Download class xii mathematics - Kendriya Vidyalaya Dhrangadhra

KENDRIYA VIDYALAYA SANGATHAN
(AHMEDABAD REGION)
GOLDEN QUESTIONS FOR 12th MATHEMATICS
UNIT-I.
RELATIONS & FUNCTIONS
(10 MARKS)
1. Define relation R on the set N of natural numbers by R={ (x,y): y= x+7, x is a natural
Number less than 4; x, y ∈ N }.Write down the domain and the range.
2. Let R be the relation in the set N given by R = {(a,b); a = b – 2 , b > 6}
Whether the relation is reflexive or not ? Justify your answer.
3. Let R be the relation on R defined as (a , b)  R iff 1+ ab > 0  a,b  R.
a. Show that R is symmetric.
b. Show that R is reflexive.
c. Show that R is not transitive.
4.
If f(x) = x 2  x 2 , then find f(1/x).
5. Let A = {-1, 0, 1} and B = {0 , 1}. State whether the function f: A → B defined by
f(x) = x2 is bijective .
6. Show the function f: R→R defined by f(x) =
, x
R is one-one and onto function.
Also find the inverse of the function f.
x 1
, then show that
x 1
1
(a) f   = – f(x) ,
x
7. If f(x) =
1
 1
(b) f    =
 x  f (x)
3x  5
. Find f -1
2
9. Show that the relation R on A ,A = { x| x  Z , 0 ≤ x ≤ 12 } ,
R = {(a ,b): |a - b| is multiple of 3.} is an equivalence relation.
8. If f: RR is defined by f(x) =
10. Let A = Set of all triangles in a plane and R is defined by
R={(T1,T2) : T1,T2  A & T1~T2 }. Show that the R is equivalence relation. Consider the
right angled ∆s, T1 with size 3,4,5; T2 with size 5, 12,13; T3 with side 6, 8, 10; Which of
the pairs are related?
11. Consider a function f :R+[-5, ∞) defined f(x) = 9x2 + 6x - 5. Show that f is invertible &
f -1(y) =
y  6 1
, where R+ = (0 , ∞).
3
12. Let * be a binary operation on the set Q of rational numbers defined as a * b =
Write the identity and inverse of *, if any.
ab
.
5
13. On R – {1} a binary operation ‘*’ is defined as a * b = a + b – ab. Prove that ‘*’ is
commutative and associative. Find the identity element for ‘*’. Also prove that every
element of R – {1} is invertible.
14. Prove the following:
\
15. Prove the following:
.
16. Prove that:
.
17.
18.
.
Prove the following :
.
19.
Solve for
20.
Solve for x:
, x>0.
21. Solve for x:
.
22. Solve for x:
23.
Prove that
 ab

 b  a cos  
2 tan 1 
tan   cos 1 

2
 a  b cos  
 ab
UNIT-II. ALGEBRA
(13 MARKS)
1. If A=
2. A=
, prove that
and I=
3. Express the matrix A=
Hence find
, find k so that
.
as the sum of a symmetric and skew –symmetric matrix.
4.Using properties of determinants, prove that
5. If x, y and z are different and
, show that xyz=-1.
6. Using properties of determinants, prove the following:
.
7. Using properties of determinants prove the following:
.
8. Using properties of determinants prove the following:
9. Using properties of determinants, solve the following for x:
10. Using properties of determinants, solve the following for x:
11. Using properties of determinants, show that
12. Using properties of determinants, prove the following
13. Using properties of determinants, show that
1 a2  b2
2ab
2ab
1 a  b
2b
 2a
2
 2b
2
2a
= 1  a 2  b2 
3
1 a2  b2
14. Solve the following system of equation by matrix method,
where x 0, y  0, z  0;
2 3 3
1 1 1
3 1 2
   10 ;    10 ;    13
x y z
x y z
x y z
15. Three shopkeepers A, B and C are using polythene, handmade bags (prepared by
persons) and newspapers envelopes as carry bags. It is found that the shopkeepers A,B
,C are using(20,30,40) ,(30,40,20) ,(40,20,30) polythene ,handmade bags and
newspapers envelopes respectively. The shopkeepers A, B,C spent Rs 250,Rs 220 and
Rs 200 on these carry bags respectively. Find the cost of each carry bags using matrices.
Keeping in mind the social and environmental conditions, which shopkeeper is better
and why?
16. In a survey of 25 richest person of three localities x, y ,z. It is found that in locality A 7
believes in honesty,12 in hard work and 6 in unfair means. While in Y 10 believes in
honesty 11 in hard work , 4 in unfair means and in Z, 8 believes in honesty ,12 in hard
work 5 in unfair means. If the income of 25 richest persons of locality x, y and z are
respectively Rs.1,19,000,Rs.1,18,0001,18,000per day then find income of each type of
person by matrix method. In your view which type of person is better for society and for
nation.
17. Two schools A and B decided to award prizes ti their students for three values honesty,
punctuality and obedience. School A decided to award a total of Rs. 11000 for the thee
values to 5,4 and 3 students respectively while school B decided to award 10700 for the
three values to 4 ,3 and 5 students respectively. If all the three prizes together amount to
Rs. 2700, then
1) Represent the above situation by a matrix equation and form linear equations using
matrix multiplications.
2) Is it possible to solve the system of equations, so obtained using matrices. If possible
solve it.
3) Which value you prefer to reward most and why?
UNIT-III. CALCULUS
(44 MARKS)
1. If f(x)=
is continuous at x=3. Find the values of a add b.
2. f(x)=
is continuous at x=0. Find the value of k also
discuss about continuity at x=1.
3. f(x)=
is continuous function.
4. Show that f(x)=
5. f(x)=
6. x=
is continuous but not differentiable at x=3
is continuous at x=0
,
y=
. Find
.
7. if
=0 then prove that
8. if
then prove that
is independent
of a and b
9. If cos y = x cos(a+y) then prove that
10. If y =
then show that (1-x2)
-x
11. If y = a sin x + b cos x, then prove that
12. If y = 500
+ 600
, prove that
13. Find the second order derivative of
14. If y =
, prove that ( 1 +
-
=0
+ y = 0.
- 49y = 0.
+ log x.
)
+ 2x
= 0.
15. Find the intervals in which the function f given by f(x) = -2
strictly decreasing .
-9
16. Find the intervals in which the function f given by f(x) =
is strictly increasing .
+
17. Find the value of x for which y =
- 12
+1 is
,0
x
2
is an increasing function.
18. Prove that the function given by f(x) = cos x is
(a) strictly decreasing in (0, ).
(b) strictly increasing in ( , 2 ).
(c) neither increasing nor decreasing in ( , 2
).
19. Find the equations of the tangent to the curve y =
coordinate is 3.
+4
+ 1 at the point whose x-
19. Find the equations of the normal to the curve y =
coordinate is 3.
20. Find the equation of tangent to the curve y =
4x -2y + 5 =0.
+4
+ 1 at the point whose x-
, which is parallel to the line
21. Find the equations of the tangent to the parabola
at the point ( a
22. Find the equations of the tangent to the hyparabola
(h,k .
23. Prove that the curves x =
-
and xy = k cut at right angles if
= 1 at the point
= 1.
24. The volume of a cube is increasing at the rate 10
increasing when the length of an edge is 15
cm?
25. A particle move along the curve 6y=
+2. Find the points on the curve at which the
Y coordinate is changing 8 times as fast as the X –coordinate.
26. The radius of an air buble is increasing at the rate of
cm/s. At what rate is the
volume of the buble increasing when the radius is 1cm?
27. Sand is pouring from a pipe at the rate of 12
.The falling sand forms, a cone on
the ground in such a way that the hight of the cone is always one –sixth of the
radius of the base .How fast is the height of the sand cone increasing when the
height is 4cm?
28.
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.
29.
A square piece of tin of side 18 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
maximumpossible.
30. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top by
cutting off squares from each corner and folding up the flaps. What should be the
side of the square to be cut off so that the volume of the box is the maximum
possible.?
31. Show that the height of cylinder of maximum volume that can be inscribed in a sphere
2a
of radius ‘a’ is
.
3
32. Show that semi-vertical angle of right circular cone of given surface area and
maximum volume is Sin-1(1/3).
33. Evaluate:
as the limit of a sum.
34. Evaluate:
35. Evaluate:
36. Evaluate:
37. Evaluate:
38.
39. Evaluate:
40. Evaluate:
41.
Evaluate:
42. Using integration, find the area of the region bounded by the parabola
the circle 4 +4 =9.
and
43. Find the area between the curves y2 = x and x2 = y.
44. Find of the area of region enclosed between the two circles
=4 and
45. Find area of region bounded by y=
and y=
.
46. Find area of region bounded by triangle whose vertices are (-1,2), (1,5) and (3,4).
47. Using integration find area of the triangular region whose sides have the equations
y=2x+1,y=3x+1 and x=4.
48. Solve the differential equation:
.
49. Solve the differential equation:
50. Form a Differential equation representing the family of curves given by
(x- a)2 + 2y2 =a2 where a is an arbitrary constant.
51. Form a Differential equation representing the family of circles touching the Y-axis at
origin.
52. Form a Differential equation representing the family of circles touching the X-axis at
origin.
53. Solve the given differential equation
54. Solve
= (1+x²) (1+y²).
= 2.
UNIT-IV. VECTORS AND THREE DIMENSIONAL GEOMETRY
(17 MARKS)
1. Show that the vectors are
and
2. Show that the vectors
3. If
and
,
is perpendicular to
are orthogonal.
,
and
and
If
3i+j are such that
perpendicular to , then find the value of
6.
If
, show that
.
4. Find the angle between the vectors
scalar product of the vectors.
5.
are orthogonal.
are unit vectors such that
.
.Find the value of
using the
is
7. If
be three vectors such that
= 3,
= 4 and
= 5 and each one of them
being perpendicular to the sum of the other two, find
8. Three vectors
.
satisfy the condition
, if
= 1,
=4 and
=
. Evaluate the quantity
= 2.
9. Find the shortest distance between the lines whose vector equations are
and







10. Find the shortest distance between the two lines r  6 i  2 j  2 k   ( i  2 j  2 k )






and r  4 i  k   (3 i  2 j  2 k ) .
11. Find the equation of the plane passing through the points (0,-1,0),(1,1,1) and (3,3,0).
12. Find the equation of the plane determined by the points (3,-1,2),(5,2,4) and (-1,-1,6).
13. Find the equation of plane passing through the line of intersection of the planes
x+2y+3z =4 and 2x+y-z+5 =0 and perpendicular to the plane 5x+3y-6z+8 =0.
14. Find the Cartesian as well as vector equations of the planes, through the intersection of
planes r.(2i  6 j )  12  0and r.(3i  j  4kˆ)  0 which are at a unit distance
from the origin.
15.
Find the co-ordinate of the foot of the perpendicular and distance of the point
from the plane
. Find also the image of the point in the
plane.
16. Find the foot of perpendicular and image of the point P(1, 2, 4) in the plane 2x + y – 2z
+ 3 = 0.
UNIT-V.
LINEAR PROGRAMMING
(6 MARKS)
1. Solve the following system of equations by matrix method
x yz 6
y  3 z  11
x  2y  z  0
2. A manufacturing company makes two models A and B of a product. Each piece of
model A requires 9 labour hours for fabricating and 2 labour hours for finishing. Each
piece of model B requires 12 labour hours for fabricating and 3 labour hours for
finishing. For fabricating and finishing, the maximum labour hours available are 180
and 30 respectively. The company makes a profit of Rs 8000 on each piece of model
A and Rs 12000 on each piece of Model B. How many pieces of model A and Model
B should be manufactured per week to realize a maximum profit? After income tax,
sale tax, service tax, the net profit of the company is Rs 1,14,000. Should owner of the
company close down the company? Discuss briefly.
3. A dealer in rural area wishes to purchase a number of sewing machine. He has only
Rs. 5760 to invest and has space for at most 20 items. An electronic sewing machine
costs him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an
electronic machine at a profit of Rs. 22 and manually operated machine at a profit of
Rs. 18. Assuming that he can sell all the items that he can buy how should he invest
his money in order to maximize his profit. Make it as a linear programming problem
and solve it graphically. Keeping the rural background justify the values to be
promoted for the selection of manually operated machine.
4. A manufacturer manufactures two types of steel trunks. He has two machines A and
B. For completing the first type of trunk, it requires 3 hours on machine A and 1 hour
on machine B; whereas the second type of trunk requires 3 hours on machine A and
two hours on machine B. Machine A can work for 18 hours and B for 8 hours only
per day. There is a profit of Rs.30 on the first type of the trunk and Rs.48 on second
type per trunk. How many trunk of each type should be manufactured every day to
earn maximum profit? Solve the problem graphically.
5. A diet for a sick person must contain at least 4000 units of vitamins, 50 units of
minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs.5
and Rs.4 per unit respectively. One unit of the food A contains 200 units of vitamins,
1 unit of minerals and 40 units of calories, while one unit of the food B contains 100
units of vitamins, 2 units of minerals and 40 units of calories. Find what combination
of the foods A and B should be used to have least cost, but it must satisfy the
requirements of the sick person. Form the question as LPP and solve it graphically.
Explain the importance of balanced diet.
6.
A cooperative society of farmers has 50 hectare of land to grow two
crops Xand Y. The profit frot from crop Xand Y per hectare are estimated as
Rs10,500 and Rs.9,000 respectively.a luquid herbicite has to be used for crop Xand Y
at rates of 20 litres and 10 litres per hectare further no harms more than 800 litres of
herbicicide should be used in order of protect fish and wildlife using pond which
collects drainage from this land.How much land should be allocated to each crop so as
to maximize the total profit of the society
UNIT-VI. PROBABILITY
(10 MARKS)
1. A fair die tossed thrice. Find the probability of getting an odd number at least once.
2. Probability of solving specific problem independently by A and B are 1/2 and 1/3
respectively. If both try to solve the problem independently, find the probability that
(1) the problem is solved (2) exactly one of them solved the problem
3. Event A and B are such that P (A) = 1/2, P (B) =7/12 and P(not A or not B) =1/4.
State whether A and B are independent.
4. Two cards are drawn at random and without replacement from a pack of 52 cards.
Find the probability that both the cards are black.
5. The probabilities of A, B and C hitting a target are 1/3,2/7 and 3/8 respectively. If all
the three tried to shoot the target simultaneously, find the probability that exactly one
of them can shoot it.
6. A husband and a wife appear for interview for two vacancies for the same post. The
probability of husband’s selection is 1/7 and that of wife’s is 1/5.What is the
probability that (1) only one of them will be selected (2) at least one of them will be
selected.
7. A coin is tossed once. If it shows a head,it is tossed again but if it shows a tail,then a
die is thrown. If all possible outcomes are equally likely, find the probability that the
die shows a number greater than 4, if it is known that the first throw of coin results a
tail.
8. Three persons A B and C fire a target in turn,starting with A. Their probability of
fitting the target are 0.5,0.3 and o.2 respectively. Find the probability of at most one
hit.
9. Two balls are drawn at random with replacement from a box containing 10 black and
8 red balls. Find the probability that (1) both balls are red (2) one of them is black
and other is red.
10. In an examination, an examinee either guesses or copies or knows the answer of
multiple choice questions with four choices. The probability that he makes a guess is
and probability that he copies the answer is . The probability that his answer is
correct, given that he copied it, is . Find the probability that he knew the answer to
the question, given that he correctly answered it.
11. A problem in statistics is given to three students A, B and C, whose chances of solving it
are ,
and
respectively. What is the probability that the problem will be solved.