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Category Archives: IITJEE Foundation mathematics
A problem of log, GP and HP…
Question: If and , pyrove that:
Solution: This is same as proving: y is Harmonic Mean (HM) of x and z;
That is, to prove that is the same as the proof for :
Now, it is given that —– I
and —– II
Let say. By definition of logarithm,
; ;
; ; .
Now let us see what happens to the following two algebraic entities, namely, and ;
Now, …call this III
Now,
Hence, ….equation IV
but it is also given that …see equation II
Hence,
Take log of above both sides w.r.t. base N:
So, above is equivalent to
But now see relations III and IV:
Hence,
Hence,
Hence, as desired.
Regards,
Nalin Pithwa
Skill Check XV: IITJEE Maths: Foundation: Powers and Roots
Reference: An old ICSE text, class VIII, Oxford Publications, India: Abhijit Mukherjea et al.
Tutorial Exercises:
I) Evaluate (you may use a numerical log table)
II) Evaluate (you may use a numerical log table).
III) Use your calculator to verify:
IV) Evaluate correct up to three decimal places. You can use a log table.
V) Find the values of the following: use a log table in case you wish:
(5a) (5b) (5c) (5d) (5e) (5f) (5g) (5h) (5i) (5j) (5k) (5l) (5m) (5n) (5o) (5p) (5q) (5r) (5s)
VI) Evaluate the following using log tables:
(6a) (6b) (6c) (6d) (6e) (6f) (6g) (6h) (6i) (6j) (6k) (6l)
Regards,
Nalin Pithwa
Skill Check: IITJEE XIV: Foundation Maths: Squares and square roots
Tutorial Exercises: (Reference: An old ICSE Text, Oxford Publishers, Class VIII, Abhijit Mukherjea and Sushil Kumar Sharma)
I) Find the value of the following:
(1a) (1b) (1c) (1d) (1e) (1f) (1g) (1h) (1i) (1j)
II) Which of the following perfect squares will have even square roots and which will have odd square roots?
(2a) 441 (2b) 2916 (2c) 3969 (2d) 21609 (2e) 389376
III) Which of the following numbers are perfect squares ?
(3a) 128 (3b) 256 (3c) 2187 (3d) 6561 (3e) 6084
IV) Find the perfect square obtained by multiplying each of the following numbers by the smallest possible number:
(4a) 882 (4b) 405 (4c) 567 (4d) 1690 (4e) 7776
V) Divide the following numbers by the smallest possible number to make each one a perfect square:
(5a) 1152 (5b) 2187 (5c) 3267 (3d) 1536 (3e) 10140
VI) Find the square roots of the following natural numbers by the prime factorization method:
(6a) 225 (6b) 1296 (6c) 1764 (6d) 2401 (6e) 2916 (6f) 5184 (6g) 7225 (6h) 10816 (6i) 14400 (6j) 13456
VII) Find the square roots of the following fractions by the prime factorisation method:
(7a) (7b) (7c) (7d) (7e) (7f) (7g) (7h) (7i) (7j)
VIII) Find the square roots of the following decimals by the prime factorization method:
(8a) 0.0001 (8b) 1.44 (8c) 3.24 (8d) 7.29 (8e) 10.24 (8f) 30.25 (8g) 0.0049 (8h) 0.1296 (8i) 15.21 (8j) 0.2025
IX) Find the square roots of the following numbers by the division method:
(9a) 4624 (9b) 11664 (9c) 16641 (9d) 44521 (9e) 426409 (9f) 550564 (9g) 840889 (9h) 917764 (9i) 1014049 (9j) 1560001
X) Find the square roots of the following decimal fractions by the division method:
(10a) 34.81 (10b) 44.89 (10c) 5.4756 (10d) 1.3225 (10e) 4.9284 (10f) 11.2896 (10g) 4.5796 (10h)36.2404 (10i) 1.304164 (10j) 4.609609
XI) Find the square roots of the following common fractions by the division method:
(11a) (11b) (11c) (11d) (11e)
XII) Find the square roots of the following numbers, correct up to 3 decimal places:
(12a) (12b) (12c) (12d) (12e) (12f) (12g) (12h) 14502 (12i) (12j) (12k) (12l) (12m) (12n) (12o) (12p) (12q) (12r) (12s) (12t)
XIII) Find the smallest number that needs to be subtracted from 66671 in order to get a perfect square.
XIV) Find the smallest number that needs to be subtracted from 1051149 in order to get a perfect square.
XV) Find the smallest number that needs to be added to 1485155 in order to get a perfect square.
XVI) Find the smallest and greatest 5digit numbers that are perfect squares.
XVII) The area of a square is 2.815684 square cm. Find its length.
XVIII) A farmer has 21126 seedlings, which he intends to plant in an equal number of rows and columns. What is the maximum number of rows of seedlings that can be planted? How many seedlings will be left unplanted?
XIX) Chairs need to be laid out in an equal number of rows and columns. If there are 1817 chairs in the stadium, find at least how many more chairs need to be brought in.
Regards,
Nalin Pithwa
Skill Check XIII IITJEE Foundation Maths: Approximation
Introduction:
(Reference: an old ICSE grade VIII text book, New Guided Mathematics, class 8, Abhijit Mukerjea and Sushil Kumar Sharma, Oxford University Press, India)
Rahul needs to divide a web page into three vertical parts. He wants it to look neat and thus wants the measurements to be perfect. But the web page is 640 pixels wide and 640/3 = 213.3 pixels. Is it necessary for Rahul to be so accurate? If he rounds off each part to 213 pixels , his page can be divided into 3 parts measuring 213, 214, and 213 pixels. Although the parts are not exactly equal, they are approximately equal to each other, a relationship that is denoted by the symbol .
Rounding Off Decimal Fractions
Rounding off a decimal fraction to its decimal places:
 If the digit in the (n+1)th decimal place is less than 5, then the digit in the (n+1)th decimal place, and all the digits after it, are simply omitted.
 If the digit in the (n+1)th decimal place is 5 or more than 5, the nth digit is increased by 1 and the digit in the (n+1)th decimal place, and all the digits are it, are omitted.
Example 1: Convert into a decimal fraction and round off your answer to:
i) 1 decimal place
ii) 3 decimal places
iii) 5 decimal places
iv) a whole number
Solution:
(i) In order to round off the answer to the tenths or 1 decimal place, we consider the (1+1)th decimal place digit. As 8>5, the digit in the tenths place is increased by 1 and all the digits after it are omitted. Thus,
(ii) As the (3+1)th decimal place digit is 7 and 7>5, the digit in the thousandths place or 5 is increased by 1 and all the digits after it are omitted. Thus,
(iii) As the digit in the (5+1)th decimal place is 4 and 4<5, the digit in the (5+1)th decimal place and all the digits after it are omitted. Thus,
(iv) As the digit in the tenths place is 2 and 2<5, it is omitted and all the digits after it are also omitted. Thus,
TRY THIS: Convert into a decimal fraction and round off your answer to 4 decimal places.
Rounding Off to a Specified Unit:
We have just learned how to round off our answers to a specified number of decimal places. But in real life we have to use our sense and figure out how much to round off such that our answer makes sense.
 Raima buys a pair of sunglasses marked at Rs. 690.75 selling at a discount of 35 %. The shopkeeper charges her Rs. 449. Should Raima complain that she has been charged Rs. 0.0125 more?
 A lady wishing to buy a gold ring asks the jeweller the market price of gold. Would it make more sense to the lady if the jeweller quoted the price of gold in kilograms or grams?
 A traveller consults a railway timetable to find out how much it would cost to travel from Kolkata to Mumbai. Would it help the traveller if the fare chart quotes fares in paise per meter rather than rupees per kilometer?
Example 2: An icecream vendor buys 60 icecreams for Rs. 210 and wishes to earn a 40% profit. At what price should he sell each icecream?
Answer 2: Cost price per ice cream is 210/60=Rs. 3.50
Profit expected per icecream is 3.5 x 40/100, that is, Rs. 1.40
Accurate SP per icecream is Rs. 3.50 + Rs. 1.40 that is Rs. 4.90
As it would be difficult for all his customers to pay exact change (Rs. 4.90), the vendro could round it off to a reasonable price of Rs. 5 per icecream.
Example 3: What is the capacity of a cubical ink pot that is 0.04 m long?
Answer 3: The calculation of capacity with length in meters would give an answer in kilolitres. But the vessel in question is an ink pot and not a tanker !! Thus, it would make more sense to convert the length to centimeters and find the capacity in cubic cm or millilitres.
0.04 m is 4 cm.
Capacity of ink pot is 4 x 4 x 4, that is, 64 cubic cm.
Significant Digits in Decimals
We know that 00002.3=2.3=2.30000
The significant or meaningful digits in the above numbers are only the digits 2 and 3 on either side of the decimal point that give us an idea of its value and location on the number line.
 All nonzero digits in a decimal number are significant digits.
(a) 5.695 has four significant digits.
(b) 58.2 has three significant digits.
2. The zeroes between nonzero digits in a decimal number are significant digits.
(a) 3.01 has three significant digits, being 3, 0 and 1.
(b) 5.2001 has five significant digits, being 5, 2, 0, 0 and 1.
3. The zeros to the left of the first nonzero digit are not significant digits.
(a) 0.09 has only one significant digit, being 9.
(b) 0.00012 has only two significant digits, being 1 and 2.
4. The zeros to the right of the last nonzero digit may or may not be significant. The condition depends on the unit of measurement or the need for approximation.
Case I: When zeros to the right of the last nonzero digits are significant.
Example : Convert 1.2000 m into centimeter.
Solution: As 1.2000m is 120 cm, so in 1.2000 m there are three significant digits, being 1, 2 and 0.
Example: Convert 3.6000000 kg into mg.
Solution: As 3.6000000 kg is 3600000 mg, in 3.6000000 kg, there are 7 significant digits being 5, 6, and give zeros.
Case II: When zeros to the right of the last nonzero digit are not significant.
Example: Express Rs. 3.60000 in paise.
Solution: As Rs. 3.60000 is 360 paise, Rs. 3.60000 has only three significant digits, being 3, 6 and 0, the three zeros after it being insignificant.
Example: Given the area of the Arctic Ocean as 13079000 square km, express its area in thousand square km.
Solution: 13079000 sq km x 1/1000 is 13079 thousand sq km. Thus, 13079000 sq km has only 5 significant digits, being 1, 3, 0, 7 and 9, the three zeros after it being insignificant.
Approximation to Signficant Digits
Example 1: Approximate 0.0003801 to 1 significant digit.
Solution 1: The given number has 4 significant digits, being 3, 8, 0 and 1. To approximate it to 1 significant digit, it will have to be rounded off to its tenthousandths place where its first significant digit is. As 8 in the hundredthousandths place is greater than 5, 0.0003801 is 0.0004.
Example 2: Approximate 3.1428571 to 3 significant digits.
Solution 2: The given number has 8 significant digits. To approximate it to 3 significant digits, it will have to be rounded off to the hundredths place. As 2 in the thousandths place is less than 5, 3.1428571 3.14
Representation of Numbers
Let us recall how we expressed a decimal number in expanded form in previous classes:
Example : Write 5873.1264 in expanded form:
Solution: 5873.1264 can be written as
5000+800+70+3+0.1+0.02+0.006+0.0004 or
TRY THIS!
Write 5840.183 in expanded form.
Thus, whatever be the place of a digit in the integral or decimal part of a number, its place value can be described as a multiple of a power of 10.
The distance between Earth and Pluto is 575,00,00,000 km. This is too big a number to communicate and remember:
(i) Thus, it is approximated to its significant digits and represented as 5 billion 750 million km. Or as 575 crore km OR
(ii) All the significant digits are considered and it is written in scientific notation as the product of a decimal number less than 10 and a power of 10, that is, km.
We know that a millilitre is onethousandth part of a metre. Scientists and industrialists are very excited, about “nanotechnology” nowadays. Do you know the relationship between a nanometre and a metre?
1 nanometre is 0.000000001 of a metre or a nanometre is a thousandmillionth part of a metre. Such a small decimal number is again written in scientific notation as the product of a decimal number less than 10 and a power of 10 as follows: m.
Example: Write 0.000200100 in scientific notation.
Solution: The significant digits in the number are 2, 0, 0, and 1. The decimal number less than 10 is 2.001 and as a product of power of 10, the scientific notation for 0.000200100 is .
Try to figure out the weight of an electron if it is given in scientific notation as equal to kg.
Exercises:
 Approximate 0.0004567 to 1 significant digit.
 Write 6245.3173 in expanded form.
 Round off 22.5% of Rs. 58.50 to the nearest piase.
 Express 7 liters 54 ml to the nearest litre.
 Write 7.000 40 23587 in scientific notation.
Regards,
Nalin Pithwa
Skill Check XII: IITJEE Foundation Math
A) Evaluate the following fractions:
a)
b)
c)
d)
B) Evaluate the following fractions:
a)
b)
c)
d)
C) Simplify:
a)
b)
c)
d)
D) In a class of 40 students, two fifth are girls. Each girl brings a ribbon of m and each boy brings . What is the total length of ribbon collected by 40 students?
E) The cost of 7.25 kg oranges is Rs. 362.50 and the cost of 4.75 kg grapes is Rs. 191.25. Himani buys 6 kg oranges and 5 kg grapes. How much moeny has Himani spent?
Regards,
Nalin Pithwa
Skill Check X: IITJEE Foundation Math
I) Arrange the following decimal fractions in ascending order:
(i) 68.95, 6.985, 9.685, 86.59, 8.695
(ii) 1.36, 1.29, 1.48, 1.26, 1.38
(iii) 5.689, 5.869, 5.896, 5.698, 5.986
(iv) 1.1001, 1.0011, 1.111, 1.0101, 1.1011
(v) 7.8899, 7.9898, 7.9889, 7.9988, 7.8998
II) Add the following decimal fractions:
(i) 7.8 +3.2
(ii) 8.91 + 1.89
(iii) 2.3 + 2.65
(iv) 1.721 + 1.892
(v) 5.867+6.719+3.0018
(vi) 3.468+2.12+1.34464
III) Subtract the following decimal fractions:
(i) 6.92.9
(ii) 3.41 – 2.21
(iii) 5.4 – 3.22
(iv) 5.216 – 3.162
(v) 2.00.9963
(vi) 8.0162.98694
IV) Multiply the following decimal fractions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
V) Divide the following decimal fractions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
VI) Find the HCF of the following decimal fractions:
(i) 0.72 and 4.8
(ii) 1.092 and 1.176
(iii) 0.21, 0.1925, and 0.175
(iv) 0.286, 0.3718, and 0.3146
VII) Find the LCM of the following decimal fractions:
(i) 4.5 and 0.42
(ii) 0.18 and 0.144
(iii) 2.52, 0.189 and 0.168
(iv) 0.112, 0.8, and 0.48
VIII) Find the greatest decimal fractions that can be divided by 0.33, 0.495 and 0.297 leaving exactly 0.15 as remainder.
IX) Simplify the following expressions:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
X) Simplify the following expressions:
(i)
(ii)
(iii)
XI) Juno buys 1.00 kg of plasticine at Rs. 35.50 per kg and 0.35 litres of paint at Rs. 142.80 per litre. If he gives the shopkeeper a 100 rupee note, how much change should he get back?
XII) 0.125 part of a peace keeping force are doctors, 0.09375 are engineers, 0.03125 parts are cooks, and the rest are armed soldiers. If the peacekeeping force has 1152 members, how many armed soldiers are in it?
XIII) Gitanjali read 0.25 part of a book on the first day, 0.35 part on the second day, and 160 pages to finish the book on the third day. How many pages were there in the book?
XIV) If i metre is equal to 3.28084 feet, how many feet will 15 metres equal to?
XV) If 2.54 cm made an inch, how many inches will 60.96 cm make?
XVI) 0.4 part of a 9.3 g ornamental chain is made of gold. If the chain is cut into 6 equal pieces how much gold will be there in each piece?
XVII) The perimeter of an isosceles trapezium is 7.07 cm. If its unequal sides measure 1.85 cm, 2.32 cm, how much do its equal sides measure?
XVIII) Mrs Rita gives 0.025 part of her salary as pocket money to her son Vinod every month. Vinod spends 0.8 part of his pocket money and saves the rest. If he saves Rs. 277.50 in three months, how much does Mrs. Rita earn in a month?
XIX) The average earning of three family members A, B and C is Rs. 11240.25. One member B leaves for another town and a new family member D starts earning. The new average earning of A, C and D is now Rs. 10520.50 . If D is earning Rs. 9886.25, how much was B earning?
XX) A petrol pump attendant lowers a 5 m long dipstick to check the oillevel in an underground tank. For every 0.1 m in height of oil on the dipstick, there is 1200 liters of oil in the tank. If ).38 part of the dipstick is wet with oil, how many litres of oil are there in the tank?
Regards,
Nalin Pithwa.
Skill Check IX: IITJEE Foundation Math
A) Arrange all the fractions in descending order:
i)
ii)
iii)
iv)
v)
B) Arrange the following fractions in ascending order:
i)
ii)
iii)
iv)
v)
vi)
III) Evaluate the following fractions:
(i)
(ii)
(iii)
(iv)
(v)
(vI)
IV) Evaluate the following fractions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
V) Evaluate the following fractions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
VI) Evaluate the following fractions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
VII) Find the HCF of the following fractions:
(i) and
(ii) and
(iii) and
(iv) and
(v) and and
(vi) and
IX) Find the LCM of the following fractions:
(i) and
(ii) and
(iii) and
(iv) and
(v)
(vi)
IX) Find the greatest fraction that divides and exactly and also find the smallest fraction that can be divided by the given fractions.
X) Simplify the following expressions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
XI) In a village consisting of 150 females and 100 males, of all females and of all males are graduates. What fraction of all the villagers are graduates?
XII) of all the money in Mr Ghoshn’s bank account is Rs 98000/. How much money does Mr Ghoshn have in his bank account?
XIII) A m long cable is cut into equal pieces measuring m each. How many such small pieces are there?
XIV) of a ship’s crew are deck officers, are engineers and stewards, and the rest are sailors. If there are 48 crew members in all, how many sailors are on board the ship?
XV) part of a road was paved on the first day, part on the second day, and part was paved on the third day. If m was paved on the fourth day to complete the road, what is the total length of the road paved?
XVI) The perimeter of an isosceles trapezium measures cm. If its unequal sides measure cm and cm, find the measure of its equal sides.
XVII) A father and his two sons construct a house for INR 525,000. The elder son contributes of his father’s contributions while the younger son contributes of his father’s contributions. How much do the three contribute individually?
XVIII) Ramu inherited of the money his grandfather left behind while his cousin Rakesh’s share was . If Ramu’s share was Rs 60000/ more than Rakesh’s share, find how much money their grandfather left behind.
XIX) A, B and C receive a total of Rs 2016 as monthly allowance from their dad such that C gets of what A gets, and B gets times C’s share. How much money do the three brothers get individually?
XX) studentts of a school come by school bus while students ride a bicycle to school. All the other students walk to school, of which walk on their own and the rest are escorted by an elder. If 196 students come to school walking on their own, how many students study in that school?
Regards,
Nalin Pithwa