Revised Questions (Maths)-XI State Board 20/21

25/11/20

1) Two equal area of two circles subtended angle 60 and 75 at the centre. Find the ratio of the radio of the two circles. 5:4

2) If cosx - sinx =√2 sinx, prove sinx + coax =√2 coax

3) If 7 coax + 5 sinx= 5, find the value of 5cosx - 7sinx. ±7

4) If secx + tanx = x show you that sinx = (x²-1/(x²+1)

5) If sinx +cosecx = 2 show that sin¹⁰ + cosec¹⁰ = 2

6) If tan⁴x + tan²x= 1 show that cos⁴x + cos²x=1

7) If co⁴x + cos²x =1, show that tan⁴x + tan²x = 1

8) If sinA, cosA,tanA are in G. P prove cot⁶A - cot²A = 1

9) If (secx -1)(secy -1)(secz -1)= (secx+1)(secy+1)(ssecz +1) show that the value of each side is

 ± tanx tany Tanz

10) If (a² - b²) sinx + 2ab cosx= a²+ b², find the of tanx. (a²-b²)/2ab

27/11/2

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11) If 100 times the 100th term of an A. P with non-zero common difference equals the 50 times it's 50th term, then find 150th term

12) If the product of the roots of the Equation 

x² - 2√2 kx + 2e² ˡᵒᵍ ᵏ -1=0 is 31, then find k.

13) Out of 64 students, the number of students taking maths is 45 and the number of students taking both maths and stats is 10. Then the number of students taking only Statistics is ?

14) Let S ᵢ denotes the sum of first n terms of an A. P. and S ₂ᵢ = 3S ᵢ. if S₃ᵢ= kS ᵢ Then find k

15) If A and B have n elements in common, then the number of elements common to AxB and BxA is ?

16) Find the greatest term in (1- x)⁻ⁿ when x= 3/4 and n=10.

17) Write down the fourth and fifth and fifth terms of (x + 1/x)⁸ in the simplified form.

18) Use the principle of Mathematical Induction Induction to prove:

1/(3.6) + 1/(6.9) + 1/(9.12) +... + 1/{3n(3n+2)} = n/{9(n+1)}.

19) Find the Quartile Deviation of the following frequency distribution:

Daily wages No. of workers

10-15 6

15-20 12

20-25 18

25-30 10

30-35 4

What is the interquartile range?

30/11/20

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20) If the pth, qth and rth terms of an AP respectively 1/a, 1/b and 1/c by show that, (q-r)bc + (r-p) ca + (p -q)ab = 0

21) How many terms of the series 1/1+ 1/3+ 1/6+...... must be taken so that the sum may be (-3/2) ?

22) Find the sum of 1-3+5 - 7 + 9 - 11 +.... to n terms.

23) How many even numbers are there between 15 and 150 ? Find the sum of all those numbers.

24) Find the sum of all the numbers between 200 and 300 which are multiples of 7.

25) If (p+1)th term of an AP be a, find the sum of first (2p+1) terms of the AP.

26) If the 11th term of an AP be 25, find the sum of first 21 terms of the AP.

27) There are (2n+1) terms in an AP. Show that the ratio of the sum of odd terms and the sum of even terms is (n+1): n.

28) Find the 99 term of the series 2+7 +14+23+34+ ........

29) How many terms are there in the series1+3+6+10+15+21+ .....+ 5050 ?

30) The sum of four numbers are in AP is and the sum of their squares is 120; find the numbers.

31) The sum of 6 numbers in AP in is 345 and the difference between the first between the first and the sixth is 55; find the numbers.

32) the 4th term of an AP is thrice the first term and the 7th term exceeds twice the third term by 2. Find the sum of first ten terms of the AP.

33) If 3rd, 7th, 12th terms of an A. P are three consecutive terms of a G. P, then find common ratio.

14) From 6 men and 4 women, the number of ways of forming a committee of 5 members, if there is no restriction on its formation, is..

15) If ¹²P ᵣ = ¹¹P₆ + 6 . ¹¹P₅ , then r is

19) If the letters of the word SACHIN are arranged in all possible ways and these words are written in dictionary order, then the word SACHIN appear at serial number ?

20) In a meetingting after every one had shaken hands with everyone else, it was found that 66 handshakes were exchanged. How many members were present at the meetings?

22) From a set of 17 balls marked 1, 2, 3, ....., 16, 17 one is drawn at random. What is the probability that its number is a multiple of 3 or 7 ?

Practice Paper (1) For JEE (Main & Advanced)

1) If A={7,8,9} and B={9,5} then (A∪B)x(A∩B) is
A){(7,9),(7,5),(8,9),(8,5),(9,9),(9,5)}
B){(5,9),(7,9),(8,9),(9,9)}
C) {(9,5),(9,7),(9,8),(9,9). D) none

2) If the number of elements in set A and in set B are m and n respectively, then the number of relations from A to B is
A) 2ᵐ⁺ⁿ  B) 2ᵐⁿ C) m+n  D) mn

3) If the relation R: A --> B, where A={1,2,3} and B={1,3,5} is defined by R= {(x,y): x< y, x ∈ B} then
A) R={(1,3),(2,3),(2,5),(3,5),(1,5)}
B) R={(1,1),(1,5),(2,3),(2,5)}
C) R⁻¹={(3,1),(5,1),(3,2),(5,3)}
D) R⁻¹={(1,1),(5,1),(3,2),(5,3)}

4) The domain of the real valued function f(x)= √(log₁₆ x² ) is
A) x> 0. B)|x|≥1  C) |x|≥4. D) x≥4

5) If x ∈ R then (x²-x+1)/(x²+x+1)
takes values in the interval.
A) (1/3, 3)  B) (-1/3,3) C) (0,3) d) n

6) In a certain town, 25% families Own a phone and 15% own a car car a car car 65% families all neither a phone nor a car 2000 families own both a car and a phone. Consider the following statements in this regard.
I) 10% family own both car and a phone.
II) 35% families own either a car or a phone.
III) 40000 families live in the town.
Which of the above statements are correct ?
A) 1 and 2.                   B) 1 and 3
C) 2 and 3.                 D)1,2 and 3

7) Let A={1,2,3} and B= {a,b}. Which of the following subset of AxB is a mapping from A to B ?
A) {(1,a),(3,b),(2,a),(2,b)}
B) {(1,b),(2,a),(3,a)}
C) {(1,a),(2,b)}.      D) none

8) A function f is defined for all positive integers and satisfies f(1) = 2014 and f(1)+ f(2) + ...+ f(n) = n² f(n), and n> 1. The value of f(2013) is
A) 2013/2014.      B) 2015/2014
C)1/2013.              D) 2/2013

9) The period the period of function cos(πx/3) + tan(πx/3)+3 is :
A) 2   B) 4       C) 6.    D) none

10) Which of the following is the domain of sin⁻¹{log₂(x³/2)} ?
A) 1<x<2, -2<x< -1
B) 1≤ x≤2, -2≤x≤-1
C) 1≤ x<2, -2≤x≤-1.  D) none

11) The domain of the function f(x)= ¹⁶⁻ˣ C ₂ₓ₋₁ + ²⁰⁻³ˣP ₄ₓ₋₅ , where the symbols have have their usual meaning meaning is the set.
A) {1,2,3,4,5}.      B) {2,3,4}
C) {2,3}.               D) none

12) The range of f(x)=
3sin√(π²/16 - x³) is
A) [-3,3].                    B) [0,3]
C) [0, √3/2].              D)[0, e/√2]

13) If A={1,3,5,7,9,11,13,15,17} , B{2,4,6,8,10,12,14,16,18} and N is the universal universal set, then
A'∪{(A∪B)∩B'} is
A) A.     B) A'.       C) B.      D) N

14) Let R be the relation on N defined by R={(a,b): a, b ∈N and a= b²}. Which of the following is True?
A) (a,a) ∈ R and a ∈ N
B) (a,b) ∈ R => (b,a) ∈ R
C) (a,b) ∈ R, (b,c) ∈ R => (a,c) ∈R
D) none of these

15) The range of
f(x)= cot⁻¹(log₄/₅ (5x²-8x+4) is
A) (0,π/2).                B) (π/4,π)
C) (-π/2, π/2).          D) (0, π/2)

16) The period of
f(x) = | sinx| + | cosx| is
A) π/2. B) π. C) 3π/2. D) 2π

17) If f(x) is an even function defined on (-5,5), then the sum of the squares the squares of all numbers satisfying the equation f(x)= f{(x+1)/(x+2)} is
A) 10. B) 12.  C) 15.   D) 8

18) If f(x)= (a - xⁿ)¹⁾ⁿ , a>0, n∈ N, then f(f(x))=
A) 1.  B) n.  C) x.    D) nx

19) If f: R --> R is defined by
f(x)= x - [x] - 1/2 and x ∈ R, where [x] denotes the greatest integer function then { x ∈ R : f(x)= 1/2} is:
A) Z, the set of all integers.
B) N, the set of all natural numbers.
C) ¢, the empty set
D) R, the set of of all real numbers.

20) The domain of f(x)=1/√(4-x²)
A) set of all real numbers.
B) set of all positive real numbers
C) (-2, 2).         D) [-2,2]

21) If A={1, 2, 3, 4, 5}, 
and B={2, 4, 6}, C={3, 4,6} then (A∪B)∩ C is :
A) {3,4,6} B) {1,2,3} C) {1,4,3} D) n

22) If A= {x:x is an even number}
B={ x:x is prime number}
C={x:x is a perfect square}
D={x:x is an odd number}
then which of the following two set are disjoint ?
A) A and B                B) B and C
C) C and D                D) D and B

23) Two sets A and B have 9 elements common. The number of common to each of the sets AxB and BxA are
A) 2⁹. B) 9².  C) 10.    D) 18

24)) The range of the function f(x)= sec⁻¹(1+ cos²x), If ([.] denotes the greatest integer function) is
A) (π/4, π/2).           B) (0, π/2). 
C) (0, sec⁻¹2).          D) (0, π/3). 
    
25) The function f(x)=(16ˣ - 1)/4ˣ
A) even function B) odd function
C) periodic function   D) none.

26) If f(x) satisfy the functional equation x² f(x)+ f(1-x) = 2x - x⁴, then f(1/3)=
A) 1/3  B) 1/9. C) 8/9. D) 10/9

27) if [x] denotes greatest integer ≤ x, and 2[x/8]² + 3[x/8]= 20, then x lies in the smallest interval [a,b] where b - a is equals to
A) 6         B) 5        C) 4      D) 8

28) The value of n belongs to I for which the function
f(x)=(sin nx)/sin(x/n) has 4π as its period is
A) 2       B) 3        C) 4        D) 5

29) Let R be a relation in N defined by R={(x,y): x+2y=8}, then range of R is
A) {2, 4,6}.           B) {1,2 ,3, 4, 6}
C) {1, 2,3}.        D) none of these

30) The graph of f(x)= cosx cos(x+2) - cos²(x+1) is
A) A straight line through (π/2, - sin²1) and parallel to x-axis.
B) a parabola with vertex (1, - sin²1)
C) a straight line passing through  origin.  D) none of these.

31) If f(x)= (1-x)/(1+x), then
f(f(1/x)) will be
A) x.   B) 1/x.    C) - x.  D) - 1/x

32) The domain of f(x)=√(log(2x-x²) is
A) 0<x≤1.               B) 0<x<2
C) 0<x≤2.               D) none

33) Let A={1,2,3} and B={2,3,4}, then which of the following relation is a function from A to B?
A) {(1,2),(2, 3),(3,4),(2,2)}
B)  {(1,2),(2, 3),(11,3)}
C)  {(1,3),(2, 3),(3,3)}
D)  {(1,1),(2, 3),(3,4)}

34) if 2f(x) - 3f(1/x)= x² (x≠0), the value of f(2) will be
A) 5/2.  B) -7/4. C) -1.  D) none

35) The range of y= 1/(2 - sin 3x) for all x is
A) 1/3 < y ≤1.       B) -1/3 < y ≤1.
C)  1/3 >y >1.       D) 1/3 >y >1. 

36) The function f (x) =
sin⁻¹[2x² -5], where [x] represents greatest integer function, has domain
A) [-√(7/2), -√2]. B) [√2, 7/2]
C) √[-7/2), √2]∪ [√2, 7/2]. D) n

37) Out of the 64 students, the number of students taking Mathematics is 55 and number of students taking both mathematics and Statistics is 10.  then the number of students taking only statistics is
A) 19.      B) 20.    C) 15.    D) 25

38) A and B are subset of the
universal set U set U such that n(U)= 800, n(A)= 300, n(B)= 400 and n(A ∩ B) = 100. The number of elements in the set A' ∩ B' is equal to:
A) 100  B) 200  C) 300    D) 400

39) If A, B and C are sets such that A∩B = A∩ C and A∪B)=A∪C then
A) A∩ B= null set.    B) A= B
C) A= C.                     D) B= C

40) If A and B are two sets, then (A - B)∪(B - A) ∪(A∩B)(A∩B) = ?
A) A∪B  B) A∩ B.   C) A.  D) B'

41) For any two sets A and
A - (A - B) equals to
A) B.   B) A - B.  C) A∩B. D) A'∩B'

42) If A and B are any two sets, then (A∪B)'∩(A'∪B)'
A) Complement of null set
B) A'.     C) B'.  D) universal set

43) Three sets A, B, C are such that A= B∩ C and B= C∩A, then
A) A is subset of B
B) B is subset of A
C) A= B          D) A subset of complement B.

44) If A, B, C are subsets of set X, then
(A'∩B'∩C)∪(B∩ C)∪(A∩ C) =?
A) A.   B) B.   C) C   D) X∩(A∪B∪)

45) If A= {a,b}, B={c,d}, C={d,c}, then {(a,c),(a,d),(a,e),(b,c),(b,d),(b,c)} is equal to
A) A∩(B∪C)          B) A∪(B∩C)
C) Ax (B∪C)          D) Ax(B∩ C)

46) In a group of 45 persons, 25 drink tea but not coffee, while 32  drink tea. How many persons drink coffee but not tea ?
A) 12.   B) 13.  C) 15.   D) 20

47) Out of 800 students in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball in hockey, 80 played cricket and basketball, 40 played cricket and hockey, 24 played all the three games. The number of students who did not play any game is
A) 128   B) 216   C) 240   D) 160

48) On the annual sports day,  school awarded 35 medal in athletics, 15 in Judo and 18 in swimming. If these medals goes to a total of 58 students and only three of them got medals in all the three sports, the number of students who received medals in exactly two of three sports are:
A)  9    B) 4     C) 5     D) 7

49) In a survey it is to be found that 70% of employees like bananas and 64% like apples. If x%  like both bananas and apples, then
A) x≥34.                          B) x≤64
C) 34≤ x ≤ 64.    D) All of these

50) A factory inspector examined the defects in hardness, finish and dimensions of an item. After examining 100 items he gave the following report; All three defects 5, defects in hardness and finish 10, defects in dimensions and finish 8, defect in dimensions and hardness 20. Defect in finish 30, in hardness 23 and in dimensions 50. The Inspector was fined because
A) The inspector took bribe.
B) the inspector's conduct towards the workers was not good.
C) the report of the Inspector was incorrect
D) none of these.

Daily Revised (Maths) X. State Board.

10/6/22

                 Group A

1) choose the correct answer from the following questions: (1x6)=6

 a) If the amount becomes ₹y for any principal the rate of simple interest of z% per annum for x years, then the principal will be

A) ₹ x y z B) ₹ xz/(100+y)

C) ₹100/(100+ xz)

D) ₹100z/(100+xz)

 b) If two roots of the equation ax²+bx+c= 0 (where a≠ 0) be equal, then 

A) c= - b²/4a B) c= b²/4a

C) c= - b/4a D) c= b/4a  

 c) The length of each of two parallel chords PQ and RS is 16cm. If the length of the diameter of the circle is 20cm, then the distance between two chords is 

A) 16 B) 14 C) 10 D) 12

2) Fill in the blanks: (any five) (1x5=5

 a) The compound interest and simple interest for one year at the fixed rate of interest on fixed sum of money are --------

 b) Determine the value of a for which the eqation, (a-p)²x² + (b-q)x + (c-r)= 0 will not be a quadratic equation. The value of a is_____.

 

 c) The region bounded by the two radii OA and OB and arc AB of a circle with Centre O is__

 3) Write true or false:(any five) 1x5= 5

  a) Ram invested ₹500 for 9 months and Memon invested ₹600 for 5 months in a business. The ratio of their profit share will be 2:3.         

  b) √3 is the smallest factors should be multiply the denominator to rationalize the denominator of 7√7÷ √48.

4) Answer the following question: (any ten). 2x10= 20

  a) If the rate of increase in population is r% per year, the population after n years is p; Find the present population.

  b) In partnership business the ratio of capitals of Bulu, Kohli and Bihan are as 1/5: 1/6 : 1/4. If they make a profit of ₹37000 at the end of the year, write by calculating profit of share of Bihan.

  c) By calculating, write the value of k for which the quadratic equation has real and equal roots. The equation is 9x²+3kx+4=0.

  d) If b∞ a³ and a increase in the ratio of 4: 3, find in what ratio b will be increased (simplified value).

e) If each edge of a cube is increased by 50%, then how much the total surface area of the cube will be increased in percent ? calculated it.

 l) If the aritmatic mean of the distribution given below is 8, find the value of p.

F: 3       5         8       10      11       13

X: 5       8         5        p        5        10

5) Answer any one question: 5x1= 5

a) Calculate the compound interest and amount on ₹16000 for 3/2 years at the rate of 10% compound interest per annum compounded at the interval of 6 months.

b) Neel, Sani and Riyama have started a small business by investing the capital of ₹65000, ₹52000 and 91000 respectively and just after one year they make a profit of ₹14400. If they divided 2/3rd of the profit equally among themselves and the remaining in the ratio of their capitals, then find the profit share of each.

6) Answer any one question: 3x1= 3

a) x/(x+1) + (x+1) = 25/12; x≠-1,0

b) If the price of one dozen pain is reduced by 6, then three more Pens will be get in ₹30. Before the reduction of price , let us calculate the price of one dozen pens.

7) Answer any one:. 3x1= 3

a) If a= (√5 +1)/(√5 - 1) and ab = 1, then what is the value of (a³+ b³)/(a³-b³) .

b) If y is a sum of two variables, one of which varies directly with x and other varies inversely with x. When x= 1 then y= -1 and when x=3 then y= 5. Calculate y when x= 6.

8) Answer any one:. 3x1= 3

a) if a, b, c and d are continued proportion, prove that, (b-c)²+ (c-a)² +(b-d)²= (a-d)²

b) if a²/(b+c) = b²/(c+a)= c²/(a+b) =1, then prove that 1/(1+a) + 1/(1+b) + 1/(1+c)= 1.

9) Answer any one: 5x1= 5

a) Draw a right angle triangle of which the length of hypotenuse is 12cm and one side of others is 5.5cn and draw a circumcircle (only traces of construction are required). 

b) draw a circle with radius of 2.8 cm length. Take a point which is at a distance of 7.5 cm from the centre. From that external point draw two tangents to the circle (only traces of construction required)

 

10) Answer any two:. 4x2= 8

b) The cross section of a rectangular parallelopiped wooden log is 2 metre length is a square and each of its side is 14 dcm in length. If this log can be converted into right circular log by wasting minimum amount of wood, then calculate what amount of wood in dcm³ will be wasted.

11) Answer any two:. 4x2= 8

a) Find the mean from the marks obtained by 64 students from the table given below:

Class-limit            Students

   0-5                           5

   5-10                        10

 10-15                        25

 15-20                        20

 20-25                        15

15/5/22

1) Solve 3x²+8x+1=0. Give your answer correct to two decimal places.

2) SOLVE: 24x² - 334x + 135=0

3) Atleast 3 person required for partnership business.        T/F

4) Capital ratio of A: B= 1/3: 2/5 and Capital ratio of B:C= 3:5. Then find the capital ratio of B and C.

5) A started a business with ₹5000. B and C joined him after 3 and 5 months with ₹6000, ₹8000 respectively. If total profit be ₹3400, then find their profits.

6) In a partnership business, Capital ratio of A: B is 5:4. They received ₹80 and ₹100. T/F

7) In partnership business the ratio of capitals of Bulu, Koli and Bihan are 1/5: 1/6: 1/4. If they make a profit of ₹37000 at the end of the year, write by calculating profit of share of Bihan.

8) Neel, Sunny, Riyama have started small business by investing the capital of ₹ 65000, ₹ 52000, and 91000 respectively and just after one year they make a profit ₹14400. If they divided 2/3 rd of the profit equally among themselves and the remaining in the ratio of their capitals, then find the profit share of each.

13/5/22

1) (x+3)/(2x+3)= (x+1)/(3x+2). -3±√6

2) ₹2000 is lent out for 10% compound interest, compounded annually. Calculate:

A) The amount at the end of the first year. ₹ 2200

B) The amount at the end of the second year. 2420

C) The total compound interest paid at the end of the 2 years. 420

3) find the difference between the simple interest and compound interest on ₹16000 at 8% p.a at the end of 1 year, if interest is compounded quarterly. 38.92

4) x/(x+1) - 4/(x+2)= 0. 3.24, -1.24

5) The bill of a party for a certain number of people is ₹19200. If there were 10 more persons, the bill each person had to pay would have reduced by ₹160. Find the number of people at the party. 30

6) A two digit number is such that the product of digits is 12. When 9 is added to the number the digits are interchanged. Find the number.

7) The sides of a right angled triangle are x cm, 4(x+1)cm, (4x+5) cm. Find x.

12/5/22

1) On what sum of money, the difference between the simple interest and Compund Intrest in 2 years at 5% per annum is Rs 15.

2) A certain sum of money is invested at 5% Intrest, compounded annually, for 3 years. If the Intrest computes to Rs.2522, determine the principal.

3) In how, many years will a sum of Rs800 at 10% per annum Compound semiannually become Rs926.10?

4) solve 

A) 3x² - x -7=0 and give your answer Correct to 2 decimal places.

B) solve: (4x²-1) - 3(2x +1) + x(2x+1)= 0

C) x² - 1/x² = 29/10(x - 1/x)

D) √(x+15) = x +3, x belongs N

E) √{x(x-3)}=√10

F) √(6x-5) - √(3x -2) = 2

5) If -3 is the root of x² - kx -27 =0, find k.

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= 0

7) 9x² - 3x - 2 = 0.

8) x² - 8x + 16 = 0.

9) (x² - 5x)/2 = 0.

10) 2x²= 3x + 35 

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

Find the nature of the roots.

1) 4x² - 4x+1=0

2) If m,n are roots of x²- px+q= 0, find

a) m²+n².                                  p²-2q

b) m³+ n².                              p³ -3pq

c) m-n.                                -√(p²-4q).

d) m⁴+ n⁴.                      p⁴- 4p²q+2q²

** Solve:

3) 3x² - x- 7 = 0.               1.70, -1.37

4) 1/(x+1) + 2/(x+2)= 4/(x+4).   2± 2√3

5) 2x² + √7x -7= 0.        √7/2, -√7

19/4/22

1) In a business Ram invested ₹1600 and Rahim invested ₹1400 for 4 months. If their profits are equal, then find the time for Ram invested.

2) In a business A and B received ₹1050 as profit. If A's and B's profit are ₹900, 630 respectively, then find the capital of B.

3) A, B started a business with ₹60000 and ₹75000 respectively. C joined them after 4 months with ₹40500. A and B draw their money as capital ratio. After 1 year they received ₹19575. Find their profits.

4) In a business, Capital ratio of A and B is 3:4 and the capital ratio of B and C is 6:5. If A's profit is ₹450. Find the C's Profit.

5) A, B, C started a business with ₹3000, ₹2000, ₹1500. After 4 months A withdraw half of his capital, after 9 months they received ₹2840 as profit. Find the profit each get.

6) Sobha and Masood purchased a car cost ₹250000. They sold them at ₹262500. If Sobha invested 3/2 times more than Masood, then find their profits.

7) In a partnership business, the ratio of their investment is 3:8:5. If 1st partner received ₹60 less than 2nd partner, then find total profit.

8) A, B, C started a business with ₹6000, ₹8000, ₹9000. After few months A invest ₹3000 more in the business. After one year, they received ₹3000 as profit. If C's profit is ₹1080, then find the time when A invest ₹3000.

10/4/22

7) The fourth proportional of 3,4,6

8) a is a positive number and if

a: 27/64 : ¾ : a, then the value of a is-

9) If a:b=m: n and b:c = p:q , then a:c is

10) If x²,4 and 9 are in Continued Proportion, find x.                              

1) A vertical stick 12m long casts a shadow 8m long on the ground. At the time a tower casts the shadow 40m long on the ground. Determine the height of the tower.      60m

2) The perimeter of two similar triangles are 30 and 20cm respectively. If one side of the first triangle is 12cm, determine the corresponding side of the second triangle.      8cm

3) The perimeter of two similar triangles ABC and PQR are respectively 36cm and 24cm. If PQ= 10cm, find AB.        15cm.

4) Two triangles BAC and BDC, right angled at A and D respectively, are drawn on the same base BC and on the same side of BC. If AC and DB intersect at P, prove AP x PC= DP x PB.

5) P and Q are points on sides AB and AC respectively of ∆ ABC. If AP= 3cm, PB=6 cm,  AQ= 5cm and QC = 10cm, show BC = 3PQ

6) Two poles of height a metres and b metres are p metres apart. Prove that the height of the point of interaction of the lines joining the top of each pole to the foot of the opposite pole is given by ab/(a+b) metres.

7) In trapezium ABCD AB//DC and DC= 2AB.  EF drawn parallel to AB  cuts AD in F and BC in E such that BE/EC = 3/4, Diagonal DB intersects EF at G. Prove that 7 FE = 10 AB.

8) A vertical stick 10cm long casts a shadow 8cm long. At the same time a tower casts a shadow 30m long. Determine the height of the tower.     37.5m

9) The perimeters of two similar triangles are 25cm and 15cm respectively. If one side of first triangles is 9cm, what is the corresponding side of the other triangle ?     5.4cm

10) in ∆ ABC and DEF, it is being given AB= 5, BC= 4cm and CA= 4.2cm, DE= 10cm, EF= 8cm and FD= 8.4cm. If AL perpendicular to BC and DM perpendicular to EF, find AL : DM.      1:2

11) D and E are the points on the side AB  and AC respectively of ∆ ABC such that AD= 8cm, DB= 12cm, AE=  6cm and CE= 9cm. prove that BC=5/2 DE.

12) DE is the midpoint of the side BC of ∆ABC. AD is bisected at the point E and BE produced cuts AC at the point X. prove that BE: EX= 3:1.

Revision (cone)

1) The base radius and height of a cone are 15cm and 20cm respectively. Find
A) the slant height.                  25cm
B) Curved surface area.        1177.5
C) Total surface area.             1884
D) volume                       4710cm³

2) A sector containing an angle of 90° is cut from a circle of radius 42cm and folded into a cone. Find the radius and the curved surface area. 10.5,1386

3) The slant height and the base radius of a cone are 17cm and 8 cm respectively. Find the volume of the cone.                        1005.71 cm³

4) The circumference of the base of a cone is 66cm. If the height is 12 cm, find the volume of the cone.      1386 cm³

5) The curved surface area of a right circular cone is 12320cm²
if the radius of the base is 56cm, find its height.                      42cm

6) The slant height of a right circular cone is 13cm and its total surface area is 90π cm². Find
A) Its radius.                      5cm
B) Its volume in terms of π.      100π

7) Two coins have their heights in the ratio 1:3 the radii of their bases in the ratio 3:1. find the ratio of their volumes.            3:1

8) the radius and height of a right circular cone are in the ratio 5:12 and its volume is 2512 cm³. Find the radius and slant height of the cone. (π=3.14).              10, 26

9) A sector of radius 35cm is  cut out of a thin cardboard with angle 180°. It is folded into a cone that of maximum size. Find the curved surface and the volume of the cone.                     1925,9724.46

10) A wooden cone has an outer radius of 60cm and an inner radius of 50cm. The outer and inner heights are 40cm and 36cm respectively. Find the volume of the wood in the cone.(π=3.14).   56556 cm³

11) How many metres of Canvas 1.25m wide will be needed to make a conical tent whose base radius is17.5 and height 6m?    814m

12) There are two cones. The curved surface area of one is twice that the other. The slant height of the later it twice that of the former. find the ratio of their radii.          4:1

13) if the radius of the base of a circular cone is halved, keeping the height same, what is the ratio of the volume of the reduced cone to that of the original cone?               1:4

14) The vertical height of a right circular cone is three times its diameter and its volume is 54π cm³. Find its height.                   18cm

15) A conical vessel whose internal radius is 5cm and height 24cm is a full of water. The water is emptied into a cylindrical vessel of height 28 to the conical vessel with internal radius 10cm. Find the height to which the water rises in the cylindrical vessel.                   2cm

16) A rectangular tank whose dimensions are 30cm, 20cm and 10cm is full of water. The water is poured into a conical vessel of height 28cm. If the conical vessel is completely filled, find its base radius.                              15cm

17) A conical tent is to accommodate 11 persons. Each person must have 4m² of the space on the ground and 20m³ of air to breathe. Find the height of the cone.                      15m

18) A cone of maximum size is carved out of the cube of edge 14cm. Find the volume of the cone and of the remaining material.      718.7cm³, 2025.3 cm³

19) A solid cone of height 8cm and a base radius 6cm is melted and recast into identical cones, each of height 2 cm and diameter 1cm. Find the number of cones formed.                       576

20) A hollow cylindrical pipe of 50 cm long, whose external diameter is 7cm and the internal diameter is 5cm, is melted and recast into a right circular cone, whose base radius is 10cm. Calculate the height of the cone.           9cm

21) A tent of height 8.25m is in the form of a right circular cylinder with diameter of base 30m the height 5.5m surmounted by a right circular cone of the same base. Find the cost of the Canvas of the tent at the rate of 44₹ per m².            54450₹

22) The interior of a building is in the form of a cylinder of base radius 12m and height 3.5m surmounted by a cone of equal base and slant height 13m. Find the internal curved surface area and the capacity of the building.   5280/7m², 16368/7m³

23) From a cubical solid of metal 42cm x 30cm x 20cm, a conical cavity of a base radius 14cm and height 20cm is drilled out, find
A) the surface area of the remaining solid.            5857.6 cm²
B) The volume of the remaining cavity.                  21093.33cm³
C)  the weight of the conical cavity of the metal weight 7gm per cm³.             147.65 kg

24) A right triangle with side 3cm and 4 cm is revolved around its hypotenuse. Find the volume and surface area of the double cone thus generated.      30.17cm³, 52.8

25) What quantity of Canvas 1.25 m wide will be required to make a conical tent whose radius is 21m and slant height is 30m?    1584m

26) A girl fills a cylindrical bucket 32cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of sand. If the height of the conical heap is 24cm, find
A) Its radius                   36cm
B) its slant height.        √1872 cm
Leave your answer in square root form)

27) water flows at the rate of 10m per minute through a cylindrical pipe 5mm in diameter. how long would it take to fill a conical vessel whose diameter at the base is 40cm and depth 24cm?          51min 12 secs

28) An exhibition tent is in the form of the cylindrical  surmounted by a cone. The height of the tent above the ground is 85 m and the height of the cylinder part is 50m. if the diameter of the base is 168m. Find the quantity of Canvas required to make the tent. Allow 20% extra for folds and for stitching. Give your answer to the nearest m².   60508 80      or 60509m³.

29) From a solid cylinder whose height is 8cm and radius 6cm, a conical cavity of height 8cm and a base radius 6cm is hollowed out. Find the volume of the remaining solid correct to four significant figures. Slso find the total surface area of the remaining solid.      603.2cm³, 603.2 cm²

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Revision Test (Partnership)

1) 

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

12/8/21

1) The compound interest on a certain sum of money at 5% per annum for 2 years is ₹246. calculate the simple interest on the same sum for 3 years at 6% per annum.                                     432

2) what sum of money amount to ₹3630 in two years at 10% p.a compound interest.                3000

3) on a certain sum of money, the difference between the compound interest for a year, payable half-yearly, and the simple interest for a year is ₹180. Find the sum left out, if the rate of interest in both the cases is 10% p.a.                   72000

4) A man borrows ₹ 5000 at 12% compound interest p.a, interest payable every six months. He pays back ₹1800 at the end of every six months. calculate the third payment he had to make at the end of six months in order to clear the entire loan.                      2897.28

5) Calculate the compound interest for the second year on ₹800 invested for 3 years at 10% p.a.          880

6) A man invests ₹5000 for 3 years at a certain rate of interest compounded annually. At the end of one year amounts ₹5600, calculate,
a)  the rate of interest per annum.
b) the interest accrued in the second year.
c) the amount at the end of the third year.             12%, 672, 7024.64

7) A man invests ₹46875 4% per annum compound interest for 3 years. Calculate
a)the interest for the first year.
b) The amount standing to his credit at the end of second year.
c) the interest for the third year.      1875, 50700, 2028

8) A person invests ₹5600 at 14% p.a. compound interest for two years. calculate:
a) the interest for the first year.
b) the amount at the end of 1st year.
c) The interest for the second year, correct to nearest rupees.        784, 6384, 894

9) the compound interest, calculated yearly, on a certain sum of money for the second year is 880 and for the third year ₹968. calculate the rate of interest and the sum of money.     10%, 8000

10) A certain sum of money amounts ₹5292 in two years and to ₹5556.60 in three years, interest being compounded annually. find the rate%.             5%

11) At what rate percent, per annum compound interest, would 80000 amounts to ₹ 88200 in 2 years; interest being compounded half yearly ?            5%

12) A sum of money is lent out at compound interest for 2 years at 20% p.a, C. I being reckoned  yearly. If the same sum of money was lent out at compound interest at the same rate per annum, C. I being reckoned half yearly. It would have fetched ₹482 more by the way of interest. calculate the sum of money lent out.                     20000

27/7/21

1) Roots of a quadratic equation are 1/2 and -14. Find the equation.  2x²+ 27x -14= 0

Solve::

2) √(3x²+x+5)= x-3.               -4, 1/2

3) 8(t²+1/t²)- 42(t- 1/t)+29=0.     15/4, 3/2

4) 5ˣ⁺¹ + 5²⁻ˣ = 126.

5) 3²ˣ - 10.3ˣ+ 9=0.                      2,0

6) 2²ˣ⁻¹ - 9. 2ˣ⁻²+1 = 0.              2, -1

7) 6x² - x -14= 0.

8) x² -8x -1280 = 0

9) 1/(2y-9) = 1/(y-3) + 4/5.

10) x⁵ +242= 243/x⁵. -3,1

**Correct up to 2 decimal places.    

11) x² -6x -16 = 0.          -1.90, 7.90

12) 2x² +11x -10 = 0. 

16) A man purchased some sheep for ₹4500. Three sheep were lost and the rest he sold for ₹30 more per sheep than he had paid. If his gain on the whole transaction is 8%, how many sheep did he buy?

17) the sum of the ages of a man and his son is 46 years and the product of their ages is 168 years. find the age of the Son.

18) The total surfaces area of a cylinder is 75.24cm² and its height is 3.6 cm. If its radius is x cm, find x 

***

24) √3 x²+ 10x - 8√3= 0.    2√3/3, -4√3

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

26) (x-2)/(x+2)+ (x+3)/(x -2)= 4.  ±2√3

27) 1/(x+1)+ 2/(x+2)= 4/(x+4).    2± 2√3

28) a(x²+1)= (a²+1)x, a≠ 0.      a, 1/a

29) 4x² - 4ax +(a² - b²)= 0.   (a±b)/2

30) x/(x+1)- 4/(x+2)=0     3.24,-1.24

31) 5/(x -1)+ 2x/(x- 2)=0   1.61,-3.11

32) 2x - 1/x= 7.              3.64, -0.138

33) 2/(x -1)+ 3/(x+2)= 4/(x+2).   0.23, -8.77

34) (x+3)/(x-3) - (1-x)/x = 17/4.   4, -2/9

1) (x+3)/(x-3) - (1-x)/x = 17/4.   4, -2/9

35) a/(ax-1) + b/(bx-1)= a+ b, a+b≠0, ab≠0.     (a+b)/ab, 2/(a+b)

Form the Quadratic equations whose roots are:

36) a) -2, 1.                         x²+x-2= 0

b) -3, -4.                          x²+7x +12=0

c) a,-b.                        x²-(a-b)x -ab=0

d)  -2/3, 4/5.                 15x²-2x -8=0

e) -3, 2/5.                  5x²+13x -6=0

f) 2/5, -1/2.              10x²+x - 2=0

37) An Aeroplane travelled a distance of 480 km at an average speed of x kmhr. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for

a)  the onward journey

b) the return journey

If the return journey took 30 minutes less than the onward journey, write an equation in x and find the value of x.           160

38) Car A travels x km for every litres of petrol, while car B travels (x+5) km for every litres of petrol.

a) write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.

b) If car A uses 4 litres of petrol more than car B in covering the 400 km, write down an equation in terms of x and solve it to determine the number of litres of petrol used by car B for the journey.     16 litres

39) In an auditorium, seats were arranged in rows and columns. The number of rows was equal to the number of seats in each row. when the number of rows was doubled and the number of seats in each row reduced by 10, The total number of seats increased by 300. Find

a) the number of rows in the original arrangement.

b) the number of seats in the auditorium after rearrangement.  30, 1200

40)  A Hotel bill for a number of people for overnight stay is ₹4800. If there were four people more, the bill each person had to pay would have reduced by ₹200. find the number of people staying overnight.                                         8

41) A trader boy x articles for a total cost of ₹600. 

a) write down the cost of one article in terms of x.

 if the cost per article were ₹5 more, the number of articles that can be bought for ₹600 would be 4 less.

b) Write down the equation in x for the above situation and solve it for x.                                                     24

42) The distance by road between two towns A and B is 216 km, and by rail it is 200 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. calculate:

a) the time taken by the car to reach town B from A, in terms of x

b) If the train takes two hours than the car, to reach town B, obtain an equation in terms of x, and solve it.

c) Hence, find the speed of the train.                                   52 km/hr

43) A train covers a distance 600 km/hr. Had the speed been (x+20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. write down an equation in terms of x and solve it to evaluate x.                                   40 

44) x/(x+1) + (x+1)/x= 34/15, x≠ 0, x≠ -1.                                  3/2, -5/2

45) (x+3)/(x -2) - (1- x)/x= 17/4,     4, -2/9

46) 4/x - 3 = 5/(2x+3)             -2, 1

47) 2x/(x -3)+ 1/(2x+3)+ (3x+9)/{(x-3)(2x+3) =0.                              -1 

48) mx²/n + n/m= 1 - 2x.    -(n±√mn)/m

49) (x-a)/(x-b)/+ (x-b)/(x-a)= a/b + b/a.                                 0, a+b  

50) 1/{(x-1)(x-2)} + 1/{(x-2)(x-3)} + 1/{(x-3)(x-4) = 1/6.              -2,7 

51) (x-5)(x-6)= 25/(24)².      145/24, 119/24

52) 7x+ 3/x= 178/5.             5, 3/35

53) a/(x-a) + b/(x-b)= 2c/(x-1).  0, (2ab -bc-ac)/(a+b+c) 

54) x² + 2ab= (2a+b)x.             2a, b 

55) (a+b)²x² - 4abx - (a-b)²= 0.     1, -{(a-b)/(a+b)}² 

56) a(x²+1) - x(a²+1)= 0.       a, 1/a 

57) x² - x- a(a+1)=0.               -a, a+1 

58) x² +(a+ 1/a)x +2= 0.       -a, -1/a 

 

** Find the values of k for which the roots are real and equal..

59) x² -2(5+2k)x+3(7+10k)= 0.    2, 1/2

60) (3k+1)x² +2(k+1)x+ k= 0.   -1/2, 1

             

10/7/21

1) Find the simple interest on ₹5500 at 7% p.a. from Marc6, 25 to August , 18 of a particular year.           154

2) What sum invested at 9% per annum will yield a simple interest of ₹630 in 7 months?          12000

3) What sum will amount ₹15250 in 2 years 9 months at 8% per annum simple interest?        12500

4) In what time would a sum of money amount to three times itself at 15% p.a. simple interest?   13 years 4 months.

5) At what rate percent p.a. simple interest, would a sum double itself in 6 years ?         50/3 % p.a.

6) Adam borrowed some money at the rate of 6% p.a for the first two years, at the rate of 9% p.a. for the next 3 years, and at the rate of 14% p.a. for the period beyond 5 years. If he pays a total interest of ₹11400 at the end of 9 years, how much money did he borrow ?         12000

7) At a certain rate of simple interest, a sum amounts to ₹4760 in 3 year and #5600 in five years. find the sum and the rate percent per annum.        3500, 12%

8) A sum of ₹30,000 was lent at simple interest, partly at 12% p.a. for 7/2 year and partly at 25/2% p a for 4 years.if the total interest earned ₹13720, find the  sum lent at each rate.        16000, 14000

9) Divide 2379 into 3 parts so that their amounts after 2 ,3 and 4 years respectively may be equal, the rate of interest being 5% per annum at  simple interest.      1098, 732, 549

9/7/21

1) Find the volume, the surface area and the diagonal of a cuboid 30cm, long 24cm wide and 18cm high. 12960cm²,3384cm²,42.42cm

2) The diagonal of a cube is √12 cm. what is its edge?              2cm

3) find the volume of the cube whose diagonal is 2.5m.       3.01m³

4) find the surface area of the cuboid whose length is 16cm, breadth 15cm and height 8.5cm. 1007cm²

5) find the length of the longest rod that can be placed in a room 12m long, 9m broad and 8m high.    17m.

6) If V is the volume of a cuboid of dimensions a, b, c and S is its surface area, then prove that 1/V= 2/S(1/a + 1/b + 1/c).

7) The area of three adjacent faces of a cuboid are x, y ,z. The volume is V, prove that V²= xyz.

8) A rectangular water reservoir contains 105m³ of water. Find the depth of the water in the Reservoir if its base measure 12m by 3.5 m.             2.5m

9) Cubes A,B,C having edges 18cm, 24cm and 30cm respectively are melted and moulded into a new cube D. find the edge of the bigger cube D.                      36cm

10) The breadth of a room is twice its height , one half of its length and the volume of the room is 512dm³. find its dimensions. 16, 8, 4

8/7/21

1) 8x² +15= 26x.                   5/2,3/4

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

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

4) 2x -3= √(2x²-2x+21).               6

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

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

7) x² - 4x -12, x belongs to N.    6

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

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

10) 1/x - 1/(x+2)= 1/24.          -8,6

3) A girl fills a cylinder bucket 32cm in height and 18cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24cm, find

A) radius. B) slant height (leave your answer in square root form).   36, √1872

4) Prove: sinA/sin(90-A) + cosA/cos(90-A) = sec(90-A) cosec(90-A).

5) X: 13   15  18  20  22  24  25

     F:  6      4   11  9   16  12    2 find a) median b) lower quartile c) upper quartile d) semi-interquartile.    21, 18, 22, 2.

6) 1/(√6-√5) - 3/(√7-√20) - 4/(√6+√2).

6/7/21

1) A hollow metallic hemisphere is uniformly 6cm thick everywhere. It has external radius of 4cm. Calculate

A) whole surface area of the hemisphere, correct to 2d.p. 187.14

B) weight of the hemisphere, if 1cm² of the metal weigh 3.2gm. answer to nearest gm. Use π= 3.142.             297gm

2) Given A= 30 and B= 60 verify sin(B-A)= sin B cos A - cos B sin A.

3) Given 2 tanx= 5 find (3sinx - 4 cosx)/(sinx + 4cosx).          7/13

4) Following table gives the basic salaries of person employed in an office.

Salary no.  Of employees

200-300        11

300-400        10

400-500        15

500-600         8

600-700         4

A) using above information Calculate cumulative frequencies of the employees.

B) estimate the median.          420

5) Rationalise: 1/(1+ √2+√3).

6) Find the value of x³ - 6x²+ 9x+8 when x= 2+√3.               10

7) If ₹20000, amounts to₹21200 after a year, at CI. Calculated yrly. Find

A) the rate of interest.              6%

B) the amount to which it would amount to at the end of the second year.                              22472

5/7/21

1) solve: √(2x+1) +√(3x+4)=7.     4

2) ₹5000, lent out on C. I amounted to₹5400 at the end of the 1st year. To what sum will it amount to at the end of the 2nd year.          5832.20

3) find: 3 sin35 sec55 + 2cos32 cosec58.       

4) find the median, mean and mode of 3,7,10,6,9,5,16,7.       

5) A cylinder whose height is equal to its base diameter has the same volume as a sphere of radius 4cm. Find the base radius of the cylinder. Give answer to 2d.p.

4/7/21

1) Calculate compound interest for the 2nd year on ₹6000, invested for 3 years at 8% p.a.                   998.40

2) evaluate : 2tan40/cot50 - cosec61/sec29.                            1

3) The hypotenuse of a right angled triangle is 17cm. And one of the other side is one less than twice the other side. Find the length of the sides.                                     8, 15cm

4) A metallic hemisphere of radius 6cm melted and recast into a cone whose base radius is 12cm. Calculate the height of the cone.  3

5) solve: 2x² - 3x -5= 0.           5/2,-1

29/6)21

1) A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter of the sphere is 14cm and the total height of the vessel is 13cm. Find its capacity.                             4928/3

2) How many spherical lead shots each 4.2cm in diameter can be obtained from a rectangular solid of lead with dimensions 66cm x 42cm x 21cm ?                                 1500

3) What sum invested for 3/2 years compounded half-yearly, at the rate of 4% p.a. will amount to ₹142651 ?        125000

4) salary (₹).    No.of people

       400                 10

       600                  8

       800                  6

      1000.               10

      1200                10

       1500                6

Find mean salary.                     892

5) Find the mean, median, and mode of: 7,4,6, 4, 5, 8, 9, 11, 10, 15, 4, 3.                                     7, 6.5, 4

28/6/21

1) The height of a cone is 5cm. Find the height of another cone whose volume is sixteen times its volume and radius equal to its diameter.  20

2) A hemispherical bowl full of water is emptied in a cone. The radii of the vessel and the cone are 12cm and 8cm respectively. Find the height of water in the cone.  54

3) If the mean of 10 Observation is 20 and that of another 15 observations is 16, find the mean of all 25 observations.                 17.6

4) The value of a flat worth ₹500000 is depriciating at the rate of 8% p a. In how many years will its value be reduced to ₹389344?      3

5) solve: 21/x² - 29/x -10= 0.    -6,-4

6) weight       no.of students

     30-35             4

     35-40             16

     40-45             40

     45-50           222

     50-55             10

     55-60              8                    44

Find median

7) class        frequency

   00-10            12

   10-20            16

   20-30             6

   30-40             7

   40-50             9                           22

27/6/21

1) Determine the ratio of the volume of a cube to that of a sphere which exactly fit inside the cube.                                         6:π

2) Money is invested for 2yrs Compound interest. Find the rate if money will have doubled?       41.4%

3)

Marks    No. Of students

0-9              5

10-19          9

20-29         16

30-39         22

40-49         26

50-59         18

60-69         11

70-79          6

80-89          4

90-99          3

A) find median.                 42.6, 10

B) the number of students who obtained more than 75% marks.

4) Find the mean of the following:

X: 200  300  400  500  600  700

F:   5      11     10    10    8       6     23

26/6/21

1) The marks scored by 40 pupils of a class in a test were as follows:

X:  0     1       2     3     4       5 

F:  2     4       5    14   11      4

Calculate mean mark.                   3

2) Solve: 4x² - 4ax +(a² - b²)= 0. 3,-4

3) Using the Step-deviation method, find mean:

Class         frequency

50-60              9

60-70             11 

60-70             10

80-90             14

90-100            8

100-110        12

110-120        11                          85.8

4) Draw the histogram and hence the mode for the following:

Class         frequency

00-10              2

10-20              8

20-30             10

30-40              5

40-50              4

50-60              3                                23

5) The diameter of an iron Sphere is 18cm. The sphere is melted and is drawn into a long wire of uniform cross section. If the length of the wire is 108m, find its diameter.  6m

6) A man invests ₹46875 at 4% p.a compound interest for 3 yrs. Calculate:

A) the amount outstanding to his credit at the end of the second year.

B) the interest for the third year.   50700, 2028

7) The mean of five numbers is 18. On excluding one number, the mean becomes 16. Find the excluded number.                                    26

8) A vessel is in the form of an inverted cone. It's height is 11cm. And the radius of its top which is open is 2.5cm. it is filled with water upto the rim. When lead shots, each of which is a sphere of radius 0.25cm are dropped into the vessel, 2/5 of water flows out. Find the number of shots dropped into the vessel.                                        440

9) Find the median

Wages           No of workers

4000-4400            8

4400-4800           12

4800-5200           20

5200-5600           25

5600-6000           17

6000-6400           10.                  4800

10) calculate the mean, the median and mode of the numbers:

3,2,5,4,1,7,2,5,4,2.            3.5, 3.5, 2

25/6/21

1) The discriminant of 2x²-5x+3 is …

2) The ratio of the sum and the product of two roots of the Equation 7x² - 12x + 18 =0 -----

3) If two roots of the Equation ax²+bx+c (a≠0) are reciprocal to each other, then c = ----

4) If ax² + bx + c then find the sum and product of the roots.

5)Marks.               Students 

      0-8                          5

      8-16                        3

    16-24                      10

    24-32                      16

    32-40                       4

    40-48                       2

Find mean

6) a cylindrical water tank, base radius 1.4 metre and height 2.1 metre is filled with with the help of a pipe of radius 7cm. calculate the time(in minutes) required to fill the tank, given that water flows at the rate of 2m/s in the pipe.

7) use graph paper for this question.

 Monthly wages of some factory workers are given in the following table .

with 2cm= Rs 400 starting the origin at Rs4000 and to 2cm=10 workers on the y-axis, draw the Ogive. estimate the median from the graph.

Wages in Rs. No. Of workers.

4000-4400                  8

4400-4800                 12 

4800-5200                 20

5200-5600                 25 

5600-6000                 17 

6000-6400                 10                

8)(i) If 7 is the mean of

         5,3,0.5,4.5,b,8.5,9.5 find b.      

(ii) if each observation is decreased in value by 1 unit,what would the new mean be ?      

9) solve by formula

  (x+3)/(2x+3) =(x+1)/(3x+2).   

10) From the following table, find:

(i) The average wage of a worker, give your answer, correct to the nearest paise.

(ii) The modal class.

Wages in Rs. No of workers

Below 10             15

Below 20             35

Below 30             60

Below 40             80

Below 50             96

Below 60           127

Below 70           190

Below 80           200.             

24/6/21

1) John sends his servant to the market to buy oranges worth Rs15. The servant having eaten three oranges on the way. John pays 25 paise per orange more than the market price. Taking x to be the number of oranges which John receives, form a quadratic equation in x. Hence, find the value of x.

2) (4x²-1) -3(2x +1) +x(2x+1)= 0

3) x² - 1/x² = 29/10(x - 1/x)

4) √(x+15) = x +3, x belongs N

5) √{x(x-3)}=√10

6) √(6x-5) - √(3x -2) = 2

7) Nature of 4x² + 4x + 1=0

8) If one root of x²+ ax +3=0 is 1 then find a.

9) x= y √(z/k) then express k in terms of x,y,z.

10) If one of the roots of the quadratic equation 7x² +kx - 3 =0 is ⅔, then the value of k is ----

23/6/21

1) The Mean of 12,18,x,13,19,22 is 16, find x.

4) 11) An open cylindrical vessel is made of steel. The internal diameter is 14cm, the internal depth is 20.6cm and the metal is everywhere 4mm thick. Calculate

a) the internal volume

b) the volume of the metal correct to the nearest cm³.

5) X            frequency

   0-5             3

  5-10            7

 10-15          15

 15-20          24

 20-25          16

  25-30           8

Find Mean correct to 2 decimal.

6)a) If 7 is the mean of 5, 3, 0.5, 4.5, b, 8.5, 9.5, find b.                           18

b) If each Observation is increased in value by 1 unit, what would be the new mean?                               6

7) The surface area of a sphere is 1256cm². It is cut in to two hemispheres. Calculate:

A) the radius of the sphere.     10cm

B) the total surface area of a hemisphere.                         942cm²

C) volume of the hemisphere, correct to 2 d.p. (π=3.14).   2093.33cm³

D) A piece of butter 3cm by 5cm by 12cm is placed in a hemispherical bowl of diameter 6.5cm. Will the butter overflow when it melts completely ?                     Overflow

10) From the following, find

A) The average wage of a worker, Give your answer, correct to the nearest paise.

  Wages in ₹.   No of workers

Below 10            15

Below 20            35

Below 30            60

Below 40            80

Below 50            96

Below 60           127

Below 70           190

Below 80           200.            44.85

6) If 4, x,36,y are in Continued Proportion, find x and y.

9) From the top of a building AB, 60 metres high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60°. Find

a) the horizontal distance between AB and CD.

b) the height of the lamp post CD.

14) A man is standing on a level ground, observes that a pole 30m away subtends an angle of 50° at his eye, which is 2.0m above the ground level. Calculate the height of the pole. (Give your answer correct to a length of a metre).

23) If a:2= b: 5= c: 8 , then 50% of a = 20% of  = ---- % of c.

24) If mean proportional of (x -2) and (x-3) is x, then the value of x is…..

30/11/20

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

25) If a:b = m; b:c = m and d:c = m then a:b:c:d = ?

 

37) prove.√{(1+cos x)/(1-cos x)} =

     Codec x + Cot x.                   

7)Simplify: 

(cos 0°+ sin²45° - sin30°) ( sin90 - cos²45 + cos²60)

Revised Questions (Maths)- XI. (20/21) ISC

25/11/20

1) Two equal area of two circles subtended angle 60 and 75 at the centre. Find the ratio of the radio of the two circles. 5:4

2) If cosx - sinx =√2 sinx, prove sinx + coax =√2 coax

3) If 7 coax + 5 sinx= 5, find the value of 5cosx - 7sinx.             ±7

4) If secx + tanx = x show you that sinx = (x²-1/(x²+1)

5) If sinx +cosecx = 2 show that sin¹⁰ + cosec¹⁰ = 2

6) If tan⁴x + tan²x= 1 show that cos⁴x + cos²x=1

7) If co⁴x + cos²x =1, show that tan⁴x + tan²x = 1

8) If sinA, cosA,tanA are in G. P prove cot⁶A - cot²A = 1

9) If (secx -1)(secy -1)(secz -1)= (secx+1)(secy+1)(ssecz +1) show that the value of each side is

 ± tanx tany Tanz

10) If (a² - b²) sinx + 2ab cosx= a²+ b², find the of tanx.         (a²-b²)/2ab

27/11/2

--------:--

11) If 100 times the 100th term of an A. P with non-zero common difference equals the 50 times it's 50th term, then find 150th term

12) If the product of the roots of the Equation 

x² - 2√2 kx + 2e² ˡᵒᵍ ᵏ -1=0 is 31, then find k.

13) If 3rd, 7th, 12th terms of an A. P are three consecutive terms of a G. P, then find common ratio.

14) From 6 men and 4 women, the number of ways of forming a committee of 5 members, if there is no restriction on its formation, is..

15) If ¹²P ᵣ = ¹¹P₆ + 6 . ¹¹P₅ , then r is

16) Out of 64 students, the number of students taking maths is 45 and the number of students taking both maths and stats is 10. Then the number of students taking only Statistics is ?

17) Let S ᵢ denotes the sum of first n terms of an A. P. and S ₂ᵢ = 3S ᵢ. if S₃ᵢ= kS ᵢ Then find k

18) If A and B have n elements in common, then the number of elements common to AxB and BxA is ?

19) If the letters of the word SACHIN are arranged in all possible ways and these words are written in dictionary order, then the word SACHIN appear at serial number ?

20) In a meetingting after every one had shaken hands with everyone else, it was found that 66 handshakes were exchanged. How many members were present at the meetings?

21) Find the greatest term in (1- x)⁻ⁿ when x= 3/4 and n=10.

22) Write down the fourth and fifth and fifth terms of (x + 1/x)⁸ in the simplified form.

24) Use the principle of Mathematical Induction Induction to prove:

1/(3.6) + 1/(6.9) + 1/(9.12) +... + 1/{3n(3n+2)} = n/{9(n+1)}.

25) Find the Quartile Deviation of the following frequency distribution:

Daily wages No. of workers

10-15 6

15-20 12

20-25 18

25-30 10

30-35 4

What is the interquartile range?

26) Calculate the Karl pearson's coefficient of a correlation for the following data:

X: 78 89 96 69 59 79 68 62

Y: 25 37 56 12 07 36 23 04

30/11/20

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

27) If the pth, qth and rth terms of an AP respectively 1/a, 1/b and 1/c by show that, (q-r)bc + (r-p) ca + (p -q)ab = 0

28) How many terms of the series 1/1+ 1/3+ 1/6+...... must be taken so that the sum may be (-3/2) ?

29) Find the sum of 1-3+5 - 7 + 9 - 11 +.... to n terms.

30) How many even numbers are there between 15 and 150 ? Find the sum of all those numbers.

31) Find the sum of all the numbers between 200 and 300 which are multiples of 7.

32) If  (p+1)th term of an AP be a, find the sum of first (2p+1) terms  of the AP.

33) If the 11th term of an AP be 25, find the sum of first 21 terms of the AP.

34) There are (2n+1) terms in an AP. Show that the ratio of the sum of odd terms and the sum of even terms is (n+1): n.

35) Find the 99 term of the series 2+7 +14+23+34+ ........

36)  How many terms are there in the series1+3+6+10+15+21+ .....+ 5050 ?

37) The sum of four numbers are in AP is and the sum of their squares is 120; find the numbers.

38) The sum of 6 numbers in AP in is 345 and the difference between the first between the first and the sixth is 55; find the numbers.

39) the 4th term of an AP is thrice the first term and the 7th term exceeds twice the third term by 2.  Find the sum of first ten terms of the AP.

Revised Questions (Maths) XII. 20/21

REVISION QUESTIONS (Class XII)

                   2020/2021

22/11/20

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

1) If A = 0   2     3   And B= 7  6  3

               2   1     4                 1  4  5

Find the value of 2A +3B

2) prove by determinant

 1     x      x²-yz

 1     y      y²-zx  = 0

 1     z      z²-xy

3) If y= (x+√(x²-1)ᵐ prove that (x²-1)(dy/dx)²= m²y²

4) you are given the following results on two variables x and y: mean of x and y are 36, 85 S.D of x and y are 11, 8 covariance of x,y is 0.66. find the two regression equation and estimate the value of x when y=25.

5) Find X and Y, if X+Y = 7     0 

                                           2     5

And X - Y = 3     0  

                    0     3

6) Solve for x:  x     1      1

                          1      x     1  =  0

                          1      1     x 

With the help of DETERMINANT.

7) Given A= 1    2 and B= 4    5

                      2   3               5    6

Calculate     AB and BA

8) Given that y= (3x+1)²+(2x-1)³, find dy/dx. And the points on the curve for which dy/dx= 0

9) If A= 1   2

                2   1 , show that

A² - 3I= 2A, with this information find inverse of A

23/11/20

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

10) If A= 1 2

                2 1 , show that

A² - 3I= 2A, with this information find inverse of A

11) Prove with the help of DETERMINANT 

 9      9     12

 1     -3     -4       = 0

 1      9     12

12) Find the value of x,y,z of

x+y+z=9 ; z+x = 5 ;y+z 7 with the help of metrices.

13) If x= t log t, y=( log t)/t , then find the value of dy/dx at t=1

14) If A= 1     0 

               -1     7 

 then find K so that A² = 8A + KI where I is unit matrix.

15) If x=eᵗ sint, y= eᵗ cos t then show (x+y)² d²y/dx² = 2(x dy/dx - y)

16) Find the symmetric part of the matrix A = 1       2        4

                    6      8        2

                    2     -2        7

17) limₓ→₀{(11-cos2x)(3+cosx)}/ (xtan4x)

18) If P= 1       k        3

                1       3        3

                2       4        4 

is the adjoint of a 3x3 matrix A and 

| A|= 4, then find the value of k.

19) Find dy/dx of log₍₂ₓ₋₃₎(x² - 2x)

20) lim θ→Π/2 {θ -π/2)/cotθ}

21) differentiate sin(x² +5)

22) If y= eˣ(sin x + cos x), then prove that d²y/dx² - 2dy/dx +2y= 0

23) If √(1- x²) + √(1-y²)= a(x-y), then prove dy/dx=√{(1-y²)/(1-x²)}

24) If y= log tan(π/4 + x/2), then show that dy/dx - sec x= 0.

25) If xʸ + yˣ = aᵇ, find dy/dx.

26) If (3x⁴-2x+6)⁴(x-3)²⁾³ find dy/dx

27) Solve

a) sin⁻¹2x + cos⁻¹2x +2tan⁻¹x = π

b) cos⁻¹{(1-x²)/(1+x²) + tan⁻¹x=π/2

28) xʸ =e⁽ˣ⁻ʸ⁾, then find dy/dx

29) Find dy/dx if y= sin²x+cos⁴x

30) If sin y= x sin(a+y) find dy/dx

31)lim ₓ→₀ {sin⁻¹(x-2)}/(x²-4).

25/11/20

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

32) If A=  2    -1

                 1     2

 then Find A² +2A - 3I

33)lim ₓ→₀ {sin⁻¹(x-2)}/(x²-4)

34) If A= 1  3  0   and B=  1     3 

                2  1  4                 0     2

                                           -2    -1

 find AB

35) Find the value of x, y, z from the equation  

4        x-z   =  4      3

2+y    xz       -1    10  

36) Without expanding at any stage, find the value of the determinant:

 2      x        y+z

 2      y        z+x 

 2      z        x+y

37) Solve  sin⁻¹cos(sin⁻¹x)= π/3

38) Find k if M=   1       2

                              2       3   

and M² - kM - I₂ = 0

39) Find dy/dx, if x= at² and y= 2at

40) Find the intervals in which the function f(x) is strictly increasing where f(x)= 10 - 6x - 2x².

41) By using property of Determinant prove     x     x²      1+px³ 

               y     y²      1+py³ 

              z      z²      1+ pz³

= (1+pxyz)(x-y)(y-z)(z-x)

42) Prove:

 tan⁻¹1/2= π/4 -  cos⁻¹(4/5)

43) If y= ₑacos⁻¹x, where -1≤x≤1 then show (1-x²)y₂ - xy - a²y = 0

44) Evaluate A=  3     -2      3

                             2      1     -1

                             4     -3      2    and 

B= -1   -5     -1

      -8   -6      9

     -10   1      7      as AB. Hence solve the equation 3x - 2y +3z= 8, 2x+y-z=1, 4x-3y +2z= 4

44) If the lines of Regression are 4x+2y -3= 0 and 3x+ 6y +5=0, find the correlation coefficient between x and y.

45) The total variable cost of manufacturing x units in a firm is ₹(3x +x⁵/25). Show that average variable cost increases with 

output x.

26/11

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46) Find the regression coefficients of x on y and y on x

X: 2     6      4        7       5 

Y: 8     8      5        6        2

47) If the demand function is given by x= (600-p)/8, where the price is ₹ p per unit and the manufacturer produces x units per week at the total cost of ₹ x²+78x +2500, find the value of x which the profit is maximum.

48) The fixed cost of a new product is ₹35000 and the variable cost per unit is ₹500. If the demand function p= 5000 - 100x, find the break-even value (s)

49) A toy company manufacturers two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for the dolls of type B is atmost half of that for dolls of type A. Further, the production level of type A can exceed three times the Production of dolls of other type by at most 600 units. If the company makes a profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?

 

50) Determine the matrices A and B, when

A+ 2B= 1      2     0 

              6     -3     3 

             -5      3      1 and

2A - B = 2      -1     5 

               2      -1     6 

               0       1     2

51) ∫ (x⁴ - x² - x -1)/(x³ - x²) dx

52) Solve the differential equation (1 - y) x dy/dx + (1+x) y =0

53) If y= 5x/(1-x)²⁾³ + cos²(2x+1) show dy/dx=(5/3)(1-x)⁵⁾³(3-x) - 2 sin(4x+2)

54) A rectangular area of 9000m² is to be surrounded by a fence, with two opposite sides made of brick and the other two of wood. One metre of wooden fencing costs Rs 25 while one metre a brick brick a brick walling costs Rs10. what is the least amount of money that must be allotted for the construction of such a fence ?

55) ∫ √{(1 - √x)/(1+√x)} dx

56) ∫ e⁻ⁿ/(1 + eⁿ) dn at (1,0)

57) find dy/dx of y= (sin x) ˡᵒᵍ ˣ dx 

58) ∫ x sin x dx

59) ∫ x² cos x dx

60) ∫ x²/(x⁴ + x² +1). dx

61) ∫ eˣ cos x dx

62) Solve: x dy/dx = y + x tan(y/x)

63) Equations of two lines of regression are 4x +3y+7=0 and 3x+ 4y+8=0. Find

A) mean of x, mean of y

B) regression Coefficient of x and y and y on x

C) Correlation Coefficient between X and Y.

64) A firm manufactures two types of paper A and B and sells them at a profit of Rs.2 on type A and also Rs.2 on type B. Each product is processed on two machines machines on two machines machines processed on two machines machines on two machines machines M and N. Type A requires on 1 minute of processing time on M and two minutes on N. Type B requires 1 minute on M and 1 minute on M on M minute on M on M 1 minute on M on M minute on M and 1 minute on M on M minute on M on M 1 minute on M on M and one minute on N. The machine M is available for not more than 6 hours 40 minutes while machine N is available for 10 hours during any working day.

Formulate the given problem as a linear programming problem and find how many products how many products products each type should the firm produce each day in order to get maximum profit.

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65) prove 

 sec²(tan⁻¹ 2) + cosec²(cot⁻¹3)=15

66) If x= √(ₐsin⁻¹t), y=√(ₐcos⁻¹t) prove dy/dx= -y/x

67)lim ₓ→π/₂ ( cos x . log (tan x)

68) ²₁∫ √(x)/{√(3-x) + √(x)}

69) If x belongs to N and Determinant. x+3        - 2     = 8

                         - 3x        2x

 then find the value of x

70) Solve (xy²+ x) dx+(x²y+y)dy=0

71) a           a+b           a+b+c

      2a       3a+2b      4a+3b+2c =a³

      3a        6a+3b    10a+6b+3c

72) Prove cos⁻¹(63/65) + 2tan⁻¹(1/5) = sin⁻¹(3/5)

73) If e ʸ(x +1) = 1 then show 

that y₂ = y₁²

74) ∫ x tan ⁻¹ x dx

75) Show that the area of the triangle formed by the tangent and the normal at the point (a,a) on the curve y²(2a - x) = x³ and the line x= 2a, is 5a²/4 sq.units

76) If the radius of a sphere is measured as 9cm with an error of 0.03cm, then find the approximate error in Calculating it's surface area.

77) solve: dy/dx - 3y cot x= sin 2x, given y=2, when x=π/2.

29/11/20

78) A printed page is to have a total area of 80 sq.cm with a margin of 1cm at the top and on each side and a margin of 1.5cm at the bottom. What should be the dimensions of the page so that the printed area will be maximum.

79) Show that of all the rectangles of a given perimeter, the square has the largest area.

80) ∫ dx/[x{6(log x)² + 7log x +2}]

81) The total cost and the total revenue of a company that produces and sells x units of a particular product are c(x)= 5x+350 and R(x)= 50x - x². Find

A) the break even values

B) the value of x that produces a profit.

82) From the Equation of the two regression lines, 4x+3y+7=0 and 3x+4y+8=0, find

A) mean of x and y

B) Regression coefficients

C) Coefficient of correation

83) The demand for a certain product is represented by the function p= 200+20x -x²(in Rs) where x is the number of units demanded and p is the price per unit.

A) find the marginal revenue

B) Obtain the marginal revenue when 10 units are sold.

84) The following results were obtained with respect to two variables x and y:

∑x=30 , ∑ y= 42, ∑xy =199, 

∑x²= 184 , ∑y²= 318 , n=6

A) The regression coefficients

B) Correation coefficient between x and y.

C) Regression Equation of y on x

D) The likely value of y when

x= 10.

85) Compute the two regression equation on the basis of the following information:                         Mean of x and y be 40, 45

S. D of x and y be 10, 9 Karl Pearson's correlation coefficient= 0.5, also estimate the value of Y on X=48, using the appropriate regression equation.

86) If total Function is given by

C= a + bx + cx², where x is the quantity of output show that: 

d/dx (AC) = (MC - AC)/x, where MC is the marginal cost and AC is the average cost.

87) If the marginal revenue Function of a commodity

MR= 9- 6x²+2x, find the total revenue and the Corresponding demand function.

88) Two tailors P and Q earn Rs150 and Rs200 per day respectively. P can stitch 6 shirts and 4 trousers a day, while Q can stitch 10 shirts and 4 trousers per day. How many days should each work to produce atleast 60 shirts and 32 trousers at minimum labour cost ?

30/11/2

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89) sin[tan⁻¹{sinx/√(cos 2x)}] 

where -π/4 < x < π/4

90) ∫ sec²x/(sec x + tan x)⁹⁾² dx

91) ∫ [x² + log{(π+x)/(π-x)}] cos x dx at (π/2, -π/2)

92) ∫x sin x²/{sin x² + sin(log 6 - x²)} at (√log(3), √log(2) )

93) (π²/log 3) ⁵⁾⁶₇₎₆∫ sec(π x) dx

94) ∫ (1 +x - 1/x) eˣ⁺ ¹⁾ˣ dx

95) lim ₓ→₀ sin(π cos²x)/x²

96) ∫ √(1 + sin²x/2 - 4 sun x/2) dx 

at (π,0)

97)sin(sin⁻¹(1/5)+cos⁻¹x) =1 find x

98) ∫ eˣ( sin x - cos x) dx at (π/2,0)

99) Solve the D.E of 

(x²-1)dy/dx +2xy = 2/(x²-1)

100) ∫ (sin⁶x + cos⁶x)/(sin²x cos²x)

 

101) ∫(x-3)√(x²+3x-18) dx

102) find the intervals in which the function f(x)=3x⁴-4x₃-12x²+5 is

a) strictly increasing

b) strictly decreasing.

103) prove

cot⁻¹{√(1+sin x)+√(1- sin x)}/

   {√(1+sin x)- √(1- sin x)}

104) 2tan⁻¹(1/5)+sec⁻¹(5√2/7)+ 2tan⁻¹(1/8) = π/4

105) If y= xˣ prove 

d²y/dx - 1/y(dy/dx)² - y/x =0 

106) prove by property of determinants

   x² +1     xy      xz

   xy       y²+1     yz   = 1+x²+y²+z²

   xz         yz     z²+1

107) Differentiate tan⁻¹[{√(1+x²) -1}/x] with respect of 

sin⁻¹{2x/(1+x²)}, when x ≠ 0

108) Solve the differential equation

dy/dx={x(2logx +1)}/(siny +ycosy) given that y= π/2,when x= 1

109) ∫ (x sin x cos x)/

       (sin⁴x +cos⁴x)dx at (π/2, 0)

110) Of all the closed right circular cylinder cans of volume 128π cm³, find the dimension of the can which has minimum surface area.

111) ∫(cos 9x + cos 5x)/(2cos 5x -1)

112) lim ₓ→₀{(1+x)¹⁾ˣ - e}/x

113) ∫ cos ᵐx sin ᵐx dx at (π/2,0)

114) ∫ x sin x/(1+cos²x)  dx at(π,0)

115) ∫ (cos 3x +1)/(2cos x -1) dx 

      at(π/2,0)

116) ∫ (1+sin 3x)/(1+2sinx) 

at(π/2,0)

  

  

4) Let Z be the set of all integers and R be the relation on Z defined as R = {a,b) : a,b ∈ Z and ( a -b) is divisible by 5}. prove that R is an equivalence relation.

7) Three persons A, B and C shoot to hit a target. If in trials, A hits the target 4 times in 5 shots, B hits 3 times in 4 shots and C hits 2 times in 3 trials, find the probability that

a) exactly two persons hit the target.

b) atleast two persons hit the target.

9) Show that the Function f: R -> R Defined by f(x)= x/(x²+1) is neither one-one nor onto.

16) In a class of 75 students. 15 are above average, 45 are average and the rest below average achievers. The probability that an above average achieving students fails is 0.005, that an average achieving students fails is 0.05 and the probability of a below average achieving students failing is 0.15. if a student is known to have passed, what is the probability that he is a below average achievers.

17) The probability that a bulb produced by a factory will fuse in 100 days of use is 0.05. find the probability that out of 5 such bulbs, after 100 days of use.

a) none fuse

b) not more than one fuse

c) more than one fuse

d) atleast one fuse.

7) The area of the region described by A= {(x,y): x²+y²≤ 1 and y²≤ 1 - x}

22) Find the area of the region in the 1st quadrant enclosed by the x-axis, the line y= x and the circle x²+y²=32