Download Most imp questions of Maths for class 12 2015

MATHEMATICS
CLASS XII
MOST IMPORTANT QUESTIONS (FOR 2015)
Relations & Functions (6 Marks)
1) Let N be the set of all natural Nos. & R be a relation on NxN defined by (a,b)R (c,d) such that
ad = bc. Prove that R is an equivalence relation
2) Let R be a relation on NxN defined by (a,b)R (c,d) : a+d = b+c. Prove that R is an equivalence relation.
3) Prove that the relation R on the set Z of all integers defined by R={(a,b): a-b is divisible by n} is an equivalence
relation.
4) A=R-{3}and B= R-{1}. Consider the function F: A
B defined by f(x)=
is f is one one and onto? Justify
5) Show that the relation R in the set R of real numbers defined as R={ (a,b): a b2} is neither reflexive nor
symmetric nor transitive.
6) Consider f: R+ [-5, ) given by f(x)= 9x2+6x-5. Show that f is invertible and find inverse of f.
7) F: R R, g: R R such that f(x)= 3x+1 and g(x)= 4x-2. Find fog and show that fog is one one and onto.
8) A binary operation * on Q-{-1} such that a*b= a+b+ab. Find the identity element on Q. Also find the inverse of
an element in Q-{-1}
9) Let A= N N and * be the binary operator 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.
10) Let A = NxN & * be a B.O. on A defined by (a,b)*(c,d) = (ac,bd) for all a, b, c, d N. Show that * is commutative &
Associative.
11) F, g : R R be a function :
F(x) =
& g(x) = 3x + 1. Find gof & prove that gof is one one & onto.
R be a function defined by f(x) = 4x2 + 12x +15. Show that f : N
12) Let f : N
range (f) is invertible.
INVERSE TRIGONOMETRY (4 Marks)
1. Write the following function in simplest form:
√
(
(i)
)
(
(ii)
√
)
(iv)
(
√
√
√
√
+=
(iii) sin-1√
.
√
2. Prove the following:
(ii) cot-1*
(i) tan-12 + tan -1 3 =
(iii) tan-1[
√
√
√
√
(v) tan- 1
(iv) tan-1*
]=
= sin-1
-1
√
√
√
√
(vi) tan- 1 + tan-1
+=
tan-1 + tan-1 =
3. Solve the following for x:
(
(i)
(iii).
)
(
(ii). tan-1 2x + tan-1 3x =
)
(
(v). sin -1(1 – x) – 2 sin -1 x =
)
(iv). 2 tan -1(cos x) = tan -1 (2 cosec x)
)
4. Prove : tan-1(
)
(
5. Prove that: tan -1( )
)
( )
.
( )
6. Prove that : 2tan-1( )+ tan-1( ) = sin-1
.
√
7. .Prove that : cos[tan-1 {sin (cot-1 x)}] = √
.
8. Solve for x : 2 tan-1 ( sin x) = tan-1 (2 sec x),
9. Prove that cot -17 + cot-1 8 + cot -1 18 = cot-13
MATRICES AND DETERMINANTS (1+4+4+4=13 Marks)
1 MARK QUESTIONS.
1. Find  if *
+ is equal to identify matrix.
2. If matrix *
+ is symmetric find x
3. (
)=(
=*
4.
) find y .
+ find A + AT .
5. Find x if |
| = |
|
6. Find area of triangle using determinants A (-3, 5) B(3, -6) & C(7, 2)
7. For what value of x, matrix *
+ is singular?
8. ‘A’ is a square matrix of order 4 : | | = 1 find (i) |
9. Find cofactor of a12 in |
10. |
11. A = *
| = |
| (ii) |
|
(iii) |
|
|
| find x.
+ write A-1 in terms of A
4 MARKS QUESTIONS
1. In a department store, a customer X purchases 2 packets of tea, 4 kg of rice & 5 dozen oranges. Customer Y
purchases 1 packet of tea, 5 kg rice and 24 oranges. Price on 1 pack of tea is Rs. 54, 1 kg of rice is Rs. 22 and that
of 1 dozen oranges is Rs. 24. Use Matrix multiplication method and calculate each individual bill.
2. If A = *
+ , Verify A2 – 4A – I = 0. Hence find A-1 .
3. If A = *
+ , then prove by P.M.I. that An = [
] .
4. The department of education runs 120 colleges and 15 universities having a strength of 300 lecturers, 100
readers and 50 professors in the universities and 5000 lecturers and 1000 readers in colleges. The monthly
salary is Rs. 9000 for professors, Rs. 8000 for readers and Rs. 5000 for lecturers. Find the monthly salary bill in
the colleges and universities.
5. Express the matrix [
] as sum of symmetric and skew symmetric matrix.
][
6. Find x if [x 4 -1] [
] =0
7. Using Elementary Row operations & column operations find A-1 whose
a. A = [
]
b. A = [
]
8. A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated in the
table :
Market
Products
X
Y
Z
I
10,000
2,000
18,000
II
6,000
20,000
8,000
If unit sale price of x, y and z are Rs. 2.50, Rs. 1.50 and Rs. 1.00 respectively, find the total revenue in each
market, using matrices.
9. Express the matrix B = [
] as sum of symmetric and skew symmetric matrices.
10. Using properties of determinants, prove that
i)
|
| = (5x + 4) (4 – x)2
ii)
|
iii)
|
iv)
|
v)
|
| = 4a2b2c2
vi)
|
| = (1 + a2 + b2 + c2)
|
| =(
= (a + b + c )3
- )( - )( - )( +
|=1
+ )
vii)
|
viii)
|
| = (a – b) (b – c) (c – a) (a + b + c)
| = 3(a + b + c) ( ab + bc + ca)
11. For the matrix A = *
+ , find the numbers a & b: A2 + aA + bI = 0. Hence find A-1.
12. Find the matrix A satisfying the matrix equation *
13. Solve the equation : |
+ A*
+ = *
+.
| = 0, a 0
14. Using properties of determinant prove :- |
| = (1 + pxyz)(x – y) (y – z) ( z – x)
15. A trust fund has Rs. 30,000 to invest in two bonds. First pays 5% interest per year & second pays 7%. Using
matrix multiplication determine how to divide Rs. 30,000 among two types of bonds if trust find must obtain
annual interest of Rs. 2000.
CONTINUITY AND DIFFERENTIABILITY (4+4=8 Marks)
| , x  R, is continuous but not differentiable at x = 2.
1. Show that the function g(x) = |
2. Differentiate log ( x sin x + cot 2 x) with respect to x.
3. If y = log [ x + √
], show that ( x2 + a2)
+x
4. For what values of a and b the function defined is continuous at x = 1, f(x) = {
5. . If √
+√
= a (x – y), Prove
= √
6. If x = a sin t and y = a ( cos t + log tan ), find
√
.
√
7. Find the value of k, for which f(x) = {
is continuous at x = 0.
8. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that
9. Differentiate tan-1 [
√
√
√
√
=
.
] with respect to cos-1 x2 .
10. y = ex tan-1 x, then show that (1 + x2)
11. Verify lagrange’s Mean value theorem:
- 2 (1 – x + x2)
+ (1 – x2 ) y = 0.
(i) f(x) = x(x – 1) (x – 2) [ 0, ½]
(ii) f(x) = x3 -5x2 – 3x [1, 3]
12. Verify Rolle’s theorem for the following functions :
(i) f(x) = x2+ x – 6[-3, 2]
13. Differentiate sin-1(
14. If x = √
15. If y =
√
(ii) f(x) = x (x – 1) (x – 2) [0, 2]
(iii) f(x) = sin x + cos x [0, ]
) w.r.t. tan-1 x .
√
.
, show that (1-
16. Differentiate cos-1 ,
)
- with respect of tan-1 ,
- .
APPLICATION OF DERIVATIVES (4+6=10 Marks)
4 Marks Questions:
touches the curve y = be –x/a at the point where it crosses the y axis.
1. Show that the line
2.
Separate the interval *
+ into sub – intervals in which f(x) = sin4 x + cos4 x is increasing or decreasing.
3. The radius of a spherical diamond is measured as 7 cm with an error of 0.04 cm. Find the approximate error in
calculating its volume. If the cost of 1 cm3 diamond is Rs. 1000, what is the loss to the buyer of the diamond?
What lesson you get?
4. Show that the curves 4x = y2 and 4xy = K cut at right angles if K2 = 512 .
5. Find the intervals in which the function f given be f(x) – sinx – cosx, 0  x  2 is strictly increasing or strictly
decreasing.
6. Find all points on the curve y = 4x3 – 2x5 at which the tangents passes through the origin.
7. Find the equation of normal to the curve y = x3 + 2x + 6 which are parallel to line x+14y+4=0.
8. Show that the curves y = aex and y = be –x cut at right angles if ab = 1.
9. Find the intervals in which the function f f(x) = x3 +
10. Prove that y =
–
,x
is an increasing function of
0 is increasing or decreasing.
in [0, ] .
11. Using differentials, find the approximate value of each of following:
(i)(401)1/2
(ii) (26)1/3
(iii) (0.009)1/3
(iv)√
(v)√
6 Marks Questions:
1. Find the area of the greatest rectangle that can be inscribed in an ellipse
= 1.
2. If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the
triangle is maximum when the angle between them is

.
3. Prove that the radius of right circular cylinder of greatest C.S.A which can be inscribed in a cone is half of that of
cone.
4. A helicopter if flying along the curve y = x2 + 2. A soldier is placed at the point (3, 2) . find the nearest distance
between the solider and the helicopter.
5. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to
diameter of base.
6. Prove that the volume of largest cone that can be inscribed in a sphere of radius R is 8/27 times volume of
sphere.
7. Show that the semi – vertical angle of cone of maximum volume and of given slant height is tan-1√ .
8. A wire of length 28 cm is to be cut into two pieces, one of the two pieces is to made into a square and other
into circle. What should be the length of two pieces so that combined area of square and circle is minimum?
9. Show that the height of cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R √ .
10. The sum of perimeter of circle and square is K. Prove that the sum of their areas is least when side of square is
double the radius of circle.
11. Find the value of x for which f(x) = [x(x – 2)]2 is an increasing function. Also, find the points on the curve, where
the tangent is parallel to x- axis.
12. Show that all the rectangles inscribed in a given fixed circle, the square has maximum area.
13. A wire of length 36 cm is cut into two pieces, one of the pieces is turned in the form of a square and other in the
form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be
minimum.
14. Prove that the surface area of a solid cuboid, of square base and given volume, is minimum when it is a cube.
15. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth in 2 m
and volume is 8 m3. If building of tank costs Rs. 70 per sq. meter for the base and
Rs. 45 per sq. Meter for sides, what is the cost of least expensive tank?
16. An open box with a square base is to be made out of a given quantity of cardboard of area c2 square units. Show
that the maximum volume of the box is
√
cubic units.
17. A window is in the form of a rectangle surmounted by a semicircular opening. Total perimeter of window is 10m.
Find the dimensions of the window to admit maximum light through whole opening.
INTEGRALS (4+4+4=12 Marks)
1. Evaluate :- ∫
dx.
2. Evaluate :- ∫
dx.
3. Evaluate :- ∫
4. Evaluate :- ∫
, using properties of definite integrals.
dx.
5.
Evaluate :- ∫
dx.
√
6. Evaluate :- ∫
7. Evaluate :- ∫
dx.
dx
8. Using properties of definite integrals, evaluate : ∫
9. Evaluate :- ∫
.
dx
10. Evaluate :- ∫
(
11. Evaluate :- ∫
) dx
12. Evaluate :- ∫
dx
13. Show that ∫ (√
)
√
√
∫
14.
15. Evaluate :- ∫ |
| dx
16. Prove that :- ∫
(√
17. Evaluate :- ∫
)
dx
APPLICATION OF INTEGRATION (6 MARKS)
1. Using integration find the area of the region { (x, y) : x2 + y2  1  x + , x , y  R} .
2. Using integration, find the area of the region enclosed between the two circles x2 + y2 = 4 and
(x – 2)2 + y2 = 4.
3. Find the area of the region { (x, y) : y2  6ax and x2 + y2  16a2} using method of integration.
4. Prove that the area between two parabolas y2 4ax and x2 = 4ay is 16 a2 / 3 sq units.
5. Using integration, find the area of the following region. ,
-.
6. Find the area of the region {(x, y) : x2 + y2
7. Find the area lying above x – axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.
8. Find the area of the region included between the curve 4y = 3x2and line 2y = 3x + 12
9. Sketch the graph of y = |
| and Evaluate ∫ |
| dx .
10. Using the method of integration, find the area of the region bounded by the lines 3x – 2y + 1 = 0,
2x + 3y – 21 = 0 and x – 5y + 9 = 0.
DIFFERENTIAL EQUATIONS (1+1+6=8 MARKS)
1 MARK QUESTIONS.
1. What is the degree and order of following differential equation?
( )
(i) y
(
2. Write the integrating factor of
) .
(ii)( ) + 3y
= 0.
(iii)
+y2 +
=0
+ 2y tan x = sin x
3. Form the differential equation of family of straight lines passing through origin.
4.
Form the differential equation of family of parabolas axis along x-axis.
6 MARKS QUESTIONS.
1. Show that the differential equation
xdy – ydx = √
dx is homogeneous and solve it.
2. Find the particular solution of the differential equation :- cos x dy = sin x ( cos x – 2y) dx, given that y = 0, when x
=
.
( )
3. Show that the differential equation *
of this differential equation, given that y =
4. Show that the differential equation x
+ dx + x dy = 0 is homogeneous. Find the particular solution
when x = 1.
( )
( )
solution of this differential equation, given that x = 1 when y =
is homogeneous. Find the particular
.
5. Solve the following differential equation :- (1 + y + x2y) dx + (x + x3)dy = 0, where y = 0 when x = 1.
6. Solve the following differential equation : √
+ xy
.
7. Find the particular solution of te differential equation :( xdy – ydx) y sin
, given that
y =  when x = 3.
8.
. given that y = 0 where x = 1, i.e.,
y(1) = 0
9. Solve the initial value problem : (x2 + 1)
- 2xy = ( x4 + 2x2 + 1) cos x, y (0) = 0.
10. Solve : (x2 + xy) dy = (x2 + y2) dx.
11. Show that the differential equation : 2y ex/y dx + (y – 2x ex/y) dy = 0 is homogenous and find its
particular solution, given x = 0 when y = 1.
12.
Solve the equation : (
)
(
13. Find the particular solution of the differential equation (1 + x3)
) x dy.
+ 6x2y = (1 + x2), given that x = y = 1.
VECTORS AND THREE DIMENSIONAL GEOMETRY(1+1+1+4+4+6=17 MARKS)
1 MARK QUESTIONS.
1. If ⃗ is a unit vector and ( ⃗
⃗
⃗
, then find | ⃗| .
⃗
𝜆 ̂ + ̂ and ⃗⃗ = ̂
2. Find the value of 𝜆 so that the vector ⃗ = 2 ̂
̂ +3 ̂ are perpendicular to each
other.
3.
⃗⃗ are : | ⃗| = 2, | ⃗⃗|
If two vectors ⃗
4. Find the angle between ̂
5.
⃗= ̂
̂ , ⃗⃗ = 3 ̂
̂
and ⃗ ⃗⃗ = 4, find | ⃗
̂
̂
̂
⃗⃗|.
̂.
̂
̂ , find a unit vector in the direction ⃗
̂
⃗⃗ .
6. Write Direction ratios of
7. Find ‘ ’ when the projection of ⃗ = 𝛌 ̂
̂ + 4 ̂ and ⃗⃗ = 2 ̂
̂ + 3 ̂ is 4 units.
8. Find the volume of the parallelopiped whose adjacent sides are represented by ⃗ , ⃗⃗ and ⃗ where
⃗=
̂
̂ , ⃗⃗ = 2 ̂
̂
̂ , ⃗ = -4 ̂
̂
9. Find the value of 𝜆 so that the vectors ̂
̂.
̂
̂,2 ̂
̂
̂
̂,-̂
̂ are coplanar.
𝜆̂
4 MARKS QUESTIONS.
1. Find a unit vector perpendicular to the plane of triangle ABC where the vertices are A (3, -1, 2), B ( 1, -1, -3) and
C ( 4, -3, 1).
2. Let ⃗ = 4 ̂
̂ , ⃗⃗ = ̂
⃗⃗
both ⃗
3.
̂
Let ⃗ ⃗⃗
̂ and ⃗ =
̂
̂
̂ . Find a vector ⃗ which is perpendicular to
̂
: ⃗ ⃗
⃗ be three vectors : | ⃗| = 3, | ⃗⃗| = 4, | ⃗| = 5 and each one of them being perpendicular to
⃗⃗
the sum of other two, find | ⃗
⃗| .
4. Using vectors, find the area of the triangle ABC with vertices A (1, 2, 3), B ( 2, -1, 4) and C ( 4, 5, -1) .
5. If ⃗
̂
̂
̂ ⃗⃗
̂
to the vector 2 ⃗ - ⃗⃗
⃗
̂
̂
̂ , find a vector of magnitude 6 units which is parallel
⃗.
⃗⃗ are unit vectors and
6. If ⃗
̂
̂
= ½ |⃗
is the angle between them, show that sin
⃗⃗| .
7. If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is √ .
8. The two adjacent side of a parallelogram are
̂
̂
̂ and ̂
̂
̂ . Find the unit vector
parallel to its diagonal. Also, find its area.
9.
The scalar product of the vector ̂
̂
10. If ̂
̂
̂ with a unit vector along the sum of vectors 2 ̂
̂ is equal to one, find the value of
̂
̂
̂,2 ̂
̂,3 ̂
̂
̂ and ̂
̂
̂ and
̂
.
̂ are the position vectors of the points A, B, C and D, find
the angle between ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ . Deduce that ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ are collinear.
11. If ⃗ ⃗⃗
⃗ are three unit vectors such that ⃗ ⃗⃗ = ⃗ ⃗ = 0 and angle between ⃗⃗
⃗⃗
⃗
12. If ⃗ ⃗⃗
⃗ is , prove that
⃗).
⃗ are three unit vectors such that | ⃗| = 5, | ⃗⃗| = 12 and | ⃗| = 13 , and ⃗
of ⃗ ⃗⃗
⃗⃗ ⃗
⃗⃗
⃗ = ⃗⃗ , find the value
⃗ ⃗.
13. Find the equation of plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1,
4, 2) and ( 2, 3, 5) . Also, find the perpendicular distance of the plane from the origin.
14. Find the shortest distance between the lines :
⃗
̂
̂
̂
( ̂
̂)
̂
⃗
̂
̂
̂
(
15. If any three vectors ⃗ ⃗⃗ and ⃗ are coplanar, prove that the vectors ⃗
̂
⃗⃗ , ⃗⃗
̂ ).
⃗ and ⃗
⃗ are also
coplanar.
16. If ⃗ ⃗⃗ and ⃗ are mutually perpendicular vectors of equal magnitudes, show that the vector ⃗
equally inclined to ⃗ ⃗⃗
⃗⃗
⃗ is
⃗ .
17. Find the position vector of a point R which divides the line joining two points P and Q whose position
vectors are (2 ⃗
⃗⃗ ) & ( ⃗
⃗⃗) respectively, externally in the ration 1 : 2. Also, show P is the mid
point of RQ .
3̂
18. . If
̂ and
̂
=2̂
̂ , then express ⃗ in the form ⃗ = ⃗1 + ⃗2, where ⃗1 is parallel
̂
to ⃗ and ⃗2 is perpendicular to ⃗ .
and the plane 10x + 2y – 11z = 3.
19. Find the angle between the line
20. Find whether the lines ⃗ = ( ̂
̂
̂ + 𝜆(2 ̂
̂) and ⃗ = (2 ̂
̂ + ( ̂
̂
̂ ) intersect or not. If
intersecting, find their point of intersection.
6 MARKS QUESTIONS.
1. Show that the lines
,
are coplanar. Also find the equation of the plane
containing the lines.
2. Find the co-ordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane determined by
points A(1, 2, 3), B (2, 2, 1) and C (-1, 3, 6) .
3. Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to the lines
And
4. Find the equation of the plane containing the lines :
⃗
̂
̂+
( ̂
̂
̂ ) and ⃗
̂
̂+
(
̂
̂
̂)
Find the distance of this plane from origin and also from the point (1, 1, 1).
5. Find the coordinates of the foot of the perpendicular drawn from the point A (1, 8, 4) to the line joining the
point B (0, -1, 3) and C ( 2, -3, -1).
6. Find the distance of the point (-1, -5, -10) from the point of intersection of the line
⃗
̂
(
̂) +
̂
(
7. Show that the lines :- ⃗
̂
̂ ) and the plane ⃗ ( ̂
̂
( ̂
̂)+
̂
̂
̂
and ⃗
̂ ) = 5.
̂
̂
(
̂ )+
(
̂ ) are coplanar. Also,
̂
find the equation of the plane containing both these lines.
8. Find the equation of the plane passing through the line of intersection of the planes ⃗
(
̂
̂
̂ - 6 = 0 and ⃗
̂ ) = 0, whose perpendicular distance from origin is unity.
̂
9. Find the image of point (1, 6, 3) in the line
=
=
.
10. Find the vector equation of the line passing through the point (2, 3, 2) and parallel to the line
⃗
̂
̂ +
(
̂
11. Find whether the lines ⃗
̂ ) . Also find the distance between the lines.
̂
( ̂
̂
̂ )+
̂
̂
and ⃗
̂
̂ +
( ̂
̂
̂ ) intersect or not. If
intersecting, find their point of intersection.
12. Find the distance between the point P (5, 9) and the plane determined by the points A(3, -1, 2), B(5, 2, 4) and C( 1, -1, 6).
13. Find the equation of plane through the points (1, 2, 3) & (0, -1, 0) and parallel to the line
⃗ .(2 ̂
̂ +5=0 .
̂
14. Prove that the image of the point (3, -2, 1) in the plane 3x-y+4z = 2 lie on plane x+y+z+4 = 0.
15. Find the vector equation of the plane through the points ( 2, 1, -1) and ( -1, 3, 4) and perpendicular to the plane
x – 2y + 4z = 10.
16. Find the coordinates of the point, where the line
intersects the plane x – y + z – 5 = 0. Also,
find the angle between the line and the plane.
17. Find the vector equation of the plane which contains the line of intersection of the planes
⃗ (̂
⃗ ( ̂
̂)
̂
̂
̂)
and ⃗ (
̂
̂
̂)
and which is perpendicular to the plane
.
PROBABILITY(4+6=10MARKS)
4 MARKS QUESTIONS.
1. Probability of solving specific problem by X & Y are and . If both try to solve the problem, find the probability
that:
(i) Problem is solved.
(ii) Exactly one of them solves the problem.
2. Three balls are drawn without replacement from a bag containing 5 white and 4 green balls. Find the probability
distribution of number of green balls.
3. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining 3rd six in 6th
throw of die.
4. A speaks truth in 60% of the cases and B in 90% of the cases. In what percentage of cases are they likely to
contradict each other in stating the same fact ?
5. Find the mean number of heads in three tosses of a fair coin.
6. Bag I contains 3 red and 4 black balls and Bags II contains 4 red and 5 black balls. One ball is transferred from
Bag I to bag II and then two balls are drawn at random ( without replacement) from Bag II. The balls so
drawn are found to be both red in colour. Find the probability that the transferred ball is red.
7. In a hockey match, both team A and B scored same number of goals up to the end of the game, so to decide the
winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain
gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respectively
probabilities of winning the match and state whether the decision of the referee was fair or not.
8. In answering a question on a MCQ test with 4 choices per question, a student knows the answer, guesses or
copies the answer. Let ½ be the probability that he knows the answer, ¼ be the probability that he guesses and
¼ that he copies it. Assuming that a student, who copies the answer, will be correct with the probability ¾ , what
is the probability that the student knows the answer, given that he answered it correctly?
Arjun does not know the answer to one of the question in the test. The evaluation process has negative marking.
Which value would Arjun violet if he resorts to unfair means? How would an act like the above hamper his
character development in the coming years?
9. The probability that a student entering a university will graduate is 0.4. find the probability that out of 3
students of the university :
a)None will graduate, b)Only one will graduate, c) All will graduate.
10. How many times must a man toss a fair coin, so that the probability of having at least one head is more than
80% ?
11. On a multiple choice examination with three possible answer (out of which only one is correct) for each of the
five questions, what is the probability that a candidate would get four or more correct answer just by guessing?
12. A family has 2 children. Find the probability that both are boys, if it is known that
(i)
At least one of the children is a boy
(ii) the elder child is a boy.
13. An experiment succeed twice often as it fails. Find the probability that in the next six trails there will be at least 4
successes.
14. From a lot of 10 bulbs, which include 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability
distribution of the number of defective bulbs.
6 Marks Questions:
1. Find the probability distribution of number of doublets in three throws of a pair of dice. Also find mean &
variance.
2. A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at
random ( without replacement) and are found to be all spades. Find the probability of the lost card being spade.
3. Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course
reduces the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a
patient can choose any one of the two options with equal probabilities. It is given that after going through one
of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient
followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is
more beneficial for the patient.
4. An insurance company insured 2000 cyclist, 4000 scooter drivers and 6000 motorbike drivers. The probability of
an accident involving a cyclist, scooter driver and a motorbike driver are 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? Which mode of
transport would you suggest to a student and why?
5. Two bags A and B contain 4 white and 3 black balls and 2 white and 2 black balls respectively. From bag A, two
balls are drawn at random and then transferred to bag B. A ball is then drawn from bag B and is found to be a
black ball. What is the probability that the transferred balls were 1 white and 1 black?
6. A biased die is twice as likely to show an even number as an odd number. The die is rolled three times. If
occurrence if an even number is considered a success, then writ the probability distribution of number of
successes. Also find the mean number of success.
7. Three bags contain balls as shown in the table below :
Bag
Number of White Balls
Number of Black
Number of Red
balls
balls
I
1
2
3
II
2
1
1
III
4
3
2
A bag is chosen at random and two balls are drawn from it. They happen to be white and red.
What is the probability that they come from the III Bag?
8. There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up
tail 25% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, it
shows heads, what is the probability that it was the two headed coin?
LINEAR PROGRAMMING (6 MARKS)
1. If a young man rides his motorcycle at 25 km/h, he has to spend Rs. 2/km on petrol. If he rides at a faster
speed of 40 km/h, the petrol cost increase at Rs. 5/km. He has Rs. 100 to spend on petrol and wishes to find
what is the maximum distance he can travel within one hour? Solve it graphically. Write two methods to save
petrol in daily life.
2. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class
ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for
executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the
executive class. Determine how many tickets of each type must be sold in order to maximise the profit for
the airline. What is the maximum profit? Do you feel that air travel is safer now than in olden days? Discuss
briefly.
3. Kellogg is new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at
least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per
kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the
minimum cost of producing this cereal if bran costs Rs. 15 per kilogram and rice costs Rs. 14 per kilogram.
4. A manufacturer produces pizza and cakes. It takes 1 hour of work on machine. A and 3 hours on machine B
to produce a packet of pizza. It takes 3 hours on machine A and 1 hour on machine B to produce a packet of
cakes. He earns a profit of Rs. 17.50 per packet on pizza and Rs. 7 per packet of cake. How many packets of
each should be produced each day so as to maximize his profits if he operates his machines for at the most
12 hours a day? Form the above as linear programming problem and solve it graphically. Why pizza and
cakes are not good for health?
5. A toy company manufactures 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
dolls of type B is at most half of that for dolls of type A. further, the production level of dolls of type A can
exceed three times the production of dolls of other type by at most 600 units. If the company makes 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 maximise the profit? Who is interested in colourful dolls?
6. A retired person has Rs. 70,000 to invest and two types of bonds are available in the market for investment.
First type of bonds yields an annual income of 8% on the amount invested and the second type of bond
yields 10% per annum. As pee norms, he has to invest a minimum of Rs. 10,000 in the first type and not
more than Rs. 30,000 in the second type. How should he plan his investment, so as to get maximum return,
after one year of investment ?
7. One kind of cake requires 300 g of flour and 15g of fat, another kind of cake requires 150g of flour and 30g
of fat. Find the maximum number of cakes which can be made from 7.5kg of flour and 600g of fat, assuming
that there is no shortage of the other ingredients used in making the cakes. Make it as an LPP and solve it
graphically.