RELATION - XI

RELATION__

Let A and B be two non-empty sets. Then, a relation R from A to B is a subset of (A x B).

Thus, R is relation from A to B 

    <=< R ⊂ (A x B).

If (a,b) ∈ R then we say that 'a is related to B' and we write, a R b.

If (a, b) not belongs to R then 'a is not related to b'.

** Let R be a relation from A to B. Then, R ⊂ (A x B).

DOMAIN:

The set of all first coordinates of elements of R is called the domain of R, written as Dom (R).

RANGE:

The set of all second coordinates of elements of R is called the range of R, denoted by range (R).

CO-DOMAIN:

The set B is called the coordinates of R.

Thus, Dom(R)= {a : (a,b) ⊂ R} and

Range (R)= {b : (a,b) ∈ R}

TYPES OF RELATION:

Let A be a non-empty set. Then, a relation R on A is said to be:

i) Reflexive: if, (a,a) ∈ R for all ∈ A, i e., a R a for all ∈ A.

ii) symmetric: if, (a, b) ∈ R 

       => (b,a) ∈ R for all a,b ∈ A, i.e, a R b

      => b R a for all a, b ∈ A.

iii) Transitive: if (a,b) ∈ R and (b, c) ∈ R => (a,c) ∈ R for all a, b, c ∈ A, i.e., a R b, b R c

  => a R c for all a, b, c ∈ A 

• EQUIVALENCE RELATION:

A relation which is Reflexive, symmetric and transitive, is called an equivalence relation.

           EXERCISE-1

               --------------------

1) Let A={3,4,5,6} and B={3,4,5}, find A x B, then find the set of all ordered pairs which satisfy:

A) is equals to.           (3,3),(4,4),(5,5)

B) is less than.           (3,4),(3,5),(4,5)

C) is greater than.              (4,3),(5,3),(5,4), (6,3),(6,4),(6,5) 

D) is two less than.                     (3,5)

E) is two more than.          (5,3),(6,4)

F) is a factor of.        (3,3),(4,4),(5,5)

2) If A={2,3,4,5,6,7,8,9}, find the set of elements of relations. L and M on the set A such that.

 a) L means "is Square of".       {(4,2)(9,3)}

 b)  M={(x,y): x+y=8}.                 {(2,6),(3,5),(4,4),(5,3),(6,2)

3) Let A{a,b,c,d} and B{x,y,z}. State which of the following are relations from A to B:

 a) {(a,y),(a,z),(c,x),(d,y)}.     

 b) {(a,x),(a,y),(a,z),(b,x)}

 c) {(x,b),(y,a),(y,c),(z,d)}.               a,b

4) Given A={a,b,c,d} and B={x,y,z}. state which of the following are relations from B to A:

 a) {(a,x),(b,y),(c,z),(d,x)}

 b) {(x,a),(y,b),(z,c),(z,d)}

 c) {(a,b),(b,z),(x,a),(z,b)}

 d) {(x,a),(x,c),(y,a),(z,b),(z,d)}.      b,d

5) Given relation R={(x,y): x> y+2}. State which of the given ordered pairs belong to this relation.

a) (3,0)   

b) (5,4) 

c) (-7,-10) 

d) (7,9).                                          a,c

6) Write the domain and range of the relation:

a) {(3,-3),(4,-4),(5,-5),....}.        

                       D:{3,4,5.....}

                       R:{-3,-4,-5,.....}

b) {(-3,1),(-2,1), (-1,1), (1,1),(2,1), (3,1)}.

                        D:{-3,-2,-1,1,2,3}

                        R:{1}

c) (1,1),(2,8),(3,27),....}

                       D:{1,2,3,......}

                       R:{1,8,27.........}

d) {2, 1/2), (3,1/3), (4,1/4),(5,1/5).

                      D:{2,3,4,5}

                      R:{1/2,1/3,1/4,1/5}

7) Find the domain and the range of the relation:

a) {(x, 1/x): x ∈ N and 2< x ≤ 6}. 

                        D:{3,4,5,6}

                      R:{1/3,1/4,1/5,1/6}

b) {(x,y): x,y ∈N and x+y < 6}.

                        D:{1,2,3,4}

                        R:{1,2,3,4}

c) {x, x-2): x ∈N and 2< x² ≤ 50}.

                      D:{2,3,4,5,6,7}

                      R:{0,1,2,3,4,5}

8) Given A={1,2,3,5,7,8,9,16,27,36}. find the relation R on A, when R means "is cube of".               

                     D:{(1,1),(8,2),(27,3)                 

9) If A={3,5,7,8} & B={4,5,6,7,8,9}. Then find:

i) K={(a,b)∈ A x B : a-b=1}.    {(5,4),(7,6),(8,7)}

ii) L={(a,b)∈  A xB : a=b-2}.      {(3,5),(5,7),(7,9)}

iii) M={(a,b)∈  A xB : 2a+3=b}  {(3,9)}

iv) N={(a,b)∈  A xB : a> b}.      {(5,4),(7,4),(7,5),(7,6),(8,4),(8,5),(8,6),(8,7)

10) For each of the following relations, determine the domain and range:

a) {(x,y): y= x², -2≤x≤3, x∈ Z}. 

                       D:{-2,-1,0,1,2,3}

                       R:{0,1,4,9}

b) {(x,y):  x= y+ √(4-y²), -2≤y≤2, y∈ Z}

                   D:{-2,-1+√3,2,1+√3}

                      R:{-2,-1,0,1,2}

c) {(x,y): y= 3x²+2, x≤3, x∈ W}.   

                           D:{0,1,2,3}

                           R:{2,5,14,29}

11) given A={4,5,6,7,8,9}. List all elements of:

I) {(x,y): ∈ A x A: x> y and x/y not ∈ to N}.     (5,4),(6,4),(6,5),(7,4),(7,5),(7,6),(8,5),(8,6),(8,7),9,4),(9,5),(9,6),(9,7) and (9,8)

ii) {(x,y): ∈  A x A: x≠ y and y/x  ∈ N}

              (4,8)

12) Let A={4,5,6,7,8,9}; B={ a∈Z: -3≤ a ≤7} and a relation R in A x B such that:

R= {(x,y): y= x-5}. Find

I) the elements of R.         (4,-1),(5,0)(6,1),(7,2),(8,3) and (9,4)

ii) domain and range of R.                                      D:{4,5,6,7,8,9}

                    R:{-1,0,1,2,3,4}

iii) the range of variable x.      Set B

               

iv) the domain of variable x.    Set A

13) If A={1,3,5,7......} and 

B={1,4,9,16,25 .....}, represent the relation from A to B {(1,1),(3,9),(5,25),(7,49),....} by means of an arrow diagram. Find the domain and range.        D:{3,6,9}, R:{1,2,3}

14) Write the domain and range of the relation  {(x,y): 3y= x}, where x and y are natural numbers less than 10}.        

15) given A={x ∈N: 2≤ x≤ 16}. list the elements of the following relations:

I) {(x,y): ∈  A x A: x+ y≤8}.    {(2,2),((2,3),(2,4),(2,5),(2,6),(3,2),(3,3),(3,4),(3,5),(4,2),(4,3),(4,4),(5,2),(5,3),(6,2)}

ii) {(x,y): ∈  A x A: x=√ y}.    {(2,4),(3,9),(4,16)}

iii) {(x,y): ∈  A x A: 2x+ y>15}.   {(2,12),(2,13),(2,14),(2,15),(2,16),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),....... So on..

iv) {(x,y): ∈  A x A: x≠ y and x/y  ∈ N}

    (4,2),(6,2),(8,2),(10,2),(12,2),(15,3),(8,4),(12,4),(16,4),(10,5),(14,5),(12,6),(14,7),(16,8)

16) Let A={0,1,2,3,4,5,} 

B={-5,-4,-3,-2,-1,0,1,2,3,4} and 

S= {(x,y): ∈  A x B: y= 2x-5}.

I) list the elements of S.       (0,-5),(1,-3),(2,-1),(3,1),(4,3)

ii) list the domain of S.     {0,1,2,3,4}

iii) list the range of S.    {-5,-3,-1,1,3}

iv) What is the domain of the variable x ?                         Set A

17) Find the linear relation between the components of the ordered pairs of the following relations:

I) {(2,0),(3,3),(4,6),(5,9)}

ii) {(0,2),(1,3),(2,4,),(3,5,)}.     y= x+2

iii) {(2,5),(3,7),(4,9),....}.       y= 2x+1

18) Express each of the following relation as the set of ordered pairs:

I) = {(x,y): 2x+y= 8; x,y ∈ W}.     {(0,8),(1,6),(2,4),(3,2),(4,0)}

ii) {(x,y): x²+y²= 25; x,y ∈ W}.     {(0,5),(3,4),(4,3),(5,0)}

iii) {(x,y): x²+y²= 4; x,y ∈ Z}.   {(-2,0),(2,0),(0,-2),(0,2)} 

          **************************

** Let R be a relation from A to B, Then, R is subset of (A x B).

1) The set of all first co-ordinates of elements of R is called the domain of R, written as dom(R).

2) The set of second coordinates of elements of R is called the range of R, denoted by range (R).

3) The set B is called the co-domain of R.

Thus dom(R)={a:(a,b) subset of R}

And range (R)={b:(a,b∈ R}.

** ARROW DIAGRAM:

 Let R be a relation from A to B.

   We puts dots to represent the elements of A and B.

   For each (a,b) ∈ R, we draw an arrow from a to b.

++++++++()+++++++()++++++++

                Exercise - 2

                 ---------------- 

1) Find x and y, if:

A)  (x+3,5)=(6, 2x+y).                 3, -1

B) (a/3 +1, b- 2/3)=(5/3,1/3).     2,1

C) (x+1,1)= 3, y-2).                        2,3

2) If the ordered pairs (x,1) and (5,y) belongs to the set {(a,b): b=2a -3}, find the value of x, y.                   2,7

3) If a ∈{2,4, 6,9} and b ∈ {4,6,18,27} then form the set of all ordered pairs (a,b) such that a divides b and a< b.          {(2,4),(2,6),(2,18),(6,18), (9,18),(9,27)}

4) Let A={1,2,3} & B{3,4}. Find AxB and B xA. Show it graphically.

5) Let A={1,2} & B{1,3}. Find AxB and B xA. Show it graphically.

6) If A={1,2,3} & B{2,4}. What are

A) AxB 

B) B xA

C) Ax A

D) Bx B

E) (AxB)∩(BxA)

7) If A and B are two sets having 3 elements in common if n(A)= 5, n(B)= 4, find 

A) n(AxB).                                        20

B) n{(AxB)∩(BxA)}.                           9

8) Let A and B be two sets such that n(A)= 3 and n(B)= 2.

If (x,1), (y,2), (z,1) are in AxB, Find A and B, where x, y, z are distinct elements.               A={x,y,z}, B={1,2}

9) If A={1,2}, form the set of AxAxA.

10) If A={1,2,4} and B={1,2,3}, represent following sets graphically:

A) AxB

B) Bx A

C) AxA

D) B x B

11) Let A={1,2,3} and B={2,4,6}. Show that R={(1,2),(1,4),(3,2),(3,4) is a relation from A to B. Find

A) dom(R).                                  {1,3}

B) co-domain (R).                    {2,4,6}

C) range (R).                               {2,4}

D) Depict the above relation by an arrow diagram.   

12) Let A={1,2,3,4,5} & B={1,4,5}

Let R be a relation ' is less than' from A to B.

A) list the element of R.          {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)}

B) domain.                            {1,2,3,4}

C) co-domain.                         {1,4,5}

D) range.                                   {4,5}

E) Depict the above relation by an arrow diagram. 

13) Let A={1,2,3,4,5,6} Define a relation R on A by R={(x,y): y=(x+1)}.

A) Depict R, using Arrow diagram.

B) Domain.                         {1,2,3,4,5}

C) co-domain.                    {1,2,3,4,5}

D) range.                            {2,3,4,5,6}

14) Let A be the set of first 10 natural numbers and let R be a relation on A defined by (x,y) ∈ R <=> x+ 2y= 10, 

i.e R={(x,y):x∈ A, y ∈ A & x+ 2y= 10}. Express R and R⁻¹ as sets of ordered pairs. Determine also 

A) domain of R and R⁻¹.

B) ranges of R and  R⁻¹.   

15) A relation R is defined from a set A={2,3,4,5,} to a set B={3,6,7,10} as follows: (x,y)∈ R <=> x divides y.  Express R as a set of ordered pairs and determine the domain and range of R . Also find R⁻¹.  

        {(2,6),(2,10),(3,3),(3,6),(5,10)}

        Dom:- {2,3,5}, Range:- {3,6,10}

     R⁻¹={(6,2),(10,2),(3,3),(6,3),(10,5}

15) If R is the relation ' less than ' from A={1, 2, 3, 4, 5} to B={1,4,5}, write down:

A) the set ordered pairs corresponding to R.       {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)}

B) Find the inverse R.    {(4,1),(5,1), (4,2),(5,2),(4,3),(5,3),(5,4)}

16) A relation R is defined on the set Z of integers as follows:

  (x,y) ∈ R <=> x²+ y²= 25.

 Express R and R⁻¹as the set of ordered pairs and hence find their respective domains.

   R={(0,5),(0,-5),(3,4),(-3,4),(3,-4), (-3,-4),(4,3),(-4,3),(4,-3),(-4,-3),(5,0), (-5,0)

  R⁻¹={(5,0),(-5,0),(4,3),(4,-3),(-4,3), (-4,-3),(3,4),(3,-4),(-3,4),(-3,-4),(0,5), (0,-5)

 Domain={0,3,-3,4,-4,5,-5}= domain of R⁻¹.

17) Let R be the relation on the set of N of natural numbers defined by R={(a,b): a+ 3b= 12, a ∈ N, b ∈ N}, Find

A) R.                         {(9,1),(6,2),(3,3)}

B) Domain.                               {9,6,3}

C) range.                                  {1,2,3} 

18) Let A={1,2,3,4,5,6}. Define a relation R on set A by R={(x,y): y= x+1}

A) Depict this relation using an arrow diagram.

B) write down the domain, co-domain and range of R.

                                   Dom:-{1,2,3,4,5}

                               Range:- {2,3,4,5,6}

19) A relation R is defined from a set A={2,3,4,5} to a set B={3,6,7,10} as follows:

 (x,y) ∈ R <=> is relatively prime to y.

Express R as a set of ordered pairs and determine its

A) domain.        

B) range.

20) Let A be the set of first five natural numbers and let R be a relation on A defined as follows: (x,y)∈ R <=> x ≤ y

Express

A)  R and R⁻¹ as set of ordered pairs.       {(1,1),(1,2),(1,3),(1,4),(1,5), (2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5), (4,4),(4,5),(5,5)}

  R⁻¹ {(1,1),(2,1),(3,1), (4,1),(5,1),(2,2),(3,2),(4,2),(5,2), (3,3),(4,3),(5,3),(4,4),(5,4)}

B) Domain of R⁻¹.               {1,2,3,4,5}

C) the range of R.              {1,2,3,4,5}

21) Find the inverse relation R⁻¹in each of the following cases:

A) R={(1,2),(1,3),(2,3),(3,2),(5,6).    {(2,1),(3,1),(3,2),(2,3),(6,5)}

B) R={(x,y): x,y ∈ N, x+ 2y= 8}.     {(3,2),(2,4),(1,6)}

C) R is a relation from {11, 12, 13} to {8, 10, 12} defined by y= x -3.     {(8,11),(10,13)}

22) let A={1, 2, 3,... 14}. define a relation on a set A by

R={(x,y): 3x-y= 0, where x,y ∈ A,

Depict this relationship using arrow diagram. write down its 

A) domain.                           {1,2,3,4}

B) co-domain.                        A

C) range.                                  {6,7,8}

23) Define relation R on the set N of natural numbers by R==(x,y): y=x+5, x is a natural number less than 4, x,y  ∈ N}. Depict this relationship using

A) roster form.        {(1,6),(2,7),(3,8)}

B) an arrow diagram.

C) domain.                               {1,2,3}

D) range.                                  {6,7,8}

24) A={1,2,3,5} and B={4,6,9}. define a relation R from A to B by R={(x,y): the difference between x and y is odd,  x∈ A, y∈ B}. write R in roster form.               {(1,4),(1,6),(2,9),(3,4),(3,6), (5,4),(5,6).

MULTIPLE CHOICE QUESTIONS

------------------------------------------------

1) If A={1,2,4}, B={2,4,5}, C{2,5}, then (AxB)x (B - C) is

A) {(1,2),(1,5),(2,5)}     B) {(1,4)}.

C) (1,4)                          D) none

2) If R is a relation on the set A={1, 2, 3,4,5,6,7,8,9} given by x R y <=>y = 3x, then 

A) {(3,1),(6,2),(8,2),(9,3).

B) {(3,1),(6,2),(9,3)}

C) {(3,1),(2,6),(3,9)}    D) none.

3) Let A={1,2,3},.B={1,3,5}. If a relation R from A to B is given by R ={(1,3),(2,5), (3,3)}. Then R⁻¹ is

A) {(3,3),(3,1),(5,2)}.

B) {(1,3),(2,5),(3,3)}

C) {(1,3),(5,2)}   D) none

4) If A={1,2,3}, B={1,4,6,9} and R is a relation from A to B to defined by ' x is greater than y'.  the range of R is

A) {1,4,6,9} B) {4,6,9} C) {1}. D) n

5) if R={(x,y): x, y ∈ Z, x²+ y²≤ 4} is a relation on Z,  then domain of R is

A) {0,1,2}           B) {0,-1,-2}

C) {-2,-1,0,1,2}.  D) none

6) A relation R is defined from {2,3,4,5} to {3,6,7, 10} by: x R y <=> x is relatively Prime to y. Then domain of R is 

A) {2,3,5} B) {3,5} 

C)  {2,3,4}      D) {2,3,4,5}.

7) A relation from ¢ from C to R is Defined by x ¢ y <=> |x| = y. Which one is correct ?

A) (2+3i) ¢13         B) 3 ¢ (-3)

C) (1+ i)¢ 2            D) i ¢ 1.

8) Let R be a relation on N Defined by x+2y = 8. The domain of R is

A) {2,4,8}                B) {2,4,6,8}

C) {2,4,6}.                D) {1,2,3,4}

9) R is a relation from {11,12, 13} to {8, 10, 12} Defined by y= x - 3.  then inverse of R is

A) {(8,11), (10,13)}. 

B){(11,8),(13,10)}

C) {(10,13),(8,11),(12,10)}     D) n

10) If the set A has p elements,  B has q elements, then the number of elements in A x B is

A) p+q  B)p+q+1 C) pq. D) p²

11) Let R be a relation from a set to B, then 

A) R = A U B          B) R= A ∩ C 

C) R ⊂ A x B.          D) R ⊂ B x A

12) If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relation from A to B is

A) 2ᵐⁿ.  B) 2ᵐⁿ -1 C) 2mn D) mⁿ

13) If R is a relation on a finite set having n elements, then the number of relation on A is

A) 2ⁿ B) 2ⁿ^². C) n² D) nⁿ

 

PROFIT, LOSS , DISCOUNT (COMPETITION)

          PROFIT AND LOSS

Exercise -1

1) Find the profit or loss%, when:

a) C. P= Rs55 and S. P= Rs72.60

A) p= 32% B) P=35% C) l=40% D) p= 40%

b) C. P= Rs490 and S.P=416.50

A) p=15% B) p=20% C) l=15% D) l= 20%

c) C.P=Rs112 and S.P= Rs94.50 overhead= Rs 14.

A) p=15% L= 25% P=25% D) l= 15%

d) C.P= 78, S. P= 89.70 P%=

A) 15 B) 17 C) 20 D) 25

e) C. P= 875, S. P= 717.50, L%=?

A) 15 B) 17 C) 18 D) 20

2) Find S. P when

a) C. P= Rs435, Loss=16%

A) 350 B) 365.40 C) 370 D) 375.60

b) C. P=Rs172, overhead=Rs61 and gain= 12%

A) 160.96 B) 260.96 C) 360.96 D) none

3) Find C. P. When

a) a cycle is sold for Rs1485 at a profit of 8%

A) 1175 B) 1275 C) 1365 D) 1475

b) a fan is sold for Rs657.60 at a loss of 4%

A) 675 B) 685 C) 695 D) 705

4) If oranges are bought at 11 for Rs 30 and sold at 10 per rupees 31 then the loss or gain percent is..

A) 13 B) 13/3 C) 40/3 D) 41/3

5) A man buys oranges at 9 for Rs8.10 and sells them at 11 for Rs12.10. then gain or loss% is..

6)  by selling an article for rupees 123, the shopkeeper losses 25%. find the gain or loss percent, if the article be sold for Rs188.60.

A) 1175 B) 1275 C) 1375 D) 1500

7) A dealer sold two almirahs for Rs 6090each, gaining 16% on one and losing and losing 16% on the other. find his net gain or loss percent in the whole transaction transaction transaction.

8)  by selling a book for book for Rs 115.20 a man loses 10%. At what price should he sell it to gaining 5%.

9) 20% more can be gained if a piece of cloth is sold for piece of cloth is sold for Rs 83 instead of 78. find the cost price of the piece of cloth.

10) the difference between selling an article at 7% profit and at 16% profit 16% profit is rupees 63. find the cost price of the article and also the two selling prices.

11) A man sales an article at 5% above cost price. If he had bought it at 5% less then what what he paid paid for if and sold it for Rs2 less, he would have gained 10%. find the cost price of the article.

12) A  grocer purchased 80 kg of rice at of rice at at rupees 13.50 per kg and mixed it with 120 kg rice 

at rupees 16 per kg. At what rate per kg must he sale the mixture to gain 16% ?

13) A manufacturer makes a profit of 15% by selling a TV set for rupees 1725. if the cost of manufacturing increases by 20% and the price paid by the the retailer is increased by 10%, find the gain percent made by the manufacturer.

14) A washing machine sold by shopkeeper at a gain of 15%. Had it been sold for Rs375 more, he would have have gained 20%. find the cost price of the washing machine.

15) A sells a watch to B at a gain of of gain of 20% and B sells it to C at a loss of of 10% and C sales it for Rs1404, gaining 4%. How much did A pay for it.

16) A man sells a TV set for

 Rs13800 and makes profit of 15%. He sells a second TV set at a loss of loss of of 10%. on the whole he neither gains nor losses. find the cost price of the second TV set.

17) A sold a watch to B at a profit of 15%. later on, B sold it back to  A at a profit of 20%. thereby gaining rupees 207 . how much did A pay for the watch the watch originally.

18) A dealer sold three-fourth of his article at a gain of gain of 20% and the remaining at cost price. find the gain earned by him in the whole transaction..

19) A man bought goods worth Rs 6000 and sold half of them at a gain of gain of 10%.  at what gain percent must he sell the remainder so as to get gain of  25% on the whole?

20) a trader purchases a watch and the wall wall clock for Rs390. He sells them making a profit of 10% on the watch and 15% on the wall clock. He earns a profit of 51.50. find the the individual cost price of the watch and wall clock.

21) A man purchases two items at a total cost of rupees 650. He sells one with 20% profit and the other at a loss of 25% and gets the same selling price for both the items. find the the respective cost price of the two items.

22) A men sells an article at a profit of 25%. If he had bought it at 20% less and sold it for 10.50 Less, he would have gained 30%.  find the cost price the articles.

Exercise- 2

1) By selling a bicycle for Rs 797.50, a man loses 8%. At what price should he sell it to gain 15% ?

2) A man goes from Agra to Delhi to buy an article which costs him ₹4350 in Delhi. If he spends ₹250 on his travels and transportation of the article and sells it in Agra for ₹5290; then his profit or loss% in the transaction.

3) By selling tea at Rs60 per kg a grocer gains 4% . At what% must he sell it in order to gain 17% ?

4) I lose 5% by selling a piece of land for Rs46740 ; what would be my gain or loss% if I sell it for RS 52890.

5) By selling 27m of clothes at a loss of 4%, I lose Rs162 altogether. What are the cost price and selling price of the cloth per metre ?

6) 15% was gained by selling a dinning table for ₹1610 and 12% was lost by selling a black and white T. V. for ₹3080. Then the total gain or loss percent.

7) A man sold two items for Rs2970 each, gaining 10% on one and losing 10% on the other. Find his total loss or gain%.

8) Smith sells two bicycle for Rs 450 each losing 20% on one and gaining 25% on the other. Calculate his total loss or profit% 

9) By selling wheat at ₹4.50 per kg, a shopkeeper gains 12.5%. find the cost of 50 kg wheat.
Also, find his profit% if he sells the wheat at Rs460 per quintal.

10) A shopkeeper buys 40 kg ghee at ₹45 per kg and another 60 kg at ₹40 per kg. If he mixes them and sells the mixture at ₹46.30 per kg, find his gain%

11) A steel almirah, coating ₹900 to the manufacturer, is sold at 15% profit. It is estimated that 5% of the selling price must be set aside to allow for overhead charges. Find the net profit, correct to one rupee.

12) Ravi buys three identical watches and sells one watch at20% profit, other at 12% profit and the third at 15% loss. If, on the whole, he earns Rs 85; find the cost price of five such watches.

13) Seventy-five unfinished articles of the same kind were bought for Rs 1275 and Rs3 per article was spent on their finishing. Find the profit percentage made if, one-third of these articles were sold at Rs26 each and the remaining at Rs22 each.

14)  A bought a radio and spent Rs110 on its repairs. He then sold it to B at 20% profit. B sold it to C at a loss of 10%. What is the amount for which A bought the radio, if C paid Rs 1080 for it ?

15) The cost of 19 apples is equal to the selling price of 16 apples. Find the profit%?

16) A reduction of 20% in the price of sugar enables Sita to buy 4 kg sugar more for Rs80 Find

a) The reduced price per kg

b) the original price per kg.

17) A dishonest dealer professes to sell his goods at the cost price, but uses a weight of 900 gm for 1 kg weight. Calculate his profit percentage.

18) A bicycle passes through the hands of these dealers each of them gain 25%. If the third dealer sells it for ₹750; what did the first pay for it.

19) John sold an article to Smith at a gain of 10%. Smith spent Rs260 on the repairing of the article and sold it to Peter at a profit of 15%. If Peter sold the article to some unknown person for Rs1966.50 and lost 5% , Calculate it's cost price for John.

20) By selling 20 pens , a shopkeeper gains equal to the selling price of 4 pens. Find his gain%.

21) A dishonest fruit-vender professes to sell his goods at cost price, but he uses a weight of 460 gm for half-kg. Find his gain%

22) When a certain quantity of tea is sold at Rs53.76 per kg, the gain% is 12% and the total gain is Rs51.84. Calculate the quantity of tea sold.

23) A fruit vendor bought some oranges at 4 for Rs 3 and an equal number at 5 for Rs3. He mixed the two lots and sold these at 90 paise each. Find the gain or loss%

24) A fruit vendor bought a certain number of oranges for Rs640 and sold half of them at 20% profit. Half of the remainder were sold at 10% profit. If on the whole he makes a profit of 15%, find the profit/loss% made on the remaining oranges.

25) A fruit vendor loses 8% by selling oranges at 5 for Rs2. How many oranges should he sells for Rs5 to gain 15% ?

26) A bicycle dealer sells 9 bicycles at a profit of 8% and 5 more at a gain of 10%;  Had he sold all the bicycle at a gain of 9%; he would have received Rs 48 more. Find the C. P. Of one bicycle if, all the bicycle cost the same.

27) A man has 15 barrels, all of the same capacity of oil. If he sells oil at Rs25 per litre, he loses Rs1500 on the whole and if he sells oil at Rs27 per litre, he gains Rs300 on the whole. Find the quantity, in litres, of oil in each barrel and the cost price of oil per barrel?

28) Sonal sold a book at a profit of 12%. Had she sold it for ₹3.60 more, the gain would have been 18%. Find the cost price of the book.

29) A certain number of same kind of articles are purchased for Rs13500 and one-third are sold at a loss of 10%. At what gain% should the remainder be sold, so as to gain 20% on the whole transaction?

30) A man bought a certain number of goods of which he sold ⅓ at a profit of 14%, ⅗ at a profit of 17.5% and the remainder at a profit of 20%. What was his profit% on the whole.

31) By selling 35m cloth at a profit of 5%, I altogether, gain ₹70. Then the cost price and selling price per metre.

32) The selling price of 10 notebooks is same as the cost of 12 notebooks. Then the profit% is..

33) By selling 20 pens , a shopkeeper gains equal to the cost price of 4 pens. Find his gain%.

34) By selling 20 pens , a shopkeeper loses equal to the cost price of 4 pens. Find his loss%. 

35) A rise of 25% in the price of sugar, compels a man to buy 5 kg sugar loss for ₹200. Calculate

A) the increased price per kg 

B) the original price per kg. 

36) A woman bought two parcels of mangoes with equal number of mangoes in each. She bought the first parcel at one rupee per mango and the second at 3 for ₹2. She mixed them all together and sold at ₹13 per dozen. Find her gain%.

37) A man bought an article and sold it at a gain of 25%. If he had bought it at 20% more and sold it for ₹150 more, the gain would have been 7.5%. Find the cost price of the article. 

***MIXED

1) A Shopkeeper sold goods for Rs 2000 at a profit of 50%. Find the cost price for the shopkeeper.

A) 1233 B) 1333.33. C) 1433 D) n

2) A man buys a shirt and a trouser for Rs 371. If the trouser costs 12% more than the shirt, find the cost of the shirt.

3) Ram sold two items at the same price. If he sells one of them at a profit of 10% and the other at a loss of 10% , find the% profit/loss.

4) For the question no. 3 Find the value of the loss incurred by Ram if the price of selling each item is Rs 160.

5) If by selling 2 items for Rs 180 each the shopkeeper gains 20% on one and loses 20% on the other, find the value of the loss.

6) By selling 15 mangoes, a fruit vendor recovers the cost price of 20 mangoes. Find the profit%.

7) A dishonest shopkeeper uses a 900 gram weight instead of 1 kg weight. Find his profit% if he sells per kg at the same price as he buys a kg.

8) Ram makes a profit of 15% by selling a colour TV for Rs 6900. If the cost of manufacturing increases by 30% and the price paid by the retailer is increased by 20%, find the profit% made by Ram.

9) Find a single discount to equal three consecutive of 10%,12%,5%.

10) A reduction in the price of petrol by 10% enables a motorists to buy 5 gallons more for $180. Find the original price of petrol ?

11) A bought an article and spent Rs110 on its repairs. He then sold it to B at a profit of 20%. B sold it to C at a loss of 10%. C finally sold it for Rs1188 at a profit of 10%. How much did A pay for the article.

12) A dishonest shopkeeper professes to sell his articles at cost price but he uses false weight with which he chits by 10% while buying and by 10% while selling. Find his profit%

13) By selling 5 articles for Rs15 a man makes a profit of 20%. Find his gain or loss% if he sells 8 articles for Rs18.4 ?

14) By selling a watch for Rs495, a Shopkeeper incurs a loss of 10%. Find the cost price of the watch for the shopkeeper.

15) By selling a cap for Rs 34.40, a man gains 7.5%. what will be the CP of the cap ?

16) A phone when sold for Rs 4600 fetches a profit of 15%. Find the cost price of the phone.

17) A machine costs Rs 375. If it is sold at a loss of 20%, what will be its cost price as a % of its selling price?

18) Ram sold goods for Rs2400 and made a profit of 25% in the process. Find his profit% if he sold his goods for Rs2040.

19) A digital diary is sold for Rs935 at a profit of 10%. What would have been the actual profit or loss on it, if it had been sold for Rs 810 ?

20) A radio when sold for Rs 4500 gives a loss of 16.66% to the merchant who sells it. Calculate his loss or gain%, if he sells it for RS 5703.75.

21) By selling bouquet for Rs 63, a florist gains 5%. At what price should he sells the bouquet to gain 10% on the cost price ?

22) Ram bought 240 chocolate at RS 9 per dozen. If he sold all of them at RS 1 each, what was his profit%

23) A feeding bottle is sold for Rs 120. Sales Tax is 1/5th of this and profit 1/3rd of the remainder. Find the cost price of the bottle.

24) Ram makes a profit of 20% by selling coal at RS 25 per quintal. If he sells the coal at RS 22.50 per quintal, what is his profit% on the whole investment ?

25) The CP of a shirt and a pair of trousers is Rs 371. If the shirt costs 12% more than the trousers, find the cost price of the trouser.

26) The marked price of a table is Rs 1200, which is 20% above the cost price. It is sold at a discount of 10% on the marked price. Find the profit%.

27) 125 toffees cost Rs 75. Find the cost of one million toffees if there is a discount of 40% on the selling price for this quantity.

28) A Shopkeeper marks the price of an article at RS 80. Find the cost price if after allowing a discount of 10% he still gains 20% on the cost price.

29) From the question no 28, what will be the selling price of the article if he allows two successive discount of 5%each.

30) A dozen pairs of gloves quoted at RS 80 are available at a discount of 10%. Find how many pairs of gloves can be bought for Rs.24.

31) Find a single discount equivalent to the discount of 20%,10%,5%

32) The M. P of a calculator is Rs180. A retailer pays Rs137.70 for it by getting successive discount of 10% and another rate which is illegible. What is the 2nd discount?

33) How much% more than the C. P should a shopkeeper mark his goods, so that after allowing a discount of 12.5% he should have a gain of 5% on his outlay ?

34) In order to maintain the price line, a trader allows a discount of 10% on the M. P of goods in his shop. However, he still gains of 17% on the C. P. Find the profit% he would have made on the selling price had he sold at the M. P.

35) A whole seller allows a discount of 20% on the list price to a retailer. The retailer sells at 5% discount on the list price. If the customer paid Rs38 for an article, what profit is made by him ?

36) In question 35, find also the retailer's% profit on his cost giving your answer correct to two decimal places.

37) The cost of production of a phone in 2009 is Rs900, Divided between material, labour and overhead in the ratio 3:4:2. If the phone set is marked at a price that gives a 20% profit on the component of accounted for by labour, what is the M. P of the set ?    If subsequently in 2007, the cost of material, labour and overhead increased by 20%,30% and 10%. Calculate the cost of producing in 2007.

What should be the new M. P if the criteria for profit is to remain the same ?

38) By selling a chain for Rs960, a man incurs a loss of 4%. At what price should he sell the chain to gain 16% ?

39) A man sells 5 articles for Rs15 and makes a profit of 20%. Find his gain or loss% if he sells 8 such articles for Rs18.40.

40) The C. P of 50 mangoes is equal to the S. P of 40 mangoes. Find profit %

41) A owns a house worth 10000. He sells it to B at a profit 15%. After some time, B sells it back to A at 15% loss. Find A’S loss/profit%

42) A bought locks at the rate of 8 locks for Rs34 and sold them at the rate of 12 locks for Rs57. Find gain%

43) Ram bought an article at Rs200 and sold it at a profit of 10%. What would have been the increase in the profit% if it was sold for Rs 230 ?

44) A makes an article for Rs120 and sells to Bat a profit of 25%. B sells it to C who sells it for 198, making a profit of 10%. What profit % did B make ?

45) A reduction of 10% in the price of petrol enables a motorists to buy 6.2 litres more for Rs279. Find the reduced price per litre

46) A man buys 50 kg of oil at RS.10 per kg and another 40 kg of oil at 12 kg and mixing them. He sells the mixture at the rate of Rs11per kg. Find his gain% if he sell the whole lot

47) If the C. P of 30 articles is equal to the selling price of 20 articles. Find the profit %

48) Ram sells sugar in such a way that the selling price of 950 gm is the same as the C. P of one kg. Find his profit%

49) A dealer buys eggs at RS 36 per gross. He sells the eggs at a profit 12.5% on the C. P. What is the S.P per egg (App.)

50) A sold a table to B at 20% profit. B sold same table to C for Rs75 thereby making profit of 25%. Find the price at which A bought the table from X if it is known that X gained 25% in the transaction.

KNOW THE SHORT CUT

Exercise-3 

 Find the cost price of an article which is sold at 630 at a profit of 12.5% find the CP of an article which is sold at 1470 at a profit of A shopkeeper sells his articles at on find the actual profit shopkeeper sells is good at honesty find actual human cells is good at 25% loss on SP find a lost person cost price of an article is equal to SP of 14 articles find the profit or loss percent find 2050 articles is equal to SP of 2500 find cost price of 12 articles is equal to SP of 9 article while the discount on 10 articles is equal to the profit earned 15 article find the difference cost price of 12 article is equals to selling price of 9 articles while the discount on 8 articles is equals to profit on 6 articles find the difference between profit percent and discount percent after selling 72 articles a man loses selling price of 9 articles find the loss percent after selling 72 articles and and answer property top selling price of 9 articles find the profit percent after 7 selling 72 articles a man earns profit of cost of 9 article find profit percent after selling 10 articles a man earns a profit of the selling price of three pains while selling 10 play some and losses selling price of 4 Candles the numerical value of property person the last person is equals to and the cost price of candle is half of the cost price of the pen find the ratio of selling price of kanto to paint the profit earned by an article is sold for 800 is 20 times the loss incurred when it is sold for 275 find at what price is sold is good if he wants to earn 25% profit profit after selling an article for 717 is more than loss incurred when it is sold at 527 what would be the selling price if we want to earn a profit of 10% A shopkeeper sells at 9% loss had he sold it 750 more than he would than 7% find the initial cost price of shopkeeper sells his goods at 7% of it had is sold in 4248 more than we would gain 13% profit find the initial cost price A shopkeeper sells is good at 20% profit had the purchase it for 10% less and sold it for rupees 18 less than he would gain 30% find the initial cost by a man sells his goods at 10% profit had a purchase it for 20% and sold it for 20 more than he would gain 40% find the initial cost price a man sells his goods at 25% had he purchased it for 900 less and sold it for 900 less than he can 5% more find the initial cost price a man sells his goods at 20% profit had he purchased for rupees 600 less and sold it for 400 less than he would gain 10% more profit find the initial Cost A man purchased some initial article at 11 article for rupees 10 and sales all the articles at 10 articles for 11 find profit percent or loss percent a man purchase some pencils at 6 pencil for rupees 5 and his says the all the materials at 5 articles for rupees 6 find overall profit or losers man purchase some oranges at 1 Orange for rupees 2 and some number of oranges at 2 oranges for Rupees 1 and sales for rupees at the rate of 5 4 Rupees 1 and some number of days at 4 for rupees find the number of articles a dishonest shopkeeper promise to sell his goods at its cost price but he uses 30% less weight find the profit percent at the dishonest shopkeeper promise to sell his goods at its it but uses 960 gram weight instead of one kg find profit percent A shopkeeper promise to sell his goods at 44% loss but he uses 30% less weight find actual loss A shopkeeper professes to sell his goods at 10% profit potential uses 20% less weight find the profit percent shopkeeper promise to sell his goods at X person pop it but he uses 20% less weight does gain find x shopkeeper professes to sell his goods at expert loss but he uses 20% less weight the temporary person 5x A shopkeeper professes to sell his goods at experts and loss but he uses 25% less weight the gain 20% find its a dishonest shopkeeper makes a cheating of 20% on the time of buying the goods and 40% sitting at the time of selling the good it promises to sell his goods at 10% loss find the profit percent A shopkeeper marks his goods 40% above the cost price and gives 25% discount to customer at the time of selling the good news is 800 gram weight instead of one kg find his profit percent A shopkeeper marks his goods 20% above the city and give 10% discount to the customer at the time of selling the boots uses 900 with instead of one kg and at the time of buying uses 1100 gram instead of one kg find his profit percent a dishonest shopkeeper makes a cheating of 10% the time of buying the ghost and 10% sitting at the time of selling to find the profit a man purchased some of the oranges at the rate of 11 or is this for Rupees 1 how many for a Rupee is did he sell to gain 10% a man purchased some number of oranges at 25 oranges for RS 1,000 many for a Rupee is did he sell to gain 20% by selling 32 oranges for a rupees a man lost 40% how many for a Rupee did he sell to earn 20% by selling 12 oranges for a man lost 20% How many for did he sell 21 2012 by selling 45 oranges for Rs 40 a man lost 20% How many dead cell for 24 2121 S make 750 articles at a cost of 65 article he thinks the selling to ise such that if only 600 article sold he would have made a profit of 40% on the outlet however 120 articles that point and he was able to sell 6:30 articles at this price find is actual profit percent is the percentage of total outlay as a name that the unsold articles are useless a man purchased some number of particles attitude 5400 and sells two third of them at 5% profit at what profit percentage he sell the remaining to give 30% over a man purchased some number of particles are to please 189000 and sales 38 of them at 12% of it at what percent is sell the remaining to get good person of overall manufacturers estimate that an inspection 12% the particles producers will be rejected we accept an order supply 22000 rupees 750 each estimate the profit on it out Le including the manufacturing of projected articles to be 20% find the cost of manufacturing it at discount bi pipe Gate 4 free + 20% more Discover by poor kid pipe 3 + 50% more discount shopkeeper allows 25% discount on the Marked price and earn 30% profit if it gets rupees 90 as appropriate find the amount of the distance by how much percent of shopkeeper marks his goods above it CP store as by giving 20% discount in a 10% by how much percent of the shopkeeper marks his goods above 8 CP so as by giving 10% discount in 1834 A shopkeeper marks his goods at such a particle price that after allowing a discount of 12.5% on the Marked price he can earn a profit of 20% of the article costing rupees 1400 then find its marked price A shopkeeper gives 20% discount is customer but he sells only smuggled goods and I as a bride he tasted person on the cost price find the what should be the Marked price if it is desired to make a profit of the cost price of an article is 2500 by selling an article for rupees 1171 allows 10% discount and 30% profit in the article is sold at a discount what should the profit by selling an article for 15609 allows 8% discount and 19.6% profit if the article is sold at a discount what should be the profit by selling an article for rupees 11 7 p.m. and allows 10% discount and an 8% profit and the article is sold at 5% discount what should be the perfect shopkeeper gives free articles free on purchase of fire particle you also allow a discount of 20% still on 25% find the ratio of cost price and shopkeeper gives one article free on the purchase of a Prit 15 article you also allowing a discount of 40% to customer and steel and 35% of to find the ratio of cost price in the Marked price of shopkeeper gives for articles free on the purchase of Pimpri to a vertical she also allows a discount of 20% to customer it still on 20% find the ratio of cost price is the Marked price of that Rakesh readers Publication public 3500 books for 350000 at cost price if 500 books free to some books of you also allowing a discount of 25% on the Marked price and given book free for every purchase of 29 book find the amount of profit or loss if the Marked price of each book is one rickshaw dealer by 1360 rupees 4025 of these 38 are four sisters and the rest are the two seater at what price must he sell the posters so that if you sell to seaters at 34 of this price in makes a profit of 40% on his outlay a man sold a book at 9% profit and a pain at 30% profit if he sold the book at 30% profit and the pain and 9% property gain a team or find the cost price of the content if we purchase both at it as a man purchased the book and pen for rupees 25000 installed the book at and pain at 17% it is sold the book at 17% profit and pain at 13% property on a team or find their individual cost price of shopkeeper but to cycle in 1600 episode first cycle of 10% profit and second at 20% profit episode first at 20% profit and second at 10% profit you are certain profit he gets 5 more the prices are both the cycles age the total cost of 8 books and 510 692 then find the cost of 83 books and to paints in the cost A5 books and 8 pen 77 Raptors at the two bats and one ball the cost of Bollywood second bat the amount received two will be less than the value of the first bad by rupees 306 what is the value of the first man Ram self driven at 5% loss and a book at 15% profit he gets 7 as profit if he says the pain at 5% profit and the book at 10% profit to get 6 more the price price of book up and pain are a man sells a table at 12% loss and a book on 19% Profit and earns a profit of 160 but he sells the table 12% of B and a book at 16% loss then he bears a loss of 40 find the price of the book Ram Sen se tape at 15% profit and a chair at 12% loss and earn 540 as profit if you sell the table at 12% loss in cheer at 15% profit then he beers no profit no loss find the price of table and chair men sales book and a table at 13% and 9% profit respectively and earns 1060 as profit but if you says the book at table at lost then the bears no problem then find the cost price of chicken egg a man sells two articles first on 20% loss and second on 50% profit find the selling price in the difference between the cost price is 3208 the selling price of both the articles is same as three articles of same price first on 20% profit s on 10% loss third on 25% loss during the whole transaction if there's a loss of 120 find the selling price of HRD a man sells two article for 1710 is said first at 10% loss and at and S at 25% find the amount of overall profit or loss if the cost price of first article is equal to the said second selling price of the second hand the selling price of A and B are 1808 calculate his profit percent on SP while be on cp find the difference between the cost price if both claims 20% profit A and B purchase an article on same prize letter on C purchase both articles from a and b at 2:40 age from A and B but the profit percent of a was prepared % while profit percent of beavers few percent csb calculate is profit on HP AC sales one of the article goes to tea at 10% profit what is the cost price of Audi If a company allow 15% discount is customer and steel and 19% of it if the production cost of the production product is increased by 12% therefore company student new list price is 10% higher than the previous list price and company still allow 15% discount to is customer find the new profit percent of the company a man purchase a home and so is sold the staff at 10% profit and home at 10% loss as selling price of both the article is same one lakh each find the amount of lost a dealer sold two TV sets for 2408 and earn 20% profit on first article and 20% loss and second article find the total profit or loss A shopkeeper bought some books at discount of 20% list price if you want to Mark them at such a price that after approval giving a discount of 20% is still makes a profit of 25% find the percent of list prices watermark on it would Seva b

DERIVATIVE AS A RATE MEASURE

1) The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8cm long ?                64 cm²/min

2) The side of a square sheet is increasing at the rate of 3 cm per minute, when heated. At what rate is the area increasing when the side is 10 cm long ?             6 cm²/min

3) The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square. 0.8 cm/sec

4) An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long ?              900cm³/s

5) A metal cube expands on heating. At the instant when each edge measures 3cm, the volume of the cube increases at the rate of 0.015 cm³/sec. At what rate is the length of the edge increasing at that instant.         0.00056 cm/sec

6) A hemisphere is constructed on a circular base. If the radius of the base is increasing at the rate of 0.5 cm/sec., Find the rate at which the volume of the hemisphere when its radius is 10cm.             314cm³/sec

7) The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference ?   1.4π cm/sec

8) The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7cm. 11.2π cm²/sec

9) The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its volume when the radius is 5cm.            20π cm³/sec

10) A balloon which always remains spherical, is being inflated by pumping in 900 cm³ of a gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15cm. 1/π cm/sec

11) A spherical balloon is inflated so that its volume increases uniformly at the rate of 40 cm³/min. How fast is its surface area increasing when the radius is 8cm. Find approximately how much the radius will increase during next 1/2 minute.      10 cm²/min, 0.0249cm

12) The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm ? 2π cm³/sec

13) A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10cm, how fast is the enclosed area increasing ? 80π cm²/sec.

14) A particle is moving in a vertical line whose equation of motion is s= 8+92t- 4.9t², when s is measured in metres and t is measured in seconds. Find the velocity and acceleration of the particle at t= 3.    62.6 m/sec, retardation= 9.8 m²/sec. 

15) If y= 7x - x³ and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x= 2 ?               48

16) A particle moves along the curve y= x³. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-cordinate. (1,1),(-1,-1).

17) A particle moves along the curve y= x² + 2x. At what point/s on the curve are the x and y coordinates of the particle changing at the same rate? (-1/2, -1/3)

18) A particle moves along the curve 6y= x³ + 2. At what point/s on the curve the y coordinates is changing 8 times as fast as x-cordinate.    (4, 11) and (-4,-31/3)

19) find an angle ¢, which increases twice as fast as its sine. π/3

20) A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h= 9 cm. 12π cm³/sec

21) The radius of a cylinder is increasing at the rate 2 cm/sec and its altitude is decreasing at the rate of 3 cm per second. find the rate of change of volume when radius is 3 cm and altitude 5 cm. 33π cm³/sec

22) The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4cm and 8 cm respectively. 1/4 cm/sec

23) A kite is 120m high and 130m of string is out. If the kite is moving away horizontally at the rate of 52 metre per second, find the rate at which the string is being paid out. 20 m/sec

24) A particle moves along the curve y= 2x³/3 +1. Find the points on the curve at which the y-coordinate is changing twice as fast as the x-cordinate. (1,5/3),(-1,1/3)

25) find the point on the curve y²= 8x for which the abscissa and ordinate change at the same rate. (2,4)

26) The volume of a cube is increasing at the rate of 9 cm³/sec. How fast is the surface are increasing when the length of an edge is 10 cm ? 3.6 cm²/sec

27) The length x of a rectangle is decreasing at the rate of 3 cm/min and the width y is increasing at the rate of 2 cm/min. When x=10cm and y = 6cm, find the rate of change of 

A) the perimeter -2 cm/min

B) the area of rectangle. 2cm/m

28) The volume of a spherical balloon is increasing at the rate of 25 cm³/sec. Find the rate of change of its surface area at the instant when the radius 5 cm. 10 cm²/sec

29) A man 180 cm tall walks at a rate of 2 m/sec away from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3m away from the base of light ? 0.5 m/sec

30) A ladder 13m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/se . How fast is the angle ¢ between the ladder and the ground is changing when the foot of the ladder is 12m away from the wall. 0.3 radian/sec

31) A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres heigh. Find the rate at which the length of his shadow increases. 5/2 km/h

32) A man 160 cm tall, walks away from a source of light situated at the top of a pole 6m heigh, at the rate of 1.1m/sec. How fast is the length of his shadow increasing when he is 1 metre away from the pole ? 0.4 m/sec

33) A pole 13 metres long rests on a vertical wall. If the foot of the pole slips away from the wall with a constant velocity of 2.5 m/sec. Find at what rate its top is moving when the foot of the pole is at a distance of 12 metres from the wall. Find also the rate of which the slope of the pole changes at that instant.    6 m/sec, 0.5868 per second.

33) The top of a ladder 6 m long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwords. At the moment when the foot of the ladder is 4m from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance ? 1/√5 m/sec, 3√2 m

34) A man 2m high walks at a random speed of 6 km/h away from a lamp-post 6 metres high. Find rate at which the length of his shadow increases.      3 km/hr

35) A man is walking at the rate of 4.8 km/hr towards the foot of a tower 40 metres high. At what rate is he approaching the top of the when he is 30 metres away from the foot of the tower ?    2.88 km/hr

36) Water is the running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 m, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 metre above the base. 0.64 m/min

37) The surface of a spherical bubble is increasing at the rate of 2 cm²/s. When the radius of the bubble is 6cm, at what rate is the volume of the bubble increasing ? 6 cm³/sec

38) Sand is being poured onto a conical pile at the constant rate of 50 cm³/min such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep. 1/2π cm/min.

39) An inverted cone has a depth of 10cms and a base radius 5 cms. Water is poured into it at the rate of 1.5 cm³ per minute. Find the rate at which the level of water in the cone is rising when the depth is 4 cms.   3/8π cm/min

INDICES (Competitive)

            INDICES

EXERCISE- 1

------------------

1) (36)¹⁾²

A) 3 B) 4 C) 6 D) none

2) (27)¹⁾³

A) 3 B) 4 C) 6 D) none

3) (32)¹⁾⁵

A) 3 B) 4 C) 6 D) none

4) (81)³⁾⁴

A) 3 B) 4 C) 6 D) none

5) (243)³⁾⁵

A) 3 B) 4 C) 6 D) none

6) (16)⁵⁾⁴

A) 3 B) 4 C) 6 D) none

7) (27)⁻¹⁾³

A) 1/3 B) 3 C) 6 D) none

8) (512)⁻¹⁾⁹

A) 1/3 B) 3 C) 1/2 D) 2

9) (625)⁻³⁾⁴

A) 1/5 B) 1/25 C) 1/125 D) none

10) (64)¹⁾⁶

A) 2 B) 1/2 C) 4 D) 1/4

11) (81)⁻¹⁾⁴

A) 1/3 B) 3 C) 6 D) none

12) (27/64)⁻²⁾³

A) 16/3 B) 16/9 C) 16/27 D) none

13) (64/125)⁻²⁾³

A) 5/16 B) 25/16 C) 16/5 D) 16/25

14) (81/16)⁻³⁾⁴

A) 8/27 B) 27/8 C) 4/27 D) 27/4

15) (256/625)⁻¹⁾⁴

A) 6/5 B) 5/16 C) 5/4 D) none

16) ⁵√(32⁴)

i) 1         ii) 6         iii) 16         iv) n

17) 7¹⁾² x 7¹⁾³

A) 7 B) 7¹⁾⁶ C) 7³⁾² D) none

18) 11¹⁾² / 7¹⁾⁴

A) 11  B) 11¹⁾⁸  C) 7¹⁾⁴  D) none

19) 7¹⁾² x 8¹⁾²

A) 7 B) 56¹⁾⁶ C) 56³⁾² D) √56

20) (3³)²⁾³ 

A) 3     B) 3²     C) 3²⁾³       D) none

21) (5³)¹⁾³ 

A) 5    B) ⁵²     C) 5²⁾³       D) none

22) (7¹⁾³)⁴

A) 7     B) 7⁴⁾³  C) 7²⁾³       D) none

23) (3¹⁾⁵)¹⁰

A) 3     B) 3²     C) 3²⁾³       D) none

24) 2¹⁾⁴ x 3¹⁾⁴

A) 6 B) 6¹⁾⁴ C) 6³⁾² D) √6

25) 3⁵⁾⁸ x 5⁵⁾⁸

A) 15 B) 15¹⁾⁸ C) 15⁵⁾⁸ D) √15

26) 7²⁾³ x 3²⁾³

A) 21 B) 21¹⁾⁶ C) 21²⁾³ D) √21

27) (32)⁻⁴⁾⁵ * (27)⁻²⁾³

i)16/9   ii) 6/9       iii) 9/16    iv) n

28) 4 x 8⁻²⁾³

i) 1         ii) 2   iii) 3           iv) 4

 

29) (64)¹⁾³ 

A) 4     B) 16      C) 8        D) 64

30) 8⁵⁾³ 

A) 2      B) 4       C) 16       D) 32

31) 9³⁾²

A) 3       B) 9       C) 27     D) 81

32) (125)⁻¹⁾³ 

A) 5    B) 1/5    C) 2/5    D) 1/25

33) (9/16)⁻¹⁾² 

A) 2/3     B) 4/3  C) 3/4   D) 8/3

34) (0.01)⁻¹⁾² 

A) -10 B) 10 C) 1/10 D) -1/10

35) (64)⁻³⁾² 

A) 1/512 B) 1/64 C) 1/8 D) 1/2

36) (125)¹⁾³ 

A) 1/5   B) 5     C) 1/25    D) -1/25

37) 8²⁾³ 

A) 2   B) 4   C) 8    D) 16

38) (1/5)⁻² 

A) 25  B) 1/25 C) 5  D) 1/5

39) (16)⁻³⁾⁴ 

A) 1/2  B) 1/4 C) 1/8    D) 1/16

40) (32)⁻⁴⁾⁵ 

A) 1/2 B) 1/4 C) 1/8 D) 1/16

41) (8/125)⁻¹⁾³ 

A) 2/5 B) 5/2 C) 4/25 D) 25/4

42) (-27)²⁾³

A) 3 B) -3 C) 9 D) - 9

43) (0.001)⁻¹⁾³ 

A) 1   B) -1 C) 10   D) - 10

44) (0.027)⁻²⁾³

A) 9/10 B) 9/100 C) 100/9 D) 10/9

45) √25 

A) 2  B)  5   C) 25 D) none

46) ³√64 

A) 2  B) 4   C) 16 D) 2³

47) ⁴√81 

A) 3  B) 9  C) 27 D) none

48) ⁵√3125. 

A) 5  B) 25 C) 625  D) none

49) √(20) 

A) 2 B) 5 C) 2√5   D) 5√2

50) ³√24. 

A) 2    B) ³√2 C) 3     D) 2 ³√3

EXERCISE --2

-------------------

1) (1³ + 2³ + 3³)¹⁾²

A) 2   B) 3     C) 6      D) N

2) (5¹⁾³ x 25¹⁾³ x 125²⁾³)/625¹⁾²

A) 5 B) 25 C) 625 D) 1

3) [{(81)¹⁾²}¹⁾⁴]².

A) 3 B) 3¹⁾² C) 3¹⁾⁶ D) none

4) (2/3)⁴ (8/3)⁻¹² (32/3)⁶

A) 4/3 B) 4/9 C) 9/4 D) 9/16

5) (81)³⁾⁴ - (1/32)⁻²⁾⁵ + 8²⁾³ x (1/2)⁻¹ x 3⁰ - (1/81)⁻¹⁾² 

A) 22    B) 24 C) 26 D) 28

6) (64/125)⁻²⁾³ + 4⁰ x 9⁵⁾² x 3⁻⁴ - √25/ ³√64 x (1/3)⁻¹ 

A) 1/2 B) 1/6  C) 13/16 D) none

7) (1/4)⁻² - 3x8²⁾³ x 5⁰ + (9/25)⁻¹⁾²

A) 1/3 B) 16/3    C) 3/16 D) none

8) √(1/4) + (0.01)⁻¹⁾²- (27)²⁾³x3⁰ 

A) 2/3 B) 3/2  C) 1/2 D) 1/3

9) 4/(216)⁻²⁾³ - 1/(256)⁻³⁾⁴

A) 4 B) 64 C) 80 D) No

10) {(2⁻¹ x 3²)/(2²x 3⁻⁴)}⁷⁾² x {(2⁻² x 3³)/(2³x 3⁻⁵)}⁻⁵⁾² 

A) 1 B) 2 C) 12 D) 21

11) [(64)²⁾³ x 2⁻² ÷ 7⁰]⁻¹⁾². 

A) 1 B) 2 C) 1/2 D) none

12) (81/16)⁻³⁾⁴ x[(25/9)⁻³⁾²÷(5/2)⁻³]

A) 1   B) 0 C) 1/2 D) none

13) (32)⁻⁴⁾⁵ / (27)⁻²⁾³

A) 9/16 B) 16/9 C) 3/4 D) 4/3

14) (8)²⁾³. (16)³⁾⁴ . (32)⁻⁴⁾⁵. 

A) 2 B) 4 C) 8 D) none

15) {(125)⁻³. (64)⁻³⁾²} ⁻¹⁾⁹. 

A) 20 B)  40 C) 60 D) none 

16) (2⁰ + 7⁰)/5⁰

A) 2 B) 7 C) 5 D)1 E) none

17) (2³⁰ + 2²⁹)/(2³¹ - 2³⁰)

A) 1 B) 2 C) 2/3 D) 3/2

18) √(3⁻²)

A) 1/9 B) 9 C) - 3 D) 1/3

19) (12)¹⁾³. (36)¹⁾⁴ /(96)⁻¹⁾⁶. 

A) 12 B) 2 C) 21 D) none

20) (256)⁻³⁾⁴

A) 25/64 B) 64/125 C) 125/64 D) 64/25

21) (32)¹⁾⁵ x (125)⁻¹⁾³

A) 16/25 B) 4/5 C) 2/5 D) 2/25

22) (2 ⁻⁴⁾³ ÷ 2⁻²)¹⁾²

A) 1/2 B) 2 C) 1/5 D) 4

23) ⁴√{³√x²}

A) x¹⁾²⁴ B) x¹⁾⁶ C) x¹⁾¹² D) x¹⁾²⁰

24) (64a⁶)¹⁾² /(64a⁶)⁻¹⁾³. 

A) 32a⁵ B) 16a⁵ C) 8a⁴ D) none

25)(8x³/27a⁻³) ²⁾³(64x³/27a⁻³)⁻²⁾³

A) 1/4 B) 1/2 C) 1/8 D) none

26) ³√[x⁴√{x⁻⁵ √(x⁶)}]. 

A) x  B) x² C) x³ D) x⁴

27) (x²y⁻²/x⁻²y²)² / (xy⁻¹/x⁻¹y)⁻².

A) x B) y C)  (x/y)⁴ D) none

28) 9⁻³ . (16¹⁾⁴/6⁻²) . (1/27)⁻⁴⁾³. 

A) 2/27 B) 4/27 C) 8/27 D) none

29) {³√4 . 1/⁹√8 . ¹²√16⁻¹}¹⁾⁴

A) 2 B) 1 C) 1/2 D) none

30) {(125)⁻⁴ . (256)⁻³⁾²)⁻¹⁾⁶

A) 0 B) 1 C) 2 D) none

31) {243³⁾⁵x 25³⁾²}/{625¹⁾²x 8⁴⁾³x 16⁵⁾⁴}

A) 135/512 B) 13/51 C) 1 D) none

32) {81⁻³⁾⁴ .16¹⁾⁴/6⁻² . (1/27)⁻⁴⁾³}¹⁾³

A) 2 B) 3 C) 6 D) none

33) (4⁴⁰+ 3³⁹)/(3⁴¹ - 3⁴⁰)

A) 2/3 B) 3/2 C) 2/9 D) none

34) (2¹⁾²x 3¹⁾³ x 5¹⁾⁴)/(10⁻¹⁾⁵x 5³⁾⁵) ÷ (3⁴⁾³ x 5⁻⁷⁾⁵)/(4⁻³⁾⁵x 6)

A) 1 B) 10 C) 12 D) none

35) 27⁻¹⁾³{27¹⁾³ - 27²⁾³}

A) 2 B) - 2 C) 3 D) none

36) (2ⁿ.6ᵐ⁺¹.10ᵐ⁻ⁿ.15ᵐ⁺ⁿ⁻²)/(4ᵐ. 3²ᵐ⁺ⁿ) . 25ᵐ⁻¹)

A) 2 B) 1 C) 3 D) none

37)[{9⁽ⁿ⁺¹/⁴⁾.√(3. 3ⁿ)}/{3 .√(3⁻ⁿ)}]¹⁾ⁿ

A) 9 B) 3 D) 3¹⁾ⁿ D) 3ⁿ E) none

38) (2ᵐ⁺².3²ᵐ⁻ⁿ.5ᵐ⁺ⁿ⁺².6ⁿ) /(6ᵐ.10ⁿ⁺². 15ᵐ)

A) 2 B) 5 C) 5¹⁾ⁿ D) none

39) (xᵇ/xᶜ)ᵃ .(xᶜ/xᵃ)ᵇ.(xᵃ/xᵇ)ᶜ

A) 0 B) 1 C) x D) none

40) (xᵃ/xᵇ)ᵃ⁺ᵇ.(xᵇ/xᶜ)ᵇ⁺ᶜ(xᶜ/xᵃ)ᶜ⁺ᵃ

A) 0 B) 1 C) x D) none

41) 3/(625)⁻¹⁾⁴ + 2/(343)⁻²⁾³ + 4/(243)⁻¹⁾⁵

A) 5 B) 25 C) 125 D) none

42) (5ⁿ⁺² - 6.5ⁿ⁺¹)/(13. 5ⁿ - 2.5ⁿ⁺¹)

A) 5/3 B) -5/3 C) 3/5 D) -3/5

43) 64⁻¹⁾³ (64¹⁾³ - 64²⁾³)

A) 3 B) -3  C) 1 D) none

EXERCISE --3

--------------------

1) (xᵐ/xⁿ)ᵐ⁺ⁿ(xⁿ/xˡ)ⁿ⁺ˡ(xˡ/xᵐ)ˡ⁺ᵐ is

A) 0 B) 1 C) 2 D) none

2) ᵐⁿ√(xᵐ/xⁿ) .ⁿˡ√(xⁿ/xˡ) . ˡᵐ√(xˡ/xᵐ)ˡ is

A) 0 B) 1 C) 2 D) none

3) 1/(1+xᵃ⁻ᵇ+xᶜ⁻ᵇ) + 1/(1+xᵇ⁻ᶜ + xᵃ⁻ᶜ) + 1/(1+xᶜ⁻ᵃ+xᵇ⁻ᵃ) is

A) 0 B) 1 C) 2 D) none 

4) (1/1+xᵃ⁻ᵇ+xᵃ⁻ᶜ) +1/(1+ᵇ⁻ᶜ+xᵇ⁻ᵃ) + 1/(1+xᶜ⁻ᵃ+xᶜ⁻ᵇ) is 

A) 0 B) 1 C) 2 D) none

5) {(x²- 1/y²)ˣ (x - 1/y)ʸ⁻ˣ}/{(y²- 1/x²)ʸ (y+1/x)ˣ⁻ʸ} is

A) 0 B) 1 C) 2 D) (x/y)ˣ⁺ʸ E) none

6) show (xᵃ)ᵇ⁻ᶜ . (xᵇ)ᶜ⁻ᵃ .(xᶜ)ᵃ⁻ᵇ is 

A) 0 B) 1 C) 2 D) none

 EXERCISE-4

--------------- 

1) If m=aˣ, n= aʸ , a² =(mʸ.nˣ)ᶻ, Then xyz is

A) 0 B) 1 C) 2 D) none  

2) If a= bˣ, b=cʸ, c=aᶻ then xyz is

A) 0 B) 1 C) 2 D) none 

3) If p=mʸ=nᶻ and mn=1 the yz + zx+ xy is

A) 0 B) 1 C) 2 D) none  

4) If aᵐ=aⁿ=(ab)ᵐⁿ, then m+n is

A) 0 B) 1 C) 2 D) none

5) If aˣ= bʸ =cᶻ, and b²=ac then 1/x + 1/z is

A) 2 B) y C)  2/y D) none

6) 2ˣ=3ʸ=12ᶻ, then z(x+2y) is

A) x B) y C) xy D) none

7) If 3ˣ= 5ʸ=75ᶻ, then 1/x+2/y is 

A) 1 B) z C) 1/z D) none

8) If 2ˣ = 4ʸ=8ᶻ, and xyz=288 then yz/x is 

A) 0 B) 1 C) 2 D) none

9) If 2ˣ = 4ʸ= 8ᶻ, and 1/2x +1/4y + 1/8z = 22/7 then x+y+z is

A) 77 B) 77/96 C) 96 D) none

10) If (1.234)ᵅ=(0.1234)ᵇ=10ᶜ, then 1/a - 1/c is

A) 1 B) b C)  1/b D) none

11) If 3ˣ =2 , 2ʸ=5 and 5ᶻ =3, then xyz is

A) 0 B) 1 C) 2 D) none

12) If 64ˣ = 48ʸ = 36ᶻ, then 1/x + 1/z is

A) 2 B) y C)  2/y D) none

13) If pᵃ= qᵇ =rᶜ and pqr=1, then 1/a + 1/b + 1/c is

A) 0 B) 1 C) 2 D) none

14) x¹⁾ᵃ= y¹⁾ᵇ = z¹⁾ᶜ and xz = y², then a+ c is

A) 2 B) b C) 2b D) none

15) If m= aˣ, n= aʸ and a²= (mʸnˣ)ᶻ, then xyz is

A) 0 B) 1 C) 2 D) none

16) If x¹⁾ᵃ = y¹⁾ᵇ =z¹⁾ᶜ and a+b+ c=0, then xyz is

A) 0 B) 1 C) 2 D) none

17) If aˣ= b, bʸ= c, cᶻ=a, then xyz is  (a,b,c positive numbers)

A) 0 B) 1 C) 2 D) none

19) if aˣ =bʸ and bˣ = aʸ (ab≠1), then a

A) 0 B) 1 C) 2 D) b  E) none

20) if aˣ=bᵖ, bʸ= c²ᵖ, cᶻ= a⁴ᵖ, then xyz  is

A) 0 B) 1 C) 2 D) 8p³ E) none

21) If xᵖ = yᵃ= (xy)ᵖᵃ, then p+q is

A) 0 B) 1 C) 2 D) none

22) If xᵃ = xᵇ⁾²zᵇ⁾² = zᶜ then 1/a + 1/c is

A) 0 B) 1 C) 2 D)  2/b E) none

23) If (44)ˣ= (4.4)ʸ=10ᶻ, then 1/y + 1/z is 

A) 1 B) x C) 1/x D) none

24) If x=2¹⁾³+2⁻¹⁾³ then 2x³- 6x is

A) 0 B) 1 C) 2 D) 5 E) none

25) If x=2¹⁾³+2²⁾³, then x³- 6x is

A) 0 B) 1 C) 2 D) 6 E) none

26) If x= 3¹⁾³- 3⁻¹⁾³, then 3x³+9x is

A) 0 B) 1 C) 2 D) 8 E) none

27) if a= 5 -5²⁾³ -5¹⁾³ then  a³ +15a²+60a is

A) 0 B) 1 C) 2 D) 20 E) none

28) If x= 3 - 3²⁾³- 3¹⁾³ then x²- 18x+12 is 

A) 0 B) 1 C) 2 D) none

29) If x=1+ 3²⁾³+3¹⁾³ then  x³-3x²-6x is

A) 0 B) 1 C) 2 D) 4 E) none

EXERCISE - 5

--------------------

SOLVE THE FOLLOWING:

1) 2ˣ⁺³ + 2ˣ⁺¹=320

2) 4ˣ - 3. 2ˣ⁺¹ + 2⁵=0

3) 2ˣ⁺²+ 2ˣ⁻¹ =9

4) 3²ˣ+ 9= 10. 3ˣ

5) x²⁾⁵ - 5x¹⁾⁵+ 6= 0

6) 9. 81ˣ = 1/(27)ˣ⁻²

7) 4ˣ⁺²+2²ˣ⁺³=96.

8) 3²ˣ⁻⁵ + 9ˣ⁻²= 4

9) 2ˣ+ 3ʸ= 7, 2ˣ - 3ʸ = 1

10) 8ˣ. 4ʸ = 128, 9ˣ⁺ʸ = 27xy

11) 2ˣ. 6ʸ= 24, 2²ˣ . 3ʸ= 48

12) 6²ˣ⁺⁴ = 3³ˣ . 2ˣ⁺⁸

13) xʸ = yˣ and x² = y³

14) √(3)²ˣ⁺¹= 243

15) 2ˣ⁺³ . 3ˣ ⁻ ³ =64

16) 4ˣ = 8³

17) 2ˣ⁺² + 2ˣ⁻¹ =9

18) (2/3)ˣ (3/2)²ˣ= 81/16.

A) 1 B) 2 C) 3 D) 4 E) none

19) (2³)⁴ = (2²)ˣ

A) 2 B) 4 C) 6 D) none

20) 27ˣ = 9/3ˣ

A) 1 B) 2 C) 1/2 D) none

21) 5ˣ⁻⁴ x 2ˣ⁻⁵ = 5

A) 1 B) 5 C) 7 D) none

22) (3/5)ˣ (5/3)²ˣ=  125/27

A) 1 B) 2 C) 3 D) none

23) √{5⁰ + 2/3}= (0.6)2²⁻³ˣ

A) 5/6 B) 6/5 C) 1 D) none

24) √(3/5)ˣ⁺¹ = 125/27

A) 7 B) -7 C) 8 D) none

25) 3²ˣ⁻⁸/225 = 5³/5ˣ.

A) 5 B) 25 C) 125 D) none

26) (2/3)ˣ(3/2)²ˣ= 81/16

A) 4 B) 8 C) 2 D) none

EXERCISE - 6

--------------------

1) If 3²ˣ = 81 then find 10⁻ˣ

2) 2³ⁿ⁺¹¹+ 2⁵ = 2⁶ find 2ⁿ

3) 3ˣ - 3ˣ⁻² =8, find xˣ

4) If 2ⁿ - 2ⁿ⁻¹ = 4 find nⁿ

5) If x= 2 and y= 3, then the value of xʸ + yˣ is

A) 15 B) 17 C) 19 D) 21

22) If p=5, a= 3, find (p+a)ᵖᵃ

23) If 3ˣ = 9ʸ, find (x/y) -1

4) Arrange in ascending order

a) ₂2³, (2³)², ₂3²

b) 2⁵⁰ , 3⁴⁰, 4³⁰

E) Find the value of

1) If a=2+√3, find a³-2a²-7a+3

                     

      

Continue.......

c) 

d) If p=5 and r= 3, find (p+r)ᵖ⁾ʳ

i) 23   ii) 32       iii) 1        iv) n

e) ₐmⁿ = (aᵐ)ⁿ Express m in terms of n.

i) n        ii) 1/n    iii) n¹⁾⁽ⁿ⁻¹⁾   iv) none

f) If xᵃ = yᵇ and yᵃ = xᵇ, then a² =

i) b    ii) b²        iii) b³      iv) none

g) If 3ˣ = 9ʸ,  x/y -1= 

i) 0         ii) 1        iii) 2      iv) none

h) If 4ˣ = 8ʸ = 16ᶻ, then  x : y: z be

i) 6:4:3.  ii) 3:4:6.  iii) 4:6:3.  iv) none

2) Arrange in ascending order of magnitude :::

(i) ₂2³ , (2³)² , ₂3²

a)₂2³, (2³),₂3²      b) ₂3², (2³)², ₂2³

c) (2³)²,₂2³, ₂3²     d) none

(ii) 2⁵⁰ , 3⁴⁰ , 4³⁰

a)2⁵⁰,4³⁰,3⁴⁰        b) 3⁴⁰ , 4³⁰,2⁵⁰

c) 4³⁰,2⁵⁰ , 3⁴⁰     d) none

j) If 3²ˣ= 81, find the value of 3⁻ˣ

a)1    b) 9      c) 1/9       d) none

k) If 10²ˣ = 25, find the value 10⁻ˣ

a)5      b) 1       c) ⅕         d) none

3) (8)²⁾³. (16)³⁾⁴ . (32)⁻⁴⁾⁵

a)1       b) 2       c) 3        d) none

4) {(125)⁻³. (64)⁻³⁾²} ⁻¹⁾⁹

a)8     b) 9          c) 10      d) none

5) (12)¹⁾³. (36)¹⁾⁴ * (96)⁻¹⁾⁶

a)10      b) 11      c) 12       d) none

6) a) (64a⁶)¹⁾² * (64a⁶)⁻¹⁾³

i) 32a³     ii) 32a⁴     iii) 32a⁵   iv) none

b) (xᵃ)ᵇ⁻ᶜ. (xᵇ)ᶜ⁻ᵃ (xᶜ)ᵃ⁻ᵇ

i) 0       ii) 1       iii) x          iv) none

c) 1/(1+xᵃ⁻ᵇ) + 1/(1+ xᵇ⁻ᵃ)

i) 0      ii) 1         iii) xᵃ       iv) xᵇ

d) (2ᵐ⁺². 3²ᵐ⁻ⁿ)/(6ᵐ . 3ᵐ⁻ⁿ⁻¹)

i) 12     ii) 2          iii) 1       iv) none

e) If (10¹¹ +25)² - (10¹¹ - 25)²= 10ⁿ, find the value of n.

i) 10      ii) 11         iii)13      iv) 14

f) If 2³ⁿ⁺¹¹ + 2⁵ = 2⁶, find 2ⁿ

i) 1        ii) ¼       iii) ½      iv) none

g) If 3ˣ - 3ˣ⁻² = 8, find xˣ

i) 4     ii) 3       iii) 2      iv) 1

h) If 2ⁿ - 2ⁿ ⁻¹ =4 find nⁿ

i) 64     ii) 8        iii) 27       iv)125

7) (8x³ * 27a⁻³) ²⁾³.(64x³ * 27a⁻³)⁻²⁾³

a)¼         b) 1/9        c) ⅙    d) none

8) ³√[x⁴√{x⁻⁵ √(x⁶)}]

a)1       b) x          c) x²       d) x³

9) (x²y⁻²/x⁻²y²)³ * (xy⁻¹/x⁻¹y)⁻²

a)(x/y)¹⁶      b) (x/y)¹⁰     c) x/y   d) n

10) 9⁻³ . (16¹⁾⁴/6⁻²) . (1/27)⁻⁴⁾³

a)0       b) 1        c) 6       d) 8

11) {³√4 . 1/⁹√8 . ¹²√16⁻¹}¹⁾⁴

a)1      b) 2       c) 4        d) none

12) {(125)⁻⁴  . (256)⁻³⁾²)⁻¹⁾⁶

a)1       b) 10       c) 100    d) 1000

13) {³√(x² y⁻⁴) . √(y³ x⁻⁵)}¹² . (x²)¹¹

a)x     b) y       c) xy      d) x²   e) y²

14) {81⁻³⁾⁴ .16¹⁾⁴/6⁻² . (1/27)⁻⁴⁾³}¹⁾³

a)6       b) 12       c) 16      d) none

15) (2ⁿ.6ᵐ⁺¹.10ᵐ⁻ⁿ.15ᵐ⁺ⁿ⁻²)/(4ᵐ. 3²ᵐ⁺ⁿ) . 25ᵐ⁻¹)

a)⅓     b) ⅓       c) ⅔      d) 3/2

16)[{9⁽ⁿ⁺¹/⁴⁾.√(3. 3ⁿ)}/{3 .√(3⁻ⁿ)}]¹⁾ⁿ

a)27    b) 72     c) 3       d) 1

17)(2ᵐ⁺².3²ᵐ⁻ⁿ.5ᵐ⁺ⁿ⁺².6ⁿ)/(6ᵐ.10ⁿ⁺². 15ᵐ)

a)1        b) 2        c) 3      d) none

18) (xᵇ/xᶜ)ᵃ .(xᶜ/xᵃ)ᵇ.(xᵃ/xᵇ)ᶜ

a)1          b) 1       c)  x      d) none

19) (xᵃ/xᵇ)ᵃ⁺ᵇ.(xᵇ/xᶜ)ᵇ⁺ᶜ(xᶜ/xᵃ)ᶜ⁺ᵃ

a)1        b) x        c) x²       d) none

20) (2²ⁿ-3.2²ⁿ⁻²)(3ⁿ-2ⁿ⁻²)/{3ⁿ⁻⁴(4ⁿ⁺³ - 2²ⁿ)

a)1    b) 4         c) ¼       d) 2

21) ᵇᶜ√(xᵇ⁾ᶜ/xᶜ⁾ᵇ).ᶜᵃ√(xᶜ⁾ᵃ/xᵃ⁾ᶜ )  . ᵃᵇ√(xᵃ⁾ᵇ/xᵇ⁾ᶜ)

a)1     b) x       c) x²       d) none

22) {(0.3)¹⁾³.(1/27)¹⁾⁴.9¹⁾⁶(0.81)²⁾³}/ {(0.9)²⁾³ .(3)⁻¹⁾².(1/3)⁻² .(243)⁻¹⁾⁴}

a)0.1    b) 0.2      c) 2      d) 0.3

23)  {(bc)ᵇ⁻ᶜ(ca)ᶜ⁻ᵃ(ab)ᵃ⁻ᵇ}/ {aᵇ⁻ᶜbᶜ⁻ᵃcᵃ⁻ᵇ}⁻¹

a)1       b) a       c) b       d) c

24) (xᵐ/xⁿ)ᵐ⁺ⁿ⁻ˡ(xⁿ/xˡ)ⁿ⁺ˡ⁻ᵐ 

(xˡ/xᵐ) ˡ⁺ᵐ⁻ⁿ

a)1     b) x       c) xᵐⁿˡ       d) none

25) ᵇᶜ√(xᵇ/xᶜ) .ᶜᵃ√(xᶜ/xᵃ). ᵃᵇ√(xᵃ/xᵇ)

a)1        b) x        c) xᵃ       d) none

26) (a¹⁾⁽ˣ⁻ʸ⁾)¹⁾⁽ˣ⁻ᶻ⁾ . (a¹⁾⁽ʸ⁻ᶻ⁾)¹⁾⁽ʸ⁻ˣ⁾ .(a¹⁾⁽ᶻ⁻ˣ⁾)¹⁾⁽ᶻ⁻ʸ⁾

a)1        b) a      c) 0     d) none

27) (xᵃ⁾⁽ᵃ⁻ᵇ⁾)¹⁾⁽ᵃ⁻ᶜ⁾ .(xᵇ⁾⁽ᵇ⁻ᶜ⁾)¹⁾⁽ᵇ⁻ᵃ⁾ (xᶜ⁾⁽ᶜ⁻ᵃ⁾)¹⁾⁽ᶜ⁻ᵇ⁾

a)1         b) x       c) xᵃᵇ     d) none

28) 1/(1+xᵐ⁻ⁿ +xᵐ⁻ᵖ) + 1/(1+ xⁿ⁻ᵖ + xⁿ⁻ᵐ) + 1/(1+xᵖ⁻ᵐ +xᵖ⁻ⁿ)

a)1        b) x        c) x⁻¹      d) none

29) {(m² - 1/n²)ᵐ (m - 1/n)ⁿ⁻ᵐ}/{n² - 1/m²)ⁿ (n + 1/m)ᵐ⁻ⁿ

a)(m/n)ᵐ⁺ⁿ  b) (m/n)ᵐ⁻ⁿ  c) mn  d) 1

30) 1/(1+xᵇ⁻ᵃ + xᶜ⁻ᵃ)  + 1/(1+ xᶜ⁻ᵇ + xᵃ⁻ᵇ) + 1/(1+ xᵃ⁻ᶜ + xᵇ⁻ᶜ) 

a)1      b) x      c) xᵃᵇᶜ     d) none

32) 1/(1+a⁻ᵐbⁿ+ a⁻ᵐ cᵖ ) +  1/(1+ b⁻ⁿ cᵖ + b⁻ⁿ aᵐ) + 1/(1+ c⁻ᵖaᵐ + c⁻ᵖbⁿ)

a)1     b) x      c) xᵐⁿ      d) none

32) {(p+ 1/q)ᵐ (p - 1/q)ⁿ}/{(q+ 1/p)ᵐ (q - 1/p)ⁿ

a)p     b) q    c) p/q      d) (p/q)ᵐ⁺ⁿ

33) multiply

a) (ₓ2ⁿ⁻¹₊ ᵥ2ⁿ⁻¹) by (ₓ2ⁿ⁻¹ ₋ ᵥ2ⁿ⁻¹)

i)x²-y²   b) x-y   c) x+y     d) none

b) (4x²⁾³ + 6x¹⁾³y¹⁾³+9y²⁾³) by (2x¹⁾³ - 3y¹⁾³)

a)4x-27y                   b) 8x -27y

c) 4x² - 27y²              c) 8x² - 27y²

34) If aˣ = bʸ = cᶻ and b²= ac, then find the value of 1/x + 1/z

a) 2/y    b) 1/y     c) 2y     d) y

35) If 3ˣ =2 , 2ʸ=5 and 5ᶻ =3, then find the value , xy

a)z       b) 1/z        c) 2/z       d) none

36) If 64ˣ = 48ʸ = 36ᶻ, then find the value of 1/x + 1/z = 

a) y        b) 1/y      c) 2/y   d) y/2

37) If pᵃ= qᵇ =rᶜ and pqr=1, then find the value if 1/a + 1/b + 1/c 

a)0           b) 1          c) 2     d) 1/2

38) x¹⁾ᵃ= y¹⁾ᵇ = z¹⁾ᶜ and xz = y², then value of  a+ c 

a)b         b) 2b         c) 3b     d) 4b

39) If m= aˣ, n= aʸ and 

a²= (mʸnˣ)ᶻ, then xyz

a)0       b) 1    c)  2        d) none

40) If x¹⁾ᵃ = y¹⁾ᵇ =z¹⁾ᶜ and 

a+b+ c=0, then value of xyz

a)0      b) 1     c)  2      d) none

41) If aˣ= b, bʸ= c, cᶻ=a, then value of xyz when (a,b,c positive numbers)

a)0     b) 1     c) 2        d) none

42) if aˣ =bʸ and bˣ = aʸ (ab≠1), then the value of  a is

a)0        b) 1        c) b.   d) 2b

43) if aˣ=bᵖ, bʸ= c²ᵖ, cᶻ= a⁴ᵖ,then the value  xyz 

a)8      b) 8p      c) 8p²      d) 8p³

44) If 2ˣ= 3ʸ= 12ᶻ, then the value of z(x+2y) be

a)x    b) y         c) xy      d) 1

45) If xᵖ = yᵃ= (xy)ᵖᵃ, then  p+q be

a)0          b) 1       c) pq    d) p   e) q

46) If xᵃ = xᵇ⁾² zᵇ⁾² = zᶜ then the value of  1/a + 1/c 

a)1/b       b) b     c)  2/b.   d) b/2

47) If (44)ˣ= (4.4)ʸ=10ᶻ, then the value of 1/y + 1/z.

a)1      b) x       c) 1/x     d) none

48) If a = xᶠ⁺ᵍ yᵉ , b= xᵍ⁺ᵉyᶠ, c= xᵉ⁺ᶠ . yᵍ, then the value  aᶠ⁻ᵍbᵍ⁻ᵉcᵉ⁻ᶠ

a)0        b) 1      c) ½      d) none

49) If (ₓn²)ⁿ = (ₓ2ⁿ)²,then ⁿ⁺¹√n³ be

a)0        b) 2            c) x       d) 1

50) If xʸ = yˣ and (x/y)ˣ⁾ʸ=(x)⁽ˣ⁾ʸ ⁻ ¹⁾ and if x= 2y then find y

a)0       b) 1        c) 2      d) none

51) If x² = y³ find (x/y)³⁾² + (y/x)²⁾³

 a)0    b) 1      c)     x²y³  iv) x¹⁾² y⁻¹⁾³

52) If x = 2 - 2¹⁾³ + 2²⁾³, find the value of x³ - 6x² +18x +11

a)0      b) 1        c) 2        d) none

53) If x=2¹⁾³ + 2²⁾³ find x³ -6x

a)1            b) 3            c) 5    d) 6

54) If a= 2¹⁾³ - 2⁻¹⁾³ find 2a³+6a

a)0           b) 1        c) 2        d) 3

55) If x= 3¹⁾³- 3⁻¹⁾³ find 3x³ +9x

a)8     b) 4      c) 2   d) 0

56) If x= 2+ 2²⁾³ +2 ¹⁾³ then find the value  x³- 6x² +6x  

a)0     b) 1        c) 2        d) 3

57) If x= 3+3²⁾³+3¹⁾³ find the value of x³- 9x² +18x -12

a)0       b) 1        c) 2        d) none

58) If x = (√2 + 1)⁻¹⁾³, value of

 (x - x⁻¹)³ + 3(x - x⁻¹) +2 

a)0        b) 1      c) x      d) x²

59) If a= (√2 -1)¹⁾³ then find the value of  (a -a⁻¹)³ + 3(a - a⁻¹) + 2 

a)0      b) 1       c) a      d) none

60) If a¹⁾³+ b¹⁾³+ c¹⁾³= 0 then the value of (a+b+c)³= 

1)27abc   b) 9abc    c) 3abc   d) none

61) If x = a¹⁾³ b⁻¹⁾³+ a⁻¹⁾³ b¹⁾³ then the value of a(bx³ - 3bx - a)

a) b²    b) b      c) 1        d) 0

62) If x= (a +√(a²+ b³)¹⁾³ +  (a - √(a²+ b³)¹⁾³ find x³+ 3bx - 2a 

a)0        b) 1        c) a       d) ab

63) simplify(e²ˣ+e⁻ˣ- eˣ-1)/(e²ˣ-e⁻ˣ+eˣ -1) 

a)(eˣ -1)/(eˣ +1)   b) 1  c) 0   d) none

*** Solve for x

65) (√3)²ˣ⁺¹ = 243

i) 9/2      ii) 2/9     iii) 9      iv) none

66) 4ˣ = 8³

i)9/2     ii) 2/9       iii) 9      iv) none

67) x√x = √(xˣ)

i)9/4      ii) 4/9      iii) 9     iv) none

68) 2ˣ⁺²+ 2ˣ⁻¹ = 9

i)1        ii) 2          iii) ½    iv) none

69) ₂2ˣ = ₁₆2³ˣ

i)1        ii) 2          iii) ½    iv) none

70) 2ˣ⁺³ . 3 ˣ ⁻³ = 64

i)1        ii) 2         iii) 3      iv) none

71) 4ˣ⁺² = 2²ˣ⁺³ + 2

i)1       ii) 2         iii) -1      iv) -2

72) ₓ√x = (ₓx)¹⁾²

i) 1       ii) 2          iii) 3    iv) 4

73) 2ˣ ⁻² + 2³⁻ˣ = 3

i) 2 or 3   ii) 1or 2     iii) 1or 3  iv) n

74) 3²ˣ + 9 = 10(3ˣ)

i) 0 or 2   ii) 1or 2     iii) 0or 3  iv) n

75) 4ˣ - 9 . 2ˣ + 2³=0

i) 0 or 1  ii) 0 or 2  iii) 0 or 3  iv) n

76) 3²ˣ  - 3ˣ ⁺¹ - 3ˣ ⁻¹ + 1 = 0

i) ±1     ii) ±2       iii) ±3      iv) none

77) 6²ˣ⁺⁴ = 3³ˣ . 2ˣ⁺⁸

i) 1     ii) 2         iii) 3     iv) 4  v) n

78) 4(4²ˣ⁺¹)= 8ˣ⁺²

i) 1       ii) 2       iii) ½       iv) none

79) ₓx√x = (x√x)ˣ

i) 4       ii) 9      iii) 4/9     iv) 9/4

80) 4ˣ - 3ˣ ⁻ ¹⁾² = 3ˣ⁺ ¹⁾² - 2²ˣ ⁻¹

i) 3        ii) 2       iii) ⅔      iv) 3/2

81) 4ˣ - 3 . 2ˣ⁺² + 2⁵= 0

i) 3       ii) 2     iii) 2,3      iv) none

82) If 2ˣ + 3ʸ = 17 and  2ˣ⁺² - 3ʸ⁺¹ = 5 then find the value of xy

i) 6      ii) 12        iii) 24       iv) none

83) If 3ˣ + 3ʸ =4 and 3⁻ˣ + 3 ⁻ʸ= 4/3 then find the value of xy

i) 0        ii) 1         iii) 2      iv) 4

84) xʸ = y² , y²ʸ = x⁴ then x, y be

i) ±1/2,2  ii) ±1, 2   iii) ±1/2, -2  iv) n

85) 3ˣ⁾⁵ + 3⁽ˣ⁻¹⁰⁾/¹⁰ = 84

i) 0     ii) 2          iii) 20       iv) none

86) (2ˣ+2⁻ˣ)/(2ˣ-2⁻ˣ)=

(16¹⁾ˣ+16⁻¹⁾ˣ) /(16¹⁾ˣ - 16⁻¹⁾ˣ)

i) ±1      ii) ±2          iii) ±3      iv) none

87) 5¹¹⁻²ˣ+ 4ˣ⁾² ⁺¹= 4 ˣ⁾² ⁻ ¹+5¹³ ⁻ ²ˣ

i) 5         ii) 3          iii) 1       iv) none

88)  If 9ˣ= 27ʸ and 81ˣ = 243 . 3ʸ then find the value of xy be

i) 2       ii) 3        iii) ⅔         iv) 3/2

89) If xʸ = yˣ and x = 2y find the value of xy be

i) 2       iii) 4      iii) 8       iv) none

90) If  5ˣ  - 3ʸ= 16 , 5ˣ⁻¹ + 3ʸ⁺¹ = 32 then find the value of x and y

i) 1,1     i) 2,2        iii) 3,3     iv) none

91) If 2ˣ + 2ʸ = 12 and  x + y = 5 find the value of xy be

i) 2      ii) 3          iii) 6         iv) none

92) xʸ = yˣ , x² = y³ find x and y

i) 1,1    ii) 27/8,1    iii) 1,9/4    iv) none

93) 2ˣ+ 3ʸ = 11, 4ˣ - 9ʸ = 55 then x, y be

i) 1, 3      iii) 3,1    iii) 2,3      iv) none

94) 2ˣ . 6ʸ= 72 , 2²ˣ . 3ʸ = 36 then x, y be

i) 1,2      b) 2,1      c) 1,1   d) 2,2

95) 5ˣ + 5ʸ= 6 , 5¹⁻ˣ + 5¹⁻ ʸ= 6 then x, y be 

i) 1, 2    ii) 2,1       iii) 1,1    iv) 2,2

96) ᵃᵇ√(pᵃ/pᵇ). ᵇᶜ√(pᵇ/pᶜ). ᶜᵃ√(pᶜ/pᵃ), where p > 0 and abc ≠ 0.

A) 0. B) 1 C) p D) abc E) 1/p

97) If 3ˣ⁺⁴ = 3ˣ⁺⁵- 6, then x is

A)2 B) -2 C) -3 . D) -4 E) 3

98) If (x³)ⁿ = ₓ3⁴ , then find n, given x ≠±1 and x≠ 0.

A) 4 B) 3 C) 9 D) 27. D)81

99) ₓmⁿ= x⁸, and x ≠±1 and x≠ 0., Where m and n are natural numbers, find the value of m

A)2 B) 3 C) 4 D) either A or C D)n.

100) ₂3ˣ = 512, find x

A) 2. B) 3 C) 4 D) 5 E) 1

101) If (2ⁿ⁾²)ⁿ⁺¹= {(2ⁿ⁾⁶)⁽ⁿ⁺¹⁾} ⁽²ⁿ⁺¹⁾. Find all the possible value of n.

A) 0 B) 1 C) -1 D) all of these. E)n

102) If aˣ= bʸ = cᶻ and 1/x+1/z =2/y, then find the relation between positive numbers a, b and c.

A) b²=ac. B) b= a/c C) b=√(a/c) D) b = a+ c E) b= ac

103) If xᵐ . x⁻ⁿ = 0 and m≠ n then which of the following is true?

A) m+n= 0 B) m=0 C) m.n=0 D) n= 0 E) x= 0.

104) If 11ˣ = 99, find the value of (11ˣ⁻²)/(11ˣ⁺²)

A) 11 B) 0 C) 1/14641. D) 99⁴ E) n

105) If 3ˣ⁺³ = 3ˣ⁺⁴, find x

A)1 B) 5 C) 3 D) 4 E) 2.

106) ₂x²- 3x +2= 1, find x.

A) 1 B) 2 C) 3 D) both B and C E) both A and B.