

Thus, ³√8 = ³√2 x 2 x 2 = 2, ³√343 = ³√7 x 7 x 7 = 7 etc.

Note:

1.√xy = √x * √y 2. √(x/y) = √x / √y = (√x / √y) * (√y / √y) = √xy / y

SOLVED EXAMPLES

Ex. 1. Evaluate √6084 by factorization method .

Sol. Method: Express the given number as the product of prime factors. 2 6084

Now, take the product of these prime factors choosing one out of 2 3042

every pair of the same primes. This product gives the square root 3 1521

of the given number. 3 507

Thus, resolving 6084 into prime factors, we get: 13 169

6084 = 2² x 3² x 13² 13 \ √6084 = (2 x 3 x 13) = 78.

Ex. 2. Find the square root of 1471369.

Sol. Explanation: In the given number, mark off the digits 1 1471369 (1213

in pairs starting from the unit's digit. Each pair and 1

the remaining one digit is called a period. 22 47

Now, 1² = 1. On subtracting, we get 0 as remainder. 44

Now, bring down the next period i.e., 47. 241 313

Now, trial divisor is 1 x 2 = 2 and trial dividend is 47. 241

So, we take 22 as divisor and put 2 as quotient. 2423 7269

The remainder is 3. 7269

Next, we bring down the next period which is 13. x

Now, trial divisor is 12 x 2 = 24 and trial dividend is

313. So, we take 241 as dividend and 1 as quotient.

The remainder is 72.

Bring down the next period i.e., 69.

Now, the trial divisor is 121 x 2 = 242 and the trial

dividend is 7269. So, we take 3as quotient and 2423

as divisor. The remainder is then zero.

Hence, √1471369 = 1213.

Ex. 3. Evaluate: √248 + √51 + √ 169 .

Sol. Given expression = √248 + √51 + 13 = √248 + √64 = √ 248 + 8 = √256 = 16.

Ex. 4. If a * b * c = √(a + 2)(b + 3) / (c + 1), find the value of 6 * 15 * 3.

Sol. 6 * 15 * 3 = √(6 + 2)(15 + 3) / (3 + 1) = √8 * 18 / 4 = √144 / 4 = 12 / 4 = 3.

Ex. 5. Find the value of √25/16.

Sol. √ 25 / 16 = √ 25 / √ 16 = 5 / 4

Ex. 6. What is the square root of 0.0009?

Sol. √0.0009= √ 9 / 1000 = 3 / 100 = 0.03.

Ex. 7. Evaluate √175.2976.

Sol. Method: We make even number of decimal places 1 175.2976 (13.24

by affixing a zero, if necessary. Now, we mark off 1

periods and extract the square root as shown. 23 75

\√175.2976 = 13.24 262 629

524

2644 10576

10576

Ex. 8. What will come in place of question mark in each of the following questions?

(i) √32.4 / ? = 2 (ii) √86.49 + √ 5 + ( ? )² = 12.3.

Sol. (i) Let √32.4 / x = 2. Then, 32.4/x = 4 <=> 4x = 32.4 <=> x = 8.1.

(ii) Let √86.49 + √5 + x² = 12.3.

Then, 9.3 + √5+x² = 12.3 <=> √5+x²= 12.3 - 9.3 = 3

<=> 5 + x²= 9 <=> x²= 9 - 5= 4 <=> x = √4 = 2.

Ex.9. Find the value of √ 0.289 / 0.00121.

Sol. √0.289 / 0.00121 = √0.28900/0.00121 = √28900/121 = 170 / 11.

Ex.10. If √1 + (x / 144) = 13 / 12, the find the value of x.

Sol. √1 + (x / 144) = 13 / 12 Þ ( 1 + (x / 144)) = (13 / 12 )²= 169 / 144

Þx / 144 = (169 / 144) - 1

Þx / 144 = 25/144 Þ x = 25.

Ex. 11. Find the value of √3 up to three places of decimal.

Sol.

1 3.000000 (1.732

27 200

189

343 1100

1029

3462 7100

6924 \√3 = 1.732.

Ex. 12. If √3 = 1.732, find the value of √192 - 1 √48 - √75 correct to 3 places

of decimal. (S.S.C. 2004)

Sol. √192 - (1 / 2)√48 - √75 = √64 * 3 - (1/2) √ 16 * 3 - √ 25 * 3

=8√3 - (1/2) * 4√3 - 5√3

=3√3 - 2√3 = √3 = 1.732

Ex. 13. Evaluate: √(9.5 * 0.0085 * 18.9) / (0.0017 * 1.9 * 0.021)

Sol. Given exp. = √(9.5 * 0.0085 * 18.9) / (0.0017 * 1.9 * 0.021)

Now, since the sum of decimal places in the numerator and denominator under the radical sign is the same, we remove the decimal.

\ Given exp = √(95 * 85 * 18900) / (17 * 19 * 21) = √ 5 * 5 * 900 = 5 * 30 = 150.

Ex. 14. Simplify: √ [( 12.1 )² - (8.1)²] / [(0.25)² + (0.25)(19.95)]

Sol. Given exp. = √ [(12.1 + 8.1)(12.1 - 8.1)] / [(0.25)(0.25 + 19.95)]

=√ (20.2 * 4) /( 0.25 * 20.2) = √ 4 / 0.25 = √400 / 25 = √16 = 4.

Ex. 15. If x = 1 + √2 and y = 1 - √2, find the value of (x² + y²).

Sol. x² + y² = (1 + √2)²+ (1 - √2)² = 2[(1)²+ (√2)²] = 2 * 3 = 6.

Ex. 16. Evaluate: √0.9 up to 3 places of decimal.

Sol.

9 0.900000(0.948

184 900

736

1888 16400

15104 \√0.9 = 0.948

Ex.17. If √15 = 3.88, find the value of √ (5/3).

Sol.

√ (5/3) = √(5 * 3) / (3 * 3) = √15 / 3 = 3.88 / 3 = 1.2933…. = 1.293.

Ex. 18. Find the least square number which is exactly divisible by 10,12,15 and 18.

Sol. L.C.M. of 10, 12, 15, 18 = 180. Now, 180 = 2 * 2 * 3 * 3 *5 = 2² * 3²* 5.

To make it a perfect square, it must be multiplied by 5.

\ Required number = (2² * 3² * 5²) = 900.

Ex. 19. Find the greatest number of five digits which is a perfect square.

(R.R.B. 1998)

Sol. Greatest number of 5 digits is 99999.

3 99999(316

61 99

626 3899

3756

143

\ Required number == (99999 - 143) = 99856.

Ex. 20. Find the smallest number that must be added to 1780 to make it a perfect

square.

Sol.

4 1780 (42

82 180

164

\ Number to be added = (43)² - 1780 = 1849 - 1780 = 69.

Ex. 21. √2 = 1.4142, find the value of √2 / (2 + √2).

Sol. √2 / (2 + √2) = √2 / (2 + √2) * (2 - √2) / (2 - √2) = (2√2 – 2) / (4 – 2)

= 2(√2 – 1) / 2 = √2 – 1 = 0.4142.

22. If x = (√5 + √3) / (√5 - √3) and y = (√5 - √3) / (√5 + √3), find the value of (x²+ y²).

Sol.

x = [(√5 + √3) / (√5 - √3)] * [(√5 + √3) / (√5 + √3)] = (√5 + √3)² / (5 - 3)

=(5 + 3 + 2√15) / 2 = 4 + √15.

y = [(√5 - √3) / (√5 + √3)] * [(√5 - √3) / (√5 - √3)] = (√5 - √3)² / (5 - 3)

=(5 + 3 - 2√15) / 2 = 4 - √15.

\ x²+ y² = (4 + √15)²+ (4 - √15)² = 2[(4)² + (√15)²] = 2 * 31 = 62.

Ex. 23. Find the cube root of 2744.

Sol. Method: Resolve the given number as the product 2 2744

of prime factors and take the product of prime 2 1372

factors, choosing one out of three of the same 2 686

prime factors. Resolving 2744 as the product of 7 343

prime factors, we get: 7 49

2744 = 2³ x 7³.

\ ³√2744= 2 x 7 = 14.

Ex. 24. By what least number 4320 be multiplied to obtain a number which is a perfect cube?

Sol. Clearly, 4320 = 2³ * 3³ * 2² * 5.

To make it a perfect cube, it must be multiplied by 2 * 5²i.e,50.

Sunday, January 16, 2011

NUMBERS

IMPORTANT FACTS AND FORMULAE

I..Numeral : In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number.

A group of digits, denoting a number is called a numeral.

We represent a number, say 689745132 as shown below :

Ten Crores (108) Crores(107) Ten Lacs (Millions) (106) Lacs(105) Ten Thousands (104) Thousands (103) Hundreds (102) Tens(101) Units(100)

6 8 9 7 4 5 1 3 2

We read it as : 'Sixty-eight crores, ninety-seven lacs, forty-five thousand, one hundred and thirty-two'.

II Place Value or Local Value of a Digit in a Numeral :

In the above numeral :

Place value of 2 is (2 x 1) = 2; Place value of 3 is (3 x 10) = 30;

Place value of 1 is (1 x 100) = 100 and so on.

Place value of 6 is 6 x 108 = 600000000

III.Face Value : The face value of a digit in a numeral is the value of the digit itself at whatever place it may be. In the above numeral, the face value of 2 is 2; the face value of 3 is 3 and so on.

IV.TYPES OF NUMBERS

1.Natural Numbers : Counting numbers 1, 2, 3, 4, 5,..... are called natural

numbers.

2.Whole Numbers : All counting numbers together with zero form the set of whole

numbers. Thus,

(i) 0 is the only whole number which is not a natural number.

(ii) Every natural number is a whole number.

3.Integers : All natural numbers, 0 and negatives of counting numbers i.e.,

{…, -3,-2,-1, 0, 1, 2, 3,…..} together form the set of integers.

(i) Positive Integers : {1, 2, 3, 4, …..} is the set of all positive integers.

(ii) Negative Integers : {- 1, - 2, - 3,…..} is the set of all negative integers.

(iii) Non-Positive and Non-Negative Integers : 0 is neither positive nor

negative. So, {0, 1, 2, 3,….} represents the set of non-negative integers, while

{0, -1,-2,-3,…..} represents the set of non-positive integers.

4. Even Numbers : A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8, 10, etc.

5. Odd Numbers : A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc.

6. Prime Numbers : A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

Prime numbers upto 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,

47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Prime numbers Greater than 100 : Letp be a given number greater than 100. To find out whether it is prime or not, we use the following method :

Find a whole number nearly greater than the square root of p. Let k > *jp. Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime.

e.g,,We have to find whether 191 is a prime number or not. Now, 14 > V191.

Prime numbers less than 14 are 2, 3, 5, 7, 11, 13.

191 is not divisible by any of them. So, 191 is a prime number.

7.Composite Numbers : Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4, 6, 8, 9, 10, 12.

Note : (i) 1 is neither prime nor composite.

(ii) 2 is the only even number which is prime.

(iii) There are 25 prime numbers between 1 and 100.

8. Co-primes : Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,

V.TESTS OF DIVISIBILITY

1. Divisibility By 2 : A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.

Ex. 84932 is divisible by 2, while 65935 is not.

2. Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible by 3.

Ex.592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3.

But, 864329 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) = 32, which is not divisible by 3.

3. Divisibility By 4 : A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

Ex. 892648 is divisible by 4, since the number formed by the last two digits is

48, which is divisible by 4.

But, 749282 is not divisible by 4, since the number formed by the last tv/o digits is 82, which is not divisible by 4.

4. Divisibility By 5 : A number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.

5. Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3. Ex. The number 35256 is clearly divisible by 2.

Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.

6. Divisibility By 8 : A number is divisible by 8, if the number formed by the last

three digits of the given number is divisible by 8.

Ex. 953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8.

But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.

7. Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible

by 9.

Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9.

But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by 9.

8. Divisibility By 10 : A number is divisible by 10, if it ends with 0.

Ex. 96410, 10480 are divisible by 10, while 96375 is not.

9. Divisibility By 11 : A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.

Ex. The number 4832718 is divisible by 11, since :

(sum of digits at odd places) - (sum of digits at even places)

- (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11.

10. Divisibility By 12 ; A number is divisible by 12, if it is divisible by both 4 and

3.

Ex. Consider the number 34632.

(i) The number formed by last two digits is 32, which is divisible by 4,

(ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.

11. Divisibility By 14 : A number is divisible by 14, if it is divisible by 2 as well as 7.

12. Divisibility By 15 : A number is divisible by 15, if it is divisible by both 3 and 5.

13. Divisibility By 16 : A number is divisible by 16, if the number formed by the last4 digits is divisible by 16.

Ex.7957536 is divisible by 16, since the number formed by the last four digits is 7536, which is divisible by 16.

14. Divisibility By 24 : A given number is divisible by 24, if it is divisible by both3 and 8.

15. Divisibility By 40 : A given number is divisible by 40, if it is divisible by both

5 and 8.

16. Divisibility By 80 : A given number is divisible by 80, if it is divisible by both 5 and 16.

Note : If a number is divisible by p as well as q, where p and q are co-primes, then the given number is divisible by pq.

If p arid q are not co-primes, then the given number need not be divisible by pq,

even when it is divisible by both p and q.

Ex. 36 is divisible by both 4 and 6, but it is not divisible by (4x6) = 24, since

4 and 6 are not co-primes.

VI MULTIPLICATION BY SHORT CUT METHODS

1. Multiplication By Distributive Law :

(i) a x (b + c) = a x b + a x c (ii) ax(b-c) = a x b-a x c.

Ex. (i) 567958 x 99999 = 567958 x (100000 - 1)

= 567958 x 100000 - 567958 x 1 = (56795800000 - 567958) = 56795232042. (ii) 978 x 184 + 978 x 816 = 978 x (184 + 816) = 978 x 1000 = 978000.

2. Multiplication of a Number By 5n : Put n zeros to the right of the multiplicand and divide the number so formed by 2n

Ex. 975436 x 625 = 975436 x 54= 9754360000 = 609647600

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VII. BASIC FORMULAE

1. (a + b)2 = a2 + b2 + 2ab 2. (a - b)2 = a2 + b2 - 2ab

3. (a + b)2 - (a - b)2 = 4ab 4. (a + b)2 + (a - b)2 = 2 (a2 + b2)

5. (a2 - b2) = (a + b) (a - b)

6. (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)

7. (a3 + b3) = (a +b) (a2 - ab + b2) 8. (a3 - b3) = (a - b) (a2 + ab + b2)

9. (a3 + b3 + c3 -3abc) = (a + b + c) (a2 + b2 + c2 - ab - bc - ca)

10. If a + b + c = 0, then a3 + b3 + c3 = 3abc.

VIII. DIVISION ALGORITHM OR EUCLIDEAN ALGORITHM

If we divide a given number by another number, then :

Dividend = (Divisor x Quotient) + Remainder

IX. {i) (xn - an ) is divisible by (x - a) for all values of n.

(ii) (xn - an) is divisible by (x + a) for all even values of n.

(iii) (xn + an) is divisible by (x + a) for all odd values of n.

X. PROGRESSION

A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression.

1. Arithmetic Progression (A.P.) : If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P.

An A.P. with first term a and common difference d is given by a, (a + d), (a + 2d),(a + 3d),.....

The nth term of this A.P. is given by Tn =a (n - 1) d.

The sum of n terms of this A.P.

Sn = n/2 [2a + (n - 1) d] = n/2 (first term + last term).

SOME IMPORTANT RESULTS :

(i) (1 + 2 + 3 +…. + n) =n(n+1)/2

(ii) (l2 + 22 + 32 + ... + n2) = n (n+1)(2n+1)/6

(iii) (13 + 23 + 33 + ... + n3) =n2(n+1)2

2. Geometrical Progression (G.P.) : A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometrical progression.

The constant ratio is called the common ratio of the G.P. A G.P. with first term a and common ratio r is :

a, ar, ar2,

In this G.P. Tn = arn-1

sum of the n terms, Sn= a(1-rn)

(1-r)

SOLVED EXAMPLES

Ex. 1. Simplify : (i) 8888 + 888 + 88 + 8

(ii) 11992 - 7823 - 456

Sol. i ) 8888 ii) 11992 - 7823 - 456 = 11992 - (7823 + 456)

888 = 11992 - 8279 = 3713-

88 7823 11992

+ 8 + 456 - 8279

9872 8279 3713

Ex. 2, What value will replace the question mark in each of the following equations ?

(i) ? - 1936248 = 1635773 (ii) 8597 - ? = 7429 - 4358

Sol. (i) Let x - 1936248=1635773.Then, x = 1635773 + 1936248=3572021. (ii) Let 8597 - x = 7429 - 4358.

Then, x = (8597 + 4358) - 7429 = 12955 - 7429 = 5526.

Ex. 3. What could be the maximum value of Q in the following equation? 5P9 + 3R7 + 2Q8 = 1114

Sol. We may analyse the given equation as shown : 1 2

Clearly, 2 + P + R + Q = ll. 5 P 9

So, the maximum value of Q can be 3 R 7

(11 - 2) i.e., 9 (when P = 0, R = 0); 2 Q 8

11 1 4

Ex. 4. Simplify : (i) 5793405 x 9999 (ii) 839478 x 625

Sol.

i)5793405x9999=5793405(10000-1)=57934050000-5793405=57928256595.b

ii) 839478 x 625 = 839478 x 54 = 8394780000 = 524673750.

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Ex. 5. Evaluate : (i) 986 x 237 + 986 x 863 (ii) 983 x 207 - 983 x 107

Sol.

(i) 986 x 137 + 986 x 863 = 986 x (137 + 863) = 986 x 1000 = 986000.

(ii) 983 x 207 - 983 x 107 = 983 x (207 - 107) = 983 x 100 = 98300.

Ex. 6. Simplify : (i) 1605 x 1605 ii) 1398 x 1398

Sol.

i) 1605 x 1605 = (1605)2 = (1600 + 5)2 = (1600)2 + (5)2 + 2 x 1600 x 5

= 2560000 + 25 + 16000 = 2576025.

(ii) 1398 x 1398 - (1398)2 = (1400 - 2)2= (1400)2 + (2)2 - 2 x 1400 x 2

=1960000 + 4 - 5600 = 1954404.

Ex. 7. Evaluate : (313 x 313 + 287 x 287).

Sol.

(a2 + b2) = 1/2 [(a + b)2 + (a- b)2]

(313)2 + (287)2 = 1/2 [(313 + 287)2 + (313 - 287)2] = ½[(600)2 + (26)2]

= 1/2 (360000 + 676) = 180338.

Ex. 8. Which of the following are prime numbers ?

(i) 241 (ii) 337 (Hi) 391 (iv) 571

Sol.

(i) Clearly, 16 > Ö241. Prime numbers less than 16 are 2, 3, 5, 7, 11, 13.

241 is not divisible by any one of them.

241 is a prime number.

(ii) Clearly, 19>Ö337. Prime numbers less than 19 are 2, 3, 5, 7, 11,13,17.

337 is not divisible by any one of them.

337 is a prime number.

(iii) Clearly, 20 > Ö39l". Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19.

We find that 391 is divisible by 17.

391 is not prime.

(iv) Clearly, 24 > Ö57T. Prime numbers less than 24 are 2, 3, 5, 7, 11, 13, 17, 19, 23.

571 is not divisible by any one of them.

571 is a prime number.

Ex. 9. Find the unit's digit in the product (2467)163 x (341)72.

Sol. Clearly, unit's digit in the given product = unit's digit in 7153 x 172.

Now, 74 gives unit digit 1.

7152 gives unit digit 1,

 7153 gives unit digit (l x 7) = 7. Also, 172 gives unit digit 1.

Hence, unit's digit in the product = (7 x 1) = 7.

Ex. 10. Find the unit's digit in (264)102 + (264)103

Sol. Required unit's digit = unit's digit in (4)102 + (4)103.

Now, 42 gives unit digit 6.

(4)102 gives unjt digit 6.

(4)103 gives unit digit of the product (6 x 4) i.e., 4.

Hence, unit's digit in (264)m + (264)103 = unit's digit in (6 + 4) = 0.

Ex. 11. Find the total number of prime factors in the expression (4)11 x (7)5 x (11)2.

Sol. (4)11x (7)5 x (11)2 = (2 x 2)11 x (7)5 x (11)2 = 211 x 211 x75x 112 = 222 x 75 x112

Total number of prime factors = (22 + 5 + 2) = 29.

Ex.12. Simplify : (i) 896 x 896 - 204 x 204

(ii) 387 x 387 + 114 x 114 + 2 x 387 x 114

(iii) 81 X 81 + 68 X 68-2 x 81 X 68.

Sol.

(i) Given exp = (896)2 - (204)2 = (896 + 204) (896 - 204) = 1100 x 692 = 761200.

(ii) Given exp = (387)2+ (114)2+ (2 x 387x 114)

= a2 + b2 + 2ab, where a = 387,b=114

= (a+b)2 = (387 + 114 )2 = (501)2 = 251001.

(iii) Given exp = (81)2 + (68)2 – 2x 81 x 68 = a2 + b2 – 2ab,Where a =81,b=68

= (a-b)2 = (81 –68)2 = (13)2 = 169.

Ex.13. Which of the following numbers is divisible by 3 ?

(i) 541326 (ii) 5967013

Sol.

(i) Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3.

Hence, 541326 is divisible by 3.

(ii) Sum of digits in 5967013 =(5+9 + 6 + 7 + 0+1 +3) = 31, which is not divisible by 3.

Hence, 5967013 is not divisible by 3.

Ex.14.What least value must be assigned to * so that the number 197*5462 is r 9 ?

Sol.

Let the missing digit be x.

Sum of digits = (1 + 9 + 7 + x + 5 + 4 + 6 +»2) = (34 + x).

For (34 + x) to be divisible by 9, x must be replaced by 2 .

Hence, the digit in place of * must be 2.

Ex. 15. Which of the following numbers is divisible by 4 ?

(i) 67920594 (ii) 618703572

Sol.

(i) The number formed by the last two digits in the given number is 94, which is not divisible by 4.

Hence, 67920594 is not divisible by 4.

(ii) The number formed by the last two digits in the given number is 72, which is divisible by 4.

Hence, 618703572 is divisible by 4.

Ex. 16. Which digits should come in place of * and $ if the number 62684*$ is divisible by both 8 and 5 ?

Sol.

Since the given number is divisible by 5, so 0 or 5 must come in place of $. But, a number ending with 5 is never divisible by 8. So, 0 will replace $.

Now, the number formed by the last three digits is 4*0, which becomes divisible by 8, if * is replaced by 4.

Hence, digits in place of * and $ are 4 and 0 respectively.

Ex. 17. Show that 4832718 is divisible by 11.

Sol. (Sum of digits at odd places) - (Sum of digits at even places)

= (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11.

Hence, 4832718 is divisible by 11.

Ex. 18. Is 52563744 divisible by 24 ?

Sol. 24 = 3 x 8, where 3 and 8 are co-primes.

The sum of the digits in the given number is 36, which is divisible by 3. So, the given number is divisible by 3.

The number formed by the last 3 digits of the given number is 744, which is divisible by 8. So, the given number is divisible by 8.

Thus, the given number is divisible by both 3 and 8, where 3 and 8 are co-primes.

So, it is divisible by 3 x 8, i.e., 24.

Ex. 19. What least number must be added to 3000 to obtain a number exactly divisible by 19 ?

Sol. On dividing 3000 by 19, we get 17 as remainder.

Number to be added = (19 - 17) = 2.

Ex. 20. What least number must be subtracted from 2000 to get a number exactly divisible by 17 ?

Sol. On dividing 2000 by 17, we get 11 as remainder.

Required number to be subtracted = 11.

Ex. 21. Find the number which is nearest to 3105 and is exactly divisible by 21.

Sol. On dividing 3105 by 21, we get 18 as remainder.

Number to be added to 3105 = (21 - 18) - 3.

Hence, required number = 3105 + 3 = 3108.

Ex. 22. Find the smallest number of 6 digits which is exactly divisible by 111.

Sol. Smallest number of 6 digits is 100000.

On dividing 100000 by 111, we get 100 as remainder.

Number to be added = (111 - 100) - 11.

Hence, required number = 100011.-

Ex. 23. On dividing 15968 by a certain number, the quotient is 89 and the remainder is 37. Find the divisor.

Dividend - Remainder 15968-37

Sol. Divisor = -------------------------- = ------------- = 179.

.Quotient 89

Ex. 24. A number when divided by 342 gives a remainder 47. When the same number ift divided by 19, what would be the remainder ?

Sol. On dividing the given number by 342, let k be the quotient and 47 as remainder.

Then, number – 342k + 47 = (19 x 18k + 19 x 2 + 9) = 19 (18k + 2) + 9.

The given number when divided by 19, gives (18k + 2) as quotient and 9 as remainder.

Ex. 25. A number being successively divided by 3, 5 and 8 leaves remainders 1, 4

and 7 respectively. Find the respective remainders if the order of divisors be reversed,

Sol.

3 X

5 y - 1

8 z - 4

1 - 7

z = (8 x 1 + 7) = 15; y = {5z + 4) = (5 x 15 + 4) = 79; x = (3y + 1) = (3 x 79 + 1) = 238.

Now,

8 238

5 29 - 6

3 5 - 4

1 - 9,

Respective remainders are 6, 4, 2.

Ex. 26. Find the remainder when 231 is divided by 5.

Sol. 210 = 1024. Unit digit of 210 x 210 x 210 is 4 [as 4 x 4 x 4 gives unit digit 4].

Unit digit of 231 is 8.

Now, 8 when divided by 5, gives 3 as remainder.

Hence, 231 when divided by 5, gives 3 as remainder.

Ex. 27. How many numbers between 11 and 90 are divisible by 7 ?

Sol. The required numbers are 14, 21, 28, 35, .... 77, 84.

This is an A.P. with a = 14 and d = (21 - 14) = 7.

Let it contain n terms.

Then, Tn = 84 => a + (n - 1) d = 84

=> 14 + (n - 1) x 7 = 84 or n = 11.

Required number of terms = 11.

Ex. 28. Find the sum of all odd numbers upto 100.

Sol. The given numbers are 1, 3, 5, 7, ..., 99.

This is an A.P. with a = 1 and d = 2.

Let it contain n terms. Then,

1 + (n - 1) x 2 = 99 or n = 50.

Required sum = n (first term + last term)

2

= 50 (1 + 99) = 2500.

2

Ex. 29. Find the sum of all 2 digit numbers divisible by 3.

Sol. All 2 digit numbers divisible by 3 are :

12, 51, 18, 21, ..., 99.

This is an A.P. with a = 12 and d = 3.

Let it contain n terms. Then,

12 + (n - 1) x 3 = 99 or n = 30.

Required sum = 30 x (12+99) = 1665.

2

Ex.30.How many terms are there in 2,4,8,16……1024?

Sol.Clearly 2,4,8,16……..1024 form a GP. With a=2 and r = 4/2 =2.

Let the number of terms be n . Then

2 x 2n-1 =1024 or 2n-1 =512 = 29.

n-1=9 or n=10.

Ex. 31. 2 + 22 + 23 + ... + 28 = ?

Sol. Given series is a G.P. with a = 2, r = 2 and n = 8.

sum = a(rn-1) = 2 x (28 –1) = (2 x 255) =510

(r-1) (2-1)

Friday, December 31, 2010

R.S.AGARWAL 09

Synonyms- E

1)Encumber:Burden

Usage:Some people encumber themselves with too much luggage
when they take short trips.

2)Endearment:Fond word or act

Usage:Your gifts and endearments can't make me forget your
earlier insolence.

3)Endure:Provide with some Quality , endow.

Usage ; He was endued with a lions courage.

4)Enduring:asting , surviving.

Usage:Keats believed in the enduring power of great art,
which would out last its creators brief lives.

5)Eminent:Lofty , Conspicious , celebrated,remarkable .

UsageV:VThis award will be given away by an eminent
person.

6) Enervate:Weaken

Usage:She was slow to recover from her illness;even
a sheet walk to the window enervated her

7)Engross:Occupy fully

Usage:John was so engrossed in the studies thst he
didn't he hear his mother call

8)Engima:Puzzle , Mystery

Usage:Their behaviour was an engima to him

9)Ennui:Borden

Usage:The monotous routine of hospital life induced
a feeling of ennui that amde her moodyard irritable.

10)Enormity:Hugeness

Usage:He didn't realize the enormity of his crime
untill he saw what suffering he had caused.

11)Enrapture:Please intensely

Usage:The audience was enraptured by the freshness
of the voices the excellent orchestration.

12)Ensconce:Settle comfortably

Usage:Now that children were ensconced safely in
the private school.

13)Ensue: Follow

Usage:he evils that ensured were direct result of
the miscalculations of the leaders.

14)Entail:Require , necessitate ;involve

Usage: Building a college level vocabulary will
entail some work on your part.

15)Entreat:Plead , ask earnestly.

Usage: She entreated her father to let her stay
out till midnight.

16)Eon:Long peroid of time , on age

Usage:It has taken Eons for our civilization to
develop.

17)Enthrall:Capture ,enslave

Usage: From the moment he saw her picture, he was
enthralled by her beauty.

18)Epiram:

Usage:Witty thought or saying usually short.poor
richards epigrams made Benjamin frankline famous.

19)Epilogue:Short speech at conclusion of dramatic
work.

Usage:The audience was so disappointed in the
play that many did not remain to heaer the epilogue.

20)Erode:Eat away

Usage:The limestoen was eroded by the dripping
water until only a thin shell remained.

21)Erotic :Pertaining to passionate love.

Usage:The erotic passages in this novel should
be removed as they are merely pornographic.

22)Erroneous :Mistaken ,wrong

Usage:I thought my answer was correct , but it
was erroneous.

23)Eshew:Avoid

Usage:Hoping to present himself to his girlfriend
as a totally reformed character ,he tried to eshew
all this vices ,especially chewing tobacco
and drinking bathtub gin.

24)Escapade:Prank , Flighty conduct.

Usage:The headmaster could not regard this latest
escapade as a byish joke and expelled the young man.

25)Espouse:adopt,support.

Usage:She was always ready to espouse a worthy cause.

R.S.AGARWAL 08

Synonyms- D

1)Dabble : Work at in a non serious fashion ,splash around

Usage:The amateur painter dabbled at art , but seldom produced
a finished pece.

2)Dally :Tritle with.

Usage:Lourtes told Ophelia that Hamlet would only dally with
her affections.

3)Dank :Damp

Usage:The walls of the dungen were dank and slimy

4)Dappled : Neat and trim

Usage:The odd couple Nimmy played Felix , an excessively
dapper soul who could not stand to have a hair out the wall.

5)Daub :Smear

Usage: From the way he daubed his paint on the canvas, I
could tell he knew nothing of oils.

6)Daunt :frighten

Usage:'Boast all you like your prowess mere word cannot
daunt me,' the hero answered the villian

7)Debris :rubble

Usage:a full year after the earthquake in Mexico city,
workers were still carting a way the debris.

8)Dawdle :loiter, waste time

Usage:we have to meet a dead line dont dwadle.Just get
down to work

9)Dearth:scarcity

Usage:The dearth of skilled labour compelled the
employers to open trade schools.

10)Decant:pour of gently

Usage:Be sure to decant this wine before serving it.

11)Decapitate:behead

Usage:They didd not hand Lady Jane ;they decapitated
her."off with her head", cried the Duches, eager to
decapitate poor alice

12)Decipher:Decode

Usage:I could not decipher the doctor's hand writing.

13)Declivity :Downward slope

Usage:The children loved to skid down the declivity.

14)Deface:Mar, Disfigure.

Usage: If you deface a library book you have to pay
a hefty fine.

15)Decoy : Lure or bait

Usage:the wild ducks were not fooled by the decoy

16)Defile:Pollute, profane

Usage:the hand looms defiled the church with their
scurrilous writings.

17)Descry :catch site of

Usage:In the distance, we could darely descry the enemy
vessels.

18)Derogatory: expressing a low opinion

Usage:I resent your derpgatory remarks.

19)Dirge :layment with music

Usage:the funeral dirge stirred us to tears.

20)Diadem:Crown

Usage:The King's diadem was on display at the museum

R.S.AGARWAL 07

Synonyms- C

1)Cacophonous Discardant,Inharmonious

Usage:Some of th students in the orchestra enjoy the
cacophonous sounds .

2)Capricious Unpredctable,steadfast

UsageThe storm was capricious changed couse constantly

3)Castigation Punishment,severe criticism,commendation

Usage:Woolf could not bear the castigation that she faced in
certain reviews.

4) Catalyst :Agent that brings about chemical change while it
remains uneffected and unchanged

Usage:Many chemical reactions can't take place without the
presenc of a catalyst

5) Cache: Hiding place

Usage : The dective followed the suspect until he led them to
the cache where he had store his loot.

6)Callow : Immature,inexperenced,Youthful

Usage : In both the cases the judge shows how callow he was.

7) Chast :Pure,modest.outspoken

Usage:The crusader had her fitted out with a chastebelt.

8)Coagulate :Thicken , clot

Usage:Even after u remove the pudding from the bushes , it
willcontinue to coagulate as it stands.

9)Coda :Concluding section of a musical composition

Usage:The piece concluded with distinctive coda that
strickingly brought together various motifs.

10)Connoisseur :Person competent to act as judge of art,A
loverof an art.

Usage:She had developed into a connoisseur of fine chdna.

R.S.AGARWAL 06

Synonyms- B

1.Bard:Poet

Shakespeare is a great poet(Bard).

2.Boretaced:Shameless,bold Shocked by Huck finn's boretaced lies,Miss Watson prayed the
good lord would give him a sense of his regenerate dickedness.

3.Bask:luxuriate,take pleasure in warmth.
Basking on the beach,she relaxed so completely that shell fell
asleep.

4.Bate:let down,restrain.
Until it was to open the presents,the children had to bate their
curiosity.

5.Beatific:giving bliss,blissful.
The beatificsmile on the child's face made us very happy.

6.Beeline:Direct,Quick route.
As soon as the movie was over,jim made a beeline for the exit.

7.Behoove:Be suited to.
In this time of crisis,it behooves all of us to remain calm
and await the instruction of our superiors.

8.Belie:Contradict,give a false impression.
His coarse,hard bitten exterior belied his innate sensitivity.
Bestow:Conter.
He wished to bestow great honors upon the hero.

9.Bilk:Swindle,Cheat.
The Con man specialized in bilking insurance companies.

10.Bland:Soothing or mild,agreeable.
Jill tried a bland ointment for her sun burn.

11.Blanch:Bleach,whiten.
Although age had blanched his hair,he was still energetic.

12. Bicker:Quarrel.
The children bickered morning,noon and night ,exasperating
their parents.

13.Blurt:utter impulsively.
Before she could stop him,he blurted out the news.

14.Boon:blessing,benefit.
The recent rains that filled our empty reservoirs
were a boon to the whole community.

15.Brawn:Muscular strenght,sturdiness.
It takes brawnto become a champion weight-lifter.

16.Brook:tolerate;endure.
The dean would brook no interference with his disciplinnary
actions.

17.Brittle:easily broken,difficult.
My employer's brittle personality made it difficult for me
to get along with her.

18.Broach:Introduce,open up.
He did not even try to broach the
subject.

19.Brazen:Insolent.
Her brazen contempt for authority angered the aooicials.

20.Brandish:Wave around,Hourish.
Doctor watson wildly brandished his gun until Holmes told him
to put the thing away before he shot himself.

21.Brunt:Main impact or shock.
Tom sawyer claimed credit for painting the fence,but the brunt
of the work fell on others.

22.Brackish:Somewhat salinc.
He found the only wells in the area were brackish.

23.Bizarre:Fantastic,violently contrating.
The plot of the novel was too bizarre to be believed.

24.Bivouac:Temporary encampment.
While in bivouac,we spent the night in our sleeping bags under
the stars.

25.Bait:harass,tease.
The school bully baited the smaller children.

26.Babble:Chatter idly.
The little girl babbled about her doll.

27.Balk:Stop short,as it faced with an obstacle and refuse
to continue.
The cheif of police balked at sending his officers into the
riot torn area.

28.Banal:hackneyed,commonplace,trite,lacking originality.
He even resorted to the banality of having someone slip on a
banana peel!

29.Beneficent:Kindly,doing good.
The overgenerous philanthropist had to curb his beneficent
impulses before he gave away all his money and left himself
with nothing.

R.S.AGARWAL 05

Synonyms- A

Synonyms- A

1.Abase = lower , degrade , humiliate

usage:Anna expected to have to curtsy to the king of Siam,
whentold to cast herself down on the ground before him
however,she refused toabase herself

2.Abate = subside ,moderate

Usage : Rather than leaving immediately ,they waited for the
storm to abate

3.Aberrant = abnormal ,deviant

Usage : Given the aberrant nature of the day,we came to
doubt the validity of the entire experiment

4.Abeyance = suspended action , not in continuation

Usage : The deal was held in abeyance until her arrival

5.Abet = Assist, usually doing something in wrong

Usage : She wall unwilling to abet him in the swindle he
had planned

6.Abbreviate = shorten

Usage : because we were running out of time , the lecture
had to abbreviate her speech

7.Abolish = cancel, put an end to

Usage : The president of the college refused to abolish to
physical education requirement

8.Ablution = Washing

Usage : His daily ablutions were accompanied by loud noises
pretence,deception, (or)fraudulent intent
fidelity-faithfullness
infidel-one who does not have the right
faith(or)religion, especially to the marriage vows
that he humorously labeled opera in the both.

9.Abominate = loathe ,hate

Usage : Moses scold the idol worshipers in the tribe because
of abominated the custom

10. Adjure = renounce upon oath, he adjured his allegiance
to the king

11. Abnegation = renunciation ,self sacrifice

Usage : Rani and Son loved one another but their love was
doomed she had to wed the king their act of abnegation was
necessary to preserve the kingdom

12.Abscission = cutting off, separation

Usage : when a flower or leaf separates naturally from the
parent ,this process is called abscission

13.Abridge = condence ,shorten

Usage : Because the publishers felt the public wanted
a shorter versionof war and peace ,they proceeded to abridge
the novel

14.Abscond = depart secretly, and hide

Usage : The teller who absconded with the bonds went
uncaptured until some one recognized him

15.Absolute = complete, certain

Usage : The king of Siam was an absolute monarch

16.Absolve = pardon

Usage : The father confessor absolved him of his sons

17.Abstinence = restraint from eating or drinking

Usage : The doctor recommended total abstinence from
salted foods

18.Abusive = coarsely insulting

Usage : An abusive parent damages a child both mentally
and physically

19.Abut = border upon

Usage : where our estates abut , we must build a fence

20.Abortive = unsuccessful, fruitless

PLACEMENT PAPERS

Thursday, January 27, 2011

R.S.AGARWAL 12

IMPORTANT FACTS AND FORMULAE

SOLVED EXAMPLES

Sunday, January 16, 2011

R.S.AGARWAL 10

Friday, December 31, 2010

R.S.AGARWAL 09

R.S.AGARWAL 08

R.S.AGARWAL 07

R.S.AGARWAL 06

R.S.AGARWAL 05

MY PROFILE

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