12th Test Math

Functions and Relations (1 Mark)

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1. Is the function f: N → N, defined by f(x) = 3x, onto? Give reasons.2. If f(x) = [ x ] and g (x) = │x │, then evaluate (f o g) (1/2) – (g o f) (-1/2).3. If tan-11 + tan-1(1/2) = tan-1α, find α.4. 1If A and B are two sets such that n (A) = m and n (B) = n then write number

of function from A → B.5. Let f : R – { - 4/3} → R be a function defined as f(x) = 4x / 3x+4. Find the

inverse of f.6. Find the value of tan-1(1) + cos-1(-1/2) + sin -1 (-1/2).7. Evaluate Sin [ π– sin-1 (-1)].8. Let set A contains 3 elements. Write the total number of binary operation

possible.9. Let ‘*’ be a binary operation defined on the set of integers as a*b=a+b+1 for a,

b € I. Find the identity element.10. Prove that cos2(tan-12)+sin2(cot-13) = 3/10.11. What is the principal value of cos-1(cos 2 π/3)+sin-13(sin 2 π/3).12. If f(x) = 8x3 and g(x) = x1/3 , find g o f.13. Find the value of sin-1(sin 3π/5).14. Show that the signum function f : R → R given by f(x) = 1, if x is greater than

0, if x = 0, - 1, if x is less than 0 is neither one-to-one nor onto.

15. If , find f o g.16. N → N, defined by f(x) = 3x onto?Give Reasons.17. If a binary operation * is defined on the set Z of integers as a*b = 3a –b, then

find the value of (2*3)*4.18. Prove that sec2 (tan -1 2) + cosec2 (cot -1 3) = 15.19. How many relations can be defined from a non-empty set A to non-empty set

B , if n(A) = 2 and n(B)=3.20. Find the principal value of tan-1[sin(sin-1x+cos-1x)] , x ε [ -1,1] .

21. Evaluate sin{ 1/2 cos -1(4/5) }.22. Find gof(-5/2) and fog(3/2) if f is identity function and g is signum function .23. Evaluate the value of cos -1(-1/2) + Sin -1(-√3/2).24. If : R → R ,a,b,c,d є R such that (a,b) *(c,d) = (ac, b + ad) Find the identity element of

the function.25. Let * be a binary operation defined by a * b =2ab-7. Is * associative?26. Write the range of one branch of sin -1 x, other than the Principal Branch.27. If f(x) = x-1 / x+1; , then find f -1(x ).28. Find the value of cosec (sec -1 ө+cosec -1 ө).29. Evaluate : Cos (π/3- sin-1(-√3/2)).30. Show that the function f: R → R such that f(x) = x2 is neither one –one nor onto.31. If f(x) = x + 7 and g(x) = x – 7, x є R, find (fog) (7).32. Let * be a binary operation on N given by a * b = H.C.F. ( a , b ) , a, b є N. Find 12 *

4.33. if f(x)=x2-1 and g(x)=2x+3 find fog(2).34. if sin(sin-1 1/5 + cos-1 x) = 1, then find the value of x.35. Evaluate: sin [ π/3 - sin-1(-1/2) ].36. Which of the following graph represent a function .

37. Write the range of one branch of cos x, other than the principal branch.38. Which one of the following graphs represents an identity function? Why?

39. Let A = {1, 2, 3) and B = {4, 5, 6}, f: A → B is a function defined on f(1) = 4, f(2)=5and f(3) = 6. Write the inverse as set of ordered pairs.

40. Let * be the binary operation on N defined by a*b = a+b+10 for all a, b є N and if 3*p=15 then find p.

41. Find the value of the parameter for which the function is the inverse of itself.42. If f(x) = ex and g(x) = logex , find fog and gof. Is fog = gof ?

Functions and Relations (4/6 Mark)

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1. Consider f : R → [ -5, ∞ ) given by f(x) = 9x2+ 6x-5. Show that f is invertible

with .

2. Prove that:

3. Consider defined by f(1) = a, f(2) = b, f(3) = c, g(a) = apple, g(b) = ball ,g(c) = cat. Show that f, g and gof are invertible. Find out f-1 , g-1 and (gof)-1 and show that (gof)-1 = f-1o g-1.

4. Let A = N x N and * be the binary operation on A defined by (a,b)*(c,d) = (a + c, b+d). Show that * is commutative and associative. Find the identity element for * on A, if any.

5. Find the value of x if

6. Let f: N→R be a function defined as f(x) = 4x2 + 12 x + 15, where S is the range of f. Show that f: N → S is invertible. Find f-1.

7. Prove that 8. Prove: 2 tan-1(1/2) + tan-1(1/7) = tan-1(31/17)9. Solve for x: tan-12x + tan-13x = π/410. Prove that the function f: R →R defined as f(x) = 2x-3 is invertible and find f-

1(x)11. Show that sin-1 12 /13 + cos-1 4/5 + tan-1 63/16 = π .12. Show that the relation are in the set a = {x : x є w, x ≤10}given by

is an equivalence relation , find the elements related to 3.

13. Examine if the following are binary operations(i) a*b =a+b/2,a,b є N(ii) a*b=a+b/2, a,b є Q .

14. Let be a function defined as , find f-1 : Range of

.

15. Prove that .16. Prove: 2 tan-11/2) + tan-1(1/7) = tan-1(31/17).17. Solve for x: tan-12x + tan-13x = π/4.18. Solve for x : 2 tan-1( cos x ) = tan-1 ( 2 cosec x ).

19. Find the value of .

20. Find the value of


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21. Prove that tan -1 (1/5) + tan -1 (1/7) + tan -1 (1/3) + tan -1 (1/8) = 1.

22. Prove that .

23. Prove that .

24. Prove that .

25. Write in simplest form: .26. Prove that tan -1 3/4 + tan -1 3/5 – tan -1 8/15 = π/4.

27. Show that the binary operation * defined by a*b = a – b, on Z is not commutative and associative..

28. Let f : x → y and g : y → z be two invertible functions. Then gof is also with (gof)-1 = f-1 o g-1.

29. Show that the function f: Z → Z given by f(x) = x² is not injective.

30. Evaluate : .

31. Solve for .

32. If .33. Check whether the operator define by a b=a+b -ab is commutative or

aasociative.

34. if .

35. Proved that .36. Let A =NxN and Let * be a binary operation on A defined by (a,b)*(c,d) =

(ad+bc,bd) for all (a,b),(c,d) € NxN.Determine if * is commutative, associative,

37. Solve for .

38. Prove that: .39. If the function f : R→R defined by ,Prove that f is one-one and onto function.

Also find the inverse of the function f and f-1(23) .

40. Solve for .


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41. If find f(f(x))

42. Solve for x:

43. If show that f(f(x)) is an identity function.44. Show that the relation R in the set A= {x ε Z ; 0 ≤ x ≤ 12} given by R =

{ ( a,b) : |a-b| is multiple of 4}is an equivalence relation . Also find the set of all elements related to 4.

45. Check whether the relation R defined in the set {1,2,3,4,5,6} as R = { (a,b): b = a +1} is reflexive, symmetric or transitive.

46. Let A = {-1,0,1,2}, B = [ -4,-2,0,2} and f , g : A →B be the function defined by f(x) = x2 – x , x ε A and g(x) = 2| x - 1/2 | , x ε A ; are f and g equal. Justify your answer.

47. Show that f [ 1, -1 ] →R given by f(x) = x/x+2 is one-one. Find the inverse of function. f: [ -1 ,1] & Range (f).

48. Express in the simplest form.

49. Let given by f(x)= x| x |is a bijection

50. If f, g : R → Rare defined respectively by Find

i) fog ii) gof iii) fof iv) gog

51. Let f, g : R → R be two functions such that

Find f(x) and g(x).52. Consider f, g : N → N and h : N → Rdefined as f(x) = 2x, g(y)= 3y+4, and

h(z)= Sin z for all x,y,z ε NShow that ho(gof) = (hog)of

53. Let and . Find

54. Let is a bijection

Vectors-3D-Lines (1Mark)

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1. Find a vector in the direction of vector that has magnitude 7 units.

2. Find , if two vectors are such that .

3. Find the value of

4. Find the projection of the vector

5. If are any two vectors such that then what is the angle

between ? 6. Find the unit vector paralle to vactors 3і+ ϳ - 3k and -2і - ϳ+ 3 k7. Find the value of λ if λі + 2ϳ + k and 5і - 9ϳ + 2k are perpendicular .

8. If find the angle between and

.9. Find the unit vector in the direction of

10. Write the vector equation and the direction ratio of the line

11. Cartesian equations of a line AB are:-

Write the equation of a line passing through (1, 2, 3) parallel to AB.

12. Write the position vector of a point dividing the line segment joining points A

and B with position Vectors externally in the ratio 1 : 3 , where

and .

13. If find a unit vector in the direction of

14. If = 2j+2 k b = 3i – 5k find

15. Find ‘λ’ so that vectors are perpendicular .

16. Find the Cartesian and vector equations of a line which passes through the

point (1,2,3) and is parallel to the line 17. Find the unit vector perpendicular to the vectors a = i + 3j + 4k and b = 2i – j –

3k. 18. What is the value of i∙ ( j x k ) + j ∙ ( k x i ) + k ∙ ( i x j ) ? 19. Write all unit vector in X Y plane. 20. Find a vector of magnitude 9 units, which is perpendicular to both the vectors


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21. Find λ , when the projection of on is .

22. If a, b, and c are unit vectors such that , then find the Value of

23. Find the value of x for which is a unit vector. 24. Find the direction cosines of a line that makes equal angle with the coordinate

axes

25. Find the angle between two vectors with magnitude and 2

respectively and such that

26. Find

27. Find 28. Find the scalar components of a unit vector which is perpendicular to the

vectors .

29. The Cartesian equations of a line AB are . Find the direction ratios of the line AB.

30. If is a unit vector

31. Find the value of 32. Find the angle made by the vector i – 4j + 8k with the z – axis. 33. Find the position vector of a point R which divides the line joining the points

P(i + 2j – k) and Q(-i + j + k) in the ratio 2 : 1 externally.34. A line makes angles 60 and 45 with x and y-axis respectively. Find the angle

which it makes with z-axis.

35. Find the angle between the line and 36. The equation of a line 3x + 1 = 6y – 2 = 1 – z. Find the d.r’s and d.c’s of the

line37. Find the equation of line which passes through the point (1, -1, 2) and parallel

to to 38. Find angle between lines r= 4i – 3j + k+ λ(3i + 2j + 6k)and r = 8i – 2k + µ(i +

2j + 2k)

39. If

40. If


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41. Three vectors satisfy the condition Evaluate the

quantity

42. The Cartesian equations of a line AB are . Find the direction ratios of the line AB.

43. Find direction cosines of a line which is equally inclined with the axes. 44. If a line makes angles ∏ /4 with each of X –axis and Y-axis, then what angle

does it makes with the Z- axis. 45. If | a + b | = | a - b | then find the angle between a and b 46. If a is a unit vector and ( x - a ).( x + a ) = 8 , then find | x |.47. What is the angle between vectors with magnitude respectively?48. If a is unit vector and (x – a). (x + a) = 8, then find |x|.

49. Find a unit vector perpendicular to each of the vectors

50. If

51. The Cartesian equation of a line AB is . Find the direction cosines of a line parallel to AB.

52. If and , find a unit vector parallel to

the vector 53. Find the direction cosine of the line which is perpendicular to lines whose

direction cosine are proportional 1,-2,-2 and 0,2,1

Vectors-3D-Lines (4-Marks )

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1. If .

2. Show that the points A, B, C with position vectors

respectively are collinear

3. If the sum of two unit vectors is a unit vector, Prove that the magnitude of

their difference is .

4. Show that the points A, B and C with position vectors,

respectively, form the vertices of a right angled triangle.

5. If a unit vector makes angles and then find θ

and hence the components of .

6. Let , and be three vectors such that and each one of them being perpendicular to the sum of other

two, Find

7. Find the area of the parallelogram whose adjacent sides are determined by the

vectors

8. The scalar product of the vector with a unit vector along the sum of

vectors Find the value of λ.

9. If is any vector in space show tha

10. If vectors , and are such | a + b + c | =0, and | a | = 6, | b | = 8 and | c | = 10. Find (a.b+b.c+c.a)

11. Find the vector of magnitude 5 units which is perpendicular to both the vectors

12. Find the values of .

13. Show that area of parallelogram having diagonals (3i + j – 2k) and (i – 3j + 4k) is 5 √3 sq unit.

14. Find the shortest distance between the lines, whose equations are

15. Find shortest distance between the lines r = i + j +λ (2i – j + k), r = 2i + j – k + μ (3i – 5j + 2k)

16. If are unit vectors such that then find the

value of .

17. By computing the shortest distance, determine whether the following pair of

lines intersect or not:

18. d1 and d2 are the diagonals of a parallelogram with sides a and b.Express a and b in terms of d1 and d2 and find the area of the parallelogram ;d1 = i + 2 j + 3 k ;d2 = 3 i - 2 j + k

19. Find the shortest distance between the following pairs of line r = (1 – t ) i + ( t – 2 ) j + ( 3 + 2t ) k and r = ( s + 1) i + ( 2s – 1) j + ( 2s + 1 ) k.

20. Find the equations of the line through the intersection of the lines

and

and parallel to the line

Vectors-3D-Lines (4-Marks)

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21. If A,B,C,D are four points in space prove that │AB x CD + BC x AD + CA x BD│ = 4 (Area ΔABC)

22. If and , and find angle between

find the angle between

23. For vectors and prove that :

24. If with reference to a right handed system of mutually perpendicular unit

vectors , we have and Express in the

form where ,is parallel to is perpendicular to

25. Find the point on line at a distance 3√2 from the point ( 1, 2, 3)

26. Dot product of vector with vectors are respectively -1,6 & 5 find the vector.

27. Find image of the point (1,0,2) in the line passing through (2,1,3) and perpendicular to the lines

28. Find the value of k so that the lines and

are at right angle.

29. . If are any two non zero vectors.then prove that

30. If

31. If Find a vector which is

perpendicular to

32. Find the unit vector perpendicular to the plane ABC, where the position vectors of A, B & C are 2i – j + k, 3 i + j + 2 k & - 2 j + 3 k respectively.

33. Dot product of a vector with the vector i + j - 3 k , i + 3 k - 2 k and 2 i + j + 4 k are 0, 5 and 8 respectively.find vector.

34. find the shortest distance between the lines =(1 - t ) i + (t - 2) j + ( 3 - 2 t ) k and r = ( s + 1 ) i +( 2 s - 1 ) j -( 2 s + 1) k

35. find the equation of the line passing through the point(1,2,3) and parpendicular

to lines and

36. The scalar product of the vector i + j + k with a unit vector along the sum of vectors 2i + 2j – 5k and λ i + 2j + 3k is equal to one. Find the value of λ .

37. If a x b = c xd =, a x c = b xd, show that (a – d) is parallel to (b – c). 38. Find the foot of perpendicular from P(1, 2, 3) on the line

Also find the equation of the plane containing the line and the point (1, 2, 3).

39. Determine whether or not the following pair of lines intersect. If these intersect, then find the point of intersection, otherwise obtain the shortest distance between them: r = i + j – k + λ (3 i – j ) and r = 4 i – k + µ ( 2 i + 3 k )

40. Find the foot of perpendicular from the point (0, 2, 7) on the line

Vectors-3D-Lines (4-Marks)

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41. Find the equations of line passing through (3, 4, 7) and perpendicular to the

lines

42. Show that the lines intersect each other: r = -i – 3j – 5k + µ(3i + 5j + 7k) and r = 2i + 4j + 6k + λ(i + 3j + 5k)

43. Find the shortest distance between the lines given by and

44. Find the shortest distance between the lines

45. Find the shortest distance between the lines

46. Find the foot of perpendicular from (1, 2, −3) to the line Also find the length of erpendicular.

47. Find the foot of perpendicular from (1,2,3) to the line Also obtain the equation of plane containing the line and the point (1,2,3)

48. Show that the lines are coplanar Also find the equation of the plane containing the lines. PROVE SD=0

49. Let the vectors be such that , then find the angle

between such that is a unit vector

50. If , find the angle between

51. Find the coordinate of the foot of perpendicular drawn from the point(1,2,1) to the line joining the points (1,4,6) and (5,4,4).

52. Find image of the point (0, 2, 3) in the line r= j+2k + ג (i+2j+3k).

53. If are two vectors such that , then

prove that .

54. Find the equation of line which passes through the point (1, -1, 2) and parallel

to .Also find the distance between the two parallel lines

55. Find the shortest distance between the lines given by

Note That They Are Parallel Lnes

56. Find the angle between the diagonals of a cube.

57. Find the area of a triangle whose vertices are (-1,2,3),(3,4,-5) and (2,0,5)

58. Show that axb+bxc+cxa is perpendicular to the plane of ABC

Assignment

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This is the easiest topic for students .All you need is practice .It contributes to about 12 marks to the whole syllabus . Following formulae may be used:-

Properties of determinants :-

1. The value of a determinant remains unchanged if its rows and columns are interchanged.

2. The sign of value of a determinant is changed if its any two rows or columns are interchanged.

3. The value of a determinant is zero if its any two rows or columns are identical.4. If a determinant is multiplied by a scalar (number) , its only one row or

column gets multiplied by that constant.5. If any two row or column of a determinant are proportional, its value becomes

zero.6. If all elements of a row or column are expressed as sum of two or more

elements ,the whole of the determinant can be expressed in sum of two or more determinants.

7. If some multiple of one row or column is added or subtracted to another row or column(elementwise) , 1 its value remains unchanged.

Tips to solve properties based problems:-

1. If a determinant is of nth order ,we can apply only n-1 propertis at a time to it.2. The format of application of properties is :- Row affected Row affected n

(Row used) Ex.

3. The format for interchanging Rows or columns :- 4. You can never multiply a number to Row affected, it is always multiplied to

Row used.5. First always try to make elements of any one row or column identical so that

you could take out common from that row or column. It makes all the elements of that Row or column unity(1) and then you make at the most two elements of that row or columns zero (0).Now expand that the determinant by that row or column.

Ex-

We shall apply

and

Taking out common b from C1 and C2

Expand by R1

6. Check the part which is required to prove ,try to take out common the factors which are given in the part.

Ex.

Here the first factor is (a-b) ,we can obtain it by R1 - R2 .Another factor is b-c which we can obtain by R2 - R3

Important questions are:-

1. Find x,y,z if

2. If = Show that and hence find A-1.

3. Using properties of determinants solve for x

: 4. Using properties prove that :

Assignment

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5. Usining properties of determinants,prove that:

6. If

7. Using properties ,prove that:

8. If A = , Using principle of mathematically induction prove that

9. Using properties of determinants,prove that

10. If A= , B = , c = Find a matrix D such that CD-AB=0.

11. Let ,Verify that .

12. If ,find k so that A-1 = KA - 2I.

13. Find X and Y if

14. If

15. Find B if

Assignment

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16. Find , find a and b such that A2 + aI = bA such that where I is unit matrix of order 2.

17. Express as a sum of a symmetric and a skew – symmetric matrix.

18. Prove using properties of determinants

19. Solve the equations by matrix method

20. If find and use it solve the system of equations:

21. Using determinants ,solve the following system of equations:

22. Find the value of for which given homogeneous system of equations have non trival solution. Also find the solution.

23. If Using A-1 solve the system of linear equations:

24. If find the product AB and use this result to solve the following system of equations:

25. Solve using matrices: 26. For what value of a and b, the following system of equations is consistent?

Matrices and Determinants (1 Mark questions)

Page 1 of 5

1. Find x if 2. How many matrices of order 3 x 3 are possible with each entry 0 or 1?

3. For any 2 x 2 matrix, if , then find |A|

4. If A is a square matrix of order 3 such that

5. If A = [1 2 3 4] and Write the order of AB and BA 6. For what value of x, the following matrix is singular?

7. A matrix A of order 3 X 3 has determinant 7, what is the value of |3A|?8. If A is a square matrix such that A² = A, then find (I + A)² - 3A.

9. If find x, 0 < x < π when A + A’ = I . 10. If A and B are matrices of the equal order and B is skew symmetric, then show

that A’BA is skew symmetric.

11. Using elementary operations, find the inverse of

12. Show that the matrix is Skew-symmetric.

13. Express the matrix as the sum of symmetric and a skew- symmetric matrix.

14. A matrix of order 3 X 3 has a determinant 15. What is the value of |5A|?15. If A is square matrix of order 3 such that |Adj A| = 289, find |A|.16. Give an example of two non-zero matrices A and B of the same order 2x2

such that AB=0.

17. Find the value of k, if the matrix is singular.18. Show that the matrix B’AB is symmetric or skew symmetric according as A is

symmetric or skew symmetric.

19. Construct a 2 x3 matrix whose elements are given by

20.


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21. If A is 3x4 matrix and B is a matrix such that A’B and BA’ are both defined, then find the order of matrix B .

22. If a square matrix A be singular find the matrix A ( adj A ).23. If A is an invertible matrix of order 2 and det (A) = 5, then find det ( A-1) .

24. Evaluate .25. If A and B are symmetric matrices of same order. Prove that AB-BA is skew

symmetric matrix.

26. For what value of k, the matrix is singular.27. If A and B are symmetric matrices then show that AB-BA will be skew-

symmetric.28. If be singular matrix, then find then find the value of ‘x’ ?

29. If A and B are two matrices such that AB =A and BA = B then what is value of B2.

30. If is additive inverse of . Find x, y, z and t .31. Construct 3x2 matrix If A = [aij] where aij = { i+j, if i > j ; i-j, if i< j }.

32. Construct a 2 x 2 matrix A= [ aij ] whose elements are given by .

33. If find the values of a and b.

34. Find value of x, If matrix is not invertible.

35. is a skew symmetric, find value of x.

36. Construct a 2x2 matrix A = [aij] whose element are given by .

37. For any 2 x 2 matrix, if , then find |A|.38. If A is a square matrix such that A2 = A. Find the value of (I+A)3 – 7A.

39. Find the cofactor of (1, 2) th entry in the given matrix .40. If a matrix is both is symmetric and skew symmetric, then show that it is a null

matrix.41. If B is a skew symmetric, write whether the matrix (ABA ) is symmetric or

skew symmetric.

42. Which type of matrix is this?


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1. If ’ find k so that A2 = 8A + kI.Hence find A-1.

2. If a, b, c are in AP, show that .

3. Without using the concept of inverse of a matrix , find the matrix such

that .4. By using properties of determinants show that

5. Using elementary row transformation find the inverse of the matrix

.6. Prove by using properties of determinant

.

7. If , then prove that , where n is any positive integer.

8. Using properties of determinants, Prove that

.9. Using the properties of determinants, show that

.

10. Express the matrix as the sum of symmetric and skew symmetric matrices.

11. Using the properties of determinants; show that .

12. For the determinant Show that (a+b+c) and (a2+b2+c2) are the factors.

13. Show that satisfies the equation x² + 4x – 42 = 0.Hence find A-1.14. Using the properties of determinants, show that

.

15. If then prove that .

16. Prove that .17. Prove by using properties of determinant

.18. If a,b,c are in A.P then find the value of the determinant

.

19. If and I is the identity matrix of order 2, show that

.

20. Show that .


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21. Using properties of determinants, prove the following

.

22. If a, b, c are positive and unequal, show that value of determinant is negative.

23. Find the matrix X and Y when it is being given that

..24. Using properties of determinants, Prove that

.

25. Using properties of determinants, Prove that .26. Using properties of determinants Prove that

.

27. If A= and I= Find x and y Such that A2= xA + yI. Hence find A-1. .

28. If Then by mathematical induction show that

Where a & b are constant.

29. If , then show that

.

30. Prove that .

31. Prove that : .32. Using properties of determinants prove that

.

33. Let show that (aX + bY)3 = a3X + 3a2b Y.

34. Prove that value of the determent is independent of .

35. Solve for

36.37. By using the properties of the determinants prove that

.

38. .

39. .

40. .

41. .

42. .

43. .

44. .

45. .

46. .

47. If a, b, c are reals, and Show that either a + b+ c =0 or a = b = c.


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1. Using matrices solve the following equations: x + y + z = 3 ; x – 2y + 3z = 2 and 2x – y + z = 2

2. If and B= ,Find AB hence solve the system of equation x-y = 3, 2x +3y +4z = 17, y+2z =7

3. Solve the following system of equations by matrix method : 5x-7y+z =11 , 6x-8y-z =15 and 3x+2y-6z =7.

4. Given that find BA. Use product solve the system of equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1

5. Solve the following system of equations by matrices: 2/x + 3/y + 10/z = 4 ; 6/x + 9/y – 20/z = 2 ; 4/x – 5/y + 5/z = 1

6. Given that Hence solve the system of equations: x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1

7. If find A-1 and solve the following system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y - 2z = - 3.

8. If , find A-1 and use it to solve the system of the equations

9. Using elementary row transformations, find the inverse of

10. Find A-1 , where . Hence, Solve the system of linear equations: x +2y-3z= - 4, 2x+3y+2z=2, 3x-3y-4z=11

11. Solve the following system of equation by matrix method

12. If Find A-1 and hence solve the system of linear equation: 3x+4y+7z=14, 2x-y+3z=4, x+2y-3z=0.

13. If prove that A2-4A-5I=0, Hence find A-1.

14. If , find A-1 using elementary row operation.15. Classify the following system of equations as consistent or inconsistent. If

consistent solve it. x – y + 3z = 6, x + 3y – 3z = – 4 and 5x + 3y + 3z = 10

Differentiability Applications (1 Mark)

Page 1 of 6

1. If a line y = x + 1 is a tangent to the curve y2= 4x ,find the point the of contact ?

2. Find the point on the curve y = 2x2– 6x – 4 at which the tangent is parallel to the x – axis

3. Find the slope of tangent for y = tan x + sec x at x = π/44. Show that the function f(x) == x3– 6x2 +12x -99 is increasing for all x.

5. Find the maximum and minimum values, if any of 6. For the curve y = 3x² + 4x, find the slope of the tangent to the curve at the

point x = -2.7. Find a point on the curve y = x2– 4x -32 at which tangent is parallel to x-axis.8. Find a, for which f(x) = a(x+sinx)+a is increasing .9. The side of a square is increasing at 4 cm/minute. At what rate is the area

increasing when the side is 8 cm long?10. Find the point on the curve y =x2-7x+12, where the tangent is parallel to x-

axis.


Page 2 of 6

1. Find the intevals in which the function f(x) = 2log(x-2) - x2 + 4x + 1is increasing or decreasing.

2. Find the intervals in which the function f ( x ) = x3 - 6x2 + 9x + 15 is(i) increasing (ii) decreasing.

3. Find the equation of the tangent line to the curve x = θ + sinθ, y = 1+cosθ a=π/4

4. Prove that is increasing in [o, π/2]5. Prove that curves y² = 4ax and xy = c² cut at right angles If c4 = 32 a4

6. A water tank has the shape of an inverted right circular cone with its axis

vertical and vertex lower most. Its semi vertical angle is . Water is poured into it at a constant rate of 5 cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10m. Find the point on the curve y =x²-7x+12, where the tangent is parallel to x-axis.

7. Discuss applicability Rolle’s Theorem for the function f(x) = cosx + sinx in [0,2π ] and hence find a point at which tangent is parallel to X axis.

8. Verify Lagrange’s mean value theorem for the function f(x) = x + 1/x in [1,3].9. Find the intervals in which f(x) = sinx + cosx , o ≤ x ≤ 2 π, is increasing or

decreasing. 10. Use differentials to find the approximate value of √25.211. Find the interval in which the function given by f(x)= (4sinx – 2x – x cosx) /

(2+cos x) is increasing.12. Find the local maximum & local minimum value of function x3– 12x2 + 36x –

413. For the curve y = 4x3 - 2x5, find all the points at which the tangent passes

throughthe origin.

14. Show that the curves 2x = y2 and 2xy = k cut at right angles if k2 = 8.15. Find the interval in which the function f(x)= 2x3 -9x2 -24x-5 is Increasing or

decreasing.16. Find the interval in which the function is increasing or decreasing. 17. Prove that the curves x = y² and xy = k cut at right angle if 8k2 = 1. 18. If f(x) = 3x² + 15x + 5, then find the approximate value of f(3.02), using

diffrentials.19. Find the local maximum and minimum values of function: f(x) = sin 2x – x,-

π/2 < x < π/220. Find the interval in which f(x) =sin 3x is increasing or decreasing in [0, π/2].


Page 3 of 6

21. A ladder 5m. long is leaning against a wall . The bottom of the ladder is pulled along the ground, away from the wall at the rate of 2 m./sec. How fast is its height decreasing on the wall,when the foot of the ladder is 4 m away from the wall.

22. Show that the function f given by f(x) = tan-1(sinx+cosx), is strictly decreasing function on (π/4,π/2)

23. Find the equation of the tangents to the curve y = √3x-2 which is parallel to the line 4x-2y+5=0.

24. Find the intervals in which the function f(x) = x3+1/ x3 increasing or decreasing.

25. Verify Rolle’s theorem for the function f(x)=x3+2x-8 ,26. Find the equation of the tangent and normal to the parabola y2 = 4 a x at ( at2,

2at).27. Using LMVT, find a point on the parabola y = ( x – 3 )2 , where the tangent is

parallel to the chord (3,0) and (4,1). 28. Verify Rolle’s theorem for f (x) = x3- 2x2- x + 3 in [0,1] .29. Find the intervals in which f(x) = x3+ 2x2–1 decreasing or increasing

30. It is given for the function of defines by f(x)=x3+ bx2–ax, Rolle’

theorem helds with 31. Find the intervals in which f(x) = - 2 x3+15x2- 36x + 1 is increasing or

decreasing.32. Show that the parabola y2 = 4 x + 4 & y2 = 4 - 4 x intersect at right angle.33. Find the interval in which the function f is given by f(x)=sinx-cosx,o ≤ x ≤ 2 π

(i) Increasing (ii) Decreasing.

34. It is given that for the function f given by f(x)=x3+ bx2–ax Rolle’s

theorem hold .Find the values of a and b 35. Find the equation of the tangent and normal to the curve: x = acost +at sint , y

= asint - atcot at any point ‘t’. Also show that the normal to the Curve is at a constants distance from the origin.

36. Using differentiate find approximate value of √51.

37. The surface area of a spherical bubble is increasing at the rate of 2m²/sec. Find the rate of at which the volume of the bubble is increasing at the instant its radius is 6cm.

38. Prove that x/a + y/b = 1 is a tangent to the curve y = be-x/a at the point where the curve cuts y-axis.

39. Find the equation of the tangent to the curve x² + 3y – 3 = 0, which is perpendicular to the line y = 4x – 5.

40. Find the approximate change in the volume V of cube of side x mts caused by increasing the side 2 %.


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41. Find the intervals in which the function f(x) = 2x³ - 15x² + 36x + 1 is strictly increasing and decreasing. Also find the points on which the tangents are || to the x-axis.

42. Find the equations of the tangent and normal to the curve 16x² + 9y² = 144 at (2, y1 > 0). Also find the point of intersection where both tangent and normal meet the x-axis.

43. A particle moves along 3y = 2x³ + 3. Find the points on the curve at which the y-coordinate changes twice as fast as x-coordinate.

44. Find the point on the parabola y = (x – 3)² where the tangent is parallel to chord joining the points (3, 0) and (4, 1).

45. The volume of a sphere is increasing at 3cm3/ s. what will be the rate at which the radius increases when radius is 2 cm

46. Verify Rolle’s theorem for the function f(x) = x2-5x+6 in the interval [2, 3].47. Water is leaking from a conical funnel at the rate of 5cm3/Sec. If the radius of

the base of funnel is 5 cm and height 10cm find the rate at which is water level dropping when it is 2.5 cm from the top.

48. The length x of a rectangle is decreasing at the rate of 2cm/s and the width y is increasing at the rate of 2cm/s. when x=12 cm and y=5cm, find the rate of change of(a) the perimeter and (b) the area of the rectangle.

49. Using differential, find the appropriate value of 3√2950. Sand is being poured at the rate of 0.3 m3/sec into a conical pile. If the height

of the conical pile is thrice the radius of the base, Find the rate of change of height when the pile is 5cm high.

51. Verify the condition of Mean Value Theorem and find a point c in the interval as statedby the MVT for the function given by f(x) = logex on [1, 2].

52. The two equal sides of isosceles triangle with a fixed base ‘b’ are increasing at the rate of 3 cm/sec. How fast is the area decreasing when the two equal sides are equal to the base?

53. Verify Rolle’s Theorem for the function f(x) = Sin x – Cos x in the interval [ π/4,5π/4]

54. The radius of a balloon is increasing at the rate of 10 cm/sec. At what rate is the volume of the balloon increasing when the radius is 10cm?

55. Find the interval in which the function f(x) = x3– 6x2 – 36x + 256. Find the intervals in which the following function is increasing : f(x)=x4– 2x2.

57. Using Rolles theorem, find the points on the curve y = 16 – x2, x є [ -1,1] where the tangent is parallel to x-axis.

58. Show that the function f given by f(x)= tan-1(sinx+cosx), is strictly decreasing function on (π/4,π/2).

59. Find the equation of the tangent to the curve x2 + 3y = 3 which is parallel to the line y – 4x + 5 = 0.

60. A man 160 cm tall; walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/s. How fast is the length of his shadow increasing when he is 1 m away from the pole?

61. A point source of light along a straight road is at a height of ‘a’ metres. A boy ‘b’ metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of ‘c’ m/min?

62. At what points of the ellipse 16x2 + 9y2 = 400 does the ordinate decrease at the same rate at which the abscissa increases?

63. The bottom of a rectangular swimming pool is 25 m by 40 m. Water is pumped out into the tank at the rate of 500 m3/min. Find the rate at which the level of the water in the tank rising.

64. An inverted cone has a depth of 40 cm and base of radius 5 cm. Water is poured into it at a rate of 1.5 cm3/ min. Find the rate at which the level of water in the cone is rising when the depth is 4 cm.

65. Water is dripping through a tiny whole at the vertex in the bottom of a conical funnel at a uniform rate of 4 cm3 / s. When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water, given that the vertical angle of the funnel is 1200.

66. Oil is leaking at the rate of 16 mL / s from a vertically kept cylindrical drum containing oil. It the radius of the drum is 7 cm and its height is 60 cm, find the rate at which the level of the oil is changing when the oil level is 18 cm.


Page 5 of 6

1. Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12cm is 16cm.

2. Find the equation of line through the point (3, 4) which cuts the 1st quadrant a triangle of minimum area.

3. Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 time the radius of the base.

4. Show that the semivertical angle of a cone of maximum volume and given slant height is tan-1√2.

5. A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum.

6. A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening.

7. An open box with a square base is to be made out of a given iron sheet of area 27 sq.m. Show that the maximum volume of the box is 13.5 m³

8. A point on hypotenuse of right angled triangle is at a distance ‘a’ and ‘b’ from the sides Show that the length of hypotenuse is at least (a2/3 + b2/3)3/2

9. A square piece of tin of side 48 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off, so that the volume of the box is the maximum possible? Also find the maximum volume.

10. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

11. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is Sin-1(1/3)

12. Awindow is in the form of rectangle surmounted by semi circular opening.The total perimeter of the window is 'p'.c.m.Show that the window will allow the

maximum possible light only when the redius of semi circle is 13. A rectangle is inscribed in a semi circle or radius a with one of its sides on the

diameter of the semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also the area .

14. Show that the surface Area of closed cuboid with square base & given volume is maximum when it is a cube.

15. Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius 5√3cm if 500∏cm3.

16. A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m find the dimensions of the rectangle that will produce the largest area of the window.

17. Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone.

18. A wire of length 36cm is cut into two pieces. One of them is turned into square and other is into equilateral triangle. Find the length of each piece so that the sum of the areas of two is minimum

19. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3

20. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.


Page 6 of 6

21. If the lengths of three sides of trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.

22. Given the sum of the perimeter of a square and a circle, show that the sum of their areas is least when the side of the square is equal to diameter of circle.

23. Find a point on the parabola y2 = 4x which is nearest to the point (2,-8) 24. Show that the right circular cone of least curved surface and given volume has

an altitude equal to√2 times the radius of the base.25. A given quantity of metal is to be cast into a solid half circular cylinder( i.e.

with rectangular base and semicircular ends).Show that the surface area may be minimum, if the ratio of the length of the cylinder to the diameter of its circular ends is π : π + 2.

26. A window is in the form of a rectangle surmounted by a semi circular opening . The total perimeter of the window is 10m. Find dimensions of the window to admit maximum light through the whole opening.

27. An open box with a square base is to be made out of a given quantity of sheet of area a2 . Show that the maximum volume of the box is a3 / 6√3.

28. An open tank with square base and vertical sides is to be constructed from a metal sheet so as to hold given quantity of water. Show that the cost of the material will be least when the depth of the tank is half of its width.

29. Show that the height of a closed circular cylinder of given total surface are and maximum volume is equal to the diameter its base.

30. A Cylindrical container with a capacity of 20 cubic feet is to be produced. The top and bottom of the container are to be made of a material that costs Rs.6 per squares foot while the side of the container is made of material costing Rs.3 per squares foot. Find the dimension that will minimize the total cost.

31. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle 300 is 4/81 πh3.

32. Find the largest possible area of a right angled triangle whose hypotenuse is 5cm long.

33. Find two positive numbers whose sum is 16 and sum of whose cube is minimum.

34. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius a is square of side √2 a.

35. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is 8m³. If building of tank cost Rs. 70 per square meters for the base and Rs. 45 per square meters for sides. What is the cost of least expensive tank?

36. Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.

37. Prove that : 38. Prove that : tan x > x for x ϵ (0,π/2).39. Find the intervals on which the following functions are

(a) strictly increasing and (b) strictly decreasing: f(x) = (x+2)e-x.

40. f(x) = x4 – 4x3 + 4x2 + 15.

Probability

Page 1 of 5

Bt - BayesTheorem;Pdt -Probability DistributionBd - Binomial Distribution

1. In a factory which manufactures bolts, machine A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by machine B?--BT

2. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it us actually six.BT

3. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident is 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?BT

4. A card from the pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of lost card beingi) diamondii) spadeBT

5. A company has two plants to manufacture scooters. Plant I manufacture 70% of the scooters and plant II manufacture 30%. At plant I, 80% of the scooters are rated as of standard quality and at plant II, 90% of the scooters are rated as of standard quality. A scooter is chosen at random and is found to be of standard quality. What is the probability that it has come from plant II. BT

6. There are 3 bags each containing 5 white balls and 3 black balls. Also there are 2 bags each containing 2 white balls and 4 black balls. A white ball is drawn at random. Find the probability that this white ball is from the bag of first group. BT

7. In a competitive examination, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess 1/3 is and probability that he copies the answer is 1/6 . The probability that the answer is correct is, given that he copied it is 1/8 . Find the probability that he knows the answer to the squestion, given that he correctly answered the question. BT

8. Find the probability distribution of number of doublets in three throws of a pair of dice.

9. Let X denotes the number of hours you study during a randomly selected school day. The probability that X can take a value x has the following form, where k is some unknown constant : a. Find the value of k. b. What is the probability that you study at least two hours? Exactly two hours? At most two hours? PDT

10. The random variable X has probability distribution P(X) of the following form, where k is some number:

1. Find the value of K.

2. PDT

11. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings. PDT

12. Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X PDT OR BD

13. Let X denotes the sum of numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X. PDT OR BD

14. In a meeting, 70% of the members favour 30% members oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X). PDT

15. 15. Find the probability distribution of number of heads when three coins are tossed. Also find the mean number of heads in the above case. PDT OR BD

16. A pair of dice is thrown 4 times. If getting a doublet is considered a success. Find the probability of two successes. PDT OR BD

17. There are 5% defective items in a large bulk of items. What is probability that a sample of 10 items will not include more than one defective item. BD

18. A fair coin is tossed 10 times. Find the probability ofa. Exactly six heads. b. At least six heads c. At most six heads. BD

19. Find the mean of binomial distribution (n=4 p=1/3 MEAN=np,VAR=npq)

20. Five dice are thrown simultaneously. If the occurrence of an even number in a single dice is considered a success. Find the probability of getting at most 3 successes. BD

Probability

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21. An unbiased dice is thrown three times. Getting 3 or 5 is considered as success. Find the probability of at least two successes. BD

22. An urn contains seven white, 5 black and 3 red balls. Two balls are drawn at random. Find the probability that :

i. Both the balls are red. ii. One ball is red and other is black. iii. One ball is white.

23. 3 cards are drawn at random from a pack of well shuffled 52 cards. Find the probability that :i. All the three cards are of same suit. ii. One is a king; the other is a queen and third is a jack.

24. There are two bags I and II. Bag I contains 3 white and 2 red balls, bag II contains2 white and 4 red balls. A ball is transferred from bag I to bag II (without seeing its colour) and then ball is drawn from bag II. Find the probability of getting a red ball.

25. Bag I contains 3 red and 4 black balls and bag II contains 4 red and 5 black balls. One ball is transferred from bag I to bag II and then a ball is drawn from bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black. BT

26. A problem in mathematics is given to three students whose chances of solving it are 1/2, 1/3, 1/4 . What is the probability in the following casesi) the problem is solvedii) Only one of them solves it correctly.

27. Five dice are thrown simultaneously . If the occurrence of an even number is considered as a “ success” , find the probability of at most 3 successes BD

28. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact present . However the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested then with probability 0.005, the test will imply that he has the disease. If 0.1 percent of the population actually has the disease , What is the probability that a person has the disease given that the test result is positive ? BT

29. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of succeses. PDT

30. If E and F are independent events prove that E and F’ are independent.31. If a machine is correctly set up, it produces 90% acceptable items. If it is

incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly set up. BT-NOTE THE 2.SEE TEXT BOOK EXAMPLE SUM

32. Let x denote a number of collages where you will apply after your results and P( X = x ) denote your probability of getting admission in x number of collage it is given that (i) Find the value of K(ii) What is the probability that you will get admission in exactly two collages (iii) Find the mean & variance of probability distributions.

k is positive constant

33. If A and B are independent events then prove that A’ and B’ are also independent events

34. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

35. In a test, and examinee either guesses or copies or knows the answer to a multiple choice question with 4 choices, the probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. the probability that his answer is correct, given that he copied it is 1/8. Find the probability that he knew the answer to the question. BT

36. Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards .find the probability distribution of the number of aces .find its mean and standard deviation.

37. In shuffling a pack of 52 playing cards, four are accidentally dropped .find chance that missing cards should be one from each suit.

38. If P(A) = 0.2, P(B) = 0.3 and P (A U B ) = 0.4, where a & b are two events associated with a random experiment. Find P (A ∩ B) and P(A / B)

39. A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other without replacement. Find the probability that of the two drawn balls, one is white and the other is black.

40. Two dice are throw together. What is the probability that the sum of the number on the two faces is divisible by 3 or 4?

Probability

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41. A and B appear for an interview for two posts. The Probability of A’s selection is 1/3 and that of B’s selection is 2/5 . Find the probability that only one of themwillselected.

42. Three bags contains 5W, 8R; 7W,6R;6W,5R balls respectively. One ball is drawn from each bag at random. Find the probability that the three balls drawn are same colour.

43. A & B through a die alternately till one of them gets a 6 and wins the games. Find their respective probabilities of winning if A starts first.

44. A bag contains 5white, 7red & 8 black balls. If 4 balls are drawn one by one with replacement. Find the probability distribution of the number of red ball drawn.

45. A bag contains 4 white and 2 black balls, and another bag contains 3white and 5 black balls. If one ball is drawn from each bag, find the probability that one is white and one is black.

46. A problem is given to three students whose chances of solving it are 1/2, 1/3, ¼ what is the probability that the problem will be solved.

47. There are two bags A and B. Bag A contains 3 white and 2 red balls. Bag B contains 2 white and 4 red balls. A ball is transferred from Bag A to Bag B

without seeing its colour and then a ball is drawn from Bag B. Find the probability of getting a redball.

48. Three bad articles are mixed with 7 good ones. Find the probability distribution of the number of bad articles, if three articles are drawn at random.

49. A bag contains 3 red, 4 black and 2 green balls. Two balls are drawn at random from the bag. Find the probability that both balls are at different colours.

50. A pair of dice is rolled. Find the probability of getting a doublet or sum of numbers to be at least 20..

51. Two cards are drawn successively (without replacement) from a well shuffled pack of playing cards. Find the probability distribution of number of spades.

52. A bag contains 3 red, 4 black and 2 green balls. Two balls are drawn at random fromthe bag. Find the probability that both balls are of different colours.

53. A bag contains 3 red, 4 black and 2 green balls. Two balls are drawn at random from the bag. Find the probability that both balls are of different colours.

54. A pair of dice is rolled. Find the probability of getting a doublet or sum of number to be at least 10.

55. Two cards are drawn from a well shuffled pack of 52 cards without replacement. Find the probability that one of the two cards is an ace & the other a queen.

56. A problem of mathematics is given to 3 students whose chances of solving it are 1/2,1/3, 1/4. What is the probability that problem is solved?

57. A bag contains 5 white and 3 black balls. Another bag contains 4 white and 5 black balls. A bag is selected at random and two balls are drawn from it. Find The probability that both balls are of different colours.

58. Two cards are drawn from a well shuffled pack of 52 cards without replacement. Find the probability that one of the two cards in an ace and the other a queen of opposite shade

59. There are two bags I and II. Bag I contains 3 white and 2 red balls, bag II contains 2 white and 4 red balls. A ball is transferred from bag I to bag II (without seeing its colour) and then a ball is drawn from bag II. Find the probability of getting a red ball.

60. Two cards are drawn successively (with replacement) from a well shuffled pack of playing cards. Find the Probability distribution of number of speeds.

Probability

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61. Two dice are thrown together, what is the probability that sum of the number on the two faces is neither 9 nor 11?

62. A bag contains 4 yellow and 5 red balls and another bag contains 6 yellow and 4 red balls. A ball is taken out from the first bag and, without seeing its colour, is put in the second bag. Find the probability that if now a ball is drawn from the second bag, it is yellow in colour

63. A bag contains 5 white, 7red and 4black balls. If four balls are drawn one by one with replacement, what is the probability that none is white? BD PUT R=0

64. A pair of dice is thrown 6 times. If getting a total of 7 is considered a success. Find the probability of getting at most 5 success. BD---R< =5

65. The mean and variance of a bionomial distribution 4/3 and 8/9 respectively. Find the probability distribution.

66. If the sum of the mean and variance of a binomial distribution for 6 trial be 10/3 , Find the distribution.

67. Three urns A, B, and C contain 6 red & 4 white 2red and 6 white balls. respectively An urn is drawn at random and a ball is drawn. If the ball drawn is found to be red. Find the probability that the ball was drawn from urn A. BT

68. A bag contains five white and three black balls. Four balls are successively drawn without replacement. What is the probability that they are alternatively of different colors

69. Two cards are drawn successively without replacement from a well shuffled pack of playing cards. Find the probability distribution of number of spades.

70. A bag contains 3 red, 4 black and 2 green balls. Two balls are drawn at random from thebag. Find the Probability that both the balls are of different colours.

71. Find the Probability that in a random arrangement of the letters of word MATHEMATICS the vowels occur together

72. A die is thrown ten times. If getting a prime number is considered a success. Find The probability of getting not more then 8 successive. BD

73. A man is known to speak truth 4 out of a 5times. He throws a pair of dice and reports that it is a doublet. Find the probability that it is actually a doublet. BT

74. A and B are two events such that P(A)=0.42, P(B)=0.48 and P(A and B) = 0.16. Determine P(AorB).

75. A bag contain 7 green , 4 white and 5 red balls. If four balls are drawn one by one with replacement . What is the probability that none is red. BT-----R=0

76. A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins.A coin is pulled out at random from one of the two purses. What is the probability that it is a silvercoin.

77. Find the probability distributation of the random variable which equals the number of heads obtained when 3 coins are tossed

78. A football match may be either won , drawn or lost by the host country’s team ,show there are three ways of for casting the result at any one match one correct and two incorrect. Find the probabilities of for at least there correct result for four matches

79. A man is known to speak truth 3 out of 4 times he thrown a die and report that it is a six find probability that it is actually a six . BT

80. In a game a man wins a rupee for a six and loses for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets six. Find the expected value of the amount he wins/ loses. PDT

Probability

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81. A pair of dice is thrown twice. If the random variable X is defined as the number of doublets, find the probability distribution of X. Also find the mean and variance for above distribution.

82. Suppose that the reliability of HIV test is specified as follows: Of the people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of HIV, 90% of the test are judged HIV –ive but 1% are diagnosed as showing HIV +ive. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV +ive. What is the probability that the person actually has HIV? BT

83. A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from (i) LONDON (ii) CLIFTON. BT

84. A and B throw a pair of die turn by turn. The first to throw 10 is awarded a prize. If B starts the game. What is the probability that A is getting prize.

85. A manufacturer has three machine operators A,B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respect$ively. A is on the job for 50% of the item, B is on the job for 30% of the time and C is on the job for the remaining time .If a defective item is produced, what is the probability that it was produced by A? BT

86. In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’, if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.

87. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

88. The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a the probability that he will buy both a shirt and a trouser. Find also the probability that he will trouser is 0.3 and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he buys a trouser given that he buys a shirt.

89. Three persons A, B, throw a die in succession till one gets a ‘six’ and wins the game. Find their respective probabilities of winning, if A begins.

90. In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. The probability that his answer is correct given that he copied it is 1/8. Find the probability that he knows the answer to the question, given that he correctly answered it. BT

91. Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red (ii) one of them is black and the other is red.

92. In a hurdle race, a race player has to cross hurdles. The probability that he will be each hurdle is 4/5. What is the probability that he will knock down fewer than 2 hurdles? BD

93. A bag X contains 2 white and 3 red balls and a bag Y contains 4 white and 5 red balls.Oneball is drawn at random from one of the bag and is found to be red. Find the probability that it was drawn from bag Y. BT

94. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six.Find the probability that it is actually a six. BT

95. In a binomial distribution, the sum of mean and variance of 5 trials is 3.75.Find the distribution.

96. Suppose that 80% of people are right handed. What is the probability that at most 6 of a random sample of 10 people are right handed? BD

97. Two cards are drawn successively with replacement from a well-shuffled of 52 cards. Find the P.D, mean and variance for the number of kings. BD

98. Two cards are drawn without replacement from a well-shuffled of 52 cards. Find the P.D, mean and variance for the number of aces. BD

99. An urn contains 5 red and 3 black balls. Find the P.D of the number of blue balls in a draw of 2 balls with replacement.

100. An insurance company insured 2000 scooters, 3000 car and 4000 track drivers. The probabilities of their meeting with an accident are 0.04, 0.06 and 0.15 respectively. If one of person meets with an accident then find the probability that he is a car driver.

101. A bag contains 6 red and 7 blue balls and another bag contains 5 red and 4 blue balls. A ball is drawn from the first bag and without noticing its color is put in the second bag. A ball is then drawn from the second bag. Find the probability that the ball is drawn is blue in color.

102. A speaks truth in 60% of the cases and B in 70% of the cases. In what percentages of cases they are likely to (i) contradict each other (ii)agree with each other, in stating same fact?

103. A problem in mathematics is given to three students who’s chancing of solving it are 1/2, 1/3 and 1/4 respectively. What is the probability that the problem will be solved?

104. A husband and wife appear in an interview for two vacancies in the same post. The probability of husband’s selection is 1/5 and that of wife’s selection is 1/3. What is the probability that (i) both are selected(ii) only one is selected (iii) none is selected(iv) at least one selected?

105. In a group, there are 3 women and 4 men. Three persons are selected at random from this group. Find the probability that 2 women and 1 man or 1 woman and 2 men are selected.

106. Two cards are drawn at random, without replacement from a pack of 52 cards, find the probability that both the cards will be red.

107. The probability that a student selected at random will pass in mathematics 2/3 and the probability that he/she passes in mathematics and c.s is 1/5. What is the probability that he/she will pass in c.s if it is known that he/she has passed in mathematics?

108. Given P(A) = 1/2, P(B) = 1/3 and P(A U B) = 2/3. Are the events A and B independent?

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12th Test Math