ASSIGNMENT 2020-21

SUBJECT- MATHEMATICS (041)

CLASS –XII

RELATION AND FUNCTIONS

1. Show that the relation S in the Set A= {5,6,7,8,9} given by S={(a, b): a − b is divisible by 2} is an
equivalence relation. Find the set of all elements related to 6.
2. Let A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (3, 2)}. Show that R is reflexive
and transitive but not symmetric
3. Prove that the function f:N→N defined by f(x) = x2 + x + 1 is one-one but not onto.
4. If f(x)=√ + 1 ; g(x)= and h(x)=2x-3 , then find f ´[h´{g´(x)}] (CBSE 2015)
5. Show that the function f:R →R defined by f(x) = ∀ x∈R , is neither one – one nor onto.
6.The relation R defined on set A={1,2,3,4,5 }by R={(a,b);|by R={ (a,b):| a2- b2|<16 } is given by
(a) {(1,1), (2,1),(3,1),(4,1),(2,3)] (b) {(2,2),(3,2),(4,2),(2,4)}
(c) {(3,3),(4,3),(5,3),(3,4)} (d) none of these
7. Show that the relation R defined by (a,b) →a+d =b+c n the set N x N is an equivalence relation.
8. Let f: N→N be de?ined by
n + 1/2, if n is odd
F(n) =
n/2, if n is even
for all n∈N. Find whether the function f is bijective.

Inverse trigonometric functions

√!
1. Show that ( sin )=
2. Solve cos (tan ) = sin (cot )

3. Prove that Cos % (&' (() **+ = ,

√ ./0 √ ./0 4
4. Prove that () - 1= , ∈ (0, )
√ ./0 √ ./0
54 5 5 √
5. Prove that 6
− &' 8 9= &' 8 9
4
6. Prove that cot( - 2cot 3) = 7
4
7. Solve : cos (;' (cos ** =
4 !4
8. What is the principal value of Cosec–18();<= > 9 + 8 9
>
5?
9. Write the value of tan (tan *.
6

10. Find the domain (a) y= sin-1(-x2) (b) y= cos-1(x2-4) (EXAMPLER)

11. If sin(sin A
+ cos-1 x ) = 1,then find the value of x.

MATRICES AND DETERMINANTS

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 (2/ B*2
1. Construct a 2x2 matrix whose element aij are given by 2 .
2. If A is a square matrix of order 3 such that | DEF| = 289 find |F|
3. If A2 = A find value of (I + A)2 – 3A (CBSE 2012)
cos G sin G cos G sin G
4. If A = - 1 then prove that An = - 1, n ∈ H
−;' G =);G − sin G cos G
1 2 −3
5. If A = I2 3 2 K, find A-1 and hence solve the system of linear equations:
3 −3 −4
x + 2y – 3z = - 4; 2x + 3y + 2z = 2; 3x – 3y – 4z = 11.
1 2 0
6. If A = I−2 −1 −2K , find F . Using F , solve the system of linear equations : x – 2y = 10 ,
0 −1 1
2x – y – z = 8 , -2y + z = 7
1 −1 2 −2 0 1
7. Use the product I0 2 −3K I 9 2 −3K to solve the following system of equations:
3 −2 4 6 1 −2
x – y + 2z = 1; 2y – 3z = 1; 3x – 2y + 4z = 2. (CBSE 2011)
−1 −2 −2
8. Find the adjoint of the matrix A =I 2 1 −2Kand hence show that A.(adj A) = ´A´I3
2 −2 1
(CBSE 2015)
2 3
9. If x = -4 is a root of N1 1N=0 , then find the two other roots. (Exampler)
3 2
10. To promote the making of toilets for women, an organization tried to generate awareness through
(i) house calls (ii) letters , and (iii) announcements. The cost for each mode per attempt is given as
: (i) Rs. 50 (ii) Rs. 20 (iii) Rs. 40
The numbers of attempts made in three villages X , Y and Z are given below :
(i) (ii) (iii)
X 400 300 100
Y 300 250 75
Z 500 400 150
Find the total cost incurred by the organisation for the three villages separately , using
matrices. Write one value generated by the organisation in the society. (CBSE 2015)
11. The management committee of a residential colony decided to award some of its members (say x)for
honesty, some (say y) for helping others (say z) for supervising the workers to keep the colony neat and
clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and
supervision added to two times the number of awardees for honesty is 33. If the sum of the number of
awardees for honesty and supervision is twice the number of awardees for helping others, using matrix
method, find the number of awardees of each category. Apart from these values, namely, honesty,
cooperation and supervision, suggest one more value which the management of the colony must include
for awards. (CBSE 2013)

Continuity & Differentiability

1. Find the constants a and b so that the function

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 3+P , >1
If f(x) = O 11 , = 1 is continuous at x = 1. (DELHI CBSE 2011)
5 − 2P , < 1
2. Find the value of k if the function

T/0
≠ V/2
If f(x) = O(4 * is continuous at x = V/2
W = V/2

T/0 XY 4
3. If y = , ,show that + sec2( − * = 0
T/0 X

4. If the function f(x) defined below in continuous at x = 0. Find the value of k. (CBSE 2010)
\]T
'Z <0
Z( * = [ W Z =0
| |
'Z > 0
_` >
, ≠2
5. Find the value of k so that the function f(x) = ^ _ > is continuous at x = 2
W, =2
T/0e 4
d f]T
'Z <
b 4 4
6. Find values of p and q , for which f(x) = g Z = is continuous at x =
ch( T/0 * 4
b 'Z >
a (4 *

ijk(l * T/0
d 'Z <0
7. If f(x) = 2 Z = 0 is continuous at x = 0 , the find the values of a and b.
c √ m
a 'Z >0

XY ijk (n o*
8. If siny = xsin(a + y) , prove that = (DELHI CBSE 2012,CBSE 2009,2013)
X ijkn

Y Y XY pqr
9. If =< , prove that = (
(CBSE 2013)
X pqr *

XY Y
10. If x = s ijktu v ,y=s wqitu v ,show that = (CBSE 2012)
X

X Y XY
11.If y = 3< + 2< , prove that X - 5X + 6y = 0. (CBSE 2009)

X Y XY
12.If x = log% + √ + +.Prove that ( + *X + = 0(DELHI CBSE 2013)
X
_`u . _
13.Differentiate w.r.t. x : Sin– 18 ( >*_
9

14.Show that the function f(x) = ´ x-1 ´ + ´ x+1 ´ , for all x ∈ R , is not differentiable at the

points x = -1 and x = 1. (CBSE 2015)

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 XY pqr v pqr v
15. Find if x = ,y= (Exampler)
X v v

XY ijk wqi 4 4
16. Find : cos ( ),- <x< (Exampler)
X √

wqi {|} _ XY Y ~nk

17. If cos , show that X = Y pqr wqi (Exampler)

APPLICATION OF DERIVATIVES

1. Prove that the curve x = y2 and xy = k cut at right angles of 8 k2 = 1. (CBSE 2008)
2. Find the equation of the tangent and the normal to the curve
4
x=1 – cos θ, y = θ – sinθ at θ = (CBSE 2010)
3. Show that the equation of normal at any point on the curve x = 3cosG – cos3 G , y = 3sin G - sin3 G is
4 ( ycos3 G - xsin3 G ) = 3sin 4G. (Exampler)
4. Find the maximum and minimum values of f(x) = secx + log cos2x , 0< < 2V
5. Prove that y = log(1 + x) – , x > – 1 is an increasing function of x through out its domain.
(CBSE 2012)
6. Find the intervals in which the function f(x) = 3x4 – 4x3 – 12x2 + 5 is strictly increasing or strictly
decreasing.
4
7. Separate the interval [0, ] in to sub interval in which f (x) = Sin4x + Cos4x is
(a) Increasing (b) decreasing
8. Find interval in which f(x) = sin 3x – cos 3x , 0< < V , is strictly increasing or decreasing (2016)

9. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of
•
radius R is . Find the volume of the largest cylinder inscribed in a sphere of radius R.
√
10. Prove that the perimeter of a right angled triangle of given hypotenuse is maximum when the
triangle is isosceles.
11. An open box with a square base is to be made out of a given quantity of card board of area c2
fe
square units. Show that the maximum volume of the box is cubic units. (CBSE 2005)
>√
Y
12. Find the area of greatest rectangle that can be inscribed in an ellipse l + m = 1 (Exampler)
13. An isosceles triangle of vertical angle 2G is inscribed in a circle of radius a. show that the area of
4
triangle is maximum when G = > (Exampler)
14. If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the
cube to the diameter of the sphere , when the sum of their volumes is minimum. (Exampler)
15. AB is a diameter of a circle and C is any point on the circle. Show that the area of triangle ABC is
maximum , when it is isosceles. (Exampler)
16. The sum of the surface areas of a rectangular parallelepiped with sides x , 2xand and a sphere is
given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times
the radius of the sphere. Also find the minimum value of the sum of their volumes. (Exampler)
Integral

Evaluate: -

1) € s „•‚ ƒ ˆ‚ 2) €
‰‚„ †Š‹‚
ˆ‚
‚ …‚ †‡
(‚ †*„
† Œ•Ž ‚ 4) € •–—‚•–—„‚•–—ƒ‚ˆ‚
3) € (‚ ˆ‚
••‘ ’“” ‚*
’›Ž ‚
5)€ s† + „˜™š‚ (Œ•Ž ‚ + ’“” ‚* ˆ‚ 6)€ („
”•’ ‚* (• ”•’ ‚*
ˆ‚
Page 4 of 12
 œ–—•‚ œ–—…‚
7)€ ’›Ž√’›Ž
‚ ”•’ ‚
„‚
ˆ‚ 8)€ dx (Exampler)
† „œ–—ƒ‚
9)€ ˜™š• ‚—‹œ… ‚ ˆ‚ (Exampler) 10)€ √”•Œ ‚ + √Œ•Ž ‚ * ˆ‚ (CBSE 2014)
ˆ‚
11)€ „—žš„ ‚ (Exampler) 12)€(ƒ − „‚* . √„ + ‚ − ‚„ (CBSE 2015)
•œ–—„ ‚
—žš‚ œ–—‚ ‚„
15) € dx (Exampler) 16) € dx
√† —žš„‚ (‚—žš‚ œ–—‚*„
„ —žš‚ „ —žš‚ ƒ œ–—‚
17) € ƒ —žš‚ … œ–—‚
dx 18) € • —žš‚ … œ–—‚
dx
† †
19) € dx 20) € dx
… œ–—‚ ƒ —žš‚ • ƒ —žš„ ‚
21) € (2 + 3*D as limit of Sum. 22) €‡„ ∣ —žš‚ − œ–—‚ ∣ ˆ‚
/„ —žš„ ‚ π
23) €‡ dx (Exampler) 4
—žš‚ œ–—‚
∫ log ( 1 + tan θ ) dθ
24) 0 (CBSE 2011)
π π
2
π 2

∫ log Sin x dx = − ∫
−1
log 2 Sin 2 x tan ( Sin x ) dx
2
0 26) 0
25) (CBSE 2008)
l • √†‡ ‚
27* €¢ &' ,l D (CBSE 2008) „•* €„ dx
√‚ √†‡ ‚
(Exampler)
„
29)€ „ ∣ ‚œ–— ‚ ∣dx (Exampler)

APPLICATIONS OF INTEGRALS

1. Using Integration, compute the area bounded by the lines, x+2y = 2 , y-x=1 and 2x+y=7.
2. Using integration find the area of triangular region whose vertices are (1,0),(2,2), (3,1).
3. Find the area of the region bounded by y2 =4x, x=1 , x=4 and x-axis in the first quadrant.
¢
4. Sketch the graph of the curve y = | + 3|. Find evaluate € >| + 3| D

5. Using integration , find the area of the triangle formed by positive x axis and the tangent and normal to
the circle + x = 4 at (1, √3) (2015)

6. Using integration , find the area bounded by the curves y = ´x - 1´ and y = 3 - ´x´ (2015)
7.Using integration, find the area of the region bounded by the parabola y2 =4x and the circle 4x2 + 4y2 =9.

8.Find the area of the region included between parabola y2 =x and the line x+ y =2.

DIFFERENTIAL EQUATION
Find the order and degree

£e o £o
1. + 5 8£¤9 + cos y = tan x
£¤e

Solve the differential equation

2. ( + x*Dx = ( + x *D
XY
3. X
= x(§)¨x − §)¨ + 1*
4. Dx + x( + x*D = 0 given that y = 1 when x = 1 (2013)
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 XY Y Y
5. = 8 + sin 9
X
6. xdy – ydx = s + x D (2005,2011, Exampler)
X
7. (y + 3x2) = (2011)
XY
8. xdy + (y – x3)dx = 0 (2011)
XY
9. xlogx + y = logx (2010,2014)
X
X
10. s1 + + x + x + xyXY = 0

XY
11. Cos2x + y = tanx (2008,2011)
X

XY tu
12.(1+x2) + y =< ~nk (2002,2014)
X

XY
13. X = cos(x + y) + sin(x + y) (Exampler)

tu X Y XY
14. Show that y = a< ~nk is a solution of the differential equation ( 1 + )X + ( 2x – 1) X = 0

VECTOR ALGEBRA

1. Find the unit vector in the direction of ©= 3ª̂–2 ¬̂+6W-

2. Find the angle between the vector ©= ª̂– ¬̂+ ®W and P̄©= ª̂+ ¬̂-W-.
3.If P(1,5,4) and Q(4,1,2) Find the direction ratios of °± ¯¯¯¯¯© and direction cosine.
4. If | ©| = √3 D²P̄©² = 2 and ©. P̄© = 3 find angle between ©and P̄© ?
5. If ¯¯¯©= ª̂+2¬̂– ®W and ¯¯¯© P =3 ª̂+ ¬̂-5 ®W ,find a vector parallel to ¯¯¯© + P ¯¯¯© and having magnitude 8 unit.
6. Find a vector whose magnitude is 3 unit & which is perpendicular to ¯¯¯©=3 ª̂+ ¬̂-4 ®W and¯¯¯¯© P = 6 ª̂+ 5¬̂-2 ®W.
7. If ¯¯¯©= ª̂+2 ¬̂-3 ®W and P
¯¯¯© = 3 ª̂– ¬̂+2 ®W.show that (¯¯¯© + P
¯¯¯©)´(¯¯¯© − P ¯¯¯©) OR orthogonal to each other.
8. Show that the area of parallelogram having diagonals ¯¯¯©= ª̂-3 ¬̂+4 ®W and P ¯¯¯© = 3 ª̂+ ¬̂-2 ®W is 5√3 sq. unit
(2008)
® ¯¯¯© ®
9. Show that the vectors ¯¯¯©= 3ª̂-2 ¬̂+ W , P = ª̂-3 ¬̂+5 Wand ¯¯© ®
= = 2 ª̂+ ¬̂-4 W. form a right angle triangle.
(2005)
10. Let ¯¯¯©= ª̂- ¬̂ , ¯¯¯©
P = 3 ¬̂- ®Wand =¯¯© = 7 ª̂ − ®W. Find a vector D ¯¯¯© which is perpendicular to both ¯¯¯© D ¯¯¯¯© P and
¯¯© ¯¯¯©
= . D =1.
11.If ¯¯¯© , ¯¯¯¯©
P D ¯¯¯©
= are three mutually perpendicular vectors of equal magnitude , prove that the angle

which ‰¯¯¯© + ¯¯¯¯© = makes with any of the vectors ¯¯¯© , ¯¯¯¯©
P + ¯¯¯©Š P )µ ¯¯¯©
= is cos– 1(1/√3) (2005,2011,2013)

12.If the ¯¯¯© + ¯¯¯¯© = =0 and | ©| = 3 ²P̄©² = 5 and |=©| = 7 show that the angle between ¯¯¯©
P + ¯¯¯© D ¯¯¯¯©
P is 600
(2008,2014)
13. If ¯¯¯© , ¯¯¯¯©
P , ¯¯¯©
= are position vectors of vertices A,B and C of a triangle ABC , show that area of triangle is
1/2²¯¯¯© ¶¯¯¯¯©
P +P ¯¯¯© ¶¯¯¯©
= + =¯¯© ¶¯¯¯¯©²
14. If ·and P- are two unit vectors and Q is the angle between them, then show that sinθ/2 = 1/2² · − P-²

15. If ¯¯¯© = ª̂ + ¬̂ + 2 ®W and P̄© = 3 ª̂+2 ¬̂- ®W find (¯¯¯© + 3P

¯¯¯©).(2¯¯¯¯© − P
¯¯¯©) (2002)

16. The two vectors ¬̂ + ®W and 3 ª̂-¬̂+ 4 ®Wrepresents two sides AB and AC respectively of a triangle ABC .

Find the length of the median through A. (Exampler)

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17. If ¯¯¯© = ª̂ + ¬̂ + ®W and ¯¯¯©
P = ¬̂ − ®W , find a vector ¯¯¯© such that ¯¯¯© ¶ ¯¯©
= = ¯¯¯©
P and ¯¯¯©. =© = 3 (Exampler)
® − 6 ¬̂ + ®W and ¯¯¯© = 2ª
18. Find the value of λ for which the vectors ¯¯¯© = 3ª ® − 4 ¬̂ + λ ®W are parallel.

¯¯¯©, ¯¯©
19. If ¯¯¯©, P ¯¯¯© + ¯¯©
= are three vectors such that ¯¯¯© + P ¯¯¯© ∣ = 3 , ∣=¯¯© ∣ = 5 , find value of
= = 0 and ∣¯¯¯© ∣ = 2 , ∣P
¯¯¯© + P
¯¯¯©. P ¯¯¯©. ¯¯©
= + ¯¯¯©. ¯¯©
= (Exampler*
¯¯¯© + 3=¯¯© , 4¯¯¯© − 7P
20. Show that the points with position vectors ¯¯¯© − 2P ¯¯¯© + 7=¯¯© ¯¯¯© − =©are
D − 2¯¯¯© + 3P

Collinear.

THREE DIMENSIONAL GEOMETRY

Y Á Y Á
1. Find p such that = = Â
and = A
= are perpendicular.
2. Using direction ratios, show that the points (2,3,4) , (-1,-2,1) and (5,8,7) are collinear.
Y Á
3.Find the image of the point (1,6,3) in the line = =
A Y Á >
6. Find the distance of the point (2,4,-1) from the line = = 5
. Also find foot of perpendicular

and image.

7. The cartesian equation of a line are 6x – 2 = 3y + 1 = 2z – 2 Find (i) The direction ratios of the line
(ii)cartesian equation of a line parallel to this line and passing through the point (2,-1,-1)
Y Á
8. Find the foot of perpendicular from the point (0,2,3) on the line A = = find the length of
perpendicular.
9. Find the shortest distance between two lines µ© = (1 − *ª̂ + ( − 2*¬̂ + (3 − 2 *W- and

µ© = (; + 1*ª̂ + (2; − 1*¬̂ + (−2; − 1*W- (2002,2011)

10.Find the coordinates of the foot of the perpendicular drawn from a point A(1,8,4) to the line
joiningthe points B(0,-1,3) andC(2,-3,-1) (Exampler)

11. Prove that the lines x = py + q , z = ry + s and x = r´y + s´ are perpendicular if pp, + rr, + 1 = 0
(Exampler)
Y Á
12. Find the equation of the two lines through the origin which intersect the line = =
Ã
at angles of each. (Exampler)
13. Find the shortest distance between the lines given by µ© = (8 + 3Å*ª̂ − (9 + 16Å*¬̂ + (10 + 7Å*W-
Ç
and µ© = 15ª̂ + 29¬̂ + 5W- + Æ(3ª̂ + 8¬̂ − 5W* (Exampler)

15.Find the equation of plane passing through the points (0,-1,0) (1,1,1) and (3,3,0)

16. Find the equation of the plane passing through the point (-1,-1,2) and perpendicular to the planes

3x + 2y – 3z =1 and 5x – 4y + z= 5.

17.Find the equation the line passing through the point P(-1,3,-2) and perpendicular to the line
Page 7 of 12
 Y Á Y Á
= = and = = (2005,2012)
A

18.Find the distance of the point (1,-2,3) from the plane x- y + z = 5 measured parallel to the line
Y Á
= = (2008)
>

!Y Á ! ! Y A > Á
19. Find the p so that the lines =
Â
= and
Â
= =
A
are at right angles.
(2014)

20.Find the equation of the plane through the intersection of planes x + 2y + 3z -4 = 0 and

2x + y – z +5 = 0 and perpendicular to the plane 5x + 3y + 6z +1 = 0. (2007)

21. Find the equation of the plane through the point (4,-3,2) and perpendicular to the line of
intersection of the planes x-y+2z-3=0 and 2x-y-3z=0 . Find the point of intersection of the line µ© = ª̂ +
2¬̂ − W® + Å(ª̂ + 3¬̂ − 9W
Ç * and the plane obtained above. (CBSE sample paper)

22. Prove that © . {(P̄© + =© )x ( © +2P̄© +3=©)} =[ ©P̄©=© ] (CBSE sample paper)
Y Á l Y
23. Find the values of a so that the following lines are skew : = = , = =z
A
(CBSE sample paper)
Y Á 4
24. Find the equation of the two lines through origin which intersect the line = = at angle of
Y Á
25. Find the image of the line = = A
in the plane 2x – y +z +3 = 0.

26. Find the length and the foot of the perpendicular from the point (1, , 2) to the plane

2x-2y+4z+5=0.
Ç+3=0
27. Find the image of the point having position vector ª̂ + 3¬̂ + 4W-in the plane µ©. (2ª̂ − ¬̂ + W*

28. Find the equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the

plane x -2y + 4z = 0 (Exampler)

29. Let © = 4ª̂ + 5¬̂ - W- , P̄© = ª̂ − 4¬̂ + 5W- and =©= 3ª̂ + ¬̂ − W- . Find a vectorD© which is perpendicular to

both =©and P̄©andD©. © = 21 (CBSE 2017)

LINEAR PROGRAMMING

1. A company uses 3 machines to manufacture and sell two types of shirts- half sleeves and full sleeves.
Machines A,B and C take 1 hr,2 hr, and 1 ¾ hr to make a half sleeve shirt and 2 hrs, 1hr and 1 3/5 hr to
make a full sleeve shirt the profit on each half sleeve shirt’s Re 1:00 and on a full sleeve shirt’s Re 1:50. No
machine can work for more than 40 hr per week. How many shirts of each type should be made to
maximize the company’s profit? Solve the problem graphically.
2. The tailors X and Y earn Rs 150 and Rs 200 per day respectively. A can stitch 6 shirts and 4 pants per
day while B can stitch 10 shirts and 4 pants per day. How many days shall each work.If it is desired to
produce at least 60 shirts and 32 pants at a minimum labor costs? Solve the problem graphically.(2005)

Page 8 of 12
3. The Principal of a school wants to buy some colour and books for giving prizes to 15 children,He wants
to by at least 4 of each.A colour box costs Rs 5 where as book costs Rs 10.How many of each should he
buy so that expenditure does not exceed Rs 100 and at the same time he can buy max. number of prize?
4. A dietician wishes to mix two Types of food in such a way that the vitamin contents of the mixture
contain at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 unit/kg of vitamin A and
1 unit/kg of vitamin C while food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C,It costs Rs
50/Kg to produce food I and Rs 20/Kg to produce food II. Find the minimum cost of such a mixture
formulate the above LPP mathematically and then solve it. (2011)
5. A grain dealer has Rs 1500 for purchase of rice and wheat A bag of rice and a bag of wheat costs Rs180
and 120 respectively .He has a storage capacity of 10 bags only. He earns a profit of Rs 11 and Rs 8 per
bag of rice and wheat respectively. How many bags of each must he buy to make maximum profit?
6. An aeroplane can carry a maximum of 200 passengers a profit of Rs 400 is made on each 1st class
ticket and a profit of Rs 300 is made on each 2nd class ticket. The airline reserve at least 20 seats for first
class. However at least four times as many passengers prefer to travel by second class then by first class.
Determine how many tickets of each type must be sold to maximize profit for the airline. Form an LPP
and solve it graphically.

7. In a mid day meal programme , an NGO wants to provide vitamin rich diet to the students of an MCD
school . The dietician of the NGO wishes to mix two types of food in such a way that vitamin contents of
the mixture contains atleast 8 units of vitamin A and 10 units of vitamin c . Food 1 contains 2 units per kg
of vitamin A and 1 unit per kg of vitamin C . Food 2 contains 1 unit per kg of vitamin A and @ units per kg
of vitamin C . It costs Rs 50 per kg to purchase food 1 and Rs 70 per kg to purchase food 2 . Formulate the
problem as LPP and solve it graphically for the minimum cost of such mixture? (CBSE sample paper)

8. If a young man drives his scooter at a speed of 25 Km/hour, he drives the scooter at a speed of 40
Km/hour, it produces more air pollution and increase his expenditure on petrol to Rs 5 per Km. He has
maximum of Rs 100 to spend on petrol and travel a maximum distance in one hour time with less
pollution. Express this problem as an LPP and solve it graphically. What value do you find here?

9. A manufacturer considers that men and women workers are equally efficient and so he pays them at
the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he
uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3units of capital are
required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced
at Rs 100 and Rs 120 per unit respectively, how should he uses his resources to maximize the total
revenue? From the above as an LPP and solve graphically. Do you agree with this view of the
manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

10. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest
and has a space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. His
expectation is that he can sell a fan at a profit of Rs. 22 and a sewing 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 the profit? From the above as an LPP and solve graphically.

PROBABILITY
1. A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the
conditional probability that the number 5 has appeared at least once. (2003)
2. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The
probability of an accident involving a scooter a car, a truck is ¢¢¢ , ¢¢ , ¢ respectively. One of the
insured persons meets with an accident. What is the probability that he is a scooter driver.
(2000,2002,2008,2012,2014)

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3. A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. Find
the probability that is a actually a six. (2005,2011,2014)
4. In an 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 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 . The
6
probability that he knew the answer to the question given he correctly answered it.
5. A card from a pack of 52 cards is lost from the remaining cards of the pack two cards are drawn and
are found to be both spades. Find the probability of the lost card being a spade. (2000,2010)
6. A bag contains 4 yellow, 5 red and another bag contains 6 yellow and 3 red balls. A ball is drawn from
the first bag and without see its colour , it is put into the second bag .Find the probability that if now is
ball is drawn from the second bag. It is yellow in colour. (2002)
7. A problem in mathematics is given to three students whose chances of solving it are 1/3, 1/5, 1/6
what is the probability that at least one of them solves the problem.
8. A pack of playing cards was found to contain only 51cards , of the first 13 cards which are examined
are all red what is the probability that the missing card is black.

9. A bag contains 4 green and 6 white balls . Two balls are drawn at random one by one without
replacement . If the second ball drawn is white , what is the probability that the first ball is also white?
(CBSE sample paper)
10. A company has two plants to manufacture motor cycles. 70% motor cycles are manufactured at the
first plant, while 30% are manufactured at the second plant. At the first plant, 80% motor cycles are
rated of the standard quality while at the second plant. 90% are rated of standard quality. A motor
cycle ,randomly picked up, is found to be of standard quality. Find the probability that it has come out
from the second plant. Why riding a motor cycle is riskier than driving other vehicles?
11. Out of a group of 30 honest people,20 always speak the truth. Two persons are selected at random
from the group. Find the probability distribution of the number of selected persons who speak the
truth. Also find the mean of the distribution. What values are described in this question?
12. A man known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number
greater than 4.
(i) Find the probability that it is actually a number greater than 4.
(ii) Write about ‘truth’ as essential human value.
13. A bag contain (2n + 1) coins. It is known that n of these coins have a head on both sides where as the
rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability
that the toss results in a head is , determine the value of n. (Exampler)
14. Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting
2 on the second die’ . Are the events A and B independent. (Exampler)
15. Three machines E1 , E2 , E3 in a certain factory produce 50% , 25% and 25% respectively, of the total
daily output of electric tubes. It is known that 4% of the tubes produced one each of machines E1 and
E2 are defective, and that 5% of those produced on E3 are defective. If one tube is picked up at random
from a day’s production, calculate the probability that it is defective.
(Exampler)
16. A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if she gets a
total of 7. If A starts the game , find the probability of winning the game by A in third throw of the pair
of dice. (Exampler)
17. A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just
two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.
(Exampler)

Q1. Let R be a relation on the set N of natural numbers defined by n R m iff n divides m. Then, R is

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 (a) Reflexive & symmetric (b) transitive & symmetric (c) Equivalence (d) Reflexive, transitive but not
symmetric.
Q2. If g(x) =x2 + x -2 and gof(x) = 4x2 – 10x +4, then f(x) is
(a) 2x-3 (b) 2x+3 (c) 2x2 +3x +1 (d) 2x2-3x-1
Q3. The value of sin-1 (cos 33π/5) is
(a) 3π/5 (b) - π/10 (c) π/10 (d) 7π/5
Q4. If 4cos-1x + sin-1x = π, then value of x is
(a) 3/2 (b) 1/21/2 (c) 31/2/2 (d) 2/31/2
Q5. If a matrix A is both symmetric and skew symmetric, then
(a) A is a diagonal matrix (b) A is a zero matrix
(c) A is a scalar matrix (d) A is a square matrix
Q6. If A and B are two matrices such that AB=A and BA=B, then B2 is equal to
(a) B (b) A (c) 1 (d) 0
Q7. If A is a matrix of order 3 and |F|=8, then | DEF| =
(a) 1 (b) 2 (c) 8 (d) 64
Q8. If A is a square matrix such that A2 = I, then A-1 is equal to
(a) A+I (b) A (c) 0 (d) 2A
Q9. If f(x)= log√ , then the value of f|(x) at x = V/4 is
(a) 1 (b) 1/2 (c) 0 (d) ∞
Q10. € vl0 dx =
(a) log(x+sin x) + C (b) log(sinx + cosx) +C (c) 2sec2x/2 + C (d) 1/2[ x + log(sinx + cosx)] +C
l 4
Q11. €¢ dx= 6 ,
π 4
(a) (b) 1/2 (c) (d) 1
Q12. The solution of the differential equation x dx + y dy = x y dy – y x dx is
2 2

(a) x2 – 1 =C (1+y2) (b) x2 + 1 =C (1-y2) (c) x3 – 1 =C (1+y3) (d) x3 +1 =C (1-y3)

Q13. If | © X P̄©|= = 4, | ©. P̄©|=2 , then | © |2 |P̄© |2 =
(a) 6 (b) 2 (c) 20 (d) 8
Q14. The image of the point ( 1, 3, 4 ) in the plane 2x – y + 3z = -3 is
(a) (3, 5, 2 ) (b) (-3, 5, 2 ) (c) (3, 5, -2 ) (d) (3,- 5, 2 )
Q15. The maximum value of Z = 4x + 3y subjected to the constraints 3x +2y ≥ 160, 5x + 2y ≥ 200,
x +2y ≥ 80; x, y ≥ 0 is
(a) 320 (b) 300 (c) 230 (d) none of these
Q16. The least number of times a fair coin must be tossed so that the probability of getting at least one
head is
(a) 7 (b) 6 (c) 5 (d) 3
Q17. If a relation R on the set {1,2,3} be defined by R ={(1,2)}, then R is :
(a) reflexive (b)symmetric (c)transitive (d) none of these
Q18. The Domain of the function cos (2x -1) is:-1

(a) [0,1] (b)[-1,1] (c)(-1,1) (d)[0,É]

0 0 4
Q19The matrix P = I 0 4 0K is a :
4 0 0
(a)square matrix (b) diagonal matrix (c)unit matrix (d)none
Q20.Let A be a square matrix of order 3 x 3 , then |kA| is equal to:
(a) k|A| (b) k2|A| (c)k3|A| (d)3k|A|
Q21. The value of cos {cos(3É/2)} is equal to:
-1

(a) É /2 (b)3 É /2 (c)5 É /2 (d)7 É/2

0 1
Q22. If A =- 1 ,then A2 is equal to:
1 0

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 0 1 1 0 0 1 1 0
(a* - 1 (b* - 1 (c*- 1 (d*- 1
1 0 1 0 0 1 0 1
Q23. Let f: R→R be de?ined by :f(x) =1/x ,∀ x ∈ R, then f is:
(a* one-one (b*onto (c* bijective (d* f is not defined
Q24. The function f(x* = tanx ‒ x:
(a* always increases (b* always decreases (c* never increases (d* remain same
Q25. The area of the region bounded by the curve y = x +1 and the lines x =2 & x=3 is:
(a* 7/2 sq units (b* 9/2 sq. units (c* 11/2 sq. units (d*13/2 sq. units
Q26. The interval on which the function f(x* = 2x3 + 9x2 + 12x ‒1 is decreasing is:
(a* %1,∞* (b* %‒2,‒1+ (c* (‒∞, ‒2+ (d* %‒1,1+
X
Q27. A homogeneous equation of the form = h(x/y* can be solved by making the substitution.
XY
(a* y =vx (b* v =yx (c* x =vy (d* x =v
Q28. The locus represented by xy + yz =0 is:
(a* A pair of perpendicular lines (b* A pair of parallel lines
(c* A pair of parallel planes (d* A pair of perpendicular planes
Key for MCQ
Q1. (d) Q2. (a) Q3. (b) Q4. (c)Q5. (b) Q6. (a) Q7. (d)Q8. (b)Q9. (a)
Q10. (d)Q11. (b)Q12. (a) Q13. (c) Q14. (b)Q15. (d) Q16. (d) Q17(c) Q18(a) Q19(a)Q20(c)
Q21 (a) Q22(d) Q23 (d) Q24(a) 25(a) Q26 (b)Q27(c) Q28(d)

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