Category Archives: IITJEE Foundation mathematics

Skill Check VI: IITJEE Foundation Math

A) Find all the factors of the following numbers: 42, 66, 88, 180, 810.

B) Use prime tree factorization to find the factors of : 1122, 2211, 2121, 8181, 8000.

C) Find the HCF of the following numbers by prime factorization: (a) 88 and 99 (b) 84 and 108 (c) 80 and 96 (d) 208 and 234

D) Find the HCF or GCD of the following numbers by Euclid’s Long Division Method: (a) 432, 540, and 648 (b) 408, 476, and 510 (c) 1350 and 1800 (d) 3600 and 5400 (e) 7560 and 8820 (f) 7920 and 8910 (g) 14112 and 12936 (h) 25740 and 24024 (i) 108, 288, and 360 (j) 1056, 1584, and 2178

E) Find the HCF or GCD of the following numbers by Euclid’s Long Division Method: (a) 1701, 1575, and 2016 (b) 4680, 4160, and 5200 (c) 3168. 3432, and 3696 (d) 4752, 5184, and 5616 (e) 8640, 10368, 12096 (f) 9072, 8400, 9744

F) Find the greatest number that divides 10368, 9504 and 11232 exactly leaving no remainders.

G) Find the greatest number that divides 7355, 8580, and 9805 leaving exactly 5 as a remainder in each case.

H) Find the greatest number that divides 9243 and 12325 leaving exactly 3 and 5 as remainders respectively.

I) What would be the length of the the longest tape that can be used to measure the length and breadth of an auditorium 204 feet wide and 486 feet long in an exact number of times.

J) A big cardboard picture 126 cm wide and 135 cm long is to be cut up into square pieces to create a jigsaw puzzle. How many small pieces would go on to make the jigsaw puzzle if each piece is to be equal and of the maximum possible size?

K) Square placards need to be cut out from a rectangular piece of cardboard 60 inches wide and 72 inches long. What is the maximum number of equal sized placards of the biggest possible size that can be cut out? What would be the length of each placard?

L) Three ribbons, 171 cm, 185 cm, and 199 cm long are to be cut into equal pieces of maximum possible length, leaving bits of ribbons 3 cm long from each. What would be the length of each piece of ribbon and how many such pieces can one get ?

M) The capacities of two emtpy water tanks are 504 litres and 490 litres. What would be the maximum capacity of a bucket that can be used an exact number of times to fill the tanks? How many bickets full of water will be needed?

Regards,

Nalin Pithwa

Skill Check V: IITJEE Foundation Math

I. Consider the following relationship amongst various number systems: $\mathcal{N} \subset \mathcal{W} \subset \mathcal{Z} \subset \mathcal{Q} \subset \mathcal{R}$.

Write the following numbers in the smallest set or subset in the above relationship:

(a) 8 (b) -8 (c) +478 (d) -2191 (e) -21.91 (f) +3.6 (g) 0 (h) +4.6 (i) $-6.\dot{7}$ (j) 8.292992999…(k) $\frac{3}{8}$ (l) $\frac{8}{2}$ (m) $0 \frac{0}{7}$ (n) $-3\frac{1}{5}$ (o) $\frac{22}{33}$ (p) $\sqrt{64}$ (r) $\sqrt{6.4}$ (s) $2+\sqrt{3}$ (t) $6\sqrt{4}$ (u) $4\sqrt{6}$

II. If $\frac{22}{7} = 3.1428571...$ is $\frac{22}{7}$ an irrational number?

III. Fill in the boxes with the correct real numbers in the following statements: (a) $2\sqrt{7}+\sqrt{7} = \Box+ 2\sqrt{7}$ (b) $3.\dot{8} + 4.65 = 4.65 + \Box$ (c) $\Box + 29 = 29 + 5\sqrt{10}$ (d) $3.\dot{9} + (4.69 +2.12) = (\Box + 4.69) + 2.12$ (e) $(\frac{7}{8} + \frac{3}{7})+\frac{6}{5} = (\frac{6}{5} + \frac{3}{7}) + \Box$ (f) $3\sqrt{2}(\sqrt{3}+2\sqrt{5}) = (3\sqrt{2}+2\sqrt{5}) + (\Box \times \Box)$ (g) $1\frac{3}{7} (2\frac{7}{11} + \Box) = (1\frac{3}{7} \times 1 \frac{8}{9}) + (1\frac{3}{7} + 2\frac{7}{11})$ (h) $2\frac{1}{3} + \Box =0$ (xi) $\frac{7}{-8} \times \Box = 1$ (i) $-7.35 + \Box = 0$

IV. Find the answers to the following expressions by using the properties of addition and multiplication of real numbers:

Before that, we can recapitulate the relevant properties here :

Properties of Real Numbers:
Closure Property: The sum, difference, product,or quotient of two real numbers is a real number.

Commutative Property of Addition and Multiplication:

A change in the order of addition or multiplication of two real numbers does not change their respective sum or product. (a) $x+y = y+x$ (b) $x \times y = y \times x$

Associative Property of Addition and Multiplication:

A change in the grouping of three real numbers while adding or multiplying does not change their respective sum or product :

$(a+b)+c = (a+b)+c$ and $a \times (b \times c) = (a \times b) \times c$

Distributive Property of Multiplication over Addition:

When a real number is multiplied by the sum of two or more real numbers, the product is the same as the sum of the individual products of the real number and each addend.

$m(a+b) = ma+mb$. Clearly, multiplication has “distributed” over addition.

Identity Property of Real Numbers
The addition of zero or the multiplication with one does not change a real number. That is,

$a+0=0+a=a$ and $a \times 1 = a = 1 \times a$

Inverse Property of Real Numbers

• Corresponding to every real number, there exists another real number of opposite sign such that the sum of the two real numbers is zero: $a+ a^{'}=0$ such that $a^{'}=-a$
• Corresponding to every (non-zero) real number, there exists a real number, known as its reciprocal, such that the product of the two real numbers is 1. That is, $a \times \frac{1}{a} = 1$, where $r \neq 0$.

Now, in the questions below, identify the relevant properties:

(a) $283 +(717 + 386)$

(b) $(2154 - 1689) + 1689$

(c) $3.18 + (6.82+1.35)$

(d) $(6.784-3.297) + 3.297$

(e) $\frac{7}{13} + (\frac{6}{13}-1)$

(e) $0.25 \times (4.17 -0.17)$

(f) $(6.6 \times 6.6) + (6.6 \times 3.4)$

(g) $(\frac{2}{3} \times 5) - (\frac{2}{3} \times 2)$

(h) $(6.\dot{8} \times 5) - (6.\dot{8} \times 4)$

(i) $\frac{6}{7} \times \frac{7}{6} \times \frac{6}{7}$

V) Which of the following operations on irrational numbers are correct?

(a) $6\sqrt{5} - 4\sqrt{3}=2\sqrt{2}$

(b) $\sqrt{7} \times \sqrt{7} = 7$

(c) $3 \sqrt{3} + 3 \sqrt{3} = 6 \sqrt{3}$

(d) $\sqrt{7} \times \sqrt{7} = 49$

(e) $\sqrt{7} + \sqrt{2} = \sqrt{9}$

(f) $2 \sqrt{8} \times 3\sqrt{2} =24$

(g) $8\sqrt{2} + 8 \sqrt{2} =32$

(h) $2\sqrt{3}= 3\sqrt{6} = \frac{2}{3\sqrt{2}}$

(i) $5+\sqrt{3} = 5\sqrt{3}$

(j) $3\sqrt{20} \div 3\sqrt{5}=2$

VI) Find the rationalizing factors of the following irrational numbers:

(a) $\sqrt{10}$

(b) $\sqrt{7}$

(c) $2\sqrt{5}$

(d) $3\sqrt{7}$

(e) $-2\sqrt{8}$

(f) $-6\sqrt{7}$

(g) $\frac{1}{\sqrt{2}}$

(h) $\frac{2}{\sqrt{3}}$

(i) $2\sqrt{3}=4\sqrt{3}$

(j) $7\sqrt{5} - 2\sqrt{5}$

(k) $1+\sqrt{2}$

(l) $3-\sqrt{5}$

(m) $3\sqrt{2}+6$

(n) $4\sqrt{7} + 6\sqrt{2}$

(o) $3\sqrt{6}-2\sqrt{3}$

VII) Rationalize the denominators of the following numbers:

(a) $\frac{1}{\sqrt{3}}$

(b) $\frac{3}{\sqrt{3}}$

(c) $\frac{3}{\sqrt{5}}$

(d) $\frac{8}{\sqrt{6}}$

(e) $\frac{3}{2\sqrt{5}}$

(f) $\frac{\sqrt{5}}{\sqrt{7}}$

(g) $\frac{3\sqrt{3}}{3\sqrt{5}}$

(h) $\frac{3}{\sqrt{5}-sqrt{3}}$

(i) $\frac{5}{\sqrt{3}+\sqrt{2}}$

(j) $\frac{17}{4\sqrt{6}+3\sqrt{5}}$

(k) $\frac{3}{3+\sqrt{3}}$

(l) $\frac{11}{3\sqrt{5}-2\sqrt{3}}$

(m) $\frac{\sqrt{5}}{3\sqrt{5}-3\sqrt{2}}$

(n) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

(o) $\frac{\sqrt{5}-sqrt{2}}{\sqrt{5}+\sqrt{2}}$

VIII. Find the additive inverse of each of the following irrational numbers:

(i) $\sqrt{7}$ (ii) $3\sqrt{5}$ (iii) $-6\sqrt{7}$ (iv) $5+\sqrt{7}$ (v) $3\sqrt{7} - 2\sqrt{8}$

IX. Find the multiplicative inverse of each of the following irrational numbers:

(i) $\sqrt{6}$ (ii) $\frac{1}{2\sqrt{7}}$ (iii) $\frac{3\sqrt{8}}{2\sqrt{7}}$ (iv) $\frac{4}{3+\sqrt{2}}$ (v) $\frac{2\sqrt{5}+3\sqrt{6}}{5\sqrt{8}-4\sqrt{7}}$

X. Illustrate the closure property of addition of real numbers using the irrational numbers $\sqrt{5}$ and $2\sqrt{5}$.

XI. Illustrate that the closure property does not apply on subtraction of real numbers using two rational numbers: $2\frac{1}{7}$ and $-3\frac{2}{5}$.

XII. Illustrate the distributive property of multiplication over addition of real numbers using three irrational numbers: $3\sqrt{7}$, $-2\sqrt{7}$ and $\sqrt{7}$.

Regards,

Nalin Pithwa

Skill Check IV: IITJEE Foundation Maths

I. State whether the following statements are True or False:

(i) 0 is to the left of all negative numbers on the number line.

(ii) 3 is greater than -3333.

(iii) $1\frac{2}{5}$ will lie to the right of the mid-point between 1 and 2 on the number line.

(iv) $1\frac{2}{5}$ will lie to the left of the mid-point between 1 and 2 on the number line.

(v) If a decimal fraction is non-terminating and non-recurring, it is known as a rational number.

(vI) The rational number $3\frac{1}{5}$ lies between $3\frac{2}{11}$ and $3 \frac{3}{11}$

II. How many natural numbers lie between 212 and 2120?

III. How many integers lie between -219 and +2190?

IV. Write the following numbers in descending order: (i) -213, +126, -212, +127, -127 (ii) $-1\frac{7}{11}, -3\frac{7}{11}$, $-2\frac{7}{11}, -5\frac{7}{11}, -4\frac{7}{11}$ (iii) $\frac{3}{5}, -\frac{2}{9}, \frac{5}{7}, -\frac{3}{10}, \frac{11}{21}$ (iv) -2.3838, -2.3388, -2.8838, -2.8833, -2.3883 (v) $3.\overline{8}, 3.\overline{6}, 3.88, 3.6\overline{8}, 3.8\overline{6}$

V. Write the following numbers in ascending order: (i) +418, -481, -418, +481, -841 (II) $-1\frac{3}{11}, -1\frac{4}{11}, -1\frac{5}{11}, -1\frac{6}{11}, -1\frac{7}{11}$ (iii) $\frac{2}{5}, \frac{11}{23}, \frac{7}{15}, \frac{9}{20}, \frac{3}{7}$ (iv) $6.7134, 6.7431, 6.7341, 6.7413, 6.7143$ (v)$7.9\dot{8}, 7.\dot{9}, 7.\overline{98}, 7.8\dot{9}, 7.\dot{8}$

VI. Insert a rational number between the following pairs of numbers: (i) -0.001 and +0.001 (ii) -8 and -3 (iii) 85 and 86 (iv) 5.5 and 6 (v) $\frac{1}{4}$ and $\frac{1}{5}$ (vi) $2\frac{2}{5}, 2\frac{3}{5}$ (vii) 3.0688 and 3.0699 (viii) 5.2168 and 5.2169 (ix) $1\frac{9}{15}, 1\frac{11}{15}$ (x) $-8\frac{6}{7}, -8\frac{5}{7}$

VII. Insert 2 rational numbers between the following numbers: (a) 3.18 and 3.19 (b) $2\frac{2}{5}, 2 \frac{3}{5}$

VIII. Represent the following rational numbers on the number line:

(i) $2\frac{1}{3}$ (ii) $-\frac{5}{7}$ (iii) $3.7$ (iv) $4.85$ (v) $6 \frac{7}{11}$

IX. Which of the following rational numbers will have a terminating decimal value? (a) $\frac{3}{5}$ (b) $-\frac{5}{7}$ (c) $\frac{1}{2}$ (d) $-\frac{7}{10}$ (v) $\frac{7}{15}$

X. Convert each of the following decimal fractions in the form $\frac{p}{q}$, where p and q are both integers, but q is not zero: (a) 0.32 (b) 0.42 (c) 0.85 (d) 1.875 (e) 0.4375 (f) $3.\dot{7}$ (g) $1.6\dot{4}$ (h) $5.\overline{23}$ (i) $7.11\dot{3}$ (j) $8.9\overline{505}$

We will continue this series later…

Regards,

Nalin Pithwa

Skill Check III: IITJEE Foundation Maths

State whether the following statements are true or false:

1. If $A = \{ x | x = 5n, 5 < n < 10, n \in N\}$, then $n(A)=4$
2. If $n(A) = n(B)$, then $set A \leftrightarrow set B$
3. If Set $A= \{ x | x \in N, x<3\}$, then A is a singleton set.
4. The intelligent students of class VIII form a set.
5. The students passing the half-yearly exams in Class VIII B of school is a set.
6. $A = \{ x | x = p^{3}, p<4, p \in \mathcal{N}\}$ and $\{ x|x = m^{2}, m < 4, m \in \mathcal{N} \}$ are overlapping sets.
7. $A = \{ x | x \in \mathcal{N}\}$ is a subset of $B = \{ x | x in \mathcal{Z}\}$
8. If we denote the universal set as $\Omega = \{ p,q,r,s,t,u,v\}$ and $A = \{ q,u,s,t\}$, then $\overline{A} = \{ p, r, v\}$
9. $A = \{ x | x =2p, p \in \mathcal{N}\}$ and $B = \{ x | x =3p, p \in \mathcal{N}\}$ are disjoint sets.
10. If $A = \{ 2,3,4\}$, then $P(A)= \{ \phi, A, \{ 2\}, \{ 3\}, \{ 4 \}, \{ 2,3\}, \{ 2,4\}, \{ 3,4\}\}$ where $P(A)$ is the power set of set A.

II. If C is a letter in the word down all the subsets of C.

III. Write down the complements of all the 8 subsets of set C above.

IV. If $Q = \{ x : x =a^{2}+1, 2 \leq a \leq 5\}$, what is the power set of Q?

V. If $x = \{ x | x<20, x \in \mathcal{N}\}$, and if $A = \{x | x = 2a, 3 < a < 8, a \in \mathcal{N} \}$, and if $B = \{ x | x = 3b, b < 5, b \in \mathcal{N}\}$, and if $C = \{x | x = c+1, 5 < c < 15, c \in \mathcal{N} \}$, then find : (i) $n(B)$ (ii) $n(C)$ (iii) $\overline{A}$ (iv) $\overline{B}$ (v) $P(B)$

VI. If $A = \{ x| x \in \mathcal{N}, 3 < x < 10\}$, and if $B = \{ x| x =4a-1, a<5, a \in \mathcal{N}\}$ and if $C = \{ x | x = 3a+2, a<7, a \in \mathcal{N}\}$, then confirm the following: (i) the commutative property of the unions of sets B and C (ii) the commutative property of intersection of two sets A and C (iii) the associative property of the union of the sets A, B and C (iv) the associative property of intersection of sets A, B and C.

VII. If $A = \{ x | x \in \mathcal{N}, 4 \leq x \leq 12\}$, and $B = \{ x| x = a+1, a<8, a \in \mathcal{N}\}$, and $C= \{ x| x =2n, 1 < n <7, n \in |mathcal{N}\}$, then find (i) $A-B$ (ii) $B-C$ (iii) $B \bigcap C$ (iv) $A - (B \bigcap C)$ (v) $B - (A \bigcap C)$ (vi) $A-C$ (vii) $A- (B-C)$ (viii) $A- (B \bigcup C)$

VIII. If $\xi = \{ x | x \hspace{0.1in}is \hspace{0.1in} a \hspace{0.1in} letter \hspace{0.1in}of \hspace{0.1in}the \hspace{0.1in} English \hspace{0.1in} alphabet \hspace{0.1in} between \hspace{0.1in} but \hspace{0.1in} not \hspace{0.1in} including \hspace{0.1in} d \hspace{0.1in} and \hspace{0.1in} o\}$, and let $A = \{ l, m , n\}$ and let $B= \{ e,f,g,h,i,j,k,l\}$, and let $C = \{ j,k,l,m\}$, find (i) $\overline{A} \bigcup \overline{B}$ (ii) $\overline{B} \bigcap \overline{C}$ (iii) $A \bigcap C$ (iv) $B - (A \bigcap C)$ (v) $\overline{B-A}$ (vi) Is $(B-C) \subset (B-A)$? (vii) Is $\overline{A} \bigcap \overline{B} = \phi$?

IX. All 26 customers in a restaurant had either drinks, snacks, or dinner. 18 had snacks, out of which 6 had only snacks, 4 had snacks and drinks but not dinner, 2 had drinks and dinner but not snacks, and 3 had snacks and dinner but not drinks. If 14 customers had drinks, find (i) how many customers had all three — drinks, snacks as well as dinner. (ii) how many customers had dinner but neither snacks nor drinks (iii) how many customers had only drinks.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101
India

Skill Check II: IITJEE foundation maths

Set Theory Primer/basics/fundamentals/preliminaries:

I. Represent the following sets in Venn Diagrams: (a) $\Xi = \{ x : x = n, n <40, n \in \mathcal{N} \}$ (b) $\mu = \{ x : x = 6n, n < 6. n \in \mathcal{N}\}$ (c) $\alpha = \{ x: x = 3n, n<8, n \in \mathcal{N} \}$

2. If $x = \{ x: x<29 \hspace{0.1in}and \hspace{0.1in}prime\}$ and $A = \{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} prime \hspace{0.1in} factor \hspace{0.1in} of \hspace{0.1in} 210\}$, represent A in Venn diagram and find $\overline{A}$.

3. 95 boys of a school appeared for a physical for selection in NCC and Boy Scouts. 21 boys got selected in both NCC and Boy Scouts, 44 boys were not selected in Boy Scouts and 20 boys were not selected only in boy scouts. Draw Venn diagram and find : (i) how many boys did not get selected in NCC and boy scouts. (ii) how many boys did not get selected only in NCC (iii) how many boys got selected in NCC (iv) how many boys got selected in boy Scouts (v) How many boys got selected in NCC not in boy Scouts?

Regards,

Nalin Pithwa.

IITJEE Foundation Maths: Tutorial Problems IV

1. Resolve into factors: (a) $2x^{2}-3ab+(a-6b)x$ (b) $4x^{2}-4xy-15y^{2}$
2. In the expression, $x^{3}-2x^{2}+3x-4$, substitute $a-2$ for x, and arrange the result according to the descending powers of a.
3. Simplify: (i) $\frac{x}{1-\frac{1}{1+x}}$ (ii) $\frac{x^{2}}{a+\frac{x^{2}}{a+\frac{x^{2}}{a}}}$
4. Find the HCF of $3x^{3}-11x^{2}+x+15$ and $5x^{4}-7x^{3}-20x^{2}-11x-3$
5. Express in the simplest form: (i) $\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x+y}{y}-\frac{y+x}{x}}$ (ii) $(\frac{x^{3}-1}{x-1} + \frac{x^{3}+1}{x+1})\div (\frac{1}{x-1} + \frac{1}{x+1})$
6. A person possesses Rs. 5000 stock, some at 3 per cent; four times as much at 3.5 %, and the rest at 4 %; find the amount of each kind of stock when his income is Rs. 178.
7. Simplify the expression: $-3[(a+b)-{(2a-3b) -( 5a+7b-16c) - (-13a +2b -3c -5d)}]$, and find its value when $a=1, b=2, c=3, d=4$.
8. Solve the following equations : (i) $11y-x=10$ and $11x-101y=110$ (ii) $x+y-z=3$, and $x+z-y=5$, and $y+z-x=7$.
9. Express the following fractions in their simplest form: (i) $\frac{32x^{3}-2x+12}{12x^{5}-x^{4}+4x^{2}}$ (ii) $\frac{1}{x + \frac{1}{1+ \frac{x+3}{2-x}}}$
10. What value of a will make the product of $3-8a$ and $3a+4$ equal to the product of $6a+11$ and $3-4a$?
11. Find the LCM of $x^{3}-x^{2}-3x-9$ and $x^{3}-2x^{2}-5x-12$
12. A certain number of two digits is equal to seven times the sum of its digits; if the digit in the units’ place be decreased by two and that in the tens place by one, and if the number thus formed be divided by the sum of its digits, the quotient is 10. Find the number.
13. Find the value of $\frac{6x^{2}-5xy-6y^{2}}{2x^{2}+xy-y^{2}} \times \frac{3x^{2}-xy-4y^{2}}{2x^{2}-5xy+3y^{2}} \div \frac{9x^{2}-6xy-8y^{2}}{2x^{2}-3xy+y^{2}}$
14. Resolve each of the following expressions into four factors: (i) $4a^{4}-17a^{2}b^{2}+4b^{4}$; (ii) $x^{8}-256y^{8}$
15. Find the expression of highest dimensions which will divide $24a^{4}b -2a^{2}b^{2}-9ab^{4}$ and $18a^{6}+a^{4}b^{2}-6a^{3}b^{3}$ without remainder.
16. Find the square root of : (i) $x(x+1)(x+2)(x+3)+1$ (ii) $(2a^{2}+13a+15)(a^{2}+4a-5)(2a^{2}+a-3)$
17. Simplify: $x - \frac{2x-6}{x^{2}-6x+9} - 3 + \frac{x^{2}+3x-4}{x^{2}=x-12}$
18. A quantity of land, partly pasture and partly arable, is sold at the rate of Rs. 60 per acre for the pasture and Rs. 40 per acre for the arable, and the whole sum obtained at Rs. 10000. If the average price per acre were Rs. 50, the sum obtained would be 10 per cent higher; find how much of the land is pasture and how much is arable.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101
India

IITJEE Foundation Maths : Tutorial Problems III

1. When $a=-3, b=5, c=-1, d=0$, find the value of $26c\sqrt[3]{a^{3}-c^{2}d+5bc-4ac+d^{2}}$
2. Solve the equations: (a) $\frac{1}{3}x - \frac{2}{7}y = 8 -2x$ and $\frac{1}{2}y - 3x =3-y$ (b) $1 = y+z =2(z+x)=3(x+y)$
3. Simplify: (a) $\frac{a-x}{a+x} - \frac{4x^{2}}{a^{2}-x^{2}} + \frac{a-3x}{x-a}$ (b) $\frac{b^{2}-3b}{b^{2}-2b+4} \times \frac{b^{2}+b-30}{b^{2}+3b-18} \div \frac{b^{3}-3b^{2}-10b}{b^{2}+8}$
4. Find the square root of : $9-36x+60x^{2}-\frac{160}{3}x^{3}+\frac{80}{3}x^{4}-\frac{64}{3}x^{5}+\frac{64}{81}x^{6}$
5. In a cricket match, the extras in the first innings are one-sixteenth of the score, and in the second innings the extras are one-twelfth of the score. The grand total is 296, of which 21 are extras; find the score in each innings.
6. Find the value of $\frac{a^{2}-x^{2}}{\frac{1}{a^{2}} - \frac{2}{ax} + \frac{1}{x^{2}}} \times \frac{\frac{1}{a^{2}x^{2}}}{a+x}$
7. Find the value of : $\frac{1}{3}(a+2) -3(1-\frac{1}{6}b) - \frac{2}{3}(2a-3b+\frac{3}{2}) + \frac{3}{2}b - 4(\frac{1}{2}a-\frac{1}{3})$.
8. Resolve into factors: (i) $3a^{2} -20a-7$ (ii) $a^{4}b^{2}-b^{4}a^{2}$
9. Reduce to lowest terms: $\frac{4x^{3}+7x^{2}-x+2}{4x^{3}+5x^{2}-7x-2}$
10. Solve the following equations: (a) $x-6 -\frac{x-12}{3}= \frac{x-4}{2} + \frac{x-8}{4}$; (b) $x+y-z=0$, $x-y+z=4$, $5x+y+z=20$; (c) $\frac{ax+b}{c} + \frac{dx+e}{f} =1$
11. Simplify: $\frac{x+3}{x^{2}-5x+6} - \frac{x+2}{x^{2}-9x+14} + \frac{4}{x^{2}-10x+21}$
12. A purse of rupees is divided amongst three persons, the first receiving half of them and one more, the second half of the remainder and one more, and the third six. Find the number of rupees the purse contained.
13. If $h=-2, k=1, l=0, m=1, n=-3$, find the value of $\frac{h^{2}(m-l)-\sqrt{3hn}+hk}{m(l-h)-2hm^{2}+\sqrt[3]{4hk}}$
14. Find the LCM of $15(p^{3}+q^{3}), 5(p^{2}-pq+q^{2}), 4(p^{2}+pq+q^{2}), 6(p^{2}-q^{2})$
15. Find the square root of (i) $\frac{4x^{2}}{9} + \frac{9}{4x^{2}} -2$; (ii) $1-6a+5a^{2}+12a^{3}+4a^{4}$
16. Simplify $\frac{20x^{2}+27x+9}{15x^{2}+19x+6} + \frac{20x^{2}+27x+9}{12x^{2}+17x+6}$
17. Solve the equations: (i) $\frac{a(x-b)}{a-b} + \frac{b(x-a)}{b-a} =1$ (ii) $\frac{9}{x-4} + \frac{3}{x-8} = \frac{4}{x-9} + \frac{8}{x-3}$
18. A sum of money is to be divided among a number of persons; if Rs. 8 is given to each there will be Rs. 3 short, and if Rs. 7.50 is given to each there will be Rs. 2 over; find the number of persons.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai , Maharastra 400101
India

IITJEE Foundation Maths: Tutorial Problems II

1. A, B, C start from the same place at the rates a, $a+b$, $a+2b$ kilometres per hour respectively. B starts n hours after A, how long after B must C start in order that they may overtake A at the same instant, and how far will they then have walked?
2. Find the distance between two towns when by increasing the speed 7 kilometres per hour a train can perform the journey in 1 hour less, and by reducing the speed 5 kilometres per hour can perform the journey in 1 hour more.
3. A person buys a certain quantity of land. If he had bought 7 hectares more each hectare would have cost Rs 80 less; and if each hectare had cost Rs. 360 more, he would have obtained 15 hectares less, how much did he pay for the land?
4. A can walk half a kilometre per hour faster than B; and three quarters of a kilometre per hour faster than C. To walk a certain distance C takes three-quarters of an hour more than B, and two hours more than A; find their rates of walking per hour.
5. A person spends Rs. 15 in buying goods; if each kg had cost 25 paise more he would have got 5 kg less, but if each kg had cost 15 paise less, he would have received 5 kg more; what weight did he buy?
6. Five silver coins weight 125 gm and are worth Rs. 6. Ten bronze coins weigh 500 gm and are worth 80 paise. A number of silver and bronze coins which are worth Rs. 134 weigh 11 kg and 250 gm. How many coins of each kind are there?
7. A and B are playing for money; in the first game, A loses one half of his money, but in the second he wins one-quarter of what B then has. When they cease playing, A has won Rs. 6 and B has still Rs. 14.50 more than A; with what amounts did they begin?
8. A, B, C each spend the same amount in buying different qualities of the same commodity. B pays 36 paise per kg less than A and obtains 750 gm more; C pays 60 paise per kg more than A and obtains one kg less; how much does each spend?

IITJEE Foundation practice or training problem sheet: I

1. If the numerator of a fraction is increased by 5, it reduces to $\frac{2}{3}$, and if the denominator is increased by 9, it reduces to $\frac{1}{3}$. Find the fraction.
2. Find a fraction such that it reduces to $\frac{3}{5}$ if 7 is subtracted from its denominator, and reduces to $\frac{3}{8}$ on subtracting 3 from its numerator.
3. If unity is taken from the denominator of a fraction, it reduces to $\frac{1}{2}$; if 3 is added to the numerator it reduces to $\frac{4}{7}$, find the required fraction.
4. Find a fraction which becomes $\frac{3}{4}$ on adding 5 to the numerator and subtracting 1 from its denominator; and, reduces to $\frac{1}{3}$ on subtracting 4 from the numerator and adding 7 to the denominator.
5. If 9 is added to the numerator a certain fraction will be increased by $\frac{1}{3}$; if 6 is taken from the denominator the fraction reduces to $\frac{2}{3}$; find the required fraction.
6. At what time between 9 and 10 o’clock are the hands of a watch together?
7. When are the hands of a clock 8 minutes apart between the hours at 5 and 6 ?
8. At what time between 10 and 11 o’clock is the hour hand six minutes ahead of the minute hand?
9. At what time between 1 and 2 o’clock are the hands of a watch in the same straight line?
10. At what times between 12 and 1 o’clock are the hands of a watch at right angles?
11. A person buys 20 m of cloth and 25 m of canvas for Rs. 22.50. By selling the cloth at a gain of 15 per cent, and the canvas at a gain of 20 per cent, he clears Rs. 3.75. Find the price of each per metre.
12. A dealer spends Rs. 6950 in buying horses at Rs, 250/- each and cows at Rs. 200/- each; through disease, he loses 20 percent of the horses and 25 % of the cows. By selling the animals at the price he gave for them, he receives Rs. 5400/-. Find how many of each kind he bought.
13. The population of a certain district is 53000, of whom 835 can neither read nor write. These consists of 2 %, of all the males and 3 % of all the females; find the number of males and females.
14. Two persons C and D start simultaneously from two places a kilometre apart, and walk to meet each other; if C walks p kilometres per hour, and D one kilometre per hour faster than C, how far will D have walked when they meet?
15. A can walk a kilometres per hour faster than B; supposing that he gives B a start of c kilometres, and that B walks a kilometres per hour, how far will A have walked when he overtakes B?

Cheers,

Nalin Pithwa

Two cute problems in HP : IITJEE Foundations\Mains, pre RMO

Problem 1:

If $a^{2}, b^{2}, c^{2}$ are in AP, show that $b+c, c+a, a+b$ are in HP.

Proof 1:

Note that a straight forward proof is not so easy.

Below is a nice clever solution:

By adding $ab+bc+ca$ to each term, we see that:

$a^{2}+ab+ac+bc, b^{2}+ab+ac+bc, c^{2}+ab+ac+bc$ are in AP.

that is, $(a+b)(a+c), (b+c)(b+a), (c+a)(c+b)$ are in AP.

Dividing each term by $(a+b)(b+c)(c+a)$.

$\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b}$ are in AP.

that is, $b+c, c+a, a+b$ are in HP.

QED.

Problem 2:

If the $p^{th}, q^{th}, r^{th}, s^{th}$ terms of an AP are in GP, show that $p-q, q-r, r-s$ are in GP.

Proof 2:

Once again a straight forward proof is not at all easy.

Below is a “bingo” sort of proof 🙂

With the usual notation, we have

$\frac{a+(p-1)d}{a+(q-1)d} = \frac{a+(q-1)d}{a+(r-1)d} = \frac{a+(r-1)d}{a+(s-1)d}$

Hence, each of the ratios is equal to

$\frac{(a+(p-1)d)-(a+(q-1)d)}{(a+(q-1)d)-(a+(r-1)d)} = \frac{(a+(q-1)d)-(a+(r-1)d)}{(a+(r-1)d)-(a+(s-1)d)}$

which in turn is equal  to $\frac{p-q}{q-r} = \frac{q-r}{r-s}$

Hence, $p-q, q-r, r-s$ are in GP.

Cheers,

Nalin Pithwa