DAILY REVISION (MATHS) (X)

3/7/22

1) Mr. Jacob has a 5 years recurring deposit account in Bank of Baroda and deposits ₹2400 a month. If he receives ₹186090 at the time of maturity, find the rate of simple interest.

2) solve and represent in nume line

A) : (4x -10)/3 ≤ (5x -7)/2

B) x - 3 < 2x -1 given {1,2,3....9}

3) Solve: 

A) 4x² - 4ax + (a² - b²)= 0

B) 3x² + 10x + 3= 0

C) A rectangle of Area 105 cm² has its length equal to x cm, write it's breadth in terms of x. Given that the perimeter is 44cm, write down an equation is x and solve it to x and solve it to determines of the rectangle.

D) Two years ago, a man's age was three times the square of his daughters age. Three years hence, his age will be four times his daughters age. Find their present ages.

4) Using remainder theorem, find the value of a if the division of x² + 5x - ax = 6 by x - 1 leaves the remainder 2a.

5) If x -2 is a factor of 2x³ - x² + px - 2, find the value of p. With this value of p, factories completely.

6) If x+ y = 7     0 & x - y = 3        0

                   2     5                0         3 find the metrices x and y.

7) If A= 3    4 B= 1     y & C= z      0

              5    x       0     1         10    5 find the value of x, y, z if the relation 2A + B= C

8) If A= 1      -2 & B= 3       2

              2      -1         -2       1 find 2B - A².

9) If A= 1      1

              x      x find the value of x so that A² = 0

10) Height (in cm) No. Of boys

150-155                              8

155-158                             20

158-160                             25

160-165                             4

165-172.                            3

Find the probability that the height of a student lies in the interval

A) 155-158

B) 158-160

C) 158-172

11) A letter is chosen from the word TRIANGLE, what is the probability that it is vowel.

12) A box contains 25 cards, numbered from 1 to 25. A card is drawn from the box at random. Find the probability that the number on the card is:

A) even

B) prime

C) multiple of 6.

  

5/6/22

1) Without performing the actual division process, find the remainder when 3x³ + 5x² - 11x - 4 is divided by 3x+1.                         1/9

2) Find the values of the constants a and b, if (x-2) and (x+3) are both factors of the expression x³ + ax² + bx - 12.                3, -4

3) use the factor theorem to factorise completely x³ + x² - 4x - 4.                 (x+1)(x+2)(x-2)

4) using the remainder theorem find the remainder when 7x² - 3x +8 is divided by x-4.            108

5) find the value of a, if (x-a) is a factor of x³ - ax² + x+2.         -2

6) Show that (x-3) is a factor of x³ -7x²+15x-9= 0. Hence, Factorize x³ -7x² + 15x -9.            (x-3)²(x-1)

7) Find the remainder when 2x³ - 3x² +7x - 8 is divided by x-1.        -2

8) Find the remainder (without division) on dividing f(x) by (x-2) where

A) f(x)= 5x² - 7x +4.                   10

B) f(x)= 2x³ - 7x²+3.                -9

9) find the remainder (without devision) on dividing 3x² +5x - 9 by (3x+2).                     -11

10) without actual division, find the remainder when p(x)= 3x² - 5x +7 is divided by (x-2).                      9

11) find out without actual division, the remainder when

A) 4x³ - 6x² + 7x - 2 is divided by x - 1/2.                         1/2

B) 3x⁴ + 2x³ - x²/3 + 2x/9+ 1 is divided by x + 2/3.            19/27

C) 5x³ - 3x² + x/5 - 3/25 is divided by 5x -3.                    0

D) x³ - 3ax² + a²x + 3a³ is divided by x - a. 2a³

12) When kx³ + 9x² + 4x - 10 is divided by (x+1), the remainder is 2. Find the value of k .            -7

13) using remainder theorem, find the value of a if the division of x³+ 5x² - ax + 6 by (x-1) leaves the reminder 2a.                      4

14) if (2x+1) is a factor of 6x³+5x² + ax -2, find the value of a.       -3

15) if (3x-2) is a factor of 3x³-kx² + 21x - 10, find the value of k.      11

16) if both ax³ + 2x² - 3 and x² - ax +4 leave the same remainder when divided by (x-2), find a.            3/10

17) if (x-1) and (x+3) are factors of x³ - ax² - 13x + b, find a,b.          3,15

18) If x² +x -2 divides 2x³ + px² + qx - 14 Completely, find p,q.         9,3

19) If (x+2) and (x-3) are factors of x³ + ax +b, find the values of a and b. with this Value of a and b, factorise the given expression.     -7, -6, (x+2)(x-3)(x+1)

20) If ax³ + 3x² + bx -3 has a factor (2x+3) and leaves remainder -3 when divided by (x+2), find the values of a and b. with these values of a and b, factorise the given expression.     2, -2, (2x+3)(x-1)(x+1) 

21) prove that x-5 is a factor of 2x² - x - 45. Hence factorise 2x² - x - 45 completely.                  (x-5)(2x+9)

22) prove that 2x-5 is a factor of 4x² - 4x - 15. Hence, factorise 4x²- 4x - 15 completely.       (2x-5)(2x+3)

23) Obtain a factor of x³ - 3x² - 4x +12 by factor theorem. Hence, factorise it completely.     (x-2), (x-2)(x+2)(x-3)

24) obtain the factor of y³ -13y -12 by factor theorem. Hence factorise completely.        y+1, (y+1)(y-4)(y+3)

25) if f(x)= 24x³ + px² - 5x +q has two factors 2x+1 and 3x-1, then find p and q. Also factorise f(x) completely.       -2,1, (2x+1)(3x-1)(4x-1)

15/5/22

1) On recurring deposit Shyam received ₹36 as interst for 8 months on ₹150 (deposited monthly). Find the rate.

2) Miss x deposited ₹350 per month for 20 months under Recurring deposit scheme. Find the total amount payable by the bank on maturity of the account if the rate of interest is 11% p.a.      

3) Ram has a cumulative time deposit account of ₹140 per month at 6% p.a. If she gets ₹7157 at the time of maturity, find the total time for which the account was held. 20 months.

4) seema opened a Recurring deposit Account with a bank and deposited ₹75 per month for 4 years. If the interest reckoned at the rate of 10% p.a., find the amount she got at the end of the maturity period.

5) Arsh has a cumulative time deposit account in a bank. She deposits ₹800 per month and gets ₹15198 as maturity value. If the rate of interest be 7% p a., Find the total time for which the account was held.                                         

6) Recurring deposit calculation are based on

A) simple interest only

B) compound interest only

C) both A and B

D) neither A nor B

7)If Rupam deposits ₹225 per month for 5 years at 9% p a. Then maturity value is..

A) 17660 B)17000 C) 17066.88 D) n

8) Amit deposited ₹150 per month in a bank for 8 months under the Recurring Deposit Scheme. what will be the maturity value of his deposits, if the rate of interest is 8% per annum and interest is calculated at the end of the year.  

12/5/22

PROBABILITY

1) In a cricket match, a boatman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary.

2) 1500 families with two children are selected randomly, and the following data were recorded:

 No of girls in a family No of fam.

          2                                 475

          1                                 814

          0                                 211

 Compute the probability of a family, chosen at random, having

A) two girls 

B) one girl 

C) no girl 

3) Three coins are tossed simultaneously 200 times with the following frequency of different outcomes:

Outcomes Frequency

3 heads 23

2 heads 72

1 head 77

No head 28

If the three are simultaneously tossed again, compute the probability of two heads coming up.

4) An organization selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed in the table below:

Monthly inc. vehicles for fam.

                            0     1      2    above2

Less than 7000 10 160   25        0

 7000-10000       0   305  27        2

 10000-13000     1   535  29        1

 13000-16000     2   469  59       25

 16000 or more   1   579 82       88

 Suppose a family is chosen. Find the probability that the family chosen is

A) earning ₹10000-13000 per month and owning exactly 2 vehicles.

B) earnings ₹16000 or more per month and owning exactly 1 vehicle.

C) earning less than ₹7000 per month and does not own any vehicle.

D) earning ₹13000-16000 per month owning more than 2 vehicles.

E) owning not more than 1 vehicle.

5) Marks No. Of students

 00-20                   7

 20-30                  10

 30-40                  10

 40-50                  20

 50-60                  20

 60-70                  15

 70 and above     8

   Total                 90

A) Find the probability that a student obtained less than 20% in Mathematics test.

B) Find the probability that a student obtained 60 marks or above.

6) To know the opinion of the students about subject statistics, a survey of 200 students was conducted. The data is recorded in the following table.

 Opinion       Number of students

      like                   135 

    Dislike                 65 

Find the probability that a student chosen at random

A) likes statistics

B) does not like it.

7) 11 bags of wheat flour, each marked 5 kg, actually contain the following weights of flour(in kg):

   4.97, 5.05, 5.08, 5.03, 5.06, 5.08, 4.98, 5.04, 5.07, 5.00

 find the probability that any of these bags chosen at random contains more than 5 kg of flour.

8) A dice is thrown 1000 times with the frequencies for the outcomes 1, 2, 3, 4, 5 and 6 as given in the following table:

Outcome            Frequency

    1                           179 

    2                           150

    3                           157

    4                           149

    5                           175

    6                           190 

9) On one page of a telephone directory three were 200 telephone numbers. The frequency distribution of their unit place digit (for example, in the number 25828573, the unit place digit is 3) is given the following table:

Digit                  frequency 

  0                          22 

  1                          26 

  2                          22 

  3                          22 

  4                          20 

  5                          10 

  6                          14 

  7                          28 

  8                          16 

  9                          20

Without looking at the page, the pencil is placed on one of these numbers, i.e., the number is chosen at random. What is the probability that the digit in its unit place is 6 ?

10) The record of a weather station shows that out of the past 250 consecutive days, its weather forecast were correct 175 times.

A) what is the probability that on a given day it was correct?

B) what is the probability that it was not correct on a given day?

11) A tyre manufacturing company kept a record of the distance covered before a tyre needed to be replaced. The table shows the results of 1000 cases.

Distances(in km).    Frequency

Less than 4000             20

4000 to 9000                210

9001 to 14000              325

More than 14000         445

 if you buy a tyre of this company, what is the probability that:

A) it will need to be replaced before it has covered 4000 km?

B) it will last more than 9000 km.

C) it will need to be replaced after it has covered some where between 4000 km and 14000 km ?

12) The percentage of marks obtained in the monthly unit tests are given by the following table:

Unit tests       % of marks obtained

  I                          69

 II                          71 

 III                         73

 IV                         68

  V                          74

Based on this data, find the probability that the student gets more than 70% marks in a unit test.

13) An Insurance Company selected 2000 drivers at random (i.e. without any preference of one driver over another) in a particular city to find a relationship between age and accidents. The data obtained are given in the following table:

Age of drivers accident in 1 yr

                         0     1    2      3  over 3

   18-28         440 160 110  61  35

   30-50         505 125  60   22   18

Above 50     360   45   35   15    9

Find the probability of the following events for a driver chosen at random from the city:

A) 18-29 years of age having exactly 3 accidents in 1 year.

B) being 30-35 years of age having one or more accidents in a year.

C) having no accidents in one 1 year.

14) 50 seeds were selected at random from each of 5 bags of seeds, and were kept under standardized conditions favorable to germinate. After 20 days, the number of seed which had germinated in each collection were counted and recorded as follows:

Bag.            No of seeds germinated

  1                     40 

  2                     48 

  3                     42 

  4                     39 

  5                     41 

what is the probability of germination of:

A) more than 40 seeds in a bag?

B) 49 seeds in a bag more than 35 seats in the bag?

C) more than 35 seeds in a bag?

15) A die is thrown once. Find the probability of getting a prime number.

16) A coin is tossed once. Find the probability of getting a head.

17) From a group of two boys and 3 girls, we select a child. Find the probability of this child being a girl.

18) if we throw a die, then the upper face shows 1 or two; or three or four; or five or six. Suppose we throw a die 150 times and get 2 for 75 times. What is the probability of getting a '2'?

19) A coin is toss 200 times and is found that a tail comes up for 120 times. Find the probability of getting a tail.

20) if a coin is tossed for a certain number of times. How many times the coin was tossed, if the probability of getting a head is 0.4 and it appeared up for 24 times ?

21) In a cricket match, if the probability (P(E)) of hitting the boundary is 0.3, then find the probability of not-hitting the boundary.

22) In a G K test a student was given 50 questions one by one. He gave the correct answer for 30 questions. Find the probability of giving correct answers.

23) A coin is tossed 150 times and it is found that heads comes 115 times and tell 35 times. If a coin tossed at random, what is the probability of a getting

A) a head 

B) a tail

24) A dice thrown 270 times and the outcomes are recorded as in the following table:

 Outcome             frequency

      1                          36

      2                          45

      3                          33

      4                          18

      5                          75

      6                          63

if a dice is thrown at random, find the probability of getting:

A) 1

B) 2

C) 3

D) 4

E) 5

F) 6

25) In a sample study of 640 people, it was found that 512 people have a high school certificate. If a person selected at random, the probability that he has a high school certificate is:

A) 0.50 B) 0.65 C) 0.80

26) In a survey of 360 children, it was found that 90 liked to eat potato chips. If a child is selected at random, the probability that he/she does not like to eat potato chips is:

A) 0.25 B) 0.50 C) 0.75

27) The probability of a sure event is:

A) 0 B) 1 c) 100

28) The probability of an event cannot be less than

A) 0 B) 1 C) -1

29) The probability of an event can not be more than:

A) 0 B) 1 C) -1

30) A common dice has

A) one face B) four faces C) six faces

31) When a die is thrown once, the least possible score must be

A) 0 B) 1 C) 6

32) when a die is thrown once, the greatest possible score must be

A) 0 B) 1 C) 6

33) the sum of the probabilities of all possible outcomes is always equal to

A) 0 B) 1 C) 100

34) if two dice are thrown together, then the least possible total score must be

A) 0 B) 1 C) 2

35) If two dies are thrown together, then the greatest possible score must be

A) 1 B) 6 C) 12

36) the probability of the occurrence of an event is 1/4. what is the probability of the non occurrence of that event?

A) 0 B) 3/4 C) 1/4

37) a coin is tossed 100 times and a head is got 63 times. The probability of getting a head is:

A) 6.3 B) 63.0 C) 0.63

38) In a medical examination of students of a class, the following blood groups are recorded:

Blood group.     No of students

      A                           15

      B                           20

      AB                        23 

      O                          12

A student is selected at random from the class. The probability that he/she has blood group B, is

A) 1/20 B) 3/4 C) 2/7

39) 80 bulbs are selected at random from a lot and their life time ( in hours is recorded as given below:

 lifetime(hrs).           No. of bulbs

        400                         10

        500                          3

        600                         12

        700                         20

        800                         14

        900                         11

one bulb is selected at random from the lot. The probability that its life is less than 800 hours is:

A) 1/80 B) 1/4 C) 11/16

40) A dice is thrown once. The probability of getting a number greater than 6 is:

A) 0. B) 1 C) 1/6

41) In a class of 10 students, 4 are or girls. The probability choosing a boy is :

A) 2/5 B) 3/5 C) 1/10

42) Cards are marked 1 to 20. The probability of drawing a card marked with a multiple of 3 is:

A) 3/10 B) 3/20 C) 1/20

43) The probability of getting a number '0' is one throw of a die is

A) 0 B) 1/6 C) 1

44) The probability of getting a number 6 is one throw of a die is

A) 0 B) 1/6 C) 1

11/5/22

31) 2(3x² - 1)  = x

32) 4x² - 2= x +1.

33) 4x² - 2x +1/4 = 0. 

34) x² + 2 √2 x - 6 = 0.

35) √3 x² + 10x + 7 √3 = 0.

36) 2x² + √7 x - 7= 0. 

37) √3 x² +10x - 8 √3 = 0. 

38) 1/x - 1/(x +2) = 1/24

39) 3/x + 5/(x +2) = 4/(x -1)

40) (x +1)/(x -1) = (3x +1)/(7x +5).

9/5/22

21) 3(x² - 6)  = x(x+7) -3.

22) x² - 4x -12 = 0. x belongs to N

23) 2x² - 8x -24 = 0. x belongs to I

24) 5x² - 8x -4 = 0. x belongs to Q

25) 2x² - 9x + 10 = 0. When 

i) x belongs to N 

ii) x belongs to Q

26) a²x² + 2ax + 1 = 0, a≠ 0.

27) 5x² + 4x - 21 = 0. 

28) 3x² - 2x - 1 = 0. 

29) x² - 4x = 32

30) y² = 10 - y

8/5/22

1) x² + 6x + 5= 0.

2) 8x² - 22x -21 = 0.

3) 8x² +15 = 26x.

4) x(2x +5)= 25.

5) (x -3)(2x+5)= 0

6) x² - 7x +10.

7) 9x² - 3x - 2 = 0.

8) x² - 8x + 16 = 0.

9) (x² - 5x)/2 = 0.

10) 2x²= 3x + 35 = 0

11) 6x² +x - 35 = 0.

12) 4= 9x² + 9x.

13) 9x = 10 - 7x²

14) 15x² = 2(x + 4).

15) 3x² = x + 4

16) 16x² = 25

17) 3x² +8 = 10x.

18) x(6x -11)= 35.

19) 6x(3x -7) = 7(7- 3x).

20) 1/7 (3x -5)²= 28.

22/4/22

1) Using the remainder Theorem, find the remainder when 7x² -3x +8 is divided by x -4.                        108

2) Find the values of p and q if g(x)= x+2 is a factor of f(x)= x³ - px + x + q and f(2)= 4.                          9/2, 2

3) (x+3)/(2x+3)= (x- 1)/(3x+ 2).    -3 ± √6

4) find the value of m if x - m is a factor of 3x³ + 2x² - 19x + 3m.    

5) Find the remainder when 5x² - 4x -1 is divided by 2x -1.   

6) Show that x - 5 is a factor of x³ - x² - 17x - 15.

20/4/22

1) SOLVE: 4x² - 4ax +(a² - b²)= 0.   3,-4

2)Find the value of m if (x-m) is a factor of 3x³ + 2x²- 19x+3m.         2

3) 5+11x-5x²=0.          (11±√221)/10

4) 3x² - 2x -1= 0.

5) 8x² +15= 26x.                    5/2,3/4

6) x(2x+5)=25.                          -5,5/2

7) (x-3)/(x+3) +(x+3)/(x+3)= 5/2, x≠ -3, 3.                                           -9, 9

8) 2x -3= √(2x²-2x+21).                  6

9) 1/7 (3x-5)²= 28.                 -3, 19/3

10) 3(y²-6)= y(y+7)-3.               5, -3/2

11) x² - 4x -12, x belongs to N.       6

12) 2x² - 8x -24= 0, x bel. to I.      6,-2

13) a²x² + 2ax +1= 0, a≠0.    -1/a,1/a

14) 1/x - 1/(x+2)= 1/24.               -8,6

19/4/22

1) x² + 6x + 5= 0.

2) 8x² - 22x -21 = 0.

3) 9x² - 3x - 2 = 0.

4) x² + 2 √2 x - 6 = 0.

5) √3 x² + 10x + 7 √3 = 0.

6) x² - 8x + 16 = 0.

7) x(x +1) + (x +1)/x = 34/15, x≠ 0, x ≠ -1.

8) (x +3)/(x -2) - (1- x)/x = 17/4.

9) 1)(x -2) + 2/(x -1) = 6/x.

10) 4/x  - 3 = 5/(2x +3)

11) 2x/(x -3)  + 1/(2x +3)+ (3x+ 9)/{(x -3)(2x +3)}= 0.

DAILY REVISION(Maths) (XII)

4/9/22

1) ∫(√x + ³√x²)²/x dx

2) (√x - 1/√x)³ dx

3) ∫dx/(3- 8x)⁵

4) ∫dx/{√(ax + b) - √(ax + c)}.

5) ∫x √(x + a) dx.

6) ∫x/√(x + a) dx

7) ∫(4x+ 3)²⁰²² dx

8) ∫(1+x+ x²)/{x²(1+ x)} dx

9) ∫(2x⁴+ 7x³+ 6x²)/(x²+ 2x) dx

10) ∫(x+ 2)/(x +1)² dx.

11) ∫(8x+ 13)/√(4x +7) dx

12) ∫ x³/(x+ 2) dx

13) ∫(x⁴+ 3)/(x² +1) dx.

14) ∫(2x+ 3)/(x -1)² dx.

15) ∫ 2x/(2x+ 1)²⁰ dx.

25/7/22

Inverse Trigo (R) 

Prove

1) 2tan⁻¹1/5 + tan⁻¹1/8=tan⁻¹4/7

2) 1/2 tan⁻¹x = cos⁻¹ √[{1√(1+x²)}/2√(1+x²)]

3) sin⁻¹4/5 +sin⁻¹16/65= π/2

4) sin (π/3 - sin⁻¹(-1/2)) = 1

5) cot (π/4 - 2 cot⁻¹3) =7

6) If cos⁻¹x+ cos⁻¹y+ cos⁻¹z=π then prove x²+y²+z²+2xyz=1

7) If tan⁻¹x+tan⁻¹y+tan⁻¹z=π/2 then prove xy +yz+zx =1

8) If tan⁻¹x+ tan⁻¹y+ tan⁻¹z = 0 then prove x+y+z=xyz

9) prove tan⁻¹(1/2 tan  2A)  + tan⁻¹(cot A)+ tan⁻¹(cot³A) =0

10) solve for x: tan⁻¹3x+ tan⁻¹2x =π/4

11) Solve:

tan⁻¹(cos x) =tan⁻¹(2 cosec x)

12) tan⁻¹(x+1) +tan⁻¹(x-1) =tan⁻¹8/31

13) sin⁻¹x/√(1+x²) - sin⁻¹1/√(1+x²) = sin⁻¹{(1+x)/(1+x²)}

14) Sin⁻¹(x/2)+cos⁻¹(x+√3/2) =π/6

15) If Sin⁻¹x +Sin⁻¹y+Sin⁻¹z=3π/2, then prove x²+y²+z²+2xyz=1

16) Simplify:

Cos⁻¹(x+ 1/2)+Cos⁻¹x+Cos⁻¹(x -1/2) = 3π/2.

17) Tan⁻¹{(2sin2θ)/1+2cos2θ)} -1/2  sin⁻¹{(3sin2θ)/(5+4cos2θ)}

18) tan⁻¹1 + tan⁻¹1/2 + tan⁻¹1/3

19) sin(sin⁻¹1/3 + sec⁻¹3) + cos(tan⁻¹1/2 + tan⁻¹2)

Simplify

20) sin{sin⁻¹√5/4 + tan⁻¹√(5/11)}

21) tan⁻¹sin cos⁻¹√(2/3)

22) cos⁻¹3/5+ cos⁻¹12/13 + cos⁻¹63/65 = π/2

23) 1/2 tan⁻¹x = cos⁻¹√{(1+√(1+x²))}/2√(1+x²)

24) prove tan⁻¹x + cot⁻¹(x+1) = tan⁻¹(x²+x+1)

25) tan(2 tan⁻¹m) = 2 tan(tan⁻¹m+ tan⁻¹m³)

26) tan⁻¹(1/2  tan 2A) + tan⁻¹(cotA)

+ tan⁻¹(cot³A) = 0

Solve

27) 3(Cos⁻¹x + 2sin⁻¹x) = 7π

28) tan⁻¹(cot2x) + tan⁻¹(-cot 3x) = x

29) sin⁻¹{2a/(1+a²)} - cos⁻¹{(1-b²)/(1+b²) = 2 tan⁻¹x

30) tan⁻¹x tan⁻¹2x tan⁻¹3x = π

31) 3 tan⁻¹{1/(2+√3)} - tan⁻¹( = 1/x)

   = tan⁻¹1/3

32) sin⁻¹x + sin⁻¹(1-x) = cos⁻¹x

33) tan⁻¹(x+1)+tan⁻¹(x-1)=tan⁻¹8/31

34) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)} = π/4

35) tan⁻¹{(1/(2x+1)}+ tan⁻¹{1/(4x+1)= tan⁻¹(2/x²)

36)  if α+β= tan⁻¹m, α-β=tan⁻¹n Express tan 2α and tan2β in terms of m,n.

37) If sin⁻¹x = tan⁻¹y then find the value of 1/x² - 1/y²

38) If r²= x²+y²+z², prove that tan⁻¹yz/xr + tan⁻¹zx/yr + tan⁻¹xy/zr = π/2

39) Solve:

 cos(2tan⁻¹1/7) =  sin(4tan⁻¹x)

40) cos(tan⁻¹x) = sin(cot⁻¹3/4).

prove :

41) cos⁻¹(63/65)+2tan⁻¹(1/5) =   sin⁻¹(3/5).

  

42) 4 tan⁻¹1/5 - tan⁻¹1/70 + tan⁻¹1/99 =π/4

43) sin⁻¹{x/√(1+x²)} + cos⁻¹{(x+1)/√(x²+2x+2)} = tan⁻¹(x²+x+1)

44) cot(π/4 - 2 cot⁻¹3) = 7

45) sin cot⁻¹ cos(tan⁻¹x) = √{(1+x²)/(2+x²)

46) sin⁻¹√3/2 + 2 tan⁻¹1/√3 = 2π/3

47) tan⁻¹1/3 +tan⁻¹1/5 + tan⁻¹1/7

    +tan⁻¹1/8 = π/4

48) tan⁻¹(1/2 tan 2A) +tan⁻¹(cot A)

   + tan⁻¹(cot³A) = 0

Solve:

49) tan⁻¹(2+x)+ tan⁻¹(2-x) =tan⁻¹2/3

50) sin⁻¹5/x + sin⁻¹12/x = π/2

51) sin⁻¹6x + sin⁻¹(6√3 x)= - π/2

52) sin⁻¹{2a/(1+a²)} + sin⁻¹{2b/(1+b²)} = 2 tan⁻¹x

53) sin{2 cos⁻¹ cot(2 tan⁻¹x)}=0

Find

54) If tan⁻¹a + tan⁻¹b + tan⁻¹c = π then Prove that a+ b + c = abc

55) If tan⁻¹a + tan⁻¹b + tan⁻¹c = π/2 prove ab + bc + ca = 2

56) If cos⁻¹a/2 + cos⁻¹b/3 = K prove 9x² - 12 xy cos K + 4y² = 36 sin²K

prove

57) tan⁻¹(1/2) + tan⁻¹(1/3) = π/4

58) 4tan⁻¹1/5 + tan⁻¹1/70 + tan⁻¹1/89 =π/4

59) tan⁻¹(1/4)+tan⁻¹2/9= 1/2 cos⁻¹3/5

60) sin⁻¹4/5 + sin⁻¹5/13 + sin⁻¹16/65 =π/2

61) 1/2 tan⁻¹x = cos⁻¹√[{(1+√(1+x²)}/2√(1+x²)]

62) tan⁻¹x + cot⁻¹(x+1) = tan⁻¹(x²+x+1)

63) tan⁻¹1+tan⁻¹2+tan⁻¹3 = 0

64) 2(tan⁻¹1+tan⁻¹1/2 +tan⁻¹1/3) =π

65) tan(2tan⁻¹a) = 2tan(tan⁻¹a+ tan⁻¹a³)

66) tan⁻¹(1/2 tan 2A)+ tan⁻¹(cot A)

 + tan⁻¹(cot³A) =0

Solve

67) tan⁻¹(x+1)+tan⁻¹(x-1) = tan⁻¹8/31

68) tan⁻¹(x-1)+tan⁻¹x+tan⁻¹(x+1)= tan⁻¹3x

69) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)} = π/4

70) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π prove x²+y²+z²+2xyz =1

71) prove sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16) = π .

72) Solve cos⁻¹(sin cos⁻¹x) = π/6

73) solve) cos (sin⁻¹x)= 1/7

74) value of tan(2 tan⁻¹ 1/5)

Solve:

75) tan⁻¹{1/(2x+1)} +tan⁻¹{1/(4x+1)} = tan⁻¹2/x²

76) tan⁻¹{(x+1)/(x-2)} + tan⁻¹{(x-1)/x} = tan⁻¹(-7)

77) If tan⁻¹x +tan⁻¹y+tan⁻¹z=π prove x+y+z=xyz

78) if tan⁻¹x+tan⁻¹y+tan⁻¹z=π/2 show that xy+yz+zx=1

79) prove tan(π/4 + 1/2cos-¹ 1/3) + Tan(π/4 - 1/2 cos-¹ 1/3) =6

 80) solve cos⁻¹(sin (cos⁻¹x))=π/3. 

                                         

81) Prove 

Sin(2tan⁻¹3/5-sin⁻¹7/25)=304/425

82) If tan⁻¹x + tan⁻¹y + tan⁻¹z=0, prove that x+ y +z = xyz.

83) tan⁻¹1/4 + tan⁻¹2/9 = 1/2 cos⁻¹3/5

19/7/22

-------:::

1) If A= 2      -1

             -1       2 and I is the unit matrix of order 2, then A² is

A) 4A - 3I. B) 4A - 3I C) A - I D) A+I

2) The multiplicative inverse matrix of    2       1     

       7       4

A) 4    -1 B) 4   -1 C) 4   -7 D) -4     -1

    -7    -2     -7   2.      7    2       7     -2

3) A is a square matrix such that A³ = I; then inverse of A is

A) A².  B) A C) A³ D) none

4) Assuming that the sums and products given below are defined which of the following is not true matrices ?

A) AB= AC does not imply B= C

B) A + B = B+ A C) (AB)'= B'A'

D) AB= O implies A= O or B= O.

5) If A=1    0    2 and Adj A=5   a    -2

            -1    1   -2                   1   1     0

             0    2    1                  -2  -2.    b then the value of a and b are

A) -4,1 B) -4,-1 C) 4,1. D) 4,-1

6) If A= -1      0

               0      2 then value of A³- A²

A) I B) A C) 2A. D) 2J

7) If A= - x     -y

                z       t then the transpose of adj A is

A) t     z B) t     y C) t     -z D) none

   -y    -x.   -z    -x      y    - x

8) If A=3      5 & B= 1       17

             2      0          0     -10 then |AB|=

A) 80 B) 100 C) -110 D) 92

9) A= 5       -2

          3        1 find inverse of A

A) -2/13      5/13            B) 1         2 

      1/13      3/13                -3         5

C) 1/11      2/11 D) 1        3

     -3/11     5/11.    -2       5

10) If A is a singular matrix of order n then A.(adj A) is

A) a null matrix.

B) a row matrix

C) a column matrix D) none

11) If A and B are two matrices and if A⁻¹ and B⁻¹ exist, then (AB)⁻¹ is

A) A ⁻¹B⁻¹ B) AB⁻¹ C) A⁻¹B D) B⁻¹A⁻¹.

12) If A= 3      -5

               -4       2 then the value of A² - 5A is

A) I B) 14I. C) O d) none

13) If A= 5      6       -3

               -4      3        2

               -4     -7        3 then the cofactors of the elements of second row are

A) 3,3,11. B) 1-3,11 C) -39,3,-11 D) 39, -3,11

14) If A= 1     2   & B= 1     2

                2     3             2    1

                3     4

Then

A) both AB and BA exist

B) neither AB nor BA exist

C) AB exist but BA dies not exist.

D) AB does not exist but BA exist

15) If A= 2     -1 & B= 1     0

                0      1         -1    -1 then (A+ B)² is not equal to

A) A²+ AB+ BA + B²

B) A²+ AB+ BA + B²I

C) A²I + AB+ BA + B²

D) A²+ 2AB + B².

16) If A be an n x n matrix and k any scalar, then det kA is equal

A) k det A B) nᵏdet A C) k ⁿdet A D) kn det A

17) If A= 1      2

                3     -5 find inverse of A

A) -5   -2 B) -5/11        -2/11 

     -3    1 C) -3/11         1/11

C) 5/11     2/11 D) 5          2

     3/11    -1/11      3        -1

18) If A= -1        2 & B= 5

                 2       -1          7 and AX= B, then X is equal to

A) 19 17 B)19/3. C) 19/3 17/3 D) 19

                    17/3                              17

19) If A= 0      1      2

                1       2     3

                3       1     1 and it's inverse B=[bᵢⱼ], then the element b₂₃ of matrix B is

A) -1. B) 1 C) -2 D) 2

20) If A= a     b    c &B= 1     2    3 

                d     e    f          2     3    4

                g     h    i          3     4    5

C= -1     -2 & D= -4     -5      -6

     -2       0           0      0       1

      0      -4

And the relation A= BCD, then value of e

A) 40. B) -40 C) -20 D) 20

21) A= 1     2 & B= 3      8

             3    4          7      2 and the relation 2X+ A = B then metrix B is

A) 2    6 B) 1    -3 C) 1    3 D) 2     -6

     4   -2      2    -1      2   -1.    4     -2

22) If A= a     2

                2     a and|A³|= 125, then the value of a is

A) ±2 B) ±3. C) ±5 D) 0

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

17/7/22

-----------

2- marks

1) y= sin(x²+1) find dy/dx

2) y= tan⁻¹x find d²y/dx²

3) y= sin 3x cos 5x find dy/dx

5- marks (any three)

4)  If 3 sin⁻¹{2x/(1+ x²)} - 4 cos⁻¹{(1- x²)/(1+ x²)+ 2 tan⁻¹{2x/(1- x²)=π/3 then find x.        

5) If xᵐ yⁿ = (x+ y)ᵐ⁺ⁿ, show dy/dx= y/x

6) If y= x log{x/(a+ bx)} then show that x³ d²y/dx² = (x dy/dx - y)²

7) Find dy/dx: y= xˢᶦⁿ ˣ + (sin x)ᶜᵒˢ ˣ

6 marks .   (Any two)

8) If x= a[cos t + log|tan (t/2)|] and y= a sin t then find dy/dx at t =π/4

9) If yˣ + xʸ + xˣ = aᵇ, find dy/dx

10) If (x - a)² + (y - b)²= c², Prove that [1+ (dy/dx)²]³⁾²/d²y/dx² is a constant independent of a and b.

5/7/22  

Question 1.                      6x2=12

i) If A= 3     1

             7     5 find x and y do that A² + xI= yA.

ii) Evaluate: tan⁻¹a/b - tan⁻¹{a -b)/(a+b)}

iii) Prove that the relation R on Z defined by (a,b) ∈ R <=> a-b is divisible by 5 is an equivalence relation on Z.

iv) Differentiate sin(x². eˣ) w.r.t.x  

v) find the derivative of the function sec⁻¹{1/(2x²-1)} w.r.t.x

vi) Find the interval in which the function f(x)=10 - 6x - 2x² are increasing or decreasing.

Question 2) Solve the following set of equation using cramer's rule: 2x-z=1;  2x + 4y -z=1; x - 8y -3 z= -2;.                 (5)

Question 3) Find dy/dx if (eˣ+1)y = (eˣ-1).                  (5)

Question 4) given two matrices A and B as 

A= 1    2     1 & B=  1   4   0

      1   -1     1          -1   2   2 

      2    3    -1           0   0   2 . Find the value of the expression AB - 2B.               (5)

Question 5) From the following data, find the best fit regression equation.

X.       Y

56     147

42     125

72     160

36     118

63     149 

47     128

55     150 

49     145 

38     115.               (5)

    

15/5/22

21) y= log(tan x/2)

22) x² w r.t. log x

23) ₘ(ax²+bx+c)

24) log(ax+ b)³

25) {f(x)}ⁿ

26) ₑf(x)

27) log{√f(x)}

28) ₃f(x)

29) sin {f(x)}

30) sec {f(x)}

11/5/22

11) y= logₓ2

12) x² + y² = a²

13) x= a cos t, y= a sin t

14) cos(logx)

15) log(sinx)

16) cot x°

17) ₁₀10ˣ

18) xˣ

19) If f(x)= √(x²+1), then f'(-1) is

20) y= 1/(3x -1) at x= 0

10/5/22

1) y= eˢᶦⁿ ˣ

2) y= sin³x.

3) y= ₂3x²

4) If y= log(x² -5) and dy/dx = ky/(x² -5) then find k

5) if y= log₁₀x= m/x then m is..

6) If d/dx(2x³ +5)¹⁰= 60x² f(x) then f(x) is .

7) if d/dx √(2x² +9)= f(x)/√(2x²+9) then f(x) is

8) y= √(x² + a²)

9) y= log f(x)

10) y= 10ᵐˣ

22/4/22

1) If A= 2       -1

             -1        2 and I is the unit matrix of order 2, then A² is

A) 4A - 3I             B) 3A - 4I    

C) A - I                 D) A + I    

2) The multiplicative inverse of matrix 2           1

            7           4

3) if A= 1    0    2                     5   a   -2

             -1    1   -2 & adj of A=1   1    0

              0    2    1                    -2  -2   b then find the values of a and b.

4) If A = -1      0

                0      2 then the value of A³ - A² is

5) If A= - x       - y

                z         t then find the transpose of adj A.

20/4/22

1) Find Adjoint of     2      3

                                  -4     -6

2) Find inverse of      2       -2

                                    4        3

3) If A= 3       1

             -1       2 then show A² - 5A + 7I = 0, hence find inverse of A.

19/4/22

1) Find Adjoint of 2       3

                              -4      -6

2) Find inverse of 2        -2

                               4         3

3) If A= 3        1

             -1        2 then show A² - 5A + 7I = 0, hence find inverse of A.

4) If             1        1       1

             A=  1        2      -3

                    2       -1       3 then show that A³ - 6A² + 5A + 11I= 0, hence find inverse of A.

18/4/22

1) Construct a 2x2 matrix whose elements are (2i + j)²/2.

2) x+3    z+4    2y-7     0        6   3y -2

   4x+6    a-1       0    =2x -3  2c      -2

     b-3     3b     z+ 2c  2b+4  -21     0 Obtain the values of a, b, c, x, y , z

3) Let A= 2      3

               -1       2 and 

f(x)= x² - 4x + 7. Show that f(A)= O. Use this result to find A⁵.

4) Show that AB= BA where

A= 5     -1 and B= 2       1

      6      7              3       4

17/4/22

1) If A= 1     2

              3     4 then A⁴ - 5A³ - A² - 4A - 2I = ?. (I is unit metrix).

A) 0      B) A     C) I        D) 2I

2) Two matrices A and B are said to be conformable for multiplication in the same order iff the number of rows of A is equal to number of columns of B.                      T/F

3) If A= 2    0     0     B= x &      C= 4

             0     3     0          y               -1

             0     0     4          z                8 then find the value of 3xyz.

4) Find the inverse of -1        4

                                       7      -20 

5) If A= a     2

              2     a and | A³|= 125, then find a.

LAST Time Revision (X)(ISC Board)

1) sec⁴x - sec²x = tan⁴x - tan²x 

2) sin⁶x + cos⁶x = 1 - 3 tan²x cos²x

3) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1

4) cosecx (secx - 1) - cotx (1- cosx)= tanx - sinx.

5) (1- sinx cosx)/{cosx(secx - cosecx) . (Sin²x - cos²x)/(sin³x + cos³x) = sinx

6) tanx/(1 - cotx ) + cotx/(1 - tanx) = (secx cosecx + 1)

7) (sin³x + cos³x)/(sinx + cosx) + (sin³x - cos³x)/(sinx - cosx)= 2

8) (secx sec y + tanx tany)² - (secx tany + tanx secy)² = 1

9) cosx/(1- sinx) = (1+ cosx + sinx)/(1- sinx + cosx).

10) √{(1- sinx)/(1+ sinx)} + √{(1+ sinx)/(1- sinx)} = -2/cosx

11) tan³x/(1+ tan²x) + cot³x/(1+ cot²x) = (1- 2 sin²x cos²x)/(sinx cosx)

12) 1 - sin²x/(1+ cotx) - cos²x/(1+ tanx)= sinx cosx.

13) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)} sin²x cos²x = (1 - sin²x cos²x)/(2+ sin²x cos²x).

14) (1+ tanx tan y)²+ (tan x - tan y)²= sec²x sec²y.

15) (1+ cotx + tanx)(sinx - cosx)/(sec³x - cosec³x)= sin²x cos²x.

16) (2 sinx cosx- cosx)/(1- sinx + sin²x - cos²x)= cotx.

17) cosx(tanx +2)(2 tanx + 1)= 2 secx + 5 sinx.

18) sin⁸x - cos⁸x = (sin²x - cos²x)(1- 2 sin²x cos²x)

19) cot⁴x + cot²x= cosec⁴x - cosec²x.

20) 2 sec²x - sec⁴x - 2 cosec²x + cosec⁴x = cot⁴x - tan⁴x.

21) (sinx + cosecx)² +(cosx+ secx)² = tan²x +cot²x+ 7

22) (1+ cotx- cosecx)(1+ tanx + secx)= 2

23) (tanx + secx- 1)/(tanx - secx +1)= (1+ sin)/cosx 

24) 3(sinx - cosx)⁴+ 6(sinx + cosx)² + 4(sin⁶x+ cos⁶x) - 13= 0

Height and Distances

1) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 50m away from the bank, he finds the angle of elevation to be 30°. Find:

A) the width of the river.           25m

B) the height of the tree.        43.3m

2) A telegraph pole is 8m high. Its shadow is 8√3m in length. Find the elevation of the sun.                  30°

3) The angle of elevation of the top of a tower at a distance of x m. from its foot on a horizontal plane is found to be 30°. If the height of the tower be 70m, find x.   121.24m

4) A tree is broken by the wind. The top struck the ground at an angle of 30° and at a distance of 30 m from the foot. Find the whole height of the tree.                                   57.96

5) The angles of elevation of the top of a tower from two points on the ground at distance a metres and b metres from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is √(ab) metres.

6) Two pillars of equal height stand on either side of a roadway which is 120m wide. At a point in the road between pillars, the elevations of the pillars are 60° and 30°. Find the height of each pillar and the position of the point.               51.96m, 30m from the pillar with elevation 60°

7) A man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this will the car reach the observation tower?              16.39m

8) A man in a boat rowing away from a tower 150m high takes 2 minutes to change the angle of elevation of the top of the tower from 60° to 45°. Find the speed of the boat.                              0.53 m/s

9) A car is travelling at 50 m/s along a road in inclined at 30° to horizontal. In what time will the vertical height of the car increase by 50 m ?                           2 sec

10) A boy standing on the ground and flying a kite with 100m of string at an elevation of 30°. Another boy is standing on the roof of a 20m high building and is flying his kite at an elevation of 45°. Both the boys are on opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet.          42.32

11) An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the angle of elevation of the two planes from the same point of the ground are 60° and 45° respectively. Find the vertical distance between the aeroplanes at that instant.  1687.86

12) The angles of elevation of an aeroplane at two consecutive kilometre posts respectively are a and b. Find the height of the aeroplane above the ground taking it to be between the two kilometre stone. (tan a tan b)/(tan a + tan b)

13) A tower subtends an angle a at a point on the same level as the foot of the tower and at a second point h metres above the first, the depression of the foot of the tower is b. Show that the height of the tower is h tan a tan b.

14) From a window (h metre high above the ground) of a house in a street, the angle of elevation and depression of the top and the foot of another house on opposite side of the street are a and b respectively. show that the height of the opposites house is h(1+ tan a cot b).

15) The angles of elevation of the Summit of a hill from the top and bottom of a tower are 45° and 60° respectively. If the height of the tower is h m. Prove that the height of the hill is {√3(1+√3)}/2 metres.

XI Model Test paper -3

Full Marks -80

Question -1.                        1x10

i) A relation R is defined from A={1,2,3,4,5} to B={1,3,4} in such a way that, (x,y) belongs to R => x > y. Express R as a ordered pairs.

ii) If 16Cr = 16C 2r+1, then value of r. 

A) 5 B) 4 C) 3 D) 2

iii) If w be the imaginary cube root of 1, then the value of (3+ w + 3w²)⁴. 

A) 16 B) w C) 16w D) 1

iv) If the difference between the roots of the quadratic equation x² + px +8= 0 be 2, then the value of p be.

A) ±3 B) ±6 C) ±4 D) none

v) If P-th term of an arithmetic progression be Q and Q-th term be P, then(P+ Q)-th term is 

A) 0 B) 1 C) P D) Q

vi) the value of lim ₓ→₀ (eˣ - e⁻ˣ)/x. 

A) 0 B) 1 C) 2 D) 3

vii) If y= cos² x/2 find dy/dx.

A) -1/2 sinx B) 1/2 sinx C) 1/2 cosx D) - 1/2 cosx

viii) The probability that a non leap year selected at random will have are 53 Sundays is. 

A) 1/2 B) 1)3 C) 1/5 D) 1/7

ix) If the variance of a distribution is 4 and coefficient of variation is 5%, then find mean. 

A) 40 B) 60 C) 30 D) 70

x) The domain of x² +3x +2 is

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

Question -2                  2x13= 26

i) y= f(x)= (px +q)/(rx - p), then show that x= f(y).

ii) Show 1/sin 10 - √3/cos 10 = 4

iii) If nPr= 504 and nCr= 84, then find the value of n and r. 

iv) Find the coefficient of 1/x² in the expansion of (2x - 1/x²)⁶.

v) solve the following in equation: {|x +2| +2x}/(x+2) > 2.

vi) lim ₓ→₀ (cos 5x - cos 7x)/(cosx - cos 5x). 

vii) If y= sec 2x find dy/dx at x=π/8

viii) If P(A∩ B)= 5/13, then find the value of P(A' U B'). 

ix) d/dx of sin(logx)

x) the number of all numbers having 5 digits, with the distinct digits is..

xi) Two finite sets A and B consist of m and n elements respectively. The number of subsets in A exceeds that of B by 112. Find the values of m and n.                     

xii) If (a,b) and (b,c) are elements of A x A, find the set A and other elements of A x A.

xiii) If f(x + 1/x)= x² + 1/x², find f(x).        

Question -3.               3x10= 30

i) For any three sets A, B and C, prove that A - (B U C)= (A - B)∩ (A - C).

ii) If z be a complex number and |z+5|≤ 6, then find the maximum and minimum value of |z +2|. 

iii) Solve: 4 sin x sin 2x sin 4x= sin 3x. 

iv) if tan x/2 = √{(1- e)/(1+ e)} tan y/2 , prove cos y = (cos x - e)/(1- e cos x).

v) If n belongs to N, then prove by mathematical induction that 7²ⁿ + 2 ³⁽ⁿ⁻¹⁾. 3 ⁿ⁻¹ is always a multiple of 25.

vi) If z= x+ iy and (z -i)/(z+ 1) is purely imaginary, then show that the point z always lies on a circle.

vii) how many odd numbers of 5 digits can be found with the digits 3,6,7,2,0 when no digit is repeated. 

viii) Show that the mid term in the expansion of (1+ x)²ⁿ is {1.3.5. ....(2n-1) 2ⁿ. xⁿ})n! (Where n belongs to N)

OR

If the ratio of the sum of first n terms of two arithmetic series is (4n -13): (3n +10), then find the ratio of their ninth terms. 55:61

ix) solve applying formula: 3x² - (2- i)x + 10 - 4i= 0. 

OR

 Out of 14 marbles 10 are red in colour and remaining 4 are of different colours. How many ways can you select 10 marbles out of these 14 marbles. 

x)  A survey shows that 75% of the Students of a school like Mathematics and 65% like Physics. If x% of the students like both Mathematics and Physics. find the maximum and minimum values of x.                    

OR

 In a survey of 35 students of a class it was found that 17 students like Mathematics and 10 like Mathematics but not Biology. Find the number of students who like 

A) Biology.                        

B) Biology but not Mathematics, it being given that each students takes at least one of the two subjects.                      

Question 4.                     3x3= 6

i) Find the equation of the directrix of the parabola x² - 4x - iy +12 = 0. y= 0

OR

 Find the equation of the lines passing through the point (4,5) making equal angles with the lines 3x = 4y +7 and 5y = 12x +6.

ii) A circle in the first quadrant touches both the axes and its centre lies in the straight line lx + my + n= 0, show that the equation of that circle is (l+m)² (x² + y²)+ 2n(l+m) (x+y) + n²= 0.

OR

 The ellipse x²/a² + y²/b² = 1 passes through the point of intersection of the lines 7x = -13y + 87 and 5x = 8y +7 and the length of its latus rectum is 32√2/5 units. Find the values of a and b. 5√2, 4√2

Question 5.             4+4

i) The standard deviation of 32 numbers is 5. If the sum of the numbers is 80, then find the sum of the squares of the numbers.      (4)

ii) calculate the mean deviation of median for the following data:

Marks.      No of students

00-10            6

10-20            5 

20-30            8 

30-40           15 

40-50             7 

50-60             6 

60-70              3                          (4)

LAST TIME REVISION (XII)

5/5/22

Probability

1) A problem in mathematics is given to three students A, B, C and their respective chances of solving the problem is 1/2,1/3,1/4, then their probability that the problem is solved.      3/4

2) The probability that a leap year will have 53 Tuesday or Saturday.    4/7

3) P(A)=2/3 , P(B)=1/2 and P(AU B)=5/6, then the events A and B are

A) mutually exclusive

B) independent as well as mutually exclusive

C) independent.

D) none

4) A fair dice is thrown till we get 6, then the probability of obtaining 6 exactly in even number of turns is.     5/11

5) A and B are two events such that P(A U B)= 3/4 , P(A ∩ B)= 1/4, P(A')=2/3 find P(A' ∩ B).      5/12

6) If A and B are two events such that P(A U B)= 5/6 , P(A ∩ B)= 1/3, then which one is the following is not correct?

A) A and B are independent

B) A and B' are independent

C) A' and B are independent

D) A and B are dependent.

7) A coin and a six faced die, both unbiased are thrown simultaneously. The probability of obtaining a head on the coin and an odd number on the die.   2/3

8) A number is chosen at random among the first 120 natural numbers. What is the probability that the number chosen being a multiple of 5 or 15 ?      1/5

9) A die is thrown. If it shows a six, we draw a ball from a bag containing 2 black balls and 6 white balls. If it does not show a six then we toss a coin. Then the number of event points in the sample space of this experiment is.      18

10) If A and B are two events such that P(A)= 1/4, P(B/A)= 1/2 and P(A /B)= 1/4 then the value of P(A' B')= ?      3/4

11) The probability that a regularly scheduled flight departs on time is 0.9, the probability that it arrives on time is 0.8 and the probability that it departs and arrives on time is 0.7. then the probability that a plane arrives on time, given that it departs on time, is...    7/9

12) A sample of 4 items is drawn at random from a lot of 10 items, containing 3 defectives. If x denotes the number of defective items in the sample, then P(0< x < 3) is equal to...   4/5

13) If A and B are two independent events such that P(A)= 1/2, P(B)= 1/3,  then the value of P(A' ∩ B').     1/3

14) The probability that in a year of 22nd century chosen at random, there will be 53 Sundays.      5/28

15) The probability that in a family of 5 members, exactly 2 members have birthday on  Sundays is.         10x6³/7⁵

16) A bag contains 5 white and 3 black balls and 4 balls are successively drawn out and not replaced. The probability that they are alternatively of different colours is ..     1/7

17) The probability that A speaks truth is 4/5, while this Probability for B is 3/4, then the probability that they will contradict each other when asked to speak on a fact, is ...    7/20

18) Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers are divisible by 2 and 3 is ..   4/1155

19) Find the probability of obtaining a total of 7 or 12 with two dice.                   7/36

20) Five horses are in a race. Mr. A selected two of the horses at random and bets on them. The probability that Mr. A selected the winning horse.    2/5

21) For three events A, B, C, if P(B)= 3/4, P(A' ∩ B ∩ C')= 1/3 and P(A ∩ B ∩ C')= 1/3 , then the value of P( B ∩ C)= ?    1/12

22) In tossing a fair coin twice, let A and B denote the events of occurrence of head on first toss and second toss respectively; then the value of P(A U B) is.    3/4

23) A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then the probability for the ball chosen be white is.     8/15

24) A five digit number is formed by writing the digits 1, 2, 3, 4, 5 in a random order without repetition. Then the probability that the number is divisible by 4 is .      1/5

25) From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability that the number on the card is divisible by 6 or 8 but not by 24 is..       1/5

26) Two persons A and B take turns in throwing a pair of dice. The first person to throw 9 from both the dice will get the prize. If A throws first then the probability of B winning the prize is..    8/17

27) A fair coin is tossed n times. The probability of obtaining head atleast once greater than 0.8. then the least value of n is....    3

28) A card is drawn from an ordinary pack of 52 cards and a gambler beta that either a spade or an ace is going to appear. Then the odds against his winning the prize are...                  9:4

29) Out of 30 consecutive natural numbers, 2 are chosen at random. The probability that their sum is odd, is..     15/29

30) The probability of having a king and a queen when two cards are drawn at random from a pack of 52 cards is.      8/663

31) A, B, C are mutually exclusive events such that P (A)= (3x+1)/3, P(B)= (1- x)/4 and P(C)= (1- 2x)/2; then the set of possible values of x are in the interval....     (1/3,1/2)

32) The probability of throwing a total of 7 or 12 with two dice is..   5/36

33) If A and B are two events and P(A U B)= 5/6, P(A ∩ B)= 1/3 and P(B')= 1/2 , then A and B are

A) dependent     B) independent.

C) mutually exclusive

D) none of these

34) If two fair coins are tossed together 5 times, then the probability of obtaining 5 heads and 5 tails is.     63/256

35) A fair coin is tossed 10 times. The probability of obtaining exactly 6 heads is.    105/512

36) A box contains 5 apples and 7 oranges and another box contains 4 apples and 8 oranges. One fruit is picked out from each box. Then the probability that the fruits are both apples or both oranges is.     19/36

37) Three numbers are selected at random from the first 20 natural numbers. The probability that their product is even, is..   15/19

38) If P(A U B)= 0.8, P(A ∩ B)= 0.3 then the value of and P(A')+P(B')= ?           0.9

 

39) 12 balls are kept in 3 different boxes; then the probability that the first box will contain 3 ball is...    (12C3. 2⁹)/3¹²

40) Four persons A, B, C, D throw an unbiased dice, turn by turn, in succession till one gets an even number and wins the game. If A starts then the probability that he wins the game is.     8/15

41) The probability of obtaining a total of atleast 6 in the simultaneously throw of 3 dice is.     103/108

42) 5 boys and 5 girls are sitting a row randomly. The probability that the boys and girls are sit alternatively is.    1/42

43) Bag A contains 4 green and 3 red balls and bag B contains 4 red and 3 green balls. One bag is taken at random and a ball is drawn and noted it is green. Then the probability that it comes from bag B 

44) If A and B are two events such that P(A U B)' = 1/6, P(A ∩ B)= 1/4 and P(A')= 1/4 , where A' stands for the complement of event A. Then events  A and B are

A) mutually exclusive and independent

B) independent but not equally likely.

C) equally likely and mutually exclusive

D) equally likely but not independent.

45) 

      X            P(X)

      1            0.15 

      2            0.23 

      3            0.12 

      4            0.10

      5            0.20

      6            0.08 

      7            0.07

      8            0.05

Let the event A and B be defined as follows:

A: X is a prime number

B: (X< 4); then find the value of P(A U B) is.      0.77

46) A person puts three cards addressed to three different people in three envelopes with three different addresses without looking. Then the probability that the cards go into their respective envelopes is.    1/6

47) If birth of a male child and that of a female child are equal probable, then the probability of having atleast one of the three children born to a couple is male, is.     7/8

48) An ordinary coin is tossed 2n times. Then the chance that the number of times one gets head is not equal to the number of times one gets tail is.    1 - 2n!)(n!)² . 1/2^(2n)

Area

1) The area of the region bounded by the curve x²= 4y, the line x= 2 and x-axis is

A) 1 B) 2/3. C) 4/3 D) 8/3

2) The area of the region bounded by the curve cos x= y and y = sinx and the ordinates x= 0, x=π/4 is

A) √2+ 1 B) √2 -1. C) 1/√2 D) (√2-1)/√2

3) 1) The area of the region bounded by the curve y²= 8x and its latus rectum is

A) 16/3 B) 25/3 C) 16√2/3 D) 32/3.

4) The area of the region bounded by the curve -3y²= x - 9 and the lines x= 0, y= 0 and y= 1 is

A) 8/3 B) 3/8 C) 8. D) 3

5) The area of the region bounded by the curve y²= 12x, the line x= 0 and y= 6 is

A) 12 B) 16 C) 3 D) 6.

6) The area of the region bounded by the curve x³= y and y = 2x² is

A) 2/3 B) 3/4 C) 4/3. D) 1/3

7) The ratio of the areas bounded by the curve cosx= y and y = cos2x between x= 0,x =π/3 and x-axis is

A) √2:1 B) 1:1 C) 1:2 D) 2:1.

8) The area in the first quadrant between y² = 4x, y²= 16x and the straight line x= 9 is

A) 36. B) 24 C) 18 D) 9

9) The area of the region bounded by the curve y= x²- 6x +10 and the lines x= 6, y= 2 is

A) 20/3. B) 16/3 C) 8 D) 32/3

10) The area of the region bounded by the curve x²= 72y and the lines y= k be 64√2 sq unit, then the value of k is

A) 2 B) 3 C) 4. D) 6

11) Two intersecting circles have their radii 1 metre and√3 metre. The distance between their centres is 2m. Then the overlapping area is

A) (19+ 6√3)/6 B) (5π+ 6√3)/6 C) π/6 D) (5π - 6√3)/6.

12) The area of the region bounded by the curve y= sinx between the ordinates x= 0, x= π and the x-axis is 

A) 2. B) 4 C) 3 D) 6

13) The area above the x-axis bounded by the curve 2ᵏˣ= y and the lines x= 0, x= 2 be 3/log 2 sq unit, then the value of k is

A) 1. B) 1/2 C) -1 D) 2

14) The area of the common region between the two parabolas y² - ax= a² and y² + ax = a² is

A) 8a²/3. B) 16a²/3 C) 4a²/3 D) 32a²/3

15) The area of the region {(x,y): x²+ y² ≤ 1 ≤ x + y} is

A) π/4 B) π/2 C) π²/4 D) π/4 - 1/2.

16) The area of the region bounded by the curve secx= y, the x-axis and the lines x= 0 and x=π/4 is

A) log(√2 -1) B) log(√2 +1) . C) 1/2 log 2 D) √2

17) The area of the smaller segment cut off from the circle x² + y² = 9 by the line x= 1 is

A) 1/2(9 sec⁻¹3 - √8) 

B) (9 sec⁻¹3 - √8).

C) √ 8 - 9 sec⁻¹3 

D) (9 sec⁻¹3 + √8) 

18) The area surrounded by the curve |x| + |y|= 1 is

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

19) The area surrounded by the curve y= √x, the straight line x= 2y+3 in the first quadrant and the x-axis is

A) 9. B) 2√3 C) 18 D) 35/3

20) The area surrounded by the curve y= ax² and x = ay², a> 0 is 1 unit, then the value of a is

A) 1 B) 1/√3. C) 1/3 D) 1/√2

21) Which of the following definite integral represents the area included between the parabola 4y = 3x² and the straight line 2y = 3x+12?

A) ∫ 3x²/4 dx at (4,-2)

B) ∫ {(3x+12)/2 - 3x²/4}dx at (4,0)

C) ∫ (3x+12)/2 - 3x²/4}dx at (4,-2).

D) ∫(3x+12)/2 - 3x²/4}dx at (2,-2)

22) A function y= f(x) is defined as follows:

 y= f(x)= x² when 0≤ x ≤ 1

            = √x when x≥ 1

Then the area above the x-axis included between the curve y = f(x) and the line x= 4 is

A) 16/3 B) 4 C) 5. D) 6

23) the area of the region bounded by the curve 9x² + 4y²= 36 is 

A) 3π. B) 9π C) 6π D) 4π

24) the area of the triangle bounded by the lines x + y = 0, y= 0 and x- 4= 0 is 

A) 4 B) 8. C) 12 D) 16

25) If A is the area of the region bounded by the curve y= √(3x+4), x-axis and the lines x= -1 and x= 4 and B is the area bounded by the curve y² = 3x + 4 and the lines x=-1 and x= 4, then the value of A: B is 

A) 1:2. B) 2:1 C) 2:3 D) 3:2

26) the curves y= sinx and, y= cosx intersect infinitely many times giving bounded regions of equal areas. The area of one such region is..

A) 4√ 2 B) 3√2. C) √2 D) 2√2

27) The area of the region enclosed between the curve y= log(x+e) and the coordinate axes is

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

28) The area bounded by the curve y= |x| - 1, and y = -|x |+ 1 is 

A) 1 B) 2. C) 2√2 D) 4

29) The area bounded by the curve 16y= x², y-axis and it's latus rectum is 

A) 32/3 B) 64/3. C) 128/3 D)16/3

30) The area bounded by the curve f(x)= 4 - |x|, and the x-axis is 

A) 16. B) 32 C) 12 D) 24

31) If the area bounded by the curve y= x - x² and the line y= mx is 9/2 sq unit, then the value of m is 

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

32) The area bounded by the curve y= 2 - x² and the line x + y= 0 is A sq unit, then the value of A is 

A) 9/2. B) 2/9 C) 1/3 D) 7/3

33) the area of the region bounded by the curve y= |x -2|, x-axis and the ordinates x= 1, x= 3 is 

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

34) the area of the region bounded by the curve 2y²= x, 3y² = x + 1 and y= 0 is 

A) 4 B) 4/2 C) 2/3. D) 2

35) the area of the region bounded by three curves y= (x -1)², y = (x + 1)² and y = 1/4, is 

A) 1/6 B) 1/3. C) 2/3 D) 5/6

36) the area between the curvey y= xeˣ, y= xe⁻ˣ, and the line x= 1 is 

A) 2(e+ 1/e) B) 2e C) 2(e - 1/e) D) 2/e.

37) the area enclosed between the curve y= x³, and y² = x is 

A) 5/12. B) 5/3 C) 5/4 D) 12/5

38) the area in the first quadrant bounded by the curve y² + x² = 8 x and its tangent at (2,2) is

A) 8-2π. B) 8+2π C) 2π-8 D) 4+2π

39) the area bounded by the curve y= - x²+ 2x+3, and y = 0 is 

A) 32 B) 32/3. C) 5/32 D) 1/3

40) the area bounded by the curve y² = 2x+ 1 and the line x - y = 1 is 

A) 16/3. B) 8/3. C) 24/3 D) 18/5

41) the area bounded by the curve y= - x² and y = x³ is 

A) 1/3 B) 1/4 C) 1/6 D) 1/12.

42) the area enclosed between the curve y²= 4a(x+a) is (a, b > 0)

A) 1/3 (a+b) √(ab)  

B) 2/3 (a+b) √(ab)  

C) 4/3 (a+b) √(ab)  

D) 8/3 (a+b) √(ab).

Tangent and Normal

1) The equation of the tangent to the curve (1+ x²)y = 2 - x where it crosses the x-axis is

A) x + 5y = 2. B) x - 5y = 2  

C) 5x - y = 2 D) 5x + y = 2

2) The equation of the tangent and the normal drawn at the point (6,0) to the curve x²/36 + y²/9 = 1 respectively are

A) x= 6, y= 0. B) x +y = 6, y-x+6= 0  

C) x = 0, y= 3 D) x = -6, y= 0

3) The straight line x+ y= a will be a tangent to the ellipse x²/9+ y²/16 = 1 if the value of a is

A) 8 B) ±10 C) ±5. D) ±6

4) The equation of the tangent to the parabola y² = 8x which is perpendicular to the line x - 3y+8= 0 is

A) 3x + y = -2 B) 3x - y = 1  

C) 9x - 3y = -2 D) 9x + 3y = -2.

5) If the slope of the normal to the curve x³ = 8a²y at P is (-2/3), then the coordinates of P are

A) (2a,a). B) (a,a) C) (2a,-a) D) n

6) If the curves y= aˣ and y= bˣ intersect at an angle p, then the value of tan p is

A) (a- b)/(1+ ab)    

B) (log a- log b)/(1+ loga logb) .

C) (a+ b)/(1- ab)    

D) (loga+ logb)/(1+ log a log b)   

7) If the straight line y = 4x - 5 touches the curve y² = px³+ q at (2,3), then the values of p and q are..

A) 2,-7 B) 2,7 C) -2,-7 D) -2,7

8) The equation of the normal to the parabola y² = 4ax at the point (at², 2at) is

A) tx +y= 2at + at³.

B) x + ty= 2at + at³

C) tx - y= at + 2at³

D) x - ty= at + at³

9) if the slope of the normal to the parabola 3y²+ 4y+2 = x at a point on it is 8, then the co-ordinate of the point are---

A) (1,-1) B) (6,2). C) (9,1) D) (2,0)

10) if the line lx + my + n= 0 is a tangent to the parabola y² = 4ax, then..

A) an² = ml           B) al² = mn

C) am² = nl.           D) a²m = nl

11) the equations of the tangents to the hyperbola 3x² - 4y² = 12 which are inclined at an angle 60° to the x-axis are ..

A) y= √3 x ±12       B) y= √3 x ±10

C) y= √3 x ±15       D) y= √3 x ±13.

12)  the equation of the tangent to the curve xy²= 4(4- x) where it meets the line y = x is..

A)  y= - x - 4          B) y=  x + 4.

C)  y=  x - 2           D) y= x + 2

13) The normal to the curve x = 3 cos a - cos³a, y= 3 sin a - sin³a at a= π/4....

A) is at a distance of 2 unit from the origin

B) is at a distance of 4 unit from the origin

C) passes through the origin.

D) passes through the point (2,3)

14) the point on the curve x² + 2y = 10 at which the tangent to the curve is perpendicular to the line 2x - 4y = 7 is..

A) (2,3). B) (-2,3) C) (4,-3) D) (-4,-3)

15) if the gradient of the tangent at any point (x,y) of a curve which passes through the point (1,π/4) is {y/x - sin²(y/x)}, then the equation of the curve is...

A) y= cot⁻¹(log x)

B) y= cot⁻¹{log (x/e)}

C) y= x cot⁻¹(log xe).

D) y= cot⁻¹(log(e/x))

16) The number of tangents that can be drawn from the point (6,2) on the hyperbola x²/9 - y²/4 = 1 is .

A) 0.        B) 1      C) 2        D) 3

17) The equation of the tangent to the curve x²⁾³+ y ²⁾³= a ²⁾³ at the point (a cos³k, a sin³k) is .

A) x cos k + y sin k= a sink cosk

B) x cos k - y sin k= a sin 2k

C) x sin k - y cos k= a sin2k 

D) x sin k + y cos k= a sink cosk.

18) the equation of the tangent to the curve y= be⁻ˣ⁾ᵃ at the point where it crosses the y-axis is..

A)  bx + ay = ab.      B) ax + by = 1

C)  bx - ay = ab       D) ax - by = 1

19) The equation of the two common tangents to the circle x² + y² = 2a² and the parabola y² = 8ax are

A) x= ±(y + 2a)        B) y= ±(x + 2a).

C) x= ±(y + a)          D) y= ±(x + a)

20)  the equation of the normal to the ellipse x²/a² + y²/b² = 1 at the point (a cosk, b sink) on it is..

A) ax sink - by cosk = a² - b²

B) ax sink + by cosk = a² - b²

C) ax cosk - by sink = (a² - b²) sink cosk

D) ax sink - by cosk = (a² - b²) sink cosk

21) the point on the curve √x + √y= √a, the normal at which is parallel to the x-axis is...

A) (0,0) B)(a,0) C)(0,a). D)a/4,a/4)

22) the slope of the tangent to the curve x= 3t²+1, y= t³ -2 at x= 1 is .

A) 1/2 B)0 C) )-2 D) undefined

23) if the line x+ y= a is a tangent to the parabola y² - y + x= 0, then the point of contact is ..

A) (0,1).   B)(a,0) B)(1,1) C)(-1,0)

24) the angle between the Curve y = Sin x and y= cos x is..

A) tan⁻¹(5√2)          B)tan⁻¹(3√3)

C) tan⁻¹(3√2)          D) tan⁻¹(2√2).

25)  the point on the curve y²= x, the tangent at which makes an angle 45° with the x-axis is

A) (0,9) B)(1/4,1/2). C)(1/2,1/4) D)(2,4) 

26) if the straight line joining the point (0,3) and (5,-2) is a tangent to the curve y(x+1)= c, then the value of c will be..

A) 3  B)-3 C) 4. D) -4

27) the equation of the normal to the hyperbola x= a sec k, y= b tank at the point (a seck, b tank) is .

A) ax cosk + by cotk = a²+ b².

B) ax cosk + by tan k = a²+ b² 

C) ax sin k - by cot k = a²- b² 

D) ax cosk - by tan k = a²- b²

28) if the straight line lx + my  = 1 is a normal to the parabola y² = 4ax, then ...

A) al² + 2 km = m²

B) al³ - 2 alm = m²

C) al³ + 2 alm = m².

D) al² + 2 am = m²

29) the equation of a tangent to the hyperbola x² - 2y¹ = 2 parallel to the line 2x - 2y+5= 0 is..

A) y= 2x+1              B) y= 2x - 1

C) x = y +1.             D) x+ y +1= 0

30) the slope of the normal in the hyperbola x²/a² - y²/b² = 2 at the point (a sec k, b tan k) is .

A) b/a  sin k       B) - a/b  sin k.

C) a/b  sin k       D) - b/a  sin k  

Maximum and minimum

1) The minimum value of f(x)= x² + 250/x.                     75

2)  The maximum value of f(x)= 1/(4x² + 2x+1) 4/3

3) If f(x)= 2x² - 3x² -12x +4 has

A) no maxima and minima

B) one maxima and one minima.

C) two maxima 

D) two minima

4) maximum value of (log x)/x in (0, ∞).                         1/e

5) Let the function f: R--> R be defined by f(x)= 2x + cosx; then f(x) is

A) has maximum value at x= 0

B) has minimum value at x=π

C) is a decreasing function

D) is an increasing function.

6) Let x and y be two variables and x > 0, xy= 1, then the minimum value of x+ y is. 2

7) The function y =a(1- cosx) is maximum when x is..                      π

8) If minimum value of f(x)= x² + 2bx+ 2c² is greater than maximum value of g(x)= x² - 2cx + b², then for real value of x is.                |c| > √|b|

9) Let f(x)= x³ + bx² + cx +d, 0 < b²< c, then f(x)..

A) has a local maximum

B) has a local minimum

C) is strictly decreasing

D) is strictly increasing.

10) If x+ y= 60, x, y > 0, then the maximum value of xy³ is.     15.(45)³

11) If the function f(x)=2x³ - 9ax² +12a²x+1, where a> 0 attains its maximum and minimum at x= p and x= q respectively, such that p²= q, then the value of a is.                    2

12) The maximum value of the function f(x)= 3 cosx - 4 sinx is.    3

13) The minimum value of f(x)=2x²+ x -1 is.                    -9/8

14) If M and m are the maximum and minimum values respectively of the function f(x)= x+ 1/x, then the value of M + m is.                    -4

15) The maximum value of xʸ when x + 2y= 8 is.                     8

16) The greatest value of the function f(x)= x² log(1/x).         1/2e

17) The minimum value of 4e²ˣ + 9e⁻²ˣ is..                                  12

18) The maximum value of (x²- x+1)/(x² + x +1) is.                        3

19) The difference between the maximum and minimum value of the function f(x)= x³/3 - 2x² + 3x +1 is.                        4/3

Increasing and decreasing

1) f(x)= kx³- 9x² + 9x+ 4 is an increasing function then

A) k< 3 B) k≤ 3 C) k> 3. D) k is indeterminate

2) If f(x)= 1/(x+ 1) - log(1+x), x> 0, then f(x) is

A) a decreasing function.

B) an increasing function

C) neither a increasing nor decreasing function

D) increasing when x> 1

3) The function f(x)=1 - x³ - x⁵ is decreasing for.                      All real values of x

4) Let f(x)= x³ + 6x² + px +2, if the largest possible interval in which f(x) is a decreasing function is (-3,-1), then the value of p is.        9

5) The function f(x)= x³+ 3x² + 2x +7 is increasing for.                       x> 0

6) The function f(x)=2x³ - 3x²+ 90x +174 is increasing in the interval.                 (-∞ < x<∞)

7) The interval in which the function f(x)= 2x² - log x (x≠0) is increasing.          -1/2< x< o or x>1/2

8) If f(x)= x³ - 6x²+ 9x +3 be a decreasing function, then x lies in.                (1,3)

9) The value of x for which the polynomial 2x³ - 9x²+12x+4 is a decreasing function of x, is.    1<x<2

10) The function f(x)=x + 2 + (x -2) eˣ is positive.                             x> 0

Differentiation 

1) Find dy/dx

2) x= ₑ tan⁻¹{(y- x²)/x².            2x(1+ tan(log x))+ x sec²(log x)

3) If x= a cos⁴ t, y= a sin⁴ t at t= 3π/4.                                               -1

4) xˣ.                                  xˣ(1+ logx)

5) ₑx³ w r t. Log x.                   3x³ ₑx³

6) √[x+ √{x+ √(x +.......∞.          2/(2y -1)

7) sin⁻¹x + sin⁻¹y =π/2.               -x/y

8) √{sin√x}.     Cos√(x/4√x√{sin√x}) 

9) If f(x)= cos⁻¹{(1- (logx)²)/(1+ (logx)²)} then f'(e) is.                   1/e

10) ₓeˣ.                      y eˣ(logx + 1/x)

11) If 2ˣ + 2ʸ =2ˣ⁺ʸ, then the value of dy/dx at x= y= 1 is.                    -1

12) siny + e ⁻ˣ ᶜᵒˢʸ = e at (1,π).      e

13) If x= 2 cost + cos 2t and y= 2 Sint - sin 2t, at t=π/4.                1 -√2

14) If y= x+ x²+ x³+...... Where |x|< 2, then for |y|< 1 the value of dx/dy is...                          1 - 2y+ 3y² - .......

15) sec⁻¹{1/(2x² -1) w r t. √(1- x²) at x= 1/2.                                              4

16) log₅(log₇x) (x>7).          1/(x log5 log 7 log₇x)

17) 2y= (x - a) √(2ax - x²) + a² sin⁻¹{(x-a)/a}.                   √(2ax - x²)

18) (secx + tanx)/(secx - tanx).       2 secx (secx + tanx)²

19) Tan⁻¹[{√x(3-x)}/(1- 3x)].        3/{2(1+x)√x}

20) y= logₐx+ logₓa + logₓx + logₐa.        1/(x loga) - loga/{x(logx)²}

21) (x+y) ᵐ⁺ⁿ = xᵐ yⁿ.                   y/x

22) sin(π/6 eˣʸ) at x= 0.      √3 π/24

23) y= log (tan x/2)+ sin⁻¹(cosx).     Cosecx +1

24) sin² cot⁻¹[√{(1-x)/(1+x)}].        1

25) x= sin⁻¹(3t - 4t³) and y = cos⁻¹√(1- t²).                               1/3

26) Cosec⁻¹{(x+1)/(x-1)} + Tan⁻¹{(x-1)/(x+1)}.                                      π

27) Y= Tan⁻¹[{√(1+x²) -1}/x] and z= Tan⁻¹{2x/(1- x²)} then find dy/dz.     1/4

28) sin⁻¹x w.r.t. cos⁻¹√(1- x²).        1

29) x= a(t+ Sint), y= a(1- cost).     tan(t/2)

30) x√(1- y²)+ y√(1- x²)= k at x= 0.     - √(1- k²)

31) tan⁻¹[√(1+ x²)- √(2- x²)}/{√(1+x²) + √(1- x²)}.                         x/√(1- x⁴)

32) tan⁻¹[{2x √(1- x²)}/(1- 2x²)] w.r.t. tan⁻¹{√(1+x²) -1}/x.                        4

33) If y= Log(x+y)= 2xy then the value of y'(0) is.                              1

34) y= sin⁻¹(x² √(1- x)) - √x √(1- x⁴).   

35) sec{(x²-y²)/(x²+ y²)}= eᵃ.       y/x

36) If f(x)= √(1+ cos²(x²)) then the value of f'(√π/2) is.              - √π/√6

Find d²y/dx²

36) a sin³t w.r.t a cos³t at t=π/4.    0

38) sinx + eˣ.         (Sinx - eˣ)/(cosx +eˣ)

39)  a cos mx - b sin MX.     -m²y

40) f(x)= sin3x cos4x at f"(π/2).  25

41) y= aˣ b²ˣ⁻¹.                y(log ab²)²

42) y= 1/(1+ x+ x²+ x³) at x= 0.     0

43) If y= sinx log(tan x/2) then the value of d²y/dx² + y is.            tanx

44) x= t²+ 2t, y= t³ - 3t at t= 1.    3/8

45) If eʸ + xy = e² at x= 0.                0

46) x= a cot t and y= 1/(x²+ a²) at t= π/6.                                            1/4a⁴

47) If dx/dy= u and d²x/dy²= v, then the value of d²y/dx² is.             v/u²

48) If y= (x + √(1+ x²))ⁿ then (1+ x²) d²y/dx² + x dy/dx =?                   n²y

49) If log x= z, then find the value of x² d²y/dx².                d²y/dx² - dy/dz

50) The value of x, at which the first derivative of x + 1/x w r t x is 3/4 is.                               ±2

51) If y= x³ then the value of d²y/dx²/{1+ (dy/dx)²}³⁾² at the point (1,1).                       3/5√10

52) If x= 1/z, y= f(x) and d²y/dx² = kz³ dy/dx + z⁴ d²y/dx², then the value of k is.                              2

53) If xy = ax² + b/x then the value of x² d²y/dx² + 2x dy/dx.              2y

54) If y= sinx° and z= log x then the value of dy/dz is.        x°sinx°/ log e 

55) If x= sin t, y= cos pt prove (1- x²) d²y/dx² - x dy/dx + p²y= 0

56) x= eᵗsin t and y= eᵗcos t, prove (x+y)² d²y/dx² - 2x dy/dx is.     -2y

57) If y= f(x²) and f'(x)=√(3x²+1) then the value of dy/dx at x= 2.   28

58) If x= sec t - cos t, y= secⁿt - cosⁿ t and (x²+4)(dy/dx)²= k(y² +4) then the value of k is.                        n²

59) If y² = 4ax, then the value of d²y/dx². d²x/dy².                    -2a/y³

60) f(x)= logₓ(logₑx), then the value of f'(e) is.                                  1/e

61) If (cos⁻¹x)², then prove (1- x²) d²y/dx² - x dy/dx = 2

62) If y= sin(x²), z= ₑy² and t= √z, then the value of dt/dx is.    2xyz/t  cos(x²)

63) If x + y= eˣ⁻ʸ Prove dy/dx = {2(x+y)}/(x+y+1)³

64) If x= 2 cos t- cos 2t and y= 2 Sint - sin2t, then the value of d²y/dx² at t= π/2 is.        -3/2

65) If (sin⁻¹x)² + (cos⁻¹x)², then prove (1+ x²) d²y/dx² - x dy/dx = 4

66) If x²+ y² = t+ 1/t and x⁴ + y⁴ = t² + 1/t², then the value of - x³y dy/dx is..       1

67) If siny = x sin(a+ y), then prove dy/dx= (sin²(a+y))/(sina)

Mean Theorem

1) let f(x)= eˣ, x ∈ [0, 1], then a number c of Lagrange's mean value theorem is..                         log(e -1)

2) If the function f(x) satisfies the conditions of Rolle's theorem in (1,2) and f'(x) is continuous in (1,2), then ²₁∫ f'(x) dx is.                        0

3) If f(x)= x(x -1)(x -2), 0≤ x ≤ 4, then the point x= c which satisfies mean value theorem satisfies.      1< c <3

4) The value of c in Rolle's theorem when f(x)= 2x³ - 5x² - 4x +3, x ∈ (1/2,3).                          2

5) The mean value theorem f(b) - f(a)= (b - a)f'(c) (a < c < b), if a= 4, b= 9 and f(x)=√x, then the value of c is.            6.25

6) if the function f(x)= 4x³ + ax² + bx - 1 satisfies all the conditions of Rolle's theorem in -1/4 ≤ x≤ 1 and f'(1/2)=0, then the value of a and b.          1, -4

7) in the mean value theorem f(a+ h)= f(a) + f'(a+ ¢h) (0<¢<1), if f(x)= √x, a=1, h= 3, then the value of ¢.         5/12

8) If the conditions of Rolle's theorem are satisfied by the function f(x)= x³ + ax² + bx -5 in 1≤ x≤3 with c= 2 + 1/√3, then the value of a and b.                  -6, 11

9) In the mean value theorem f(b) - f(a)= (b - a) f'(c) (a< c< b), if a=π/6, b= 5π/6 and f(x)= log(sinx), then the value of c is.                         π/2

10) if the function f(x)= x⁴ + ax² - bx +4 defined in -2≤ x ≤ 2 satisfies Rolle's theorem when c= 1/3 (1+ √3), then the value of a and b.     -1, 4

INTEGRATION (Definite and Indefinite)

1) ∫ eˣ(1- cotx + cot²x) dx.          - eˣ cosecx

2) ∫ dx/√(e²ˣ -1).               Sec⁻¹(eˣ)

3) ∫ sinx/sin(x -a).            (x-a) cosa + sina log|sin(x -a)|

4) ¹₀∫ d/dx [sin⁻¹{2x/(1+ x²)}].     π/2

5) ²₋₂ ∫ |1 - x²|dx.                            4

6) ∫ sin 2x log(tanx) dx at (π/2,0).   0

7) ∫ dx/√(2x - x²).               sin⁻¹(x -1)

8) ∫ xeˣ/(x+1)² dx.                 eˣ/(x+1)

9) ∫ cos⁻¹(1/x) dx.               x sec⁻¹x - log|x + √(x² -1)|

10) ∫ √x/{√(a-x)+ √x} at ((an-1)/n, 1/n).                (an-2)/2n

11) ∫ x/(x² +4x+5).            1/2 log|x²+ 4x+5| - 2 tan⁻¹(x+2)

12) ∫ x|x| dx at (1,-1).                   0

13) ∫ {1+ x +√(x+x²)}/{√x + √(1+x)}.    2/3 √(1+x)³

14) ∫ ₑ√x.                     2(√x - 1) ₑ√x

15) If ∫ x sinx dx= - x cos x+ m, then the value of m is.                    Sinx

16) ∫ 1/√x . ₐ√x dx.      2.ₐ√x /(loga)

17) ∫ ₑtan⁻¹x /(1+ x²).           ₑtan⁻¹x 

18) ∫ dx/(1+ cotx) at (π/2,0).      π/4

19) ∫ U= dx/logx at (e², e) and V= ∫ eˣ/x dx at (2,1), then find the value of U - V is.                  0

20) ∫ {2x(1+ sinx)}/(1+ cos²x) at (π, - π).                  π²

21) ∫ dx/(sinx - cosx +√2).        -1/√2 cot(x/2 +π/8)

22) ∫ cos³x ₑ sin²x. dx at (π,0).        0

23) ∫ dx/(5+ 3 cosx) at (π,0).       π/4

24) ∫ ₑ log(tanx).               Log(secx)

25) ∫ (1+ x - 1/x)ₑ x+ 1/x.           x. ₑ x+ 1/x

26) ∫ x dx/(a² cos²x + b² sin²x) at (π,0).              π²/ab

27) ∫ √tanx/sinx cosx.          2√tanx

28) ∫ x sinx at (π/2,0).                 1

29) ∫ log(tanx) at (π/2,0).               0

30) ∫ log(sin²x) at (π,0).               2π log(1/2)

31) ∫ (logx)² at (e,1).                    e -2

32) ∫ dx/(eˣ+ 1/eˣ).                tan⁻¹eˣ

33) ∫ x(1- x)ⁿdx at (1,0).         1/(n+1) - 1/(n+2)

34) ∫ {log (1+ x)}/(1+ x²) at (1,0).    π

35) ∫ sinx/(sinx + cosx) at (π/2,0).   π/4

36) ∫ x³ sin²x at (π/7, -π/7).             1

37) ∫ cosec⁴x.          - cotx -1/3 cot³x

38) ∫ dx/{2√x(x +1)}.              tan⁻¹√x

39) ∫ dx/(x² + 2x cos a +1) at (1,0).        a/2sina

40) ∫ sin³x cosx.            1/4 sin⁴x

41) ∫ sin[2 tan⁻¹√{(1+x)(1-x)}] at (1,0).                π/4

42) ∫ √(1+ sin(x/4)).          8(sin x/8 - cos x/8)

43) ∫ | log x|dx at (e, 1/e).    2(1- 1/e)

44) ∫ | sinx| dx at (10π, π).           18

45) ∫ dx/(x² +4x +13).      1/3 tan⁻¹(x +2)/3

46) ∫ cosx log{(1-x)/(1+x)} at (1/2,-1/2).                0

47) ∫ (3x+1)/(x²+9) at (3,0).      Log(2√2)+ π/12

48) ∫ eˣ{(1+ sinx)/(1+ cosx).    eˣ tan x/2

49) ∫ tan⁻¹{1/(x²- x+1)} at (1,0).      π/2 - log 2

50) ∫ aˣ⁾² /√(a⁻ˣ - a ˣ).       1/log a sin⁻¹ aˣ

51) ∫ √{1-x)/(1+x) at (1,0).        π/2 -1

52) ∫ |1- x²|dx at (3,-2).              28/3

53) ∫ x³ log x.           1/16 (4x⁴ log x - x⁴)

54) ∫ x dx/{(1+x)(1+ x²)} at (∞,0).    π/4

55) ∫ eˣ/{(eˣ+2)/(eˣ+1)}.    Log{(eˣ+2)/(eˣ+1({

56) ∫ dx/(cosx - sinx).        1/√2 log| tan (x/2 - π/8)|

57) ∫ (sinx + cosx)²/√(1+ sin2x) at (π/2,0).           2

58) ∫ ³√(x - x³)/x⁴.               -3/8. ³√(1- 1/x²)⁴

59) ∫ eˣ ˡᵒᵍ ᵃ eˣ dx.       (ae)ˣ/log(ae)

60) ∫ {(log x -1)/(1+ (lox)²)}².     x/(1+ (logx)²)

DIFFERENTIAL EQUATIONS

A) Find the Order and degree of:

1) √dy/dx - 4 dy/dx - 7x = 0.       1, 2

2) (d²y/dx²)⁵ + 4 {d²y/dx²)³/(d³y/dx³)}  + d³y/dx³ = x² -1.                                                 3,2

3) [a + (dy/dx)⁶] ⁷⁾⁵= b d²y/dx².   2,5

4) dy/dx - x = (y - x dy/dx)-⁴

5) +1+ 3 dy/dx)⁴⁾³= 4 d³y/dx³.   3,3

6) (d³y/dx³)²⁾³ - 3 d²y/dx² + 5 dy/dx + 4y = 0.                         2

B) Write the Integrating factor of:

1) (x+1) dy/dx - ny = eˣ(x +1)ⁿ⁺¹.  1/(x+1)ⁿ

2) x dy/dx +(x -1) y = x².           eˣ/x

3) dy/dx + y tanx = secx.         secx

4) 3 dy/dx + 3y/x = 2x⁴y⁴.    1/x³

5) cosx dy/dx + y sinx = 1.       secx

6) cos²x dy/dx - y tan 2x = cos⁴x.             (1- tan²x)

Solve:

1) (x+y) dx + x dy = 0.      x²+ 2xy= c

2) dy/dx= xy + 2y at (1, 1).     y= ₑ(2x+ x²/2 - 5/2)

3) dy/dx= ʸ ⁻ ˣ.       2 ⁻ ˣ - 2 ⁻ʸ = c

4) dy/dx = eˣ ⁻ʸ + 1.    e ʸ⁻ ˣ = x+ c

5) x log x dy/dx + 2y = log x.     (Log x)²

6) dy/dx - y tan x = - 2 sinx.     y cosx = 1/2 cos 2x + c

7) cosx siny dx + sinx cosy dy= 0.    |sinx siny| = c

8) dy/dx = (x-y)/(x+y).        x² - y² - 2xy = c

9) y - x dy/dx = a(y² + dy/dx).      |(x+ a)(1- ay)|= c|y|

10) dy/dx = y/x + p(y/x)/p'(y/x).      |p(y/x|=k|x|

11) (x +2y³) dy/dx = y.      x= y(y² + c)

12) dy/dx = (1+ y²)/(1+ x²).    y - x = c(1+ xy)

13) dy/dx = √(1- x² - y² + x²y²).   sin⁻¹y = x/2  √(1- x²) + 1/2 sin⁻¹x+ c

14) dy/dx= (x log x² + x)/(siny + y cosy).      y siny = x² log x + c

15) y dx + (x + x²y)dy = 0.       -1/xy  + log y + c

16) ₑdy/dx = x +1, when y(0)= 3.    y = (x+1) log|x +1| - x +3

17) dy/dx + y = e⁻ ˣ , y(0)= 0.    y = xe⁻ ˣ 

18) cot y dx = x dy.          |x|=|secy|

19) y² dx +(x² - xy + y²) dy= 0.    tan⁻¹(x)y) + log y + c= 0

20) cos²x dy/dx + y = tan x.    y= tanx - c ₑ- tanx

21) tan y d/dx = sin(x+y)+ sin(x -y).    secy + 2 cosx = c

22) (2y-1) dx - (2x +3) dy = 0.    |(2x +3)/(2y -1)| = c

23) (2x - y+1) dx + (2y - x +2) dy = 0.    x² + y² - xy + x + y = c

24) (x +y)(dx - dy) = dx + dy.    Log|x + y|= x - y + c

25) dy= cosx(2 - y cosecx) dx where y= 3/√2 when x= π/4.         y = sinx + cosecx

26) dy/dx + y = sinx.       y= ce⁻ ˣ 

27) sin⁻¹x + sin⁻¹y = c.        √(1- x²) dy + √(1- y²) dx = 0

28) tany sec²x dx + tanx sec²y dy = 0.       |tan x tan y| = k

29) dy/dx + (1+ cos 2y)/(1- cos 2x) = 0.                  tany - cot x= c

30) (1+ y²)+ (x - ₑ tan⁻¹y) dy/dx = 0.      2 x ₑ tan⁻¹y = ₑ 2tan⁻¹y + k

31) x dy - y dx + ax (x² + y²) dx = 0.       2 tan⁻¹(y/x) + ax² = c

32) 2(y +3)- xy dy/dx = 0, with y= -2 when x= 1.        |x²(y +3)³|= ʸ⁺²

33) (x +y)(dx - dy)= dx+ dy.      x+ y = ˣ⁻ʸ

34) dy/dx + y/x = sin x.        x(y + cosx) = sinx + c