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INTERNATIONAL INDIAN SCHOOL, RIYADH
WORKSHEET (2020-2021)
GRADE - XII – MATHEMATICS – PRE-BOARD
CHAPTER I: RELATIONS AND FUNCTIONS
1. Let * be a binary operation defined by a * b = 3a +4 b -2. Find 5*4.
2. Show that the relation R defined by R = { ( a b ) : a – b is divisible by 3,
a, b € N } is an equivalence relation.
3. If f : R → R defined by f ( x ) = 3𝑥+5
2 is an invertible function. Find f – 1 .
4. If the function f : R → R is given by f ( x ) = x2 + 3 x + 1 and g : R → R is given
by g ( x ) + 2 x – 3. Find f o g and go f .
5. If f ( x ) = 27 x 3 and g ( x ) = x 1/3. Find g o f .
6. Consider f : R + -- [ - 5 ,∞ ) given by f ( x ) = 9 x2 + 6 x -5. Show that f is invertible .and also
find f -- 1 .
7. Let A = N X N and * be a binary operation on A defined by ( a b ) * ( c d ) = (a + c , b + d ).
Show that * is commutative, associative. Also find the identity element foe * on A, if any.
8. Show that the relation S defined on the set N X N by ( a b ) * ( c d ) =( a + d , b + c ) is an
equivalence relation.
9. State the reason for the relation R in the set { 1, 2, 3 } given by R = { (1 2) , ( 2 1 ) } not to
be transitive.
10. Consider the binary operation * on the set { 1, 2, 3, 4, ,5 } defined by a * b = minimum of a
and b . write the operation table for *.
11. Prove that the relation R in the set { 5, 6, 7, 8, ,9 } given by R = { ( a b ) : | a – b | is divisible
by 2 } is an equivalence relation. Find all elements related to the element 6 .
12. Let f : W → W be defined as f ( x ) = x – 1 if x is odd, and f ( x ) = x + 1 if x is even. Show that
f is invertible. Find the inverse of f.
13. Let N denote the set of all natural numbers and R be the relation on N X N defined by ( a b )
R ( c d ) iff ad ( b + c ) = bc ( a + d ). Show that r is an equivalence relation.
14. If f : R + -- [ 4 ,∞ ) given by f ( x ) = x2 + 4. Show that f is invertible , also find f -- 1 .
15. If f : R → R defined by f ( x ) = 3 x + 2 . Find f ( f ( x ) ).
CHAPTER 2: INVERSE TRIGONOMETRIC FUNCTION
1. Find the principal values of the following
(a) sin−1 − 1 ( b) sec−1 2
√3 ( c ) sin−1 −
√3
2 .
2. Evaluate the following
(a) sin−1 sin5𝜋
6 (b) cos [cos−1 ( −
√3
2 ) +
𝜋
6 ]
3. Prove that cos −1 𝑥 = 2 sin−1 √1−𝑥
2
4. Prove that 𝑡𝑎𝑛−1 𝑥 = sin−1 𝑥
√𝑥2+1
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5. Simplify Prove that cos−1 𝑥 = 2 sin−1 √1−𝑥
2
6. Write in the simplest form
tan −1√1 + 𝑥2 − 1
𝑥
7. Write in the simplest form
tan −1√1 + 𝑥2 + √1 − 𝑥2
√1 + 𝑥2 − √1 − 𝑥2
8. Write the following in the simplest form
sin−1[ 𝑥 √1 − 𝑥 − √𝑥 √1 − 𝑥2 ]
9. Prove that tan−1 5 - tan−1 3+ tan−1 7
9 =
𝜋
4 .
10. Prove that tan−1 1
4 - tan−1 2
9=
1
2 cos−1 3
5 .
11. Prove that tan−1 63
16 - sin−1 5
13= cos−1 3
5 .
12. Prove that tan−1 27
11 - cos−1 4
5= tan−1 3
5 .
13. Solve for x ,
tan−1 1
1+2𝑥 - tan−1 1
4𝑥+1= tan−1 2
𝑥2 .
14. Solve for x ,
tan−1 𝑥−1
𝑥+1 - tan−1 2𝑥−1
2𝑥+1= tan−1 23
36 .
15. Ifcos−1 𝑥
𝑎 + cos−1 𝑦
𝑏 = 𝜃 then Prove that
𝑥2
𝑎2 -
2𝑥𝑦
𝑎𝑏cos 𝜃 +
𝑦2
𝑏2=(𝑠𝑖𝑛𝜃)2
CHAPTER 3 – MATRICES
1. Let A = [2 3
−1 2 ], then show that A2 – 4A + 7I = 0. Using the above, calculate A5 also.
2. Find non zero values of x satisfying
x [2𝑥 2 3 𝑥
] + 2 [8 5𝑥4 4𝑥
] = 2 [𝑥2 + 8 24 10 6𝑥
]
3. If A is a square matrix X such that A2 = A show that
(I + A)3 = 7A + I
4. Find the matrix X such that
[ 2 − 1 0 1 −2 4
] X = [
−1 − 8 − 10 3 4 0
10 20 10
]
5. If A = [−1 2 3
] and B = [−2 − 1 − 4]
Verify that (AB)-1 = B1A1
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6. If A and B are symmetrical, then show that (AB + BA) is symmetric and (AB – BA) is
skew symmetric.
7. If f(x) = x2- 4x + 1 , find f(A) given A = [2 31 2
]
8. Find x if [ x 1 ] [1 0
−2 − 3 ] [
𝑥3
] = 0
9. Find the inverse by elementary transformations of
A = [−1 1 2 1 2 3 3 1 1
]
10. If A = [1 3 5
−2 5 7 ] and 2A – 3B = [
4 5 − 91 2 3
]
Find the matrix B.
11. If A = [ 𝐶𝑜𝑠 ∝ 𝑆𝑖𝑛 ∝
−𝑆𝑖𝑛 ∝ 𝐶𝑜𝑠 ∝ ], show that
A2 = [ 𝐶𝑜𝑠 2 ∝ 𝑆𝑖𝑛 2 ∝
−𝑆𝑖𝑛 2 ∝ 𝐶𝑜𝑠 2 ∝ ]
12. Given A = [3 17 5
], find a and b such that A2 + aI=bA
13. If A = [0 1
−1 1 ], find p and q so that ( pI + qA )2=A
14. If A = [−2 3 1 2
] and B = [−1 0 1 2
], find (A + 2B)1
15. If A = [ 0 1 0 0
], prove that (aI+ bA)3 = a3I + 3a2bA
16. Show that all the elements of the main diagonal of a skew symmetric matrix are zero.
17. Express as the sum of a symmetric and a skew symmetric matrix and verify, given
A = [ 3 − 2 − 4 3 − 2 − 5−1 1 2
]
18. If 2[ 1 3 0 𝑥
] + [ 𝑦 0 1 2
] = [ 5 6 1 8
], find x + y
19. Using elementary operations, find the inverse of the matrix
A = [ 1 3 − 2 −3 0 − 1 2 1 0
]
20. If A = [ 0 1 0 0
], show that (aI+ bA)n = anI + nan-1bA using mathematical induction.
CHAPTER 4 – DETERMINANTS
1. If for matrix A , |A| = 3, find |5A| where A is of
order 2×2
2. A is a non-singular matrix of order 3 and |A| = -4, find |adj A |
3. Given A is of order 3×3 and |A| = 12, find |A. Adj A |
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4. If A = [ 1 𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑥 1
] , show that A1A-1 = 𝐶𝑜𝑠 2𝑥 − 𝑆𝑖𝑛 2𝑥
𝑆𝑖𝑛 2𝑥 𝐶𝑜𝑠 2𝑥
5. Prove using properties of determinants
|
𝑦 + 𝑧 𝑥 𝑦𝑧 + 𝑥 𝑧 𝑥𝑥 + 𝑦 𝑦 𝑧
| = (x+y+z) (x-z)2
6. Find the value of 𝑝
𝑝−𝑎 +
𝑞
𝑞−𝑏 +
𝑟
𝑟−𝑐 given |
𝑝 𝑏 𝑐𝑎 𝑞 𝑐𝑎 𝑏 𝑟
| = 0
7. Prove that |𝑏 + 𝑐 𝑎 − 𝑏 𝑎𝑐 + 𝑎 𝑏 − 𝑐 𝑏𝑎 + 𝑏 𝑐 − 𝑎 𝑐
| = 3abc – a3 – b3 – c3
8. Prove that |𝑎 + 𝑏 + 𝑐 − 𝑐 − 𝑏 −𝑐 𝑎 + 𝑏 + 𝑐 − 𝑎
−𝑏 − 𝑎 𝑎 + 𝑏 + 𝑐| = 2(a+b)(b+c)(c+a)
9. Prove using properties of determinants
|𝑎2 + 2𝑎 2𝑎 + 1 12𝑎 + 1 𝑎 + 2 1 3 3 1
| = (a – 1)3
a. Use the product [1 − 1 20 2 − 3 3 − 2 4
] [−2 0 1
9 2 − 3 6 1 − 2
] to solve the system of
equations x – y + 2z = 1 ; 2y – 3z = 1 ; 3x – 2y + 4z = 2
10. Without expanding show that
| 𝑏 + 𝑐 𝑎 𝑎 𝑏 𝑐 + 𝑎 𝑏
𝑐 𝑐 𝑎 + 𝑏 | = 4abc
11. Show that the points A(a, b+c ), B(b, c+a) and C(c, a+b) are collinear
12. Find the value of θ satisfying | 1 1 𝑆𝑖𝑛3𝜃 −4 3 𝐶𝑜𝑠2𝜃 7 − 7 2
| = 0
13. Find equation of the line joining (1,2) and (3,6) using determinants.
14. Using matrices, solve x – y + 2z = 7 ; 3x + 4y – 5z = – 5 ; 2x – y + 3z = 12
15. Show that A = | 3 1−1 2
| satisfies A2 – 5A + 7I = 0. Hence find A-1
16. Let A = [aij]n×n . Write |2A| where |A| = 4 and n = 3.
17. For what value of x, [5 − 𝑥 𝑥 + 1
2 4] is singular.
18. If A = |1 𝑥 𝑥2
1 𝑦 𝑦2
1 𝑧 𝑧2
| and A1 = | 1 1 1
𝑦𝑥 𝑧𝑥 𝑥𝑦 𝑥 𝑦 𝑧
| show that A + A1 = 0
19. If a, b, c are in A.P , then find the value of
|
2𝑦 + 4 5𝑦 + 7 8𝑦 + 𝑎3𝑦 + 5 6𝑦 + 8 9𝑦 + 𝑏
4𝑦 + 6 7𝑦 + 9 10𝑦 + 𝑐|
Commented [aa1]:
Page 5 of 12
CHAPTER 5 & 6: CONTINUITY AND DIFFERENTIABLITY, APPLICATIONS OF
DERIVATIVES
1) Differentiate: cos x3. sin2x5 w.r.t.x
2) Find: 𝑑𝑦
𝑑𝑥 if ax2+2hxy+by2=0
3) Find: 𝑑𝑦
𝑑𝑥 if y= tan-1 (
3𝑥−(𝑥)3
1−3𝑥2)
4) Differentiate: √〖(𝑥−3)𝑥2+4)
√3x2+4x+5 w.r.t.x
5) Find: 𝑑𝑦
𝑑𝑥 if yx+xy+xx=ab
6) Differentiate: xcosx+ (cosx)sinx w.r.t.x
7) If x=√𝑎𝑠𝑖𝑛−1t ; y=√𝑎𝑐𝑜𝑠−1
t show that 𝑑𝑦
𝑑𝑥=
−𝑦
𝑥
8) If x=cosθ- cos2θ; y= sinθ- sin2θ find 𝑑𝑦
𝑑𝑥
9) If y= 3 cos(10gx)+ 4 sin(10gx) show that x2y2+ xy1+y=0
10) If y= (tan-1x)2 show that (x2+1)2y2+ 2x(x2+1)y1=2
11) Differentiate: cos2x w.r.t. esinx
12) Differentiate: cot-1[√1+𝑠𝑖𝑛𝑥1
√1+𝑠𝑖𝑛𝑥1±
√1−𝑠𝑖𝑛𝑥1
√1−𝑠𝑖𝑛𝑥1] w.r.t.x
13) If x√1 + 𝑦1+ y√1 + 𝑥1=0 prove that 𝑑𝑦
𝑑𝑥=
−1
(1+𝑥)2
14) Find the interval in which the fuction f given by f(x)= sinx+cosx; 0≤x≤2π is strictly
increasing or strictly decreasing.
15) A water tank has the shape of an inverted right circular cone with its axis vertical and
vertex lowermost. Its semi-vertical angle is tan (0.5). Water is poured into it at a constant
rate of 5 cubic meters per hour. Find the rate at which the level of the water is rising at the
instant when the depth of water in the tank is 4m.
16) Show that the function f given by f9x)= tan-1(sinx+ cosx); x≥0 always an strictly increasing
function in (0,𝜋
4)
17) A point on the hypotenuse of a triangle is at distance ‘a’ and ‘b’ from the sides of the
triangle. Show that the maximum length of the hypotenuse is a2/3+b2/3)3/2
18) Using differentiation find the approximate value of (33)-1/5
19) Find the equation of the normal to the curve x2= 4y which passes through the point (1,2)
20) Find the equation of tangent to the curve y= cos(x+y); -2π≤x≤2π that are parallel to
x+2y=0.
CHAPTER 7, 8 & 9: INTEGRALS, AREAS & DIFFERENTIAL EQUATIONS
1. ∫𝑙𝑜𝑔𝑥
𝑥 dx
2. ∫tan−1 𝑥
1+𝑥2 dx
3. ∫𝑥3+𝑥2+𝑥+1
𝑥+1 dx
4. ∫ sin 𝑥 sin 2𝑥 dx
5. ∫ cos 5𝑥 sin 4𝑥 𝑑𝑥
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6. ∫2𝑥
(𝑥2 + 1)(𝑥2 + 3) 𝑑𝑥
7. ∫5𝑥+3
√𝑥2+4𝑥+10 dx
8. ∫𝑑𝑥
√5−4 𝑥−2𝑥2 dx
9. ∫ 𝑥 sin−1 𝑥 𝑑𝑥
10. ∫1
1+ cot 𝑥 dx
11. ∫2 𝑥 tan−1 𝑥2
1+ 𝑥4 dx
12.∫𝑒𝑥( 𝑥2+ 1)
(𝑥+1)2 dx
13, ∫( 𝑥−4)𝑒𝑥
(𝑥−2 )2 dx
14. ∫(√tan 𝑥 + √cot 𝑥)𝑑𝑥
15. ∫cos 𝑥 𝑑𝑥
( 1+ sin 𝑥 ) ( 2+ sin 𝑥 )
𝜋
20
𝑑𝑥
16. ∫ log(2+𝑥
2−𝑥
1
−1) dx
17. ∫ log tan 𝑥 𝑑𝑥𝜋
20
18. ∫ 𝑥 √2 − 𝑥2
0 dx
19. ∫𝜋
40
log( 1 + log 𝑥) 𝑑𝑥
20. Solve x dy - ( y + 2 𝑥2) dx = 0
21. Solve x dy - y dx = √𝑥2 + 𝑦2
22. Evaluate ∫ 𝑥 (1 − 𝑥 )𝑛1
0 𝑑𝑥
23. Find the area of triangle formed by the vertices ( -1 , 0), ( 1, 3 ), and ( 3, 2)
24. Find the area bounded by 𝑥2
9 +
𝑦2
4 = 1
25. Find the area bounded by 𝑥2 = 4 𝑦 and x = 4y - 2
CHAPTER 10 VECTORS
1. Find the direction ratios and direction cosines of the vector b= 2i+2j-k and verify (i) cos2α + cos2β
+ cos 2 γ = 1 (ii). Sin2α + sin2β +sin2γ = 2
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2. Show that the three points A ( -2i + 3j + 5k ) , B ( i + 2j + 3k ) and C (7i – k ) are collinear.
3 . If vectors a = 2i + j + k and b = 3i +2j – 3k , find the direction ratios and direction cosines of the
vector r = 3a + 2b ( drs 12,7,-3 dcs 12/√202 ,7/√202, -3/√202
4. Find the value of λ for which the angle between the vectors a = 2λi + 4λj + k and b = 7i – 2j + λk is
obtuse. (hint – get λ such that cosθ < 1 )
5. If a= i + 3j – 5k and b = 3i – j + 2k , find the projection of p = 2a +3b on q = 3a – 2b
6.If a and b are two unit vectors such that their sum is also a unit vector, then find the angle between
them.
7. If a, b and c are three vectors of magnitudes 3 , 4 and 5 respectively, such that each one is
perpendicular to the sum of the other two vectors , prove that magnitude of ( a + b + c ) = 5√2
8. If a = i- j +2k and b = 2i + j – k find r = ( 2a – b ) x ( a + 2b )
9. If a = I + 4j + k and b = 3i + 4j + 5k ,find (i) a unit vector perpendicular to a and b,how many such
vectors are there? (ii). Find a 7 unit vector perpendicular to a and b
10.If a = i + j + k ,b = j – k ,find a unit vector c such that a x c = b and a . c = 3
11. If a = 3i – j anb b = 2i + j – 3k then express b in the form b = b1 + b2 where b1 parallel to a and b2
perpendicular to a
12. Find a vector p which is perpendicular to both α = 4i + 5j –k and β = I – 4j + 5k and p . q =21
where q = 3i + j – k
13.If r = xi + yj + zk then find (r x i) . ( r x j ) + xy
14. If a = 2i + j + 3k , b = - I + 2j + k and c = 3i + j + 2k find a. ( bxc)
15.show that the vectors a = - 2i – 2j + 4k, b = - 2i + 4j - 2k and c = 4i – 2j – 2k are coplanar
16. Find the value of λ if the following four points are coplanar; A ( -6, 3 , 2 ) ,
B ( 3, λ,4 ) , C ( 5, 7, 3 ) and D ( - 13,17 ,- 1 )
17. If a ,b, c are three nonzero vectors , prove that [ a +b , b + c , c + a ] = 2 [ a , b, c ].
18. If a ,b , c are three nonzero vectors, show that a +b , b +c , c + a are coplanar if a ,b , c are
coplanar.
19. If a x b = a x c , b not equal to c and a not equal to zero vector , then show that b = c + λa
20. If a , b, c are three non zero vectors and a + b + c = 0 ,then prove that a x b = b x c = c
x a
21. If the sum of two unit vectors is a unit vector , prove that magnitude of their difference is √3
22.If the difference of two unit vectors is a unit vector, prove that magnitude of their sum is √3
Page 8 of 12
23.If a and b are two vectors such that |𝑎 + 𝑏| = |a |. Prove that (2a + b) is perpendicular to b
24.If a + b + c =0 and |a| =3 , |𝑏|=5 , |c| =7 , show that the angle between a and b is 𝝅/3
25. Dot product of a vector with vectors ( i – j ), (i + k )and ( i + j – 2k) are respectively 1 ,-6, 3,find the
vector.
CHAPTER 11 THREE DIMENSIONAL GEOMETRY
1. Find the equation of a line passing through the point ( α ,β ,γ ) and parallel to z axis.
2.The vertices of a triangle ABC are A ( 1 ,1 , 3 ) , B ( 2 ,3 ,4 ) and C ( 4, 5 ,-2 ). Find the equation
of the median AD in Cartesian and vector form.
3. The equation of a line are 5x—3 = 15y +7 = 3 – 10z. Write the direction cosines of the line.
4. Find the point on the line 𝑥+2
3 =
𝑦+1
2 =
𝑧−3
2at a distance of 5 units from a point P(1, 3, 3 )
5. Find the value of p so that the lines 1−𝑥
3 =
7𝑦−14
𝑝 =
𝑧−3
2 and
7−7𝑥
3𝑝 =
𝑦−5
1 =
6−𝑧
5 are
perpendicular to each other.
6. A line makes angles α ,β , γ and δ with the diagonals of a cube. Prove that cos2 α + cos2β
+ cos2γ +cos2δ = 4
3
7. Find the equation of the line passing through a point A ( 1 , -1 , 1 ) and perpendicular to the line
joining the points B (4 ,3 ,2 ), C ( 1, -1 0 )and D ( 1 , 2 , -1 ) , E ( 2,1 ,1 )
8 . There is a point A ( 5 ,8 , 1 ) and a line BC: 𝑥−1
2 =
𝑦−3
4 =
𝑧−8
1 . P is the foot of the
perpendicular from A on BC. (i). Find the coordinates of P (ii). Find the image of A in
the line BC. (iii) Find the distance of the point A from the line BC.(iv) Find the equation of AP.
9. Find the shortest distance between the lines 𝑥−3
1 =
𝑦−5
− 2 =
𝑧−7
1 and
𝑥+1
7=
𝑦+1
− 6=
𝑧+1
1
10. Show that the lines r = ( 3i +2j – 4k ) + λ( i + 2j +2k ) and r = ( 5i -
2j ) + λ( 3i + 2j +6k) intersect. Also find their point of intersection.
11. Show that the lines r = ( i +j – k ) + λ(3 i - j) and
r = (4i - 2) + μ( 2i +3k) are co-planar.
12. Find the point of intersection of a line 𝑥+1
2 =
𝑦+2
3 =
𝑧+3
4 and the plane, x+y+z=3
13. Find the intersection of the line r = (i+2j+3k) + λ(2i+j+3k) and the plane
r.(2i-6j+3k) + 5 = 0
14. Find the distance of the point P (3, 4, 4) from the point where the line joining the points A (3,-4,-
5) and B (2,-3, 1) intersects the plane 2x+y+z=7.
15. Find the angle between the line 𝑥+1
2 =
𝑦
3 =
𝑧−3
6 and the plane 10x + 2y – 11z = 3.
Page 9 of 12
16. Find the equation of a plane passing through a point A (1,2,1) and perpendicular to the line
joining the points P (1,4,2) and Q (2,3,5). Also find the distance of the plane from the line 𝑥+3
2
= 𝑦−5
−1 =
𝑧−5
−1.
17. Show that the following two lines are co-planar 𝑥+3
−3 =
𝑦−1
1 =
𝑧−5
5 and
𝑥+1
1 =
𝑦−2
2 =
𝑧−5
5. Also find the equation of the plane containingType equation here. the two
lines.
18. Find the Cartesian equation of the plane containing the two lines
r = ( i+2j-3k) + λ(i+2j+5k) and r = (2i+5j+2k) + μ(3i-2j+5k).
19. Find the equation of a plane containing a point A (2i+4j-k) and a line
r = (i+j+k) + λ(3i-j+4k)
20. Find the equation of a plane passing through a point P (1,1,1) and containing a line r = (-
3i+j+5k) + λ(3i-j-5k). Also show that the plane contains the line
r = (-i+2j+5k) + μ(i-2j-5k).
21. Find a vector equation of a plane through the point A(2,1,-1) and B(-1,3,4) perpendicular to the
plane x-2y+4z = 10.
22. Find the equation of the plane containing a line 𝑥+1
2 =
𝑦+1
−1 =
𝑧−3
4 and perpendicular to the
plane x+2y+z = 12.
23. Find the equation of a plane through the line of intersection of two planes
3x+2y+z+2 = 0 and x+y+z+4 = 0 and:
1. Perpendicular to a plane 5x-y-3z+7 = 0
2. Parallel to a line 𝑥−1
1 =
𝑦+2
3 =
𝑧−1
−3
3. Parallel to y-axis
4. Perpendicular to XY plane
5. Perpendicular to ZX plane
24. Find the equation of the plane through the line of intersection of the planes
x+y+z = 1 and 2x+3y+4z = 5 which is perpendicular to the plane x-y+z = 0. Also find the
distance of the plane so obtain from the origin.
25. Find the equation of a line passing through A (1,2,3) and parallel to two planes
r.(i-j+2k) = 5 and r.(3i+j+k) = 6
CHAPTER 12: LINEAR PROGRAMMING
1. Graph the solution set of the following inequations
3x +4y ≤ 18, x -6y ≤ 3, 2x +3y ≥3, - 7 x +14y ≤ 14, x ≥ 0 and y ≥ 0.
Page 10 of 12
2. A furniture dealer deals in only two items tables and chairs. He has Rs 5000
to invest and a space to store at most 60 pieces. A table costs him Rs 250 and a chair Rs 50.
He can sell a table at a profit of Rs 50 and a chair at a profit of Rs 15. Assuming that he
Sell all the items that he buys, how should he invest his money in order that he may
Maximize his profit?
3. If a young man rides his motorcycle at 25 k.m per hour, he has to spend Rs 2 per k.m on
Petrol, if he rides it at a faster speed of 40 k.m per hour, the petrol cost increases to Rs 5
Per k.m. He has Rs 100 to spend on petrol and wishes to find the maximum distances he
Can travel within one hour. Express this as a linear programming problem and then solve it.
4. Suppose every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates, and
The corresponding values for rice are 0.05 g and 0.5 g respectively. Wheat costs Rs 5 and rice
Rs 20 per k.g. The minimum daily requirements of proteins and carbohydrates for an average
man are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed
In the daily diet to provide the minimum daily requirements of proteins and carbohydrates
At minimum cost, assuming that both wheat and rice are to be taken in the diet?
5. A firm manufactures two type of products, A and B and sells them at a profit of Rs 5 per
Unit of type A and Rs 3 per unit of type B. Each product is processed on two machines M1 and
M2. One unit of type A requires 1 minute of processing time on M1 and 2 minutes of processing
Time on M2 whereas one unit of type B requires 1 minute of processing time on M1& 1 minute
On M2. Machines M1 and M2 are respectively available for at most 5 hours and 6 hours in a
day.
Find out how many units of each type of product should the firm produce a day in order to
maximize the profit. Solve the problem graphically.
6. An aeroplane of an airline can carry a maximum of 200 passengers. A profit of Rs 400 is
Made on each first class ticket and a profit of Rs 300 is made on each economy class ticket
The airline reserves at least 20 seats for first class. However, at least 4 times as many
passengers prefer to travel by economy class than by first class. Determine how many of each
type of tickets
must be sold in order to maximize the profit for the airline. What is the maximum profit?
7. A chemical industry produces two compounds A & B. The following table gives the units of
Ingredients C and D per kg of compounds A and B as well as minimum requirements of C and
D
And costs per kg of A and B.
Compound
A
( in units )
Compound B
( in units )
Minimum
requirements
Ingredient C(per kg ) 1 2 80
Ingredient D(per kg ) 3 1 75
Cost per kg (in Rs) 4 6
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Variables x y
Find the quantities of A and B which would minimize the cost?
8. A company makes two type of belts A and B profits on these belts being Rs 4 and Rs 3
respectively. Each belt of type A requires twice as much time as a belt of type B, and
If all belts were of B, the company could make 1000 belts per day. The supply of leather
Is sufficient for only 800 belts per day (both A and B combined). At the most 400 buckles
For belts of type A and 700 for those of type B are available per day. How many belts of
Each type should the company make per day so as to maximize the profit?
9. A company has factories located at each of the two places P and Q. From these locations
A certain commodity delivered to three depots situated at A, B and C. The weekly
requirements
Of these depots are respectively 7/6 and 4 units of the commodity while the weekly production
Capacities of the factories at P and Q are respectively 9 and 8 units. The cost of transportation
Per unit is given below.
From /To Cost (in
Rs )
A B C
P 16 10 15
Q 10 12 10
How many units should be transported from each factory to each depot in order that that the
Transportation cost is minimum? Formulate the above LPP mathematically and solve it.
10. A retired person has Rs 70 000 to invest and two types of bonds are available in the market
For investment. First type of bond yields and annual income of 8% on the amount invested
And the second type of yields 10% per annum. As per norms, he has to invest 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?
Chapter 13: PROBABILTY
1. Find the mean, variance and standard deviation of the number of sixes in two tosses of a die.
2. Two cards are drawn simultaneously ( or successively without replacement) from a well-
shuffled pack of 52 cards.Find the mean and variance of the number of aces.
3. Three defective bulbs are mixed with 7 good ones. Let X be the number of
defective bulbs when 3 bulbs are drawn at random. Find the mean and variance of X.
4. The bulbs produced in a factory are supposed to contain 5% defective bulbs. What is the
probability that a sample of 10 bulbs will contain not more than 2 defective bulbs?
5. Faaiz and Faaiq appear for an interview for 2 vacancies in a company. The probability of
Faaiz selection is 1/5 and that of Faaiq is 1/6. What is the Probability that (i) both of them are
selected (ii) only one of them is selected
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(iii) none of them is selected.
6. Three persons A,B,C through a die in succession till one gets 6 and wins the game. Find their
respective probability of winning if A starts first.
7. A die is thrown twice and the sum of the numbers appearing is observed to to be 8 what is
the conditional probability that the number 5 has appearedatleast once?
8. A factory has 3 machines X, Y and Z producing 1000, 2000 and 3000 bolts per day
respectively. The machine X produces 1% defective bolts, Y produces 1.5% defective bolts and
Z produces 2% defective bolts. At the end of the day abolt is drawn at random and is found to
be defective. What is the probability that this defective bolt has been produced by the machine
X.
9.A doctor is to visit a patient. From past experience, it is known that the Probability that he
will come by train, bus, scooter or by car are respectively 3
10,
1
5,
1
10,
2
5 . The probability that he
will be late are 1
4
1
3
1
12 , if he comes by train, bus and scooter respectively but if he comes by car,
he will not be late when he arrives, he is late. What is the probability that he has come by train.
10. In an examination, an examinee either guesses or copies or knows the answer to a multiple
choice 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 prob that his answer is correct, given that he copied it, is 1/8.
The prob that his answer is correct, given that he guessed, is ¼. Find the prob that he knows the
answer to the question, given that he answered it correctly.
11. By examining chest x-ray the probability that a person is diagnosed with TB when he is
actually suffering from it, is 0.99. The probability that the Doctor incorrectly diagnoses a
person to behaving TB, on the basis of x-ray reports, is 0.001. In a certain city, one in thousand
persons suffers from TB. A person is selected at a random and is diagnosed to have TB. What
is the chance that he actually has TB?
12. Bag A contains 5 red balls and 4 white balls, another bag B contains 5 red balls and 5
white balls. 2 balls are transferred from bag A to bag B and A ball is drawn from bag B and
found to be red ball. What is the probability that both the transferred balls are white.
13. Six coins are tossed simultaneously. Find the probability of getting (i)3 heads (ii) no head
(iii) at least one head (iv) not more than 3 heads.
Prepared by:
XI – XII Boys Section
