simplification and roots
TRANSCRIPT
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PREREQUISITES:
(I) SIMPLIFICATION
(i)'BODMAS' Rule:
This rule depicts the correct sequence in which the operations are to be executed, so as tofind out the value of given expression.
Here B - Bracket,O - of,D - Division,M - Multiplication,
A - Addition andS - Subtraction
Thus, in simplifying an expression, first of all the brackets must be removed, strictly inthe order (), {} and ||.
After removing the brackets, we must use the following operations strictly in the order:
(i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction.
(ii)Modulus of a Real Number:
Modulus of a real numbera is defined as
|a| =a, ifa > 0
-a, ifa < 0
Thus, |5| = 5 and |-5| = -(-5) = 5.
(iii)Virnaculum (or Bar):
When an expression contains Virnaculum, before applying the 'BODMAS' rule, wesimplify the expression under the Virnaculum.
(II)ROOTS
Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo"
a power with a radical, and a radical can "undo" a power. For instance, if you square 2,
you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and
if you "take the square root of 9", you get 3: Copyright Elizabeth Stapel 1999-2011
All Rights Reserved
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The " " symbol is called the "radical"symbol. (Technically, just the "check mark" part
of the symbol is the radical; the line across the top is called the "vinculum".) The
expression " " is read as "root nine", "radical nine", or "the square root of nine".
You can raise numbers to powers other than just 2; you can cube things, raise them to the
fourth power, raise them to the 100th power, and so forth. In the same way, you can take
the cube root of a number, the fourth root, the 100th root, and so forth. To indicate some
root other than a square root, you use the same radical symbol, but you insert a number
into the radical, tucking it into the "check mark" part. For instance:
The "3" in the above is the "index" of the radical; the "64" is "the argument of the
radical", also called "the radicand". Since most radicals you see are square roots, the
index is not included on square roots. While " " would be technically correct, I've
never seen it used.
a square (second) root is written as
a cube (third) root is written as
a fourth root is written as
a fifth root is written as:
You can take any counting number, square it, and end up with a nice neat number. Butthe process doesn't always work going backwards. For instance, consider , the square
root of three. There is no nice neat number that squares to 3, so cannot be simplified as
a nice whole number. You can deal with in either of two ways: If you are doing a word
problem and are trying to find, say, the rate of speed, then you would grab your calculator
and find the decimal approximation of :
Then you'd round the above value to an appropriate number of decimal places and use a
real-world unit or label, like "1.7 ft/sec". On the other hand, you may be solving a plain
old math exercise, something with no "practical" application. Then they would almostcertainly want the "exact" value, so you'd give your answer as being simply " ".
Simplifying Square-Root Terms
To simplify a square root, you "take out" anything that is a "perfect square"; that is, you
take out front anything that has two copies of the same factor:
(III)
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Note that the value of the simplified radical ispositive. While either of +2 and2 might
have been squared to get 4, "the square root of four" isdefinedto beonly the positive
option, +2. When you solve the equationx2 = 4, you are trying to findall possible values
that might have been squared to get 4. But when you are just simplifying the expression
, the ONLY answer is "2"; this positive result is called the "principal" root. (Other
roots, such as2, can be defined using graduate-school topics like "complex analysis"
and "branch functions", but you won't need that for years, if ever.)
Sometimes the argument of a radical is not a perfect square, but it may "contain" a square
amongst its factors. To simplify, you need to factor the argument and "take out" anything
that is a square; you find anything you've got a pair of inside the radical, and you move it
out front. To do this, you use the fact that you can switch between the multiplication ofroots and the root of a multiplication. In other words, radicals can be manipulated
similarly to powers:
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SOLVED PROBLEMS ON SIMPLIFICATION:
1. A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupeenotes. The number of notes of each denomination is equal. What is the total number of notesthat he has ?A.45 B.60C.75 D.90
Answer & Explanation
Answer: Option D
Explanation:
Let number of notes of each denomination bex.
Thenx + 5x + 10x = 480
16x = 480
x = 30.
Hence, total number of notes = 3x = 90.
2.There are two examinations rooms A and B. If 10 students are sent from A to B, then thenumber of students in each room is the same. If 20 candidates are sent from B to A, then thenumber of students in A is double the number of students in B. The number of students inroom A is:A.20 B.80C.100 D.200
Answer & ExplanationAnswer: Option C
Explanation:
Let the number of students in rooms A and B be x and y respectively.Then, x - 10 = y + 10 x - y = 20 .... (i)
and x + 20 = 2(y - 20) x - 2y = -60 .... (ii)Solving (i) and (ii) we get: x = 100 , y = 80.
The required answer A = 100.
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3. The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables togetheris Rs. 4000. The total price of 12 chairs and 3 tables is:
A.Rs. 3500 B.Rs. 3750C.Rs. 3840 D.Rs. 3900
Answer & Explanation
Answer: Option D
Explanation:
Let the cost of a chair and that of a table be Rs.x and Rs.y respectively.
Then, 10x = 4y or y =5x.2
15x + 2y = 4000
15x + 2 x5x = 4000
2
20x = 4000
x = 200.
So,y =5x 200 = 500.
2
Hence, the cost of 12 chairs and 3 tables = 12x + 3y
= Rs. (2400 + 1500)
= Rs. 3900.
4.If a - b = 3 and a2 + b2 = 29, find the value of ab.A.10 B.12C.15 D.18
Answer & ExplanationAnswer: Option AExplanation:2ab = (a2 + b2) - (a - b)2
= 29 - 9 = 20ab = 10.
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5. The price of 2 sarees and 4 shirts is Rs. 1600. With the same money one can buy 1 saree and 6shirts. If one wants to buy 12 shirts, how much shall he have to pay ?
A.Rs. 1200 B.Rs. 2400C.Rs. 4800 D.Cannot be determinedE.None of these
Answer & Explanation
Answer: Option B
Explanation:
Let the price of a saree and a shirt be Rs.x and Rs.y respectively.
Then, 2x + 4y = 1600 .... (i)
andx + 6y = 1600 .... (ii)
Divide equation (i) by 2, we get the below equation.
=> x + 2y = 800. --- (iii)
Now subtract (iii) from (ii)
x + 6y = 1600 (-)x + 2y = 800----------------
4y = 800----------------
Therefore, y = 200.
Now apply value of y in (iii)
=> x + 2 x 200 = 800
=> x + 400 = 800
Therefore x = 400
Solving (i) and (ii) we getx = 400,y = 200.
Cost of 12 shirts = Rs. (12 x 200) = Rs. 2400.
6.A sum of Rs. 1360 has been divided among A, B and C such that A gets of what B gets and
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B gets of what C gets. B's share is:
A.Rs. 120 B.Rs. 160C.Rs. 240 D.Rs. 300
Answer & Explanation
Answer: Option C
Explanation:
Let C's share = Rs.x
Then, B's share = Rs.x, A's share = Rs. 2xx = Rs.x4 3 4 6
x+x
+x = 13606 417x
= 136012
x =1360 x 12
= Rs. 96017
Hence, B's share = Rs.960
= Rs. 240.4
7.One-third of Rahul's savings in National Savings Certificate is equal to one-half of his
savings in Public Provident Fund. If he has Rs. 1,50,000 as total savings, how much has hesaved in Public Provident Fund ?A.Rs. 30,000 B.Rs. 50,000C.Rs. 60,000 D.Rs. 90,000
Answer & ExplanationAnswer: Option CExplanation:Let savings in N.S.C and P.P.F. be Rs. x and Rs. (150000 - x) respectively. Then,1x =
1(150000 -x)
3 2
x+x
= 750003 2
5x= 75000
6
x =75000 x 6
= 900005
Savings in Public Provident Fund = Rs. (150000 - 90000) = Rs. 600008.A fires 5 shots to B's 3 but A kills only once in 3 shots while B kills once in 2 shots. When B
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has missed 27 times, A has killed:A.30 birds B.60 birds
C.72 birds D.90 birds
Answer & ExplanationAnswer: Option AExplanation:Let the total number of shots be x. Then,
Shots fired by A =5x
8
Shots fired by B =3x8
Killing shots by A =1
of5x=
5x
3 8 24
Shots missed by B =1of
3x=
3x
2 8 16
3x= 27 orx =
27 x 16= 144.
16 3
Birds killed by A =5x
=5
x 144 = 30.24 24
9. Eight people are planning to share equally the cost of a rental car. If one person withdrawsfrom the arrangement and the others share equally the entire cost of the car, then the share ofeach of the remaining persons increased by:
A.17
B.18
C.19
D.78
Answer & Explanation
Answer: Option A
Explanation:
Original share of 1 person =18
New share of 1 person =17
Increase = 1-1 =1
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7 8 56
Required fraction =(1/56)
=1
x8
=1
(1/8) 56 1 7
10. To fill a tank, 25 buckets of water is required. How many buckets of water will be required tofill the same tank if the capacity of the bucket is reduced to two-fifth of its present ?A.10 B.35C.62.5 D.Cannot be determinedE.None of these
Answer & Explanation
Answer: Option C
Explanation:
Let the capacity of 1 bucket =x.
Then, the capacity of tank = 25x.
New capacity of bucket =2x5
Required number of buckets =25x
(2x/5)=
25xx5
2x
=1252
= 62.5
11.In a regular week, there are 5 working days and for each day, the working hours are 8. Aman gets Rs. 2.40 per hour for regular work and Rs. 3.20 per hours for overtime. If he earns
Rs. 432 in 4 weeks, then how many hours does he work for ?A.160 B.175C.180 D.195
Answer & ExplanationAnswer: Option BExplanation:Suppose the man works overtime for x hours.Now, working hours in 4 weeks = (5 x 8 x 4) = 160.
160 x 2.40 + x x 3.20 = 4323.20x = 432 - 384 = 48
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x = 15.Hence, total hours of work = (160 + 15) = 175.
12.Free notebooks were distributed equally among children of a class. The number ofnotebooks each child got was one-eighth of the number of children. Had the number ofchildren been half, each child would have got 16 notebooks. Total how many notebooks weredistributed ?A.256 B.432C.512 D.640E.None of these
Answer & ExplanationAnswer: Option C
Explanation:Let total number of children be x.
Then,x x1
=x
x 16 x = 64.8 2
Number of notebooks =1x
2 =1x 64 x 64 = 512.
8 8
13.A man has some hens and cows. If the number of heads be 48 and the number of feetequals 140, then the number of hens will be:A.22 B.23
C.24 D.26
Answer & ExplanationAnswer: Option DExplanation:Let the number of hens be x and the number of cows be y.Then, x + y = 48 .... (i)and 2x + 4y = 140 x + 2y = 70 .... (ii)
Solving (i) and (ii) we get: x = 26, y = 22.The required answer = 26.
14.(469 + 174) - (469 - 174) = ?(469 x 174)A.2 B.4C.295 D.643
Answer & ExplanationAnswer: Option BExplanation:
Given exp. =(a +b) - (a -b)ab
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=4abab
= 4 (where a = 469, b = 174.)
15.David gets on the elevator at the 11th floor of a building and rides up at the rate of 57floors per minute. At the same time, Albert gets on an elevator at the 51st floor of the samebuilding and rides down at the rate of 63 floors per minute. If they continue travelling atthese rates, then at which floor will their paths cross ?A.19 B.28C.30 D.37
Answer & ExplanationAnswer: Option C
Explanation:Suppose their paths cross after x minutes.Then, 11 + 57x = 51 - 63x 120x = 40
x =13
Number of floors covered by David in (1/3) min. =1x 57 = 19.
3
So, their paths cross at (11 +19) i.e., 30th floor.
SOLVED PROBLEMS ON ROOTS:
1.Simplify
There are various ways I can approach this simplification. One would be by factoring and then
taking two different square roots:
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The square root of 144 is 12.
You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But
my steps above show how you can switch back and forth between the different formats
(multiplication inside one radical, versus multiplication of two radicals) to help in the
simplification process.
2.Simplify
Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?
3.Simplify
This answer is pronounced as "five, root three". It is proper form to put the radical at the end of
the expression. Not only is " " non-standard, it is very hard to read, especially when hand-
written. And write neatly, because " " is not the same as " ".
You don't have to factor the radicand all the way down to prime numbers when simplifying. As
soon as you see a pair of factors or a perfect square, you've gone far enough.
4.Simplify
Since 72 factors as 236, and since 36 is a perfect square, then:
Since there had been only one copy of the factor 2 in the factorization 266, that left-over 2
couldn't come out of the radical and had to be left behind.
5.Simplify
Variables in a radical's argument are simplified in the same way: whatever you've got a pair of
can be taken "out front".
6.Simplify
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7.Simplify
The 12 is the product of 3 and 4, so I have a pair of 2's but a 3 left over. Also, I have two pairs of
a's; three pairs ofb's, with oneb left over; and one pair ofc's, with onec left over. So the root
simplifies as:
You are used to putting the numbers first in an algebraic expression, followed by any variables.
But for radical expressions, any variables outside the radical should go in front of the radical, as
shown above.
8.Simplify
Writing out the complete factorization would be a bore, so I'll just use what I know about
powers. The 20 factors as 45, with the 4 being a perfect square. Ther18 has nine pairs ofr's; the
s is unpaired; and thet21 has ten pairs oft's, with onetleft over. Then:
Technical point: Your textbook may tell you to "assume all variables are positive" when you
simplify. Why? The square root of the square of anegative number is not the original number.
For instance, you could start with2, square to get +4, and then take the square root (which is
definedto be thepositive root) to get +2. You plugged in a negative and ended up with a
positive. Sound familiar? It should: it's how the absolute value works: |2| = +2. Taking the
square root of the square is in fact the technical definition of the absolute value. But this
technicality can cause difficulties if you're working with values of unknown sign; that is, withvariables. The |2| is +2, but what is the sign on |x |? You can't know, because you don't know
the sign ofx itselfunless they specify that you should "assume all variables are positive", or
at least non-negative (which means "positive or zero").
Multiplying Square Roots
The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.
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Simplifying multiplied radicals is pretty simple. We use the fact that the product of two radicals
is the same as the radical of the product, and vice versa.
9.Write as the product of two radicals:
Copyright Elizabeth Stapel 1999-2011 All Rights Reserved
Okay, so that manipulation wasn't very useful. But working in the other direction can be helpful:
10.Simplify by writing with no more than one radical:
11.Simplify by writing with no more than one radical:
12.Simplify by writing with no more than one radical:
The process works the same way when variables are included:
13.Simplify by writing with no more than one radical:
Just as with "regular" numbers, square roots can be added together. But you might not be able to
simplify the addition all the way down to one number. Just as "you can't add apples and
oranges", so also you cannot combine "unlike" radicals. To add radical terms together, they have
to have the same radical part.
14.Simplify:
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Since the radical is the same in each term (namely, the square root of three), I can combine the
terms. I have two copies of the radical, added to another three copies. This gives me five copies:
That middle step, with the parentheses, shows the reasoning that justifies the final answer. You
probably won't ever need to "show" this step, but it's what should be going through your mind.
15.Simplify:
The radical part is the same in each term, so I can do this addition. To help me keep track that the
first term means "one copy of the square root of three", I'll insert the "understood" "1":
Don't assume that expressions with unlike radicals cannot be simplified. It is possible that,after
simplifying the radicals, the expression can indeed be simplified.
EXERCISEPROBLEMS WITH SOLOUTION:
. The sum of the smallest six-digit number and the greatest five-digit number is:
a. 199999 b. 201110 c. 211110 d. 1099999
Answer with Explanation:
The smallest six digit number = 1,00,000
The greatest five digit number = 99,999
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Therefore the sum of the smallest = 1,00,000 + 99,999
six digit number and the greatestfive digit number = 1,99,999
2. Which of the following has most number of divisors?
a. 99 b. 101 c. 176 d. 182
Answer with Explanation:
Divisiors of 99 = 1, 3, 9, 11, 33, 99
Divisors of 101 = 1,101
Divisors of 176 = 1, 2, 4, 8, 11, 22, 44, 88, 176
Divisors of 182 = 1, 2, 7, 13, 14, 26, 91, 182
Therefore 176 has most number of divisors.
3. Which of the following has fractions in ascending order?
a.9
8,
11
9,
9
7,
5
3,
3
1b.
9
8,
9
7,
11
9,
3
2,
5
3c.
5
3,
3
2,
9
7,
11
9,
9
8d.
3
2,
5
3,
9
7,
11
9,
9
8
Answer with Explanation:
LCM of (1,3,7,9,8) = 1512 and
LCM of (2,3,7,8,9) = 3024
Therefore For (a) we have 4536
1512, 2515
1512, Not in ascending order
For (b) we have5040
3024,4536
3024, Not in ascending order
For (c) we have3402
3024,3696
3024,3888
3024,4536
3024,5040
3024
The fractions are in ascending order.
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For (d), we have3402
3024,3696
3024,3888
3024,5040
3024,4536
3024
Not in ascending order.
Therefore the answer is (c)
4. If one-third of one-fourth of a number is 15, then three-tenth of that number is:
a. 35 b. 36 c. 45 d. 54
Answer with Explanation:
Given one third of one fourth of a number (x) is 15.
(3
1(4
1xx)) = 15
x = 15 x 12
x = 180
Three benth ofx =10
3xx
=10
3x 180 = 54
5. (8 88) 8888088 =?
a. 808008 b. 808080 c. 808088 d. 8008008
Answer with Explanation:
88
8x 8888088 =
11
8888088
= 808008
Note :
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808008 x 11 = 8888088
6. The value of 2251541082510 is:
a. 4 b. 5 c. 8 d. 10
Answer with Explanation:
2251541082510
= 151541082510 Therefore 225 = 15
= 1691082510 Therefore 169 =13
= 1212510 Therefore 121 = 11
= 112510 Therefore 36 = 6
= 610
= 16
=4
7. 9548 + 7314 = 8362 + ?
a. 8230 b. 8410 c. 8500 d. 8600
Answer with Explanation:
9548 + 7314 = 9548
7314
___________
16862
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(-) 8362
___________8500
____________
Ans = 8500
(i.e) 9548 + 7314 = 8362 + 8500
8. The H.C.F of 24 33 55 24, 23 32 52 7 and 24 34 5 72 11 is
a. 22
32
5 b. 22 32 5 7 11 c. 24 34 54 d. 24 3455777
Answer with Explanation:
Among the three factors
HCF of 24 x 24, 23, 24 is = 23
HCF of 33, 32, 34 is = 32
HCF of 55, 52, 5 = 5
7 & 11 are not in three terms
So we leave the values 7, 72, 11
HCF of three terms = 23 x 32 x 5
9. What is the difference between the biggest and the smallest fraction among6
5
5
4,
4
3,
3
2and ?
a.6
1b.
12
1c.
20
1d.
30
1
Answer with Explanation:
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The biggest among3
2,4
3,5
4,6
5is =
6
5
The Smallest among3
2,
4
3,
5
4,
6
5is =
3
2
The difference between the biggest and the smallest fraction =6
5-
3
2
=18
1215
=18
3
=6
1
10. Find the number which when multiplied by 15 increased by 196.
a. 14 b. 20 c. 26 d. 28
Answer with Explanation:
15 x 14 = 210
Which is increased by 196 21014
196 since we multiply it by 15
So the number is 14
11. ?51466123
861844
a. 1 b. 2 c. 6.65 d. 7.75
Answer with Explanation:
51466123
86)184(4
xx
x
=
)5146()6123(
86724
xx
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=730738
86724
=8
62
= 7.75
12. In the following sum, ? stands for which digit?
? + 1? + 2? + ?3 + ?1 = 21?
a. 4 b. 6 c. 8 d. 9
Answer with Explanation:
8+81+28+83+81 = 218
Answer is 8
13. Which of the following is a pair of co-primes?
a. (16, 62) b. (18, 25) c. (21, 35) d. (23, 92)
Two numbers are said to be co-prime if their H.C.F is 1.
Answer with Explanation:
H.C.F of 16, 62 is 2
H.C.F of 18, 25 is 1
H.C.F. of 21, 35 is 7
Co-prime = (18, 25).
14. 34.95 + 240.016 + 23.98 = ?
a. 298.0946 b. 298.111 c. 298.946 d. 299.09
Ordinary Addition
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Answer with Explanation:
34.95
240.016
23.98 (+)
_________________
298.946
_________________
15. If the sum of one-half and one-fifth of a number exceeds one-third of that
number by3
17 , the number is:
a. 15 b. 18 c. 20 d. 30
Answer with Explanation:
Let the unknown number bex
2
1x +
5
1x(7
3
1) =
3
1x
2
x+
5
x-
3
22=
3
x
2
x+
5
x-
3
x=
3
22
30
10615 xxx =
3
22
30
11x=
3
22
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Answer with Explanation:
9
1
81
25 =
81
925
=81
16= 4/9
18. The value of 112 54 is:
a. 6700 b. 70000 c. 76500 d. 77200
Answer with Explanation:
54 = 5 x 5 x 5 x 5
= 25 x 25
= 625
112 x 54 = 112 x 625
_______________
560
224
672
________________
70000
_________________
The Answer is 70000
19. The L.C.M of 23 32 5 11, 24 34 52 7 and 25 33 53 72 11 is :
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5625.1 = 1.25
31. If -1 < x < 2 and 1 < x < 3, then least possible value of (2y3x) is:
a. 0 b. -3 c. -4 d. -5
Answer with Explanation:
-1 x 2 1 y 3
-3 3x 6 2 2y 6
3 -3x -6
i.e. -6 -3x 3
-4 2y - 3x 9
The least possible value of
(2y3x) is -4.
32. The G.C.D. of 1.08, 0.36 and 0.9 is
a. 0.03 b. 0.9 c. 0.18 d. 0.108
Answer with Explanation:
0.2 1.08, 0.36, 0.9 (or) 2 108, 36, 90
3 54, 18, 45
0.9 5.4, 1.8, 4.5 3 18, 6, 15
6, 2, 5 6, 2, 5
G.C.D = 0.2 x 0.9 = 0.18 2 x 3 x 3 =100
18= 0.18
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Basic Formula:
(x + y)2
= (x-y)2
+ 4xy
Answer with Explanation:
Givenx +y = 25 & x-y = 13
(x + y)2 = (x-y)2 + 4xy
252 = 132 + 4xy
4xy = 252 - 132
=625-169
4xy = 456
xy =4
456
xy = 114
40. On simplification, 3034(100220.04) is equal to:
a. 2543 b. 2984 c. 2993 d. 3029
Answer with Explanation:
3034(1002 -20.04)
= 3034 -04.20
1002
= 30341002 x 100
20.04 x 100
= 30341002 x 100
2004
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Since we have the sum of multiples of 33 are 528, we will combine these multiples as to get their
sum as 528.
we get (33, 495) (66, 462) (99, 429) (132, 396) (165, 363) (198, 330)
(231,297)
But we have a condition that HCF of (x, y) = 33 Eliminating the ones which does not satisfies
the condition we get (33, 495) (99, 429) (165, 363) (231, 297)
The number of pairs of numbers satisfying the above condition is 4 .
44. 1.0?
009.
a. .0009 b. 009 c. 9 d. 9
Answer with Explanation:
Letx
009.= 0.1
x =1.0
009.
x =1.0
009.x
10
10
x =1
09.0
x = 0.09
45. Two numbers differ by 5. If their product is 336, then the sum of the two numbers is:
a. 21 b. 28 c. 37 d. 51
Basic Formula:
(x+y)2 = (x-y)2 + 4xy
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Answer with Explanation:
Givenx-y = 5 andxy =336
We know that
(x+y)2 = (x-y)2 + 4xy
= 52 +4 (336)
= 25+1344
(x+y)2
= 1369
x+y = 37
46. 3[1.6{3.2(3.2 + 2.25 X)}] = .65. The value of X is:
a. 0.3 b. 0.7 c. 3 d. 7
Answer with Explanation:
Given 3-[1.6-{3.2-(3.2+2.25 x)}] = 0.65
-[1.6-{3.2-(3.2+2.25x)}] = 0.65-3
-1.6+{3.2-(3.2+2.25x)} = 0.65-3
3.2- (3.2+2.25x) = - 0.75
- 3.2-2.25x = -0.75-3.2
-2.25x = -0.75-3/2 + 3/2
-2.25x = - 0.75
x
25.2= -0.75
x =75.0
25.2
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x = 3
47. If 02.0441 x , then the value of x is:
a. 0.1764 b. 1.764 c. 1.64 d. 2.64
Answer with Explanation:
x 441 = 0.02
441
x= 0.02
Squaring on both sides
441
x= (0.02)2
441
x= 0.004
x = 0.004 x 441
x = 0.1764
48. In dividing a number by 585, a student employees the method of short division. He divided
the number successively by 5, 9 and 13 (factors of 585) and got the remainders 4, 8 and 12. If he
had divided the number by 585, the remainder would have been:
a. 24 b. 144 c. 292 d. 584
Basic Formula:
x
x+(x-1) =x where (x-1) is remainder here.
Answer with Explanation:
Given a unknown numberx when divided by 5,9,13. We get remainders 4, 8 &12 .
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(i.e.)5
x+4 =x x = 5 & 4 =x-1 (remainder)
9
x+8 =x x = 9 & 8 =x-1 (remainder)
13
x+12 =x x =13 & 12 =x-1 (remainder)
IIIly
585
x+(585-1) =x (i.e.)
585
x+(x-1) =x x-1 = 584 (remainder)
The remainder would have been 584.
49. The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when
divided by 9 leaves no remainder, is:
a. 1677 b. 1683 c. 2523 d. 3363
Answer with Explanation:
336 28 24 21 187
5 1683 6 1683 7 1683 8 1683 9 1683
15 12 14 16 9
18 48 28 8 78
15 48 28 8 72
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33 3 3 3 63
30 63
3 0
Ans: L.C.M of 5,6,7,8 is 840
Let the required number be 840k+3,
Which is divisible by 9.
Least value of k for which (90k+4)
is divisible by 9 is k=2
840 x 2 +3 = 1680 +3
= 1683
50. The value of15.009.05.05.0
027.0125.0
is:
a. 0.08 b. 0.2 c. 0.8 d. 1
Basic Formula:
22
33
baba
ba
=
22
22 ))((
baba
bababa
= a+b
Answer with Explanation:
15.009.05.05.0
027.0125.0
x=
22
33
)3.0(3.05.0)5.0(
)3.0()5.0(
x
=22
22
)3.0(3.05.0)5.0(
])3.0(3.05.0)5.0)[(3.05.0(
x
x
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= 0.5 + 0.3
= 0.8
51.Simplify:
To simplify a radical addition, I must first see if I can simplify each radical term. In this
particular case, the square roots simplify "completely" (that is, down to whole numbers):
52.Simplify:
I have three copies of the radical, plus another two copies, giving meWait a minute! I cansimplify those radicals right down to whole numbers:
Don't worry if you don't see a simplification right away. If I hadn't noticed until the end that the
radical simplified, my steps would have been different, but my final answer would have been the
same:
53.Simplify:
I can only combine the "like" radicals, so I'll end up with two terms in my answer:
There is not, to my knowledge, any preferred ordering of terms in this sort of expression, so the
expression should also be an acceptable answer.
54.Simplify: Copyright Elizabeth Stapel 1999-2011 All Rights Reserved
I can simplify the radical in the first term, and this will create "like" terms:
55.Simplify:
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63.Simplify:
64.Simplify:
This looks very similar to the previous exercise, but this is the "wrong" answer. Why? Because
the denominator contains a radical. The denominator must contain no radicals, or else it's
"wrong". (Why "wrong" in quotes? Because this issue may matter to your instructor right now,
but it probably won't later on. It's like when you were in elementary school and improper
fractions were "wrong" and you had to convert everything to mixed numbers instead. But now
that you're in algebra, improper fractions are fine, even preferred. Once you get to calculus or
beyond, they won't be so uptight about where the radicals are.)
To get the "right" answer, I must "rationalize" the denominator. That is, I must find some way to
convert the fraction into a form where the denominator has only "rational" (fractional or whole
number) values. But what can I do with that radical-three? I can't take the 3 out, because I don't
have a pair of threes.
Thinking back to those elementary-school fractions, you couldn't add them unless they had the
same denominators. To create these "common" denominators, you would multiply, top and
bottom, by whatever the denominator needed. Anything divided by itself is just 1, and
multiplying by 1 doesn't change the value of whatever you're multiplying by the 1. But
multiplying that "whatever" by a strategic form of 1 could make the necessary computations
possible, such as:
We can use the same technique to rationalize radical denominators.
I could take a 3 out of the denominator if I had two factors of 3 inside the radical. I can create
this pair of 3's by multiplying by another copy of root-three. If I multiply top and bottom by root-
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This multiplication made the radical terms cancel out, which is exactly what I want. This "same
numbers but the opposite sign in the middle" thing is the "conjugate" of the original expression.
By using the conjugate, I can do the necessary rationalization.
Do not try to reach inside the numerator and rip out the 6 for "cancellation". The only thing that
factors out of the numerator is a 3, but that won't cancel with the 2 in the denominator. Nothing
cancels!
67.Simplify:
I'll multiply by the conjugate in order to "simplify" this expression. The denominator's
multiplication results in a whole number (okay, a negative, but the point is that there aren't any
radicals):
The numerator's multiplication looks like this:
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Then the simplified (rationalized) form is:
It can be helpful to do the multiplications separately, as shown above. Don't try to do too much at
once, and make sure to check for any simplifications when you're done with the rationalization
Operations with cube roots, fourth roots, and other higher-index roots work similarly to squareroots.
Simplifying Higher-Index Terms
68.Simplify
Just as I can pull from a square (or second) root anything that I have two copies of, so also I can
pull from a fourth root anything I've got four of:
If you have a cube root, you can take out any factor that occurs in threes; in a fourth root, take
out any factor that occurs in fours; in a fifth root, take out any factor that occurs in fives; etc.
69.Simplify the cube root:
70.Simplify the cube root:
71.Simplify: Copyright Elizabeth Stapel 1999-2011 All Rights Reserved
72.Simplify:
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QUESTION BANK PROBLEMS:
I. Among the two numbers, the bigger number is greater than the smaller number by 6
II. 40% of the smaller number is equal to 30% of the bigger number
III. The ratio between half of the bigger number and one-third of the smaller number is 2 : 1
a. I and II only b. II and III only c. All I, II and III d. None of these
2. The taxi charges in a city comprise of a fixed charge, together with the charge of the distance
covered. For a journey of 16 km, the charges paid are Rs.156 and for a journey of 24 km, the
charges paid are Rs.204. What will a person have to pay for traveling a distance of 30 km?
a. Rs.236 b. Rs.240 c. Rs.248 d. Rs.252
3. ?125
1243
a.521 b.
531 c.
541 d.
522
4. A number when divided by 114 leaves the remainder 21. If the same number is divided by 19,
then the remainder will be:
a. 1 b. 2 c. 7 d. 21
5. The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is:
a. 74 b. 94 c. 184 d. 364
6. The value of09.69.3.2
027.3.22
3
is:
a. 0 b. 1.6 c. 2 d. 3.4
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7. What is the two-digit number?
I. Sum of the digits is 7.
II. Difference between the number and the number obtained by interchanging the digits is 9.
III. Digit in the tens place is bigger than the digit in the units place by 1.
a. I & II only b. II & III only c. All I, II & III d. None of these
8. In and examination, a student scores 4 marks for every correct answer and loses 1 mark for
every wrong answer. If he attempts in all 60 questions and secures 130 marks, the number of
questions he attempts correctly, is:
a. 35 b. 38 c. 40 d. 42
9. ?503212235
63
a. 3 b. 23 c. 6 d. None of these
10. On dividing a number by 999, the quotient is 366 and the remainder is 103. The number is:
a. 364724 b. 365387 c. 365737 d. 366757
11. The smallest number which when diminished by 7, is divisible by 12, 16, 18, 21 and 28 is:
a. 1008 b. 1015 c. 1022 d. 1032
12. )5.25.25.375.75.7( is equal to:
a. 30 b. 60 c. 80 d. 100
13. 54 is to be divided into two parts such that the sum of 10 times the first and 22 times the
second is 780. The bigger part is:
a. 24 b. 34 c. 30 d. 32
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a. 5 b. 6 c. 15 d. 30
22. If x and y are positive integers such that (3x+7y) is a multiple of 11, then which of the
following will also be divisible by 11?
a. 4x + 6y b. x + y + 4 c. 9x + 4y d. 4x9y
23. Which of the following fraction is the largest?
a.8
7b.
16
13c.
40
31d.
80
63
24. The value of (4.7 13.26 + 4.7 9.43 + 4.7 7731) is:
a. 0.47 b. 47 c. 470 d. 4700
25. The denominator of a fraction is 3 more than the numerator. If the numerator as well as the
denominator is increased by 4, the fraction becomes5
4. What was the original fraction?
a. 11
8
b. 8
5
c. 13
10
d. 10
7
26. A sum of Rs.1360 has been divided among A B and C such that A gets3
2of what B gets and
B gets4
1of what C gets. Bs share is:
a. Rs.120 b. Rs.160 c. Rs.240 d. Rs.300
27. if 5 = 2.236, then the value of 1255
10
2
5 is equal to :
a. 5.59 b. 7.826 c. 98.994 d. 10.062
28. The sum of three consecutive odd numbers is always divisible by:
I. 2 II. 3 III. 5 IV. 6
a. Only I b. Only II c. III. Only I & II d. Only II and IV
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29. A rectangular courtyard 3.78 metres long and 5.25 metres wide is to be paved exactly with
square tiles, all of the same size. What is the largest size of the tile which could be used for the
purpose?
a. 14 cms b. 21 cms c. 42 cms d. None of these
30. ?7.05.140073.0
92.20203.0
a. 0.8 b. 1.45 c. 2.40 d. 3.25
31. The different between a tow-digit number and the number obtained by interchanging the
digits is 36. What is the difference between the sum and the difference of the digits of the
number if the ratio between the digits of the number is 1 : 2 ?
a. 4 b. 8 c. 16 d. 18
32. A person travels 3.5 km form place A to place B. Out of this distance, he travels3
21 km on
bicycle,
6
11 km on scooter and the rest on foot. What portion of the whole distance does he
cover on foot?
a.19
3b.
11
4c.
21
4d.
6
5
33. The value of1.01
01.01
is close to:
a. 0.6 b. 1.1 c. 1.6 d. 1.7
34. The digits indicated * and $ in 3422213*$ so that this number is divisible by 99, are
respectively:
a. 1, 9 b. 3, 7 c. 4, 6 d. 5, 5
35. The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85
cm, 12 m 95 cm is :
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a. 15 cm b. 25 cm c. 35 cm d. 42 cm
36. If 1.5 x = 0.04y, then the value of
xy
xyis:
a.77
730b.
77
73c.
77
3.7d. None of these
37. The difference between a two-digit number and the number obtained by interchanging the
two digits is 63. Which is the smaller of the two numbers?
a. 29 b. 70 c. 92 d. None of these
38. A pineapple costs Rs.7 each. A watermelon costs Rs.5 each. X spends Rs.38 on these fruits.
The number f pineapples and purchased is:
a. 2 b. 3 c. 4 d. 5
39. Which one of the following numbers has rational square root?
a. 0.4 b. 0.09 c. 0.9 d. 0.025
Ans is 0.09
40. Which of the following numbers is divisible by 3, 7, 9 and 11?
a. 639 b. 2079 c. 3791 d. 37911
41. About the number of pairs which have 16 as their H.C.F. and 136 as their L.C.M., we can
definitely say that:
a. No such pair exists b. Only one such pair exists
c. Only two such pairs exist d. Many such pairs exist
42. ?963.8654.9
3.8964.965
63.8954.96
63.8954.96
a. 10-2 b. 10-1 c. 10 d. None of these
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