INCREASING & DECREASING FUNCTIONS

Solution of Rational Algebraic Inequations

The following results are very useful in solving rational algebraic operations:

a) ab>0=> (a>0 and b>0) or (a<0 & b<0)

b) ab<0=> (a>0 and b<0) or (a<0 & b>0)

c) ab>0 and a>0 => b>0

d) ab<0 and a<0 =>  b>0

If P(x) and Q(x) are polynomial, then the Inequations P(x)/Q(x) >0, P(x)/Q(x) <0, P(x)/Q(x) ≥ 0  and P(x)/Q(x) ≤0 are known as rational algebraic Inequations. These Inequations can be solved by using the following STEPS

Step 1: Factorize P(x) and Q(x) into linear factors.

Step 2: Make Coefficient of x positive in all factors.

Step 3: Equate all the factors to zero and find the corresponding values of x. These values are generally known as critical points.

Step 4: Plot the critical points on the number line. Note that n critical points will divide the number line in (n+1) regions.

Step 5: In the right most region, the expression will be positive and in other regions it will be alternatively positive and negative. So, mark positive in the right most region and than mark alternatively positive and negative signs in the remaining regions.

Step 6: Obtain the solution set of the Inequation by selecting the appropriate region in step 5.

EXERCISE -1

1) 4x³- 24x²+ 44x - 24> 0.        (1,2) U (3,∞)

2) 1/(x+1) - 4/(2+ x)² > 0, x≠ -1, -2.        (-1,0) U (0,∞)

3) (1- x²)/(5x - 6 - x²) < 0.        x∈ (-1,1) U (2,3)

4) (8x²+16x -51)/(2x²+5x-12) > 3.          x∈(- ∞-4) U(-3, 3/2)U (5/2,∞)

5) (x² -2x +5)/(3x² -2x- 5) > 1/2.           x∈(-5, -1)U (5/3,3)

6) (x² -2x +24)/(x²-3x+4) ≤ 4.          x∈(- ∞-2/3) U[4,∞)

7) (x² -4x +7)/(x²-7x+12) ≤ 2/3.          x∈[-3,1] ∈ (3,4)

8) Show that the function f(x)= 2x +3 is strictly increasing function on R.

9) Show that the function f(x)= -3x +12 is strictly decreasing function on R.

10) Show that the function f(x)= 7x - 3 is strictly increasing function on R.

11) Show that the function f(x)= -ax + b, where a, b are constants and a > 0 is an increasing function on R.

12) Show that the function f(x)= ax +b, where a, b are constant and a< 0 is a decreasing function on R.

13) Show that the function f(x)= 1/x is a decreasing function on (0,∞)

14) Show that the function f(x)= 1/(1+ x²) decreasing in the interval [0,∞) and increasing in the interval (-∞, 0].

15) Show that f(x)= 1/(1+ x²) is neither increasing nor decreasing on R.

16) Show that the function f(x)= x² is strictly increasing function on [0, ∞).

17) Show that the function f(x)= x² is a strictly decreasing function on (- ∞, 0].

18) Show that the function f(x)= x² is neither strictly increasing nor decreasing on R.

19) Show that the function f(x)= aˣ, a > 1 is strictly increasing on R.

20) Show that the function f(x)= aˣ, 0 <a < 1 is strictly decreasing on R.

21) Show that the function f(x)= logₑx is increasing on (0, ∞).

22) Show that the function f(x)= logₐx is increasing on (0, ∞) if a> 1 and decreasing on (0, ∞), if 0 < a < 1.

23) Show that the function f(x)= |x| is 

   a) strictly increasing (0, ∞) 

   b) strictly decreasing in (-∞, 0).

********* 

• For sin - 1st +ve 2nd & 3rd -ve

In order to find the interval in which a given function is increasing or decreasing, use the following steps

Step 1:  Obtain the function and put it equal to f(x).

Step 2: Find f'(x).

Step 3: Put f'(x)>0 and solve this Inequation.

For the values of x obtained in step 2 f(x) is increasing and for the remaining points in its domain it is decreasing.

**************************

EXERCISE -2

A) Find the intervals of the following is increasing or decreasing:

1) 10-6x-2x².                (-∞,-3/2), (-3/2,∞)

2) 6 - 9x- x².              (-∞,-9/2), (-9/2,∞)

3) x²+ 2x-5.               (-1,∞),(-∞,-1)

4) f(x)= - x²-2x+25.           (-∞,-1), (-1,∞)

5) f(x)= x³-6x²-36x+2. (-∞,-2) U (6,∞), (-2,-6)

6) 2x³ -9x²+12x+15.         (-∞,-1) U (2,∞), (1,2)

7) f(x)= 2x³+9x²+12x+20.           (-∞,-2) U (-1,∞), (-2,-1)

8) f(x)= 2x³+9x²+12x+20. (-∞,-2) U (-1,∞), (-2,-1)

9) f(x)= 2x³-15x²+36x+1.            (-∞,2) U (3,∞), (2,3)

10) f(x)= 2x³- 24x+107.           (-∞,-2) U (2,∞), (-2,2)

11) f(x)= 2x³-24x+7.           (-∞,-2) U (2,∞), (-2,2)

12) 5+ 36x + 3x²-2x³.           (-2,3), (-∞,-2) U (3,∞),

13) 8+ 36x + 3x²-2x³.            (-2,3), (-∞,-2) U (3,∞),

14) 6+ 12x + 3x²-2x³.           (-1,2), (-∞,-1) U (2,∞),

15) f(x)= x⁴ - x³/3.           (1/4,∞), (-∞,1/4)

16) f(x)= x⁴- 8x³+ 22x²-24x+21.         (1,2) U(3,∞), (1,2) U(3,∞)

17) 3x⁴/10 - 4x³/5 -3x²+ 36x/5 +11.          (-2,1) U(3,∞), (-∞,-2)U(1,3)

18) x⁴/4 +2x³/3 -5x²/2-6x +7.           (-3,-1) U(2,∞), (-∞,-3)U(-1,2)

19) {x(x-2)}².                        (0,1)U(1,∞)

20) f(x)= log(1+x) - 2x/(2+x)          (-1,∞)

21)  f(x)= x/2 + 2/x.          (-∞,-2) U(2,∞), (-2,0)U(0,2)

22) f(x)= x³+ 1/x³, x≠0         (-∞,-1) U(-1,∞), (-1,0)U(0,1)

23) f(x)= (4x²+1)/x. (-∞,-1/2), (1/3,∞)

24) f(x)= (x +1)³(x -3)³.    (1,3) U(3,∞), (-∞, -1) U (-1,1)

25) f(x)= (x -1)³(x -2)².     (-∞,1) U(1, 8/5) U(2,∞), (8/5,2)

EXERCISE - 3

A) prove that the function is increasing in R:

1) f(x)= x³- 3x²+ 3x -100.

2) f(x)= x³-15x²+75x -50.

3) f(x)= x³- 6x²+ 12x - 18.

4) f(x)= x⁹+ 4x⁷+ 11.

5) f(x)= e²ˣ

6) f(x)= (x -1)eˣ + 1 for all x> 0.

7) f(x)=Tan⁻¹(sinx + cosx) at (0,π/4)

8) f(x)= 4sinx/(2+ cosx) at (0,π/2).

9) x - sinx.

B) Prove that the function is decreasing in 

1) f(x)= e¹⁾ˣ, x ≠ 0 for all x≠ 0.

2) f(x)= kogₐx, 0 < a < 1 for all x > 0.

3) Prove that the function is neither increasing nor decreasing

 f(x)= x²- x+1 on (0,1)

EXERCISE - C

1) Find the least value of a such that the function :

a) x²+ ax +1 is increasing on [1,2].      -2

b) x³- ax is an increasing on R.        a≤0

c) sinx - ax + c is a decreasing on R.  a≥1

d) x + cosx - a is an increasing on R.    

2) Determine the values of x for which the function f(x)= x²- 6x + 9 is increasing or decreasing. Also, find the coordinates of the point in the curve y = x²- 6x + 9 where the normal is parallel to the line y= x +5.    Inc:(3,∞), dec: (-∞,3); (5/2,1/4)

3) Show that f(x)= sinx is increasing on (0,π/2) and decreasing on (π/2,π) and neither increasing nor decreasing in (0,π).

4) Separate [0,π/2] into subinterval in which f(x)= sin 3x is increasing or decreasing. (0,π/6), ((π/6,π/2)

5) Find the interval in which the function f given by f(x)= sinx+ cosx, 0≤ x≤ 2π is increasing or decreasing. (0,π/4) U (5π/4,2π), (π/4,5π/4)

6) Find the interval in which the function f given by f(x)= (4sinx-2x - x)/(2+ cosx) 0≤ x≤ 2π is increasing or decreasing. (0,π/2) U (3π/2,2π), (π/2,3π/2)

7) Show that f(x)= log sinx is increasing on (0,π/2) and decreasing on (π/2,π).

8) Determine the values of x for which it is increasing or decreasing 

a) f(x)= xˣ, x>0. (1/e, ∞), (0,1)e)

b) x/log x. (e,∞), ((0,e)-{1}

c) f(x)= e¹⁾ˣ, x≠0 is a decreasing function for all x≠0

d) cos²x is a decreasing on (0,π/2)

PERCENTAGE (C)

EXERCISE -A

1) Express as percentage as fractios:

a) 356%

b) 5/2%

c) 16-2/3%

2) Express the following Fractions as percentage:

a) 3/2

b) 9/20

c) 1-1/4

d) 17:20

e) 13:18

f) 93:80

3) Express the following Fractions as decimals. Then express them as percentage:

a) 3/4

b) 5/8

c) 3/16

4) Express the percentage as decimals:

a) 20%

b) 2%

c) 3-1/4%

5) Find the value of

a) 27% of ₹50

b) 25/4% of 250kg

6) What percent is:

a) 300g of 2 kg

b) ₹7.50 of ₹6.

7) What percent of:

a) 50kg is 65

b) ₹9 is ₹4.

8) If 50/3% of a number is 25, find the number.

9) If 13.25% of a number is 159, find the number.

10)a) Increase the number 60 by 30%.

b) What number when increased by 15% becomes 299?

c) Price of an article increases from ₹960 to ₹1080. Find the percentage increase in the price.

d) On increasing the price of an article by 16%, it becomes ₹1479. What was its original price?

11)a) Decrease the number 750 by 10%.

b) On decreasing a number by 18%, it becomes 697. Find the number.

c) Prem reduced his weight by 15%. If now his weights 59.5kg, what was his earlier weight?

d) In a sale, a shop reduces all its prices by 15%. Find

 i) the cost of an article which was orginally priced at ₹40.

b) The original price of an article which was sold for ₹20.40.

12) Mr. Raj spent 83% of his salary and saved ₹1870. Calculate his

a) monthly salary

b) Yearly expenditure 

13) In a school, 38% of the students are girls. If the number of boys is 1023, find

a)  the total strength of the school.

b) number of girls

14) In a straight contest, the loser polled 42% votes and lost by 14400 votes. Find the total number of votes polled. If the total number of eligible voters was 1 lakh, 

Find what percent of voters did not vote.

15) Out of 8000 candidates, 60% were boys. If 80% of the boys and 90% of the girls passed the exam, find the number of candidates who failed.

16) Increase a price of ₹200 by 10% and then decrease the new price by 10%. Is the final price same as the original one?

17) Gyan purchased some parrots. 20% flew away and 5% died. Of the remaining, 45% were sold. Now 33 parrots remain. How many parrots had Gyan purchased?

18) A candidate who gets 36% marks in an examination fails by 24 marks but another candidate, who gets 43% marks, gets 18 more marks than the minimum pass marks. Find the maximum marks and the percentage of pass marks 

19) The sides of a rectangle are 20cm and 15 cm. If each side is increased by 20%, find the percentage increase in the area.

20) Imran gives 1% his monthly income to his two sons as pocket money. The elder son gets 80% of the total amount given and he spends 80% of his share. If he saves ₹60 per month, determine Imran's monthly income.

EXERCISE -B

1) A man is 18% of his monthly income. if he saves ₹ 3780 per month. what is his monthly income ?

a) ₹12000 b) ₹21000 c) ₹20000 d) ₹2100

2) A football team will 7 games, which is 35% the total games played. How many games were played in all ?

a) 10 b) 15 c) 30 d) 25

3) Amit was given on increment of 20% on his salary. If his new salary is ₹30600, what is his salary before the increment.

a) 20500 b) 25000 c) 25500 d) 22500

4) Sonal attended her school on 204 days in a full year. If her attendance is 85%, Find the number of days on which the school was opened ?

a) 200 b) 210 c) 220 d) 230 e) 240

5) A's income is 20% less than that of B. By what percent is B's income more than A's

a) 20% b) 25% c) 30% d) 32% e) 27%

6) The price of petrol goes up by 10%. By how much percent must a motorists reduce the consumption of petrol so that the expenditure on its remains unchanged ?

a) 100/11% b) 109/9% c) 100/7% d) 200/21% e) none 

7) The population of a town increases by 8% annually. If the present population is 54000, what was it a year ago?

a) 500 b) 5000 c) 50000 d) 500000 e) n

8) The value of a machine depreciates every year by 20%. If the present value of the machine be ₹160000,  What was its last year ?

a) 2000 b) 20000 c) 200000 d) 2000000

9) An alloy contains 40% copper, 32% Nickel and rest zinc. Find the mass of zinc in 1 kg of the alloy ?

a) 280g b) 270g c) 260g d) 250g e) n

10) Balanced diet should contain 12% the proteins , 25% the fats and 63% of carbohydrates. If a child needs 2600 calories in his food daily, find in calories the amount of proteins of these in his daily intake.

a) 121 b) 112 c) 212 d) 312 e) 412

11) Gunpowder contains 75% nitre and 10% sulphur. Find the amount of gunpowder which carries 9kg nitre. What amount of gunpowder would contain 2.5 kg sulphur ?

a) 12, 25 b) 14,30 c) 21,25 d) none

12) Divided ₹7000 among A,B and C such that A gets 50% of what B gets and B gets 50% if what C gets.

a) 1000,2000,4000 

b) 2000,1000,4000

c) 2000,4000,1000

d) 2000,1000,4000

13) Find the percentage of pure gold in 22 carat is 25 carat gold. If 24-carat gold is 100% pure.

a) 295/3%  b) 298/5% c) 297/3% d) n

14) The salary of an officer is increased by 25% . By what percent should the new salary be decreased to restore the original salary ?

a) 12% b) 15% c) 20% d) 22% e) 25%

15) 3/5=?

a) 30% b) 40% c) 50% d) 60%

16) 0.8% when expressed as a decimal, is

a) 0.08 b) 0.008 c) 8 d) 0.8

17) In an examination. Nitin gets 98 msrks. This amounts to 56% of the maximum marks. What are the maximum marks

a) 75 b) 150 c) 175 d) 225

18) Saru spends 65% of her salary and the rest she saves. If she saves ₹2940 per month. What is her monthly salary?

a) 4000 b) 4800 c) 8400 d) 4200

19) In a class, the girls are 60% of the total number of students and the boys are 18 in number. How many students are there in the class?

a) 40 b) 45 c) 54 d) 64 e) 56

20) A number is increased by 20% and then decrease by 20% m. Find the net increase or decrease percent?

a) 2 b) 3 c) 4 d) 5 

21) In an examination, one required 40% marks to pass. Rahul gets 185 marks and fails by 15 marks. What are the maximum marks ?

a) 50 b) 500 c) 5000 d) 50000

22) Out of her total monthly salary. Tony spends 30% on house rent and 60% of the rest on household expenditure . If she saves ₹10500, what is her total monthly salary ?

a) 30000 b) 37000 c) 37500 d) 40000

23) The price of sugar goes up by 20%. By how much percent must a housewife reduce the consumption of sugar so that the expenditure on sugar remains the same ?

a) 50/3% b) 50% c) 100/7% d) none 

24) The population of a town increased by 6% annually. If the present population is 17490, what was it a year ago?

a) 16000 b) 1600 c) 16500 d) 1650

25) The value of a machine depreciates every year by 10%. If the present value of the machine be 99000. What was its value last year ?

a) 1100 b) 11000 c) 110000 d) none 

26) A is income is 60% more than that B. By what percent is B's income less than A's ?

a) 30% b) 35% c) 37% d) 37.5% e) n

27)  One-third of 1206 is what % of 134 ?

a) 3% b) 30% c) 20% d) 300%

28) x% of y is y% of

a) x b)209x c) x/100 d) y/100

29) What percent of 2/7 is 1/35?

a) 2.5% b) 10% c) 20% d) 25%

30) An alloy contains 30% copper, 42% nickel and rest zinc. Find the mass of zinc in 1 kg of alloy

a) 250g b) 250g c) 280g d( 300g

31) Which is largest in 24/3%, 4/25, 0.15?

32) A number exceeds 25% of itself by 60. The number is

a) 75 b) 45 c) 80 d) 65 e) none 

33) 10% of 20% of 30% of 20000

Compact Questions - Mg.A.R-1

1) If 50/3% of a number is added with itself than result becomes 4956. Find the original number .                         4248

2) If 20/3% of a number is substracted from itself then result becomes 5670. Find the original number.                 6075

3) If 100/9% of a number is added with itself then result becomes 900. Find the original number.                               810

4) If 64 is added in a number then number becomes 1100/7% of itself. Find the number .                                         112

5) If 930 is added to a number then number becomes 4000/9% of itself. Find the original number.                     270

6) If 16 is added in a number then result becomes 350/3% of itself. Find the number.                                         96

7) If 21 is added in a number then result becomes 275/2% of a number. Find the number.                                       56

8) If the length of a rectangle is increased by 75/2% and its breath is decreased by 20%. Find the % change in its area.

9) The price of sugar is increased by 50/3% and the consumption of a family is decreased by 20%. Find the percentage change in his expenditure.     20/3%

10) The sale of a cinema ticket is increased by 400/7% and price of tickets increased 50/3%. Find the percentage change in his revenue .            250/3%

11) If the sides of a square is increased by 40%. Find the % change in its area.      96%

12) The population of town is 189000. 4/9 of them are males and rest females . 50% males are married.

a) Find the percentage of married population.                           400/9%

b) find the % of married female.        40%

13) A man multiply a number by 7/4 instead of 3/5. Find the % change in his revenue.                             375/3%

14) A student multiplied a number by 3/5 Instead of 5/3. Find the % error in the calculation.                        64%

15) A student multiply a number by 3/4 instead of 4/3. Find the percentage error in calculation.                      175/4%

16) If x earns 25% more than y. What % less does y earns then x. 20%

17) If Kishan's salary is 50/3% less than Radha. By how much % does Radha's salary is more than Krishan.           20%

18) P is 6 times as large as Q. By how much percentage is A less than P .    250/3%

19) If 60% of the students in a school are boys and the number of girls is 972. How many boys are there in the school. 1458

20) If a student 25% marks then he is failed by 210 marks. But if he scores 55% marks then he is passed with 240 marks. Find the passing %.            39%

21) If a student score 36% marks then he is failed by 32 marks. But when hr score 48% marks he is passed by 64 marks. Find the passing %.                         49%

22) When a student score 40% marks then he is failed by 50 marks. but when he score 50% marks then he is passed with 40 marks. Find passing%. 410/9%

23) When a student score 30% marks then he fails by 5 marks. but when he scores 40% marks, he got 10 more marks than passing %. Find

a) maximum marks.                     150

b) passing percent.                       100/3%

24) The population of a town is 6000. If males are increased by 5% and female are increased by 9% then population will become 6500 after 1 year. Find the numbers males and females .      1000, 5000

25) The population of a village was 9600. If the males and females are increased by 8 and 5%, the population will becomes 1072 after 1 year. Find the number of males at present.        6400

26)In an election two candidates participated. 20% votes declare invalid and the winners gets 70% of the valid votes, and won by 9600 vote. Find the number of

a) voting list .                 30000

b) valid votes.           24000

27) In an election two candidates participated. 20% votes did not vote . 25/2% votes declared invalid and winner get 60% of the valid vote and won by 5600 votes. Find the number of voter list.                       40000

28) In an election 10% voters did not vote and 100/9% votes declared invalid and the winners get 75% of the valid votes . If he win by 2000 votes then Find the voting list. 5000

29) In an election two candidates participated, 10% voters did not vote , out of which 10% votes declare invalid and the winner got 70% of the valid votes and win by 7290 votes. Then find the voting list.                           22500

30) In an election two candidates participated. 20% voters did not cast their votes, out of which 600 votes declared invalid and the winners get 75% of valid votes and wins by 1500 votes. Find the number of voting list.       4500

31) In an election two Candidates participated, 10% voters did not vote, 2500 votes declared invalid and the winner get 55% the valid votesand wins by 2000 votes. Find the number of voters in the voting list. 25000

32) If the price of sugar is increased by 30%. Then by how much percent the consumption increase so that the expenditure remains same .       300/7%.

33) The price of sugar increased by 20% by how much percent the consumption is decrease so that expenditure will not increase.                 50/3%

34) The price of sugar is decreased by 30%. By how much % the consumption is increased so as the expenditure will decreased by 10% only.              200/7%

35) The price of sugar is increased by 20%. By how much kg of consumption is decreased so as the expenditure will increased by 5% only when the originally consume 280 kg per sugar.          36kg

36) The price of sugar is increased by 30% due to this a family use 40 kg less sugar so as expenditure will increased by 10% only. Find original consumption.     260kg

37) The price of sugar decreased by 30% due to this a family purchase 32 kg more sugar so as the expenditure will decreased by 16% only. Find the current consumption.                192

38) The price of sugar is increased by 20% due to this family purchased 12 kg less sugar for ₹300. Find 

a) the original consumption .       72kg

b) current consumption.                 60kg

c) original price.                           25/6 @

d) current price.                             ₹5@

39) In an election two candidates participated. 10% did not vote. 300 votes declared invalid and the winners get 60% of the voting list and win by 900 votes . Find the number of valid votes.     1500

40) In an election two Candidates participated. 20% voters did not vote and 120 votes declared invalid. winner wins by 200 votes and winners gets 41% of the voter list . Find the voting list.    3200

41) A man spends 5% of his total income in travelling and 20% of the rest spend in food and then it donate ₹120 and he stills left with ₹1400. Find his income.       2000

42) In a library 20% of the books are in Hindi, 50% of the remaining in English and 30% the remaining are in French and rest 6300 books are in rigional language. then find the number of books in library. 22500

43) A spends 50% of his income on household items and of the remaining 50% on transport, 25% on entertainment, 10% on sports and remaining amount of ₹900 is saved. What A's monthly income. 12000 

44) Mr x spent 20% of income on food and 15% on childrens education, 40% of the remaining is spent on entertainment and transport together and 30% on medical. He is left with an amount of ₹8775 after all these expenditures. What is Mr X monthly income. 45000

45) In a class of 60 children, 30% children can speaks only English, 20% only Hindi and rest of the children can speak both the languages . How many childrens can speak Hindi ? 42

46) In an exam 65% student passed in maths, 48% passed in physics and 30% passed in both. How much percent of students failed in both the subjects . 17%

47) in an examination 35% the student failed in maths and 25% in English. If 10% failed in both math and English, then how much percent passed in both the subjects . B 50%

48) if the numerator of a fraction is increased by 200% and the denominator is increased by 350%, the resultant fraction is 5/12. What was the original fraction ?           5/8