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12th Maths – Public Exam 2019 – Unit Wise Exam Tips!
1. APPLICATIONS OF MATRICES AND DETERMINANTS
1. Matrix Cofactor Adjoint Inverse Matrix matrix matrix [Aij] (adj. A) (A-1)
2. Any matrix A=(
𝑎11 𝑎12 𝑎13𝑎21 𝑎22 𝑎23𝑎31 𝑎32 𝑎33
)
3. Aij = (
𝐴11 𝐴12 𝐴13𝐴21 𝐴22 𝐴23𝐴31 𝐴32 𝐴33
) ie. (+ − +− + −+ − +
) Cofactor sign=(-1)i+j
4. adj. A= [Aij]T 5. A (adj.A)=(adj.A) A=|𝐴|. In 6. If |𝐴| = 0 then A is a singular matrix. 7. If |𝐴| ≠ 0 then A is a non-singular matrix. (∴ 𝐴-1 exists)
8. A-1 = 1
|𝐴| (𝑎𝑑𝑗. 𝐴)
9. Solving by matrix inversion method : X= A-1 B 10. Determinant method/ Cramer’s rule method: Linear equations of non-homogeneous system of 2 variables
a1x+b1y=c1 ∆= |𝑎1 𝑏1𝑎2 𝑏2
| ∆𝑥= |𝑐1 𝑏1𝑐2 𝑏2
| ∆𝑦= |𝑎1 𝑐1𝑎2 𝑐2
|
a2x+b2y=c2 2 x 2 ∆≠ 0 ∆= 0 [consistent- unique solution] 𝑖) ∆= ∆𝑥= ∆𝑦= 0 𝑏𝑢𝑡 𝑎𝑛𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
of ∆ (aij≠ 0) then the system is consistent By Cramer’s rule, and has many solutions.
x =∆𝑥
∆ y=
∆𝑦
∆ Reduction: 1 equ. Put y=k
ii) ∆=0 ; ∆𝑥(𝑜𝑟)∆𝑦≠0 then the system is
inconsistent and has no solution. Non-homogeneous system of linear equations of 3 variables:
a1x+b1y+c1z=d1 a2x+b2y+c2z=d2 a3x+b3y+c3z=d3
∆= |
𝑎1 𝑏1 𝑐1𝑎2 𝑏2 𝑐2𝑎3 𝑏3 𝑐3
| ∆𝑥= |
𝑑1 𝑏1 𝑐1𝑑2 𝑏2 𝑐2𝑑3 𝑏3 𝑐3
|
∆𝑦= |
𝑎1 𝑑1 𝑐1𝑎2 𝑑2 𝑐2𝑎3 𝑑3 𝑐3
| ∆𝑧= |
𝑎1 𝑏1 𝑑1𝑎2 𝑏2 𝑑2𝑎3 𝑏3 𝑑3
|
3 x 3
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∆≠ 0 ∆= 0 [consistent- unique solution] 𝑖) ∆= ∆𝑥= ∆𝑦= ∆𝑧= 0 𝑏𝑢𝑡 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 2 𝑥 2 𝑚𝑖𝑛𝑜𝑟𝑠
of ∆ ≠ 0 then the system is consistent and has many solus. By Cramer’s rule, Reduction: 2 equs. Put z=k
x =∆𝑥
∆ y=
∆𝑦
∆ z=
∆𝑧
∆ again apply Cramer’s rule…
ii) ∆= ∆𝑥= ∆𝑦= ∆𝑧= 0 𝑎𝑛𝑑 𝑎𝑙𝑙 2 𝑥 2 𝑚𝑖𝑛𝑜𝑟𝑠
of ∆, ∆𝑥, ∆𝑦, ∆𝑧 are zero but aij≠ 0 𝑖𝑛 ∆ then the
system is consistent and has many solutions. Reduction: 1 equ. Put y=s; z=t iii) ∆=0 ; ∆𝑥(𝑜𝑟)∆𝑦(𝑜𝑟)∆𝑧≠0 then the system is
inconsistent and has no solution. iv) ∆= ∆𝑥= ∆𝑦= ∆𝑧= 0 𝑎𝑛𝑑 𝑎𝑙𝑙 2 𝑥 2 𝑚𝑖𝑛𝑜𝑟𝑠
of ∆ are zero but atleast one 2x2 minor ≠0 𝑖𝑛 ∆𝑥(𝑜𝑟)∆𝑦(𝑜𝑟)∆𝑧 then the system
is inconsistent and has no solution. Rank of the matrix 𝜌(𝐴):
The order of any highest order non-vanishing minor of the matrix. Echelon form: I) the first entry of first non-zero row must be 1. ii) non-zero row after zero row iii) no of zeros in the row is less than no of zeros in the next row iv) triangular form zeros in any corner. Rank method: Non-homogeneous system 3 x 3 𝜌(𝐴, 𝐵) = 𝜌(𝐴) ≤ 3 𝜌(𝐴, 𝐵) ≠ 𝜌(𝐴) [consistent-solvable] [inconsistent-no solution]
𝜌(𝐴,𝐵) = 𝜌(𝐴) = 3 = 𝑛𝑜 𝑜𝑓 𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑠 𝜌(𝐴, 𝐵) = 𝜌(𝐴) = 2 𝜌(𝐴, 𝐵) = 𝜌(𝐴) = 1
(unique solution) (consistent-many solutions) Put z=k put y=s; z=t Homogeneous system
Trivial solution (x,y,z)=(0,0,0) Non-trivial solution (x,y,z)≠ 0
2 x 2
{
Det.method
∆≠ 0____________
𝑅𝑎𝑛𝑘 𝑚𝑒𝑡ℎ𝑜𝑑𝜌(𝐴) = 2
The system is consistent and has a unique trivial solution
∆= ∆𝑥= ∆𝑦= 0 (aij≠ 0) in ∆ 𝜌(𝐴) = 1 K 𝜖 𝑅 − {0}
The system is consistent and has many non-trivial solutions
3 𝑥 3 {
𝐷𝑒𝑡.𝑚𝑒𝑡ℎ𝑜𝑑∆≠ 0
𝑅𝑎𝑛𝑘 𝑚𝑒𝑡ℎ𝑜𝑑𝜌(𝐴, 𝐵) = 𝜌(𝐴) = 3
∆= ∆𝑥= ∆𝑦= ∆𝑧=0
𝜌(𝐴, 𝐵) = 𝜌(𝐴) = 2 (or) 1 K 𝜖 𝑅 − {0} (or) s, t 𝜖 𝑅 − {0}
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The system is consistent and has a unique trivial solution
The system is consistent and has many non-trivial solutions
UNIT-1 TEST I. Choose the best answer of the following: 10 x 1 =10 1. If A is a square matrix of order n, then I adj.A I is a) IAI2 b) IAIn c) IAIn-1 d) IAI
2. The inverse of (3 15 2
) is
a) (2 −1−5 3
) 𝑏) (−2 51 −3
) 𝑐) (3 −1−5 −3
) 𝑑) (−3 51 −2
)
3. If the equations -2x+y+z=𝑙 , x-2y+z=m, x+y-2z=n such that l+m+n=0 then the system has a) a non-zero unique solution b)trivial solution c) many solutions d) no solution 4. (AT)-1 is equal to a) A-1 b) AT c) A d) (A-1)T
5. In a system of 3 linear non-homogeneous equations with three unknowns, if ∆= 0 𝑎𝑛𝑑 ∆𝑥= 0, ∆𝑦≠ 0
and ∆𝑧=0, then the system has a) unique solution b) two solutions c) many solutions d) no solution 6. If (𝐴, 𝐵) = 𝜌(𝐴) = 3 = 𝑛𝑜 𝑜𝑓 𝑡ℎ𝑟𝑒𝑒 𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑠 , then the system is a) consistent and has infinitely many solutions b) consistent and has a unique solution
c) consistent d) inconsistent 7. If A= (2 0 1 ) then the rank of AAT a) 1 b) 2 c) 3 d) 0 8. If A is a matrix of order 3, then det(kA) a) k3 det(A) b) k2det(A) c)k det(A) d) det(A)
9. If the matrix (−1 3 21 𝑘 −31 4 5
)has inverse then
a) k is any real number b) k=-4 c) k≠ 4 d) k≠ −4 10. Every homogeneous system a) is always consistent b) has only trivial solution c) has infinitely many solutions d) need not be consistent II. Answer the following questions: (any 5) 5 x 6 = 30
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11. If A=(5 27 3
) 𝑎𝑛𝑑 𝐵 = (2 −1−1 1
) 𝑣𝑒𝑟𝑖𝑓𝑦 (AB)-1 = B-1A-1
12. If A=(−1 2 −24 −3 44 −4 5
) prove that A=A-1
13. If A= (1 23 −5
)verify that A (adj.A)=(adj.A)A=IAI . I
14. If any non-singular matrix A, show that (A-1)T=(AT)-1 15. Solve by determinant method 2x-3y=7; 4x-6y=14
16. Find the rank of (−2 1 3 40 1 1 21 3 4 7
)
III. Answer the following questions: (any 6) 6 x 10 = 60
17. Find the adjoint of the matrix A=(3 −3 42 −3 40 −1 1
) and A (adj.A)=(adj.A)A=IAI . I
18. Solve by matrix inversion method: x-3y-8z+10=0, 3x+y=4, 2x+5y+6z=13 19. Solve the system of equations x+2y+z=2, 2x+4y+2z=4, x-2y-z=0 using determinant method. 20. A bag contains 3 types of coins namely Re.1, Rs.2 and Rs.5. There are 30 coins amounting to 100 in total. Find the number of coins in each category. 21. Solve by determinant method: x+y+2z=0, 3x+2y+z=0, 2x+y-z=0 22. Solve by rank method: 2x+5y+7z=52, x+y+z=9, 2x+y-z=0 23. For what values of k, the system of equations kx+y+z=1, x+ky+z=1, x+y+kz=1 have i) unique solution ii) more than one solution iii) no solution.
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2. VECTOR ALGEBRA
1. Scalar product (or) dot product: ��.�� = |��| |��|𝑐𝑜𝑠𝜃 = 𝑎𝑏 𝑐𝑜𝑠𝜃 ��
2. Angle between the vectors �� and �� : 𝜃 = cos−1 (��.��
|��| |��|) 𝜃 ��
3. Projection of �� on �� : ��.��
|��|
4. Projection of �� on �� : ��.��
|��| y
5. ��.�� = ��.�� (commutative) . p(x,y,z)
6. ��.�� = 0 (�� is perpendicular to ��) 𝑗
7. Unit vectors of axes: (𝑖, 𝑗, �� )
By properties, 𝑖. 𝑖 = 𝑗. 𝑗 = 𝑘. ��=1 �� o 𝑖 x
𝑖. 𝑗 = 𝑗. �� = 𝑘. 𝑖=0
8. �� is parallel to ��, then ��=m�� z
9. Properties: 𝑜𝑝 = 𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧��
i) (�� + ��)2=��2 + 2 ��.�� +��2 magnitude, |𝑟| = √𝑥2 + 𝑦2 + 𝑧2
ii) (�� − ��)2=��2 - 2 ��.�� +��2
iii) (��+��) (��-��)= ��2-��2
iv) (��+��+ 𝑐)2= ��2+��2+ 𝑐2+ 2 ��.�� + 2 ��. 𝑐 + 2 𝑐.�� 𝛽 𝛼
v) | �� + �� | ≤ |��| + |��| 𝛾 directional ratios=(x,y,z)
vi) (��+��)2+ (��-��)2=2 (|��| + |��|)2 directional cosines=(cos 𝛼, cos 𝛽, cos 𝛾)
vii) �� = ± ��
|��| ; unknown vector=𝜇�� =(
𝑥
𝑟,
𝑦
𝑟,𝑧
𝑟)
10. Work done by the force, W = ��.𝑑
11. Volume of parallelepiped = [��,��,𝑐]=( �� x ��). 𝑐 = �� .( �� x 𝑐) (scalar product of triple vectors)
12. Collinear (or) Coplanar (or) any two vectors are parallel: [��,��,𝑐]=0
13. Vector product, �� x �� = |��| |��|𝑠𝑖𝑛𝜃 ��
14. Angle between the vectors, 𝜃 = sin−1 (|�� x ��|
|��| |��|) { �� = |��| = 1
15. Area of parallelogram = |�� x ��|
16. Vector area of parallelogram = �� x ��
17. Area of triangle = 1
2 |�� x ��|
18. �� x �� = - �� x �� (not commutative)
19. �� x �� = 0 (vectors are parallel)
20. Unit vectors of axes: (𝑖, 𝑗, �� )
By properties,𝑖 𝑥𝑖 = 𝑗 𝑥𝑗 = ��𝑥��=0 𝑖x𝑗 = ��; 𝑗x�� = 𝑖; ��x𝑖 = 𝑗 𝑖x�� = −𝑗; 𝑗x𝑖 = −��; ��x𝑗 = −𝑖
21. Unit vector : ��=± (�� x ��
|��| |��|)
22. Vector of magnitude 𝜇 normal to plane containing �� and ��, = ± 𝜇 (�� x ��
|��| |��|)
23. |�� x ��|2+(��.��)2=|��|2|��|2
24. If |�� x ��|= |��. ��| → 𝜃 =𝜋
4
25. Momentum of force (or) Torque of force about a point: ��=𝑟 x ��
26. Vector product of triple vectors: �� x (�� x 𝑐)=( �� . 𝑐) �� - (�� . ��) 𝑐
27. [�� +��, �� + c, 𝑐 +a]= 2 [�� �� 𝑐] 28. [�� -��, �� - c, 𝑐 - a]= 0 29. [�� x��, ��xc, 𝑐 x a]=[�� �� 𝑐]2
30. Vector product of four vectors: : (𝑎 x ��)x (𝑐 x 𝑑)= [�� 𝑏 𝑑]𝑐 - [�� �� 𝑐]𝑑
31. Scalar product of four vectors: (𝑎 x ��). (𝑐 x 𝑑)=|��. 𝑐 ��. 𝑑
��. 𝑐 ��. 𝑑|
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32. Equation of line, i) Given that one point and IIel vector,
vector equation:: 𝑟 = �� + 𝑡�� Cartesian equation:: 𝑥−𝑥1
𝑙=
𝑦−𝑦1
𝑚=
𝑧−𝑧1
𝑛
Non-parametric form: 𝑟 × �� = �� × �� ii) Two passing points,
vector equation:: 𝑟 = �� + 𝑡(�� − ��) Cartesian equation:: 𝑥−𝑥1
𝑥2−𝑥1=
𝑦−𝑦1
𝑦2−𝑦1=
𝑧−𝑧1
𝑧2−𝑧1
Non-parametric form: (𝑟 − 𝑎) × (�� − 𝑎) = 0 33. Angle between the two lines,
𝑟 = 𝑎1 + 𝑡��; 𝑟 = 𝑎2 + 𝑡�� 𝜃 = cos−1 (��.��
|��||��|)
34. Skew lines: Two lines in a plane which are not coplanar are called skew lines.
35. Distance between two skew lines, d=|[(𝑎2 −𝑎1 ), 𝑢, ��]|
|����|
36. Condition for two lines of intersection:
d = [(𝑎2 − 𝑎1), 𝑢, ��] = |
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
|=0 (𝑢, �� 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙)
37. Distance between two IIel lines, d = | 𝑢 ×(𝑎2 −𝑎1 )]|
|��|
38. Equation of plane: i) Given that three passing points,
vector equation: 𝑟 = �� + 𝑠(�� − ��) + 𝑡(𝑐 − ��) Cartesian equation: |
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑥3 − 𝑥1 𝑦3 − 𝑧1 𝑧3 − 𝑧1
|=0
Non-parametric form: [(𝑟 − 𝑎), (�� − 𝑎), (𝑐 − ��)] = 0
ii) Two points and one parallel vector,
vector equation: 𝑟 = �� + 𝑠(�� − ��) + 𝑡 𝑢 Cartesian equation: : |
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙 𝑚 𝑛| = 0
Non-parametric form: [(𝑟 − 𝑎), (�� − 𝑎), 𝑢 ] = 0
iii) One point and two IIel vectors:
vector equation: 𝑟 = �� + 𝑠 𝑢 + 𝑡 𝑣 Cartesian equation: |
𝑥 − 𝑥1 𝑦 − 𝑦1 𝑧 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
|=0
Non-parametric form: [(𝑟 − 𝑎), 𝑢 , 𝑣 ] = 0
39. Intercept form of plane equation: 𝑥
𝑎+
𝑦
𝑏+
𝑧
𝑐= 1
40. Vector equation of plane in the normal form, 𝑟. �� = 𝑝
41. Angle between two planes, 𝜃 = cos−1 (𝑛1 .𝑛2
|𝑛1 ||𝑛2 |)
42. Angle between the line and plane , 𝜃 = sin−1 (��.��
|��||��|)
43. Distance between the two parallel lines, 𝐷 =|𝑑1−𝑑2|
√𝑎2+𝑏2+𝑐2
44. Equation of plane passing through the point of intersection of two planes, 𝜋1 + 𝜆𝜋2=0
45. Distance between the point and the plane, d= |𝑎𝑥1+𝑏𝑦1+𝑐𝑧1+𝑑|
√𝑎2+𝑏2+𝑐2
46. Equation of sphere: i) Given that the centre(𝑐) and radius(𝑎),
Vector equation: |𝑟 − 𝑐| = 𝑎 Cartesian equation: (use 𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧��) ii) Two extremities of diameter
vector equation: (𝑟 − 𝑎). (𝑟 − 𝑏) = 0 Cartesian equation: (𝑥 − 𝑥1)( 𝑥 − 𝑥2)+( 𝑦 − 𝑦1)( 𝑦 − 𝑦2)+( 𝑧 − 𝑧1)( 𝑧 − 𝑧2)=0 47. General equation of sphere: 𝑥2 + 𝑦2 + 𝑧2 + 2𝑢𝑥 + 2𝑣𝑦 + 2𝑤𝑧 + 𝑑 = 0
Centre, C(-u,-v,-w) Radius, r=√𝑢2 + 𝑣2 + 𝑤2 − 𝑑
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2. Vector algebra test-100 marks
Answer the following questions:(any 5)
6 marks:- 5 * 6=30
1. Forces 2𝑖 + 7𝑗 , 2𝑖 − 5𝑗 + 6�� , −𝑖 + 7𝑗−𝑘 , act at a point P whose position vector is 4𝑖 + 3𝑗 − 2��.
Find the moment of the resultant of three forces acting at P about the point Q whose position
vector is 6𝑖 + 𝑗 − 3�� .
2. Show that the points (3,-1,-1)(1,0,-1) and (5,-2,-1) are collinear.
3. Find the shortest distance between the parallel lines.
𝑟 = (𝑖 − 𝑗) + 𝑙(2𝑖 − 𝑗 + ��) 𝑎𝑛𝑑 𝑟 = (2𝑖 − 𝑗 + ��) + 𝑠(2𝑖 − 𝑗 + ��).
4. Prove that the diagonals of a rhombus bisect each other at right angles by vector method .
5. Prove that [�� × �� , �� × 𝑐 , 𝑐 × ��] = [����𝑐]2
6. Prove that the area of the quadrilateral ABCD , having AC and BD diagonals is 1
2|𝐴𝐵 × 𝐵𝐷|.
7. Find the vector and Cartesian equation of the line through the point (3,-4,-2) an parallel to the
Vector 9𝑖 + 6𝑗 + 2�� .
8. Find the vector and Cartesian equation of the sphere on the join of the points A and B having
position vectors 2𝑖 + 6𝑗 − 7�� 𝑎𝑛𝑑 − 2𝑖 + 4𝑗 + 3�� respectively as a diameter . Find
also the centre and radius of the sphere .
9. (i) If ��, 𝑏 are any two vectors , then prove that |�� × ��|2
+ (��. ��)2 = |��|2 |��|2
(ii) For any vector ��, prove that 𝑖 × (�� × 𝑖) + 𝑗 × (�� × 𝑗) + �� × (�� × ��) = 2�� .
10. (i) Find 𝜆 if the planes 𝑟. (2𝑖 + 𝜆𝑗 − 3��) = 10and 𝑟. (𝜆𝑖 + 3𝑗 + ��) = 5 are perpendicular .
(ii) Find the angle between the planes 2x-y+z=4 and x+y+2z=4.
Answer the following questions:( any 7)
10 marks:- 7 * 10 =70
1. Prove by vector method cos( 𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵.
2. Find the vector and Cartesian equation of the plane passing through the points (-1,1,1) and
(1,-1,1) and perpendicular to the plane x+2y+2z=5
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3. Show that the lines 𝑥−1
1=
𝑥+1
−1=
𝑧
3 𝑎𝑛𝑑
𝑥−2
1=
𝑦−1
2=
−𝑧−1
1 intersect and hence find the point of
intersection .
4. Prove by vector method that altitudes of a triangle are concurrent .
5. If �� = 𝑖 + 𝑗 + �� , �� = 2𝑖 + �� , 𝑐 = 2𝑖 + 𝑗 + �� , 𝑑 = 𝑖 + 𝑗 + 2�� , Prove that
(�� × ��) × (𝑐 × ��) = [����𝑑]𝑐 − [����𝑐]𝑑
6. Find the vector and Cartesian equation of the plane passing through the points (1,2,-2) ,
parallel to 𝑥+2
3=
𝑦+1
−2=
𝑧−4
−4 and perpendicular to 2x+3y+3z=8 .
7. Find the vector and Cartesian equation of the plane containing the line 𝑥−1
2=
−𝑦
3=
𝑧+1
1and
perpendicular to the plane x-2y+3z-2=0 .
8. Show that the lines 𝑥−1
3=
𝑦−1
−1`=
𝑧−1
0and
𝑥−4
0=
𝑦
0=
𝑧−1
3 intersect and hence find the point of
intersection .
9. Prove sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 by vector method.
10. Derive the equation of the plane in the intercept form .
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3. COMPLEX NUMBERS
1. Complex form (or) Standard form: Z = a+ib (or) x+iy Real part- ‘a’ or ‘ x’ imaginary part- s‘b’ or ‘y’
2. i2 =-1 ⟸ √−1 = i 3. Conjugate of Z, �� = a –ib or x-iy 4. Properties:
i) Z. �� = real ix) |𝑍1
𝑍2| =
|𝑍1|
|𝑍2|
ii) �� = Z x) |𝑍1 + 𝑍2| ≤ |𝑍1| + |𝑍2|
iii) Z+�� = 2 re(z) xi) |𝑍1 − 𝑍2| ≥ |𝑍1| − |𝑍2|
iv) Z-�� = 2 im(z) xii) Z . �� = ��. Z = |𝑍|2
v) 𝑍1 + 𝑍2 = 𝑍1
+ 𝑍2 xiii) If |𝑍| = 1 and only if �� =
1
𝑍
vi) 𝑍1 − 𝑍2
= 𝑍1 - 𝑍2
xiv) Re(Z) ≤ |𝑍|; Im (Z) ≤ |𝑍|
vii) 𝑍1𝑍2 = 𝑍1 𝑍2
xv) |𝑍| = |��| viii) |𝑍1𝑍2| = |𝑍1| |𝑍2| xvi) 𝑍𝑛 = ( 𝑧 ) n
5. Modulus, r = |𝑍| = √𝑥2 + 𝑦2 (or) √𝑎2 + 𝑏2
6. Argument (or) Amplitude, arg(z) or amp(z) = tan−1(𝑦
𝑥) -𝜋 < 𝜃 ≤ 𝜋
7. arg(𝑍1𝑍2)=arg(𝑍1)+arg(𝑍2)
8. arg(𝑍1
𝑍2)= arg(𝑍1)-arg(𝑍2)
9. Polar form, x+iy = r (cos𝜃+ i sin𝜃)
10. Exponential form, 𝑒𝑖𝜃 =(cos𝜃+ i sin𝜃) ; 𝑒−𝑖𝜃 =(cos𝜃- i sin𝜃) 11. De moivre’s theorem: (cos𝜃+ i sin𝜃)n= cos𝑛𝜃+ i sin𝑛𝜃 12. Some polar form equalities:
i) cos0°+ i sin0°=1 iii) cos𝜋
2+ i sin
𝜋
2 = i
ii) cos𝜋+ i sin𝜋= -1 iv) cos𝜋
2 - i sin
𝜋
2 = -i
13. Graph: 2nd Y 1st
𝜃 = 𝜋 + tan−1 𝑦
𝑥 𝜃 = tan−1 𝑦
𝑥
(- , +) (+, +) X
𝜃 = −𝜋 + tan−1 𝑦
𝑥 𝜃 = tan−1 𝑦
𝑥
(-,-) (+,-) 3rd 4th 14. For z = 0, the argument is indeterminate.
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15. |𝑧1.𝑧2.𝑧3………….𝑧𝑛| = |𝑍1||𝑍2||𝑍3| … … … |𝑍𝑛|
16. All values of (cos𝜃 + i sin𝜃)𝑝
𝑞; No fo values = q
17. nth root of unity, 11
𝑛 = 𝜔 ⇒ 𝜔𝑛 = 1
18. Cube root of unity, 11
3 = 𝜔 ⇒ 𝜔3 = 1
Roots are { 1, 𝜔, 𝜔2} ie. 1+𝜔 + 𝜔2=0 {1 + 𝜔 = −𝜔2
𝜔 + 𝜔2 = −11 + 𝜔2 = −ω
19. nth roots of unity are {1, 𝜔, 𝜔2, 𝜔3,………… 𝜔𝑛−1} 20. Sum of the roots =0 21. Products of the roots=(−1)𝑛+1 22. Roots are in G.P (common ratio= 𝜔)
23. Arguments are in A.P (common difference=2𝜋
𝑛)
24. Distance between the complex numbers= |𝑍1 − 𝑍2| 25. Trigonometry hints:
𝟎° 𝟑𝟎° 𝟒𝟓° 𝟔𝟎° 𝟗𝟎° Degrees Radians Notes
𝐬𝐢𝐧 𝜽 0 𝟏
𝟐
𝟏
√𝟐 √𝟑
𝟐
1 𝟏𝟖𝟎°
𝝅 Sin(-𝜽) - sin𝜽
𝐜𝐨𝐬 𝜽 1 √𝟑
𝟐
𝟏
√𝟐
𝟏
𝟐
0 𝟑𝟔𝟎° 𝟐𝝅 cos(-𝜽) cos𝜽
𝐭𝐚𝐧 𝜽 0 𝟏
√𝟑
1 √𝟑 ∞ 𝟎° 0 tan(-𝜽) - tan𝜽
𝐜𝐬𝐜 𝜽 ∞ 2 √𝟐 𝟐
√𝟑
1 𝟑𝟎° 𝝅
𝟔
𝐬𝐞𝐜 𝜽 1 𝟐
√𝟑 √𝟐 2 ∞ 𝟒𝟓° 𝝅
𝟒 𝐬𝐢𝐧−𝟏(−𝒙) -𝐬𝐢𝐧−𝟏 𝒙
𝐜𝐨𝐭 𝜽 ∞ √𝟑 1 𝟏
√𝟑
0 𝟔𝟎° 𝝅
𝟑 𝐜𝐨𝐬−𝟏(−𝒙) 𝝅 −𝐜𝐨𝐬−𝟏 𝒙
𝟗𝟎° 𝝅
𝟐 𝐭𝐚𝐧−𝟏(−𝒙) -𝐭𝐚𝐧−𝟏 𝒙
𝟐𝟕𝟎° 𝟑𝝅
𝟐 𝐭𝐚𝐧−𝟏 𝒙 − 𝐭𝐚𝐧−𝟏 𝒚 = 𝐭𝐚𝐧−𝟏 (
𝒙 − 𝒚
𝟏 + 𝒙𝒚)
26. Graph: 2nd 90° 1st
Sin 𝜃/𝑐𝑠𝑐𝜃 All
90° + 𝜃/180° − 𝜃 90° − 𝜃/360° + 𝜃 (- , +) (+, +)
180° 0°/360°
Tan 𝜃 /cot 𝜃 Cos 𝜃/𝑠𝑒𝑐𝜃
180° + 𝜃/270° − 𝜃 270° + 𝜃/360° − 𝜃 (-,-) (+,-) 3rd 270° 4th
(HINT: After School To College / All Silver Tea Cups)
Working rules:
180°/360° 90°/270° sin𝜃 → 𝑠𝑖𝑛𝜃 cot𝜃 → 𝑐𝑜𝑡𝜃 sin𝜃 → 𝑐𝑜𝑠𝜃 cot𝜃 → 𝑡𝑎𝑛𝜃 cos𝜃 → 𝑐𝑜𝑠𝜃 csc𝜃 → 𝑐𝑠𝑐𝜃 cos𝜃 → 𝑠𝑖𝑛𝜃 csc𝜃 → 𝑠𝑒𝑐𝜃 tan𝜃 → 𝑡𝑎𝑛𝜃 sec𝜃 → 𝑠𝑒𝑐𝜃 tan𝜃 → 𝑐𝑜𝑡𝜃 sec𝜃 → 𝑐𝑠𝑐𝜃
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3. COMPLEX NUMBERS I. Choose the correct answer: 10 X 1 =10
1. The Polar form of the complex number (𝑖25)3is
a) 𝑐𝑜𝑠𝜋
2+ 𝑖 𝑠𝑖𝑛
𝜋
2 𝑏)𝑐𝑜𝑠
𝜋
2− 𝑖 𝑠𝑖𝑛
𝜋
2 𝑐) 𝑐𝑜𝑠𝜋 + 𝑖 𝑠𝑖𝑛𝜋 d) 𝑐𝑜𝑠𝜋 − 𝑖 𝑠𝑖𝑛𝜋
2. The value of 𝑖 + 𝑖22 + 𝑖23 + 𝑖24 + 𝑖25 is a) i b) –i c) 1 d) -1 3. If 𝜔 is a cube root of unity then the value of (1 − 𝜔 + 𝜔2)4 + (1 + 𝜔 − 𝜔2)4 is a) 0 b) 32 c) -16 d) -32 4. If a + ib = (8-6i)-(2i-7) then the values of a and b respectively are a) 8,-15 b) 8,15 c) 1,4 d) 15,-8 5. If P represents the variable of complex number z and if |2𝑧 − 1| = 2|𝑧|, then the locus of p is
a) the straight line x=1
4 b) the straight line y=
1
4
c) the straight line z=1
2 d) the circle x2+y2-4x-1=0
6. The arguments of nth roots of a complex number differ by
a) 2𝜋
𝑛 b)
𝜋
𝑛 c)
3𝜋
𝑛 d)
4𝜋
𝑛
7. If z1=4+5i and z2= -3 + 2i then 𝑧1
𝑧2 is
a) 2
13−
22
13𝑖 b) −
2
13+
22
13𝑖 c) −
2
13−
23
13𝑖 d)
2
13+
22
13𝑖
8. The value of √𝑧 𝑧 is a) |𝑧|2 b) |𝑧| c) 2|𝑧| d) 2|𝑧|2 9. The equation having 4-3i and 4+3i as roots is a) x2+8x+25=0 b) x2+8x-25=0 c) x2-8x-25=0 d) x2-8x+25=0
10. The value of 𝑒𝑖𝜃 − 𝑒−𝑖𝜃is a) 𝑠𝑖𝑛𝜃 b) 2𝑠𝑖𝑛𝜃 c) 𝑖 𝑠𝑖𝑛𝜃 d) 2𝑖 𝑠𝑖𝑛𝜃
II. Answer the following questions: (any 5) 5 x 6 = 30 11. Find the square roots of -8-6i.
12. Simplify: (𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃)4
(𝑠𝑖𝑛𝜃+𝑖𝑐𝑜𝑠𝜃)5
13. Prove that (1 + 𝑖)𝑛 + (1 − 𝑖)𝑛 = 2𝑛+2
2 𝑐𝑜𝑠𝑛𝜋
4; 𝑛 ∈ 𝑁
14. Solve the equation : x4-8x3+4x2-32x+20=0 if 3+i is a root. 15. State and prove triangle inequality of complex numbers. 16. Show that the points representing the complex numbers (7+9i), (-3+7i), (3+3i) form a right-angled triangle on the argand plane .
17. If x+1
𝑥= 2𝑐𝑜𝑠𝜃 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑖) 𝑥𝑛 +
1
𝑥𝑛 = 2 cos 𝑛𝜃 𝑖𝑖) 𝑥𝑛 −1
𝑥𝑛 = 2 i sin 𝑛𝜃
III. Answer the following questions: (any6) 6 x 10 = 60
18. P represents the variable of complex number Z. Find the locus of p if Im [2𝑧+1
𝑖𝑧−1] = −1
19. If 𝛼 𝑎𝑛𝑑 𝛽 are the roots of the equation x2-2px+(p2+q2)=0 and tan𝜃 =𝑞
𝑦+𝑝, show that
(𝑦+𝛼)𝑛−(𝑦+𝛽)𝑛
𝛼−𝛽= 𝑞𝑛−1 sin 𝑛𝜃
𝑠𝑖𝑛𝑛𝜃
20. Find all values of (1
2− 𝑖
√3
2)
3
4. Prove that the product of the values is equal to 1.
21. Solve: x4-x3+x2-x+1=0 22. If a=cos 2𝛼 + 𝑖 sin 2𝛼, b= cos 2𝛽 + 𝑖 sin 2𝛽, c= cos 2𝛾 + 𝑖𝑠𝑖𝑛2𝛾. Prove that
i) √𝑎𝑏𝑐 +1
√𝑎𝑏𝑐= 2𝑐𝑜𝑠(𝛼 + 𝛽 + 𝛾) 𝑖𝑖)
𝑎2𝑏2+𝑐2
𝑎𝑏𝑐= 2 cos 2 (𝛼 + 𝛽 − 𝛾)
23. P represents the variable of complex number Z. Find the locus of p if arg (𝑧−1
𝑧+1) =
𝜋
3
24. If 𝑥 +1
𝑥= 2 𝑐𝑜𝑠𝜃, 𝑦 +
1
𝑦= 2 𝑐𝑜𝑠∅ 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
𝑖) 𝑥𝑚
𝑦𝑛 +𝑦𝑛
𝑥𝑚 = 2 cos(𝑚𝜃 − 𝑛∅) 𝑖𝑖) 𝑥𝑚
𝑦𝑛 −𝑦𝑛
𝑥𝑚 = 2 i sin (𝑚𝜃 − 𝑛∅)
25. If 𝛼 𝑎𝑛𝑑 𝛽 are the roots of x2-2x+4=0, prove that 𝛼𝑛 − 𝛽𝑛 = 𝑖 2𝑛+1𝑠𝑖𝑛𝑛𝜋
3 𝑎𝑛𝑑 𝑑𝑒𝑑𝑢𝑐𝑒 𝛼9 − 𝛽9, 𝑛 ∈ 𝑁
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4. ANALYTICAL GEOMETRY
Conical figures
D
Parabola Ellipse Hyperbola P(x,y) M
F
Curves of sliced surfaces
Eccentricity = e = 𝐹𝑃
𝑃𝑀
e = 1 e <1 e >1
b2-4ac=0 b2-4ac<0 b2-4ac>0
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PARABOLA: e = 1 / b2-4ac = 0
Type:1 y x
parabola is open rightward
Type:2 y x
parabola is open leftward
Type:3 y x
parabola is open upward
Type:4 y
x parabola is open downward
1. General form: (y-k)2 = 4a (x-h) 2. centre/vertex , C (or) V (0 , 0) 3. Equation, y2 = 4ax 4. symmetrical axis: x axis (y=0) 5. Focus, F (a,0) 6. Equation of Latusrectum: x = a 7. Equation of Directrix: x = -a 8. Length of Latus rectum: ‘4a’ 9. Distance between vertex (or) centre and focus: ‘a’ units 10. Distance between focus and Directrix : ‘2a’ units
(y-k)2 = - 4a (x-h)
C (or) V (0,0)
y2 = - 4ax
x axis (y=0)
F (-a,0)
x = -a
x = a
4a units
a units
a units
(x-h)2 = 4a (y-k)
C (or) V (0,0)
x2 = 4ay
y axis (x=0)
F (0,a)
y = a
y = - a
4a units
a units
a units
(x-h)2 = - 4a (y-k)
C (or) V (0,0)
x2 = - 4ay
y axis (x=0)
F (0,- a)
y = - a
y = a
4a units
a units
a units
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ELLIPSE: e < 1 / b2-4ac < 0 a2 > b2 ; a > b
Type: 1 y x
Type: 2 y x
1. General form: (𝑥−ℎ)2
𝑎2 + (𝑦−𝑘)2
𝑏2 = 1
2. centre, C (0,0)
3. Equation, 𝑥2
𝑎2 + 𝑦2
𝑏2 = 1
4. Vertices , V (±𝑎, 0) 5. Foci, F (±𝑎𝑒, 0) 6. Equation of Latusrectums: x = ± 𝑎𝑒
7. Equation of Directrices: x = ±𝑎
𝑒
8. Major axis : x axis (y=0) Length : 2a units 9. Minor axis : y axis (x=0) Length: 2b units
10. Length of the Latus rectum: 2𝑏2
𝑎 𝑢𝑛𝑖𝑡𝑠
11. Eccentricity, e =√1 −𝑏2
𝑎2
1. General form: (𝑥−ℎ)2
𝑏2 + (𝑦−𝑘)2
𝑎2 = 1
2. centre, C (0,0)
3. Equation, 𝑥2
𝑏2 + 𝑦2
𝑎2 = 1
4. Vertices , V (0, ±𝑎) 5. Foci, F (0, ±𝑎𝑒) 6. Equation of Latusrectums: y = ± 𝑎𝑒
7. Equation of Directrices: y = ±𝑎
𝑒
8. Major axis : y axis (x=0) Length : 2a units 9. Minor axis : x axis (y=0) Length: 2b units
10. Length of the Latus rectum: 2𝑏2
𝑎 𝑢𝑛𝑖𝑡𝑠
11. Eccentricity, e =√1 −𝑏2
𝑎2
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HYPERBOLA: e > 1 / b2 – 4ac > 0
Type: 1 y x
Type : 2 y x
1. General form: (𝑥−ℎ)2
𝑎2 − (𝑦−𝑘)2
𝑏2 = 1
2. centre, C (0,0)
3. Equation, 𝑥2
𝑎2 −𝑦2
𝑏2 = 1
4. Vertices , V (±𝑎, 0) 5. Foci, F (±𝑎𝑒, 0) 6. Equation of Latusrectums: x = ± 𝑎𝑒
7. Equation of Directrices: x = ±𝑎
𝑒
8. Transverse axis : x axis (y=0) Length : 2a units 9. Conjugate axis : y axis (x=0) Length: 2b units
10. Length of the Latus rectum: 2𝑏2
𝑎 𝑢𝑛𝑖𝑡𝑠
11. Eccentricity, e =√1 +𝑏2
𝑎2
1. General form: (𝑦−𝑘)2
𝑎2 − (𝑥−ℎ)2
𝑏2 = 1
2. centre, C (0,0)
3. Equation, 𝑦2
𝑎2 − 𝑥2
𝑏2 = 1
4. Vertices , V (0,±𝑎) 5. Foci, F (0, ±𝑎𝑒) 6. Equation of Latusrectums: y = ± 𝑎𝑒
7. Equation of Directrices: y = ±𝑎
𝑒
8. Transverse axis : y axis (x=0) Length : 2a units 9. Conjugate axis : x axis (y=0) Length: 2b units
10. Length of the Latus rectum: 2𝑏2
𝑎 𝑢𝑛𝑖𝑡𝑠
11. Eccentricity, e =√1 +𝑏2
𝑎2
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PARABOLA ELLIPSE HYPERBOLA
1. Parametric form x= at2 −∞ < 𝑡 < ∞ y = 2at
x = a cos𝜃 0≤ 𝜃 ≤ 2𝜋
(or) x = 𝑎(1−𝑡2)
1+𝑡2
y= b sin𝜃 y= 2𝑏𝑡
1+𝑡2
x=a sec𝜃 0≤ 𝜃 ≤ 2𝜋 y= b tan𝜃
2. Any point on the curve (x1,y1) (at2,2at)
(a cos𝜃,bsin𝜃) or (𝑎(1−𝑡2)
1+𝑡2 ,2𝑏𝑡
1+𝑡2 )
(a sec𝜃, 𝑏 𝑡𝑎𝑛𝜃)
3. Equation of tangent at (x1, y1) at ‘t’ or ‘𝜃′
y y1 = 2a (x+x1) yt = x +at2
𝑥 𝑥1
𝑎2+
𝑦 𝑦1
𝑏2= 1
𝑥
𝑎 𝑐𝑜𝑠𝜃 +
𝑦
𝑏𝑠𝑖𝑛𝜃 = 1
𝑥 𝑥1
𝑎2−
𝑦 𝑦1
𝑏2= 1
𝑥
𝑎 𝑠𝑒𝑐𝜃 −
𝑦
𝑏𝑡𝑎𝑛𝜃 = 1
4. Equation of normal at (x1, y1) at ‘t’ or ′𝜃′
xy1+2ay = x1y1+2ay1 tx + y =2at + at3
𝑎2𝑥
𝑥1−
𝑏2𝑦
𝑦1=a2-b2
𝑎𝑥
𝑐𝑜𝑠𝜃−
𝑏𝑦
𝑠𝑖𝑛𝜃= a2-b2
𝑎2𝑥
𝑥1+
𝑏2𝑦
𝑦1=a2+b2
𝑎𝑥
𝑠𝑒𝑐𝜃+
𝑏𝑦
𝑡𝑎𝑛𝜃= a2+b2
Rule to form tangent or chord equation:
x 2----- x x1
y2 ------- y y1
x --------- 𝑥+𝑥1
2
y -------- 𝑦+𝑦1
2
x y ------- 𝑥𝑦1+𝑦𝑥1
2
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If y=mx+c may be a tangent to the conics then the condition :
Condition Point of contact Equation of any tangent
Parabola c=𝑎
𝑚 (
𝑎
𝑚2 ,2𝑎
𝑚)
y=mx + 𝑎
𝑚
Ellipse c2=a2m2+b2 (
−𝑎2𝑚
𝑐 ,
𝑏2
𝑐) y=mx ±√(a2m2+b2)
Hyperbola c2=a2m2-b2 (
−𝑎2𝑚
𝑐 ,
−𝑏2
𝑐) y=mx ±√(a2m2-b2)
ASYMPTOTES :-
An asymptote to a curve is the tangent , such that the point of contact is at infinity.
Hyperbola:- 𝑥
𝑎−
𝑦
𝑏= 0 y
x
𝑥
𝑎+
𝑦
𝑏= 0
1. Standard form of hyperbola ,
𝑥2
𝑎2−
𝑦2
𝑏2 = 1
2. Combined equation of asymptotes ,
𝑥2
𝑎2 −𝑦2
𝑏2 = 0
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3. Separate equations are ,
𝑥
𝑎+
𝑦
𝑏= 0
𝑥
𝑎−
𝑦
𝑏= 0
4. The standard equation of hyperbola and combined equation of asymptotes differs only by a constant.
5. Angle between the asymptotes ,
(i) 𝜃 = 2𝛼 = 2 tan−1(𝑏
𝑎)
(ii) ) 𝜃 = 2 sec−1(𝑒)
(iii) tan 𝜃 = |𝑚1−𝑚2
1+𝑚1𝑚2|
(iv) tan𝜃 = |2√ℎ2−𝑎𝑏
𝑎+𝑏| xy=c2
Rectangular hyperbola:
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1. Equation of rec. hyperbola, xy=c2 where 𝑐2 =𝑎2
2
2. Eccentricity, e=√2
3. 𝑏2 = 𝑎2(𝑒2 − 1), a=b
4. C(0,0) or (h,k)
5. V1(𝑎
√2,
𝑎
√2) V2(
𝑎
√2,
𝑎
√2)
6. F1(a,a) F2(-a,-a)
7. Transverse axis, y=x
8. Conjugate axis, y=-x
9. Parametric form, x=ct, y=𝑐
𝑡
10 . Any point on the curve (ct, 𝑐
𝑡)
11. Equation of tangent, xy1+yx1=2 c2
X+yt2 = 2ct
12. Equation of normal, x x1-y y1= x1 2-y1 2
y- xt2 = 𝑐
𝑡- ct3
13. Normal to rec.hyperbola meets at ‘t1’ ant ‘t2’ , t1 3 t2 =-1
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5. APPLICATION OF DIFFERENTIAL CALCULUS
Ex:5.2
1. Equation of tangent line , y-𝒚𝟏=m(x-𝒙𝟏)
Note:
m ------> slope of tangent , m => 𝑑𝑦
𝑑𝑥
recall , {m = tan 𝜃
m = −𝑐𝑜.𝑒𝑓𝑓𝑖 𝑜𝑓 𝑥
𝑐𝑜.𝑒𝑓𝑓𝑖 𝑜𝑓 𝑦
2. Equation of normal line , y-𝒚𝟏=−𝟏
𝒎(𝒙 − 𝒙𝟏)
−1
𝑚 => slope of normal
3. Angle between two curves ,
𝐭𝐚𝐧 𝜽 = |𝒎𝟏−𝒎𝟐
𝟏+𝒎𝟏𝒎𝟐|
𝜽 = 𝐭𝐚𝐧−𝟏(𝒎𝟏−𝒎𝟐
𝟏+𝒎𝟏𝒎𝟐)
4. When the curves cut at right angles , 𝒎𝟏𝒎𝟐 = −𝟏 (the cures are orthogonal)
Ex:5.3:
Rolle’s theorem
If f(x) is a real valued function such that ,
(i) 𝑓(𝑥) is continuous in the closed interval (a,b)
(ii) 𝑓(𝑥) is differentiable in the open interval (a,b)
(iii) 𝑓(𝑎) = 𝑓(𝑏)
Then there exist atleast one point x=c in (a,b) such that f’(c)=0
Ex:5.4:
Lagrange’s theorem
If f(x) is a real valued function such that ,
(i) 𝑓(𝑥) is continuous in the closed interval (a,b)
(ii) 𝑓(𝑥) is differentiable in the open interval (a,b)
(iii) 𝑓(𝑎) ≠ 𝑓(𝑏)
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Then there exist atleast one point x=c in (a,b) such that f’(c)=𝒇(𝒃)−𝒇(𝒂)
𝒃−𝒂
Ex:5.5:
Maclaurin’s series:
𝒇(𝒙) = 𝒇(𝟎) + 𝒇′(𝟎)𝐱
𝟏!+
𝐱𝟐
𝟐! 𝒇′′(𝟎) +
𝐱𝟑
𝟑! 𝒇′′′(𝟎)+……..
Ex:5.6:
Indeterminate forms:
𝟎
𝟎 , ∞ ,
∞
∞ , ∞ ± ∞ , ∞∞ , 𝟎𝟎 , ∞𝟎 , 𝟏∞
L’H𝒐pital’s Rule :
If lim𝑥→𝑐
𝒇(𝒙)
𝒈(𝒙)=>
0
0 𝑓𝑜𝑟𝑚
Apply L’H��pital’s Rule ,
lim𝑥→𝑐
𝒇′(𝒙)
𝒈′(𝒙)=>
0
0 𝑓𝑜𝑟𝑚
Again apply L’H��pital’s Rule ,
Lim𝑥→𝑐
𝒇′′(𝒙)
𝒈′′(𝒙) = (𝑠𝑜𝑚𝑒 𝑙𝑖𝑚𝑖𝑡)ss
Ex:5.7:
Monotonic function :
A function that is increasing everywhere (or) decreasing everywhere is called a monotonic
function .
Conditions:-
a , b 𝑓(𝑎) , 𝑓(𝑏) 𝑑𝑦
𝑑𝑥 (𝑜𝑟) 𝑓′(𝑥)
Increasing func. a<b 𝑓(𝑎) ≤ 𝑓(𝑏) 𝑑𝑦
𝑑𝑥≥ 0 (+𝑣𝑒)
decreasing func. a<b 𝑓(𝑎) ≥ 𝑓(𝑏) 𝑑𝑦
𝑑𝑥≤ 0 (−𝑣𝑒)
strictly increasing a<b 𝑓(𝑎) < 𝑓(𝑏) 𝑑𝑦
𝑑𝑥> 0 (+𝑣𝑒)
strictly decreasing a<b 𝑓(𝑎) > 𝑓(𝑏) 𝑑𝑦
𝑑𝑥< 0 (−𝑣𝑒)
Monotonicity:-
Either a function increasing or decreasing in the given interval.
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Critical numbers:-
𝑓′(𝑐) =0 (or) 𝑓′(𝑐) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 at x
Stationary numbers/points :-
𝑓′(c) = 0 at x
Ex:5.9 :-
Maximum and minimum
Absolute maximum and minimum Local maximum and minimum
A function 𝑓(𝑥) has an absolute maximum at c if 𝑓(𝑐) > 𝑓(𝑥) for all x in the domain of f.
Local maximum:
𝑑𝑦
𝑑𝑥= 0 𝑎𝑛𝑑
𝑑2
𝑑𝑥2 < 0
A function 𝑓(𝑥) has an absolute maximum at c if 𝑓(𝑐) < 𝑓(𝑥) for all x in the domain of f.
Local minimum:
𝑑𝑦
𝑑𝑥= 0 𝑎𝑛𝑑
𝑑2
𝑑𝑥2 > 0
Ex:5.11:-
Concavity and convexity and points of inflection :-
1. A curve that lies above its tangents is called concave.
𝑑2𝑦
𝑑𝑥2 > 0 (+𝑣𝑒)
2. A curve that lies below its tangents is called convex.
𝑑2𝑦
𝑑𝑥2 < 0 (−𝑣𝑒)
3. Separates concave and convex --- a point of inflection at 𝑥 = 𝑐
(i) 𝑑2𝑦
𝑑𝑥2 = 0
(i) 𝑑3𝑦
𝑑𝑥3 ≠ 0
Ex:5.1:-
1. Rate of changes of y. → 𝒅𝒚
𝒅𝒙
2. Rate of changes of displacement , v= 𝒅𝒙
𝒅𝒕 (𝑥 − 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒) → 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
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3. Rate of changes of velocity ,
a= 𝒅𝒗
𝒅𝒕=
𝑑2𝑥
𝑑𝑡2 (𝑥 − 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒) → 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
4. Initial velocity → 𝑎𝑡 𝑡 = 0
𝒗 = (𝒅𝒙
𝒅𝒕)𝒕=𝟎
5. Particle comes to rest → 𝑣 = 0 (𝑟𝑒𝑎𝑐ℎ𝑒𝑠 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 ℎ𝑒𝑖𝑔ℎ𝑡)
6. Acceleration is zero → 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
Fermat’s theorem ;-
If 𝒇(𝒙) has local maximum (or) minimum at c if 𝑓′(𝑐) exists then 𝑓′′(𝑐) = 0
Monotonicity :-
1𝑠𝑡 derivative test
Concavity and Convexity :-
2𝑛𝑑 derivative test
Extreme value theorem :-
Absolute maximum and minimum.
Differentiation:-
Differentiation is a variation .
y 𝑑𝑦
𝑑𝑥
x 1
𝑥𝑛 𝑛𝑥𝑛−1 𝑒𝑥 𝑒𝑥
log 𝑥 1
𝑥
sin 𝑥 cos 𝑥 cos 𝑥 −sin 𝑥 tan 𝑥 𝑠𝑒𝑐2𝑥 cot 𝑥 −𝑐𝑜𝑠𝑒𝑐2𝑥 sec 𝑥 sec 𝑥 tan 𝑥
cosec 𝑥 cosec 𝑥 cot 𝑥 constant 0
sin−1 𝑥 1
√1 − 𝑥2
cos−1 𝑥 −
1
√1 − 𝑥2
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tan−1 𝑥 1
1 + 𝑥2
sec−1 𝑥 1
𝑥√𝑥2−1
cosec−1 𝑥 −1
𝑥√𝑥2 − 1
𝑎𝑥 𝑎𝑥 log 𝑥
√𝑥 1
2√𝑥
cot−1 𝑥 −1
1 + 𝑥2
7x 7
𝑥5 5𝑥4
1
𝑥
−1
𝑥2
6.Differential calculus-II :-
1. Relative error = ∆𝑦
𝑦
2. Percentage of error = ∆𝑦
𝑦∗ 100
3. Approximation ,
𝑓(𝑥 + ∆𝑥) ≅ 𝑦 + ∆𝑦
4. Tracing the graph ,
Symmetry:-
(i) x axis only if 𝑓(𝑥, −𝑦) = 𝑓(𝑥, 𝑦)
(ii) y axis only if 𝑓(−𝑥, 𝑦) = 𝑓(𝑥, 𝑦)
(iii) Origin only if 𝑓(−𝑥, −𝑦) = 𝑓(𝑥, 𝑦)
(iv) Symmetrical about a line y=x if 𝑓(𝑥, 𝑦) = 𝑓(𝑦, 𝑥)
(v) 𝑦 = −𝑥 If 𝑓(𝑥, 𝑦) = 𝑓(−𝑦, −𝑥)
Passes through the origin :-
(i) No constant term - Passes through the origin
(ii) A constant term - Does not pass
Asymptotes :-
As 𝑥 = 𝑎 then 𝑦 → ∞
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u= 𝑓(𝑥, 𝑦) is a real valued function then 𝜕𝑢
𝜕𝑥 and
𝜕𝑢
𝜕𝑦 are its partial derivatives .
(i) 𝜕2𝑢
𝜕𝑥𝜕𝑦=
𝜕
𝜕𝑥(
𝜕𝑢
𝜕𝑦)
(ii) 𝜕2𝑢
𝜕𝑦𝜕𝑥=
𝜕
𝜕𝑦(
𝜕𝑢
𝜕𝑥)
(iii) 𝜕2𝑢
𝜕𝑥2 =𝜕
𝜕𝑥(
𝜕𝑢
𝜕𝑥)
(iv) 𝜕2𝑢
𝜕𝑦2=
𝜕
𝜕𝑦(
𝜕𝑢
𝜕𝑦)
(v) 𝑈𝑥 =𝜕𝑢
𝜕𝑥
(vi) 𝑈𝑦 =𝜕𝑢
𝜕𝑦
Euler’s Theorem :-
f is a homogeneous function of degree n. then ,
𝑥 𝜕𝑓
𝜕𝑥+ 𝑦
𝜕𝑓
𝜕𝑦= 𝑛𝑓
7.Integral calculus :-
Integration is a summation .
𝒚 = 𝒇(𝒙)
∫ 𝒇(𝒙)𝒅𝒙 + 𝒄
1. ∫ 𝑥𝑛𝑑𝑥
𝑥𝑛+1
𝑛 + 1+ 𝑐
2. ∫ 𝑑𝑥
𝑥 + 𝑐
3. ∫ 5 𝑑𝑥
5𝑥 + 𝑐
4. ∫
1
𝑥
log 𝑥 + 𝑐
5. ∫ 𝑒𝑥
𝑒𝑥 + 𝑐
6. ∫ 𝑒𝑎𝑥 𝑑𝑥
𝑒𝑎𝑥
𝑎+ 𝑐
7. ∫ sin 𝑥
−cos 𝑥 + 𝑐
8. ∫ cos 𝑥
sin 𝑥 + 𝑐
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9. ∫ tan 𝑥
log(sec 𝑥) + 𝑐
10. ∫ cot 𝑥
log(sin 𝑥) + 𝑐
11. ∫ sec 𝑥 tan 𝑥 𝑑𝑥
sec 𝑥 + 𝑐
12. ∫ 𝑠𝑒𝑐2𝑥
tan 𝑥 + 𝑐
13. ∫ 𝑐𝑜𝑠𝑒𝑐 𝑥 cot 𝑥 𝑑𝑥
− cosec 𝑥 + 𝑐
14. ∫ 𝑐𝑜𝑠𝑒𝑐2𝑥
− cot 𝑥 + 𝑐
15. ∫
1
1 + 𝑥2
tan−1 𝑥 + 𝑐
16. ∫
1
𝑎2 + 𝑥2
1
𝑎tan−1(
𝑥
𝑎) + 𝑐
17. ∫
1
√1 − 𝑥2
sin−1 𝑥 + 𝑐
18. ∫ 𝑎𝑥𝑑𝑥
𝑎𝑥
log 𝑎+ 𝑐
19. ∫
1
√𝑎2 + 𝑥2 log(𝑥 + √𝑎2 + 𝑥2) + 𝑐
20. ∫
1
√𝑎2 − 𝑥2 sin−1(
𝑥
𝑎) + 𝑐
21. ∫
1
√𝑥2 − 𝑎2 log(𝑥 + √𝑥2 − 𝑎2) + 𝑐
22. ∫ √𝑥2 + 𝑎2 𝑑𝑥
𝑥
2√𝑥2 + 𝑎2 +
𝑎2
2log (𝑥 + √𝑥2 + 𝑎2) + 𝑐
23. ∫ √𝑥2 − 𝑎2 𝑑𝑥
𝑥
2√𝑥2 − 𝑎2 −
𝑎2
2log (𝑥 + √𝑥2 − 𝑎2) + 𝑐
24. ∫ √𝑎2 − 𝑥2 𝑑𝑥
𝑥
2√𝑎2 − 𝑥2 +
𝑎2
2sin−1
𝑥
𝑎+ 𝑐
25. ∫
1
𝑥2 − 𝑎2𝑑𝑥
1
2𝑎log(
𝑥 − 𝑎
𝑥 + 𝑎) + 𝑐
26. ∫
1
𝑎2 − 𝑥2𝑑𝑥
1
2𝑎log(
𝑎 + 𝑥
𝑎 − 𝑥) + 𝑐
27. ∫ 𝑒𝑎𝑥 cos 𝑏𝑥 𝑑𝑥
𝑒𝑎𝑥
𝑎2 + 𝑏2[𝑎 cos 𝑏𝑥 + 𝑏 sin 𝑏𝑥] + 𝑐
28. ∫ 𝑒𝑎𝑥 sin 𝑏𝑥 𝑑𝑥
𝑒𝑎𝑥
𝑎2 + 𝑏2[𝑎 sin 𝑏𝑥 − 𝑏 cos 𝑏𝑥] + 𝑐
29. ∫
2𝑥
1 + 𝑥2𝑑𝑥
log(1 + 𝑥2) + 𝑐
30. ∫
𝑓′(𝑥)
𝑓(𝑥)𝑑𝑥
log(𝑓(𝑥)) + 𝑐
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31. ∫ cot 𝑥 𝑑𝑥
log(sin 𝑥) + 𝑐
Integration by parts:-
∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢
Reduction formula:-
∫ 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥 = 𝜋
20
∫ 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 𝜋
20
(i) If ‘n’ is odd → ∫ 𝑠𝑖𝑛𝑛𝑥 (𝑜𝑟) 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 = 𝑛−1
𝑛 𝑛−3
𝑛−2… .
2
3 .1
𝜋
20
(ii) If ‘n’ is even → ∫ 𝑠𝑖𝑛𝑛𝑥 (𝑜𝑟) 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 = 𝑛−1
𝑛 𝑛−3
𝑛−2 𝑛−5
𝑛−4… .
1
2 .
𝜋
2
𝜋
20
Integration:-
Properties :-
1. ∫ 𝑓(𝑥)𝑑𝑥 𝑏
𝑎= − ∫ 𝑓(𝑥)𝑑𝑥
𝑎
𝑏
2. ∫ 𝑓(𝑥)𝑑𝑥 𝑏
𝑎
= ∫ 𝑓(𝑎 + 𝑏 − 𝑥)𝑑𝑥𝑏
𝑎
3. ∫ 𝑓(𝑥)𝑑𝑥 𝑎
0= ∫ 𝑓(𝑎 − 𝑥)𝑑𝑥
𝑎
0
4. ∫ 𝑓(𝑥)𝑑𝑥 = {0 𝑖𝑓 𝑓 𝑖𝑠 𝑜𝑑𝑑
2 ∫ 𝑓(𝑥) 𝑖𝑓 𝑓 𝑖𝑠 𝑒𝑣𝑒𝑛𝑎
0
𝑎
−𝑎
5. ∫ 𝑠𝑖𝑛𝑛𝑥 = ∫ 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥 𝜋
20
= {
𝑛−1
𝑛 𝑛−3
𝑛−2 𝑛−5
𝑛−4… .
2
3 𝑤ℎ𝑒𝑛 ′𝑛′𝑖𝑠 𝑜𝑑𝑑
𝑛−1
𝑛 𝑛−3
𝑛−2 𝑛−5
𝑛−4… .
1
2.
𝜋
2 𝑤ℎ𝑒𝑛 ′𝑛′𝑖𝑠 𝑒𝑣𝑒𝑛
𝜋
20
6. ∫ 𝑓(𝑥)2𝑎
0𝑑𝑥 = ∫ 𝑓(𝑥)
𝑎
0𝑑𝑥 + ∫ 𝑓(2𝑎 − 𝑥)
𝑎
0𝑑𝑥
{𝑖𝑓 𝑓(2𝑎 − 𝑥) = 𝑓(𝑥) → ∫ 𝑓(𝑥)
2𝑎
0𝑑𝑥 = 2 ∫ 𝑓(𝑥)
𝑎
0𝑑𝑥
𝑖𝑓 𝑓(2𝑎 − 𝑥) = −𝑓(𝑥) → ∫ 𝑓(𝑥)2𝑎
0𝑑𝑥 = 0
Bernoulli’s Theorem:-
∫ 𝑢 𝑑𝑣 =uv- u’𝑣1+u”𝑣2-u’’’𝑣3 + ⋯
Gamma integral:-
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∫ 𝑥𝑛∞
0𝑒−𝑎𝑥𝑑𝑥 =
𝑛!
𝑎𝑛+1 (𝑜𝑟)⌊𝑛
𝑎𝑛+1
Area of the bounded region :-
1. 𝑦 = 𝑓(𝑥) ; 𝑥 𝑎𝑥𝑖𝑠 , 𝑥 = 𝑎 , 𝑥 = 𝑏 above the 𝑥 − 𝑎𝑥𝑖𝑠
𝐴 = ∫ 𝑦𝑏
𝑎
𝑑𝑥
2. 𝑦 = 𝑓(𝑥) ; 𝑥 𝑎𝑥𝑖𝑠 , 𝑥 = 𝑎 , 𝑥 = 𝑏 below the 𝑥 − 𝑎𝑥𝑖𝑠
𝐴 = ∫ (−𝑦)𝑏
𝑎
𝑑𝑥
3. Right side of y axis , 𝑦 = 𝑐 ; 𝑦 = 𝑑
𝐴 = ∫ 𝑥𝑑
𝑐
𝑑𝑦
4. Right side of y axis , 𝑦 = 𝑐 ; 𝑦 = 𝑑
𝐴 = ∫ (−𝑥)𝑑
𝑐
𝑑𝑦
Length of the arc :-
1. about 𝑥 − 𝑎𝑥𝑖𝑠 and its limits
𝐿 = ∫ √1 + (𝑑𝑦
𝑑𝑥)2𝑏
𝑎 . 𝑑𝑥
2. about 𝑦 − 𝑎𝑥𝑖𝑠 and its limits
𝐿 = ∫ √1 + (𝑑𝑥
𝑑𝑦)2𝑑
𝑐 . 𝑑𝑦
3. Between the limits 𝑡1and𝑡2
𝐿 = ∫ √(𝑑𝑥
𝑑𝑡)2 + (
𝑑𝑦
𝑑𝑡)2𝑡2
𝑡1 . 𝑑𝑡
Volume of solid:-
1. Resolves about −𝑎𝑥𝑖𝑠 :-
𝑣 = 𝜋 ∫ 𝑦2𝑏
𝑎 𝑑𝑥
2. Resolves about −𝑎𝑥𝑖𝑠 :-
𝑣 = 𝜋 ∫ 𝑥2𝑑
𝑐 𝑑𝑦
Surface area of solid generated:-
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1. Resolves about −𝑎𝑥𝑖𝑠 :-
𝑠 = 2𝜋 ∫ 𝑦√1 + (𝑑𝑦
𝑑𝑥)2
𝑏
𝑎
𝑑𝑥
2. Resolves about −𝑎𝑥𝑖𝑠 :-
𝑠 = 2𝜋 ∫ 𝑥√1 + (𝑑𝑥
𝑑𝑦)2
𝑑
𝑐
𝑑𝑦
3. Resolves about 𝑥 − 𝑎𝑥𝑖𝑠 with limits 𝑡1 𝑎𝑛𝑑 𝑡2
𝑠 = 2𝜋 ∫ 𝑦√(𝑑𝑥
𝑑𝑡)2 + (
𝑑𝑦
𝑑𝑡)2
𝑡2
𝑡1
𝑑𝑡
Prepared By: B.MOHANKUMAR M.Sc.,B.Ed , Maths Academy, Madurantakam. Cell: 9787754734
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9. DISCRETE MATHEMATICS 6 MARKS:
1. Construct a truth table for (p 𝛬 q ) V ( ~ 𝑟 ) 2. Construct a truth table for (p 𝑉 q ) 𝛬 r
3. Construct a truth table for the statement (p 𝛬 q) V [~(𝑝 𝛬𝑞 )] 4. (i) Show that (( ~p) V (~ q)) V p is a tautology.
(ii) Show that (( ~q) 𝛬 p) 𝛬 q is a contradiction.
Use the truth table to establish which of the following statements are
tautologies and which are contradictions.
5. (p 𝛬 (~ q)) V ((~𝑝) 𝑉 𝑞 )
6.(p 𝛬 (~ p)) 𝛬 ((.(p 𝛬 (~ p)) q ) 𝛬 p )
7. Show that p→ 𝑞 ≡ (~𝑝) 𝑉 q
8. Show that p ↔ q ≡ (p →q) 𝛬( q → p)
9. Show that p ↔ q ≡ ((~𝑝) V q ) 𝛬 ((~𝑞) 𝑉 𝑝)
10. Show that ~(𝑝 𝛬 𝑞 ) ≡ ((~𝑝)𝑉 (~𝑞 ))
11. Show that p →q and q →p are not equivalent .
12. Prove that ( Z , +) is an infinite abelian group .
13. Show that ( R , - {0} , . ) is an abelian group . Here '. ' denotes usual
multiplication.
14. Show that the cube roots of unity forms a finite abelian group under usual
multiplication.
15. Prove that the set of all 4th roots of unity forms an abelian group under
multiplication.
16. Prove that ( C , +) is an infinite abelian group .
17. Show that the set of all non – Zero complex numbers is an abelian group under
the usual multiplication of complex numbers.
18. State and prove Cancellation laws.
19. State and prove Reversal law.
20. Prove that the matrices (1 00 1
) , (0 11 0
) forms a group under matrix
multiplication.
Prepared by,
B.MOHANKUMAR M.Sc., B.Ed
Maths academy, Madurantakam
Cell:9787754734
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9. DISCRETE MATHEMATICS.
TEN MARKS
1. Show that (Z , * ) is an infinite abelian group where * is defined as a * b = a+b+2.
2. Show that the set of four matrices(1 00 1
) , (−1 00 1
) , (1 00 −1
) , (−1 00 −1
) form an abelian
group , under multiplication of matrices.
3. Show that the set G of all matrices of the form (𝑥 𝑥𝑥 𝑥
) , where x € R – {0} , is a group under
matrix multiplication.
4. Let G be the set of all rational numbers except 1 and * be defined on G by a * b = a + b – ab
for all a, b € G . Show that (G,*) is an infinite abelian group .
5.Prove that the set of four function f1 , f2 , f3 ,f4 on the set of non – zero complex number C
– {0} defined by
f1 (z) = z , f2 (z) = -z , f3 (z) = 1\z and f4 (z) = - 1\z z €C- {0} forms an abelian group with
respect to the composition of functions.
6.Show that the set G of all positive rational forms a group under the composition * defined by
a * b = ab/3 for all a, b € G
7. Show that {(1 00 1
) , (𝜔 00 𝜔2) , (𝜔2 0
0 𝜔) , (
0 11 0
) , (0 𝜔2
𝜔 0) , ,(
0 𝜔𝜔2 0
)} Where
𝜔3 = 1, ω ≠ 1 , form a group with respect to matrix multiplication.
8. Show that the set G of all rational numbers except -1 forms an abelian group with respect to
the operation * given by a * b = a+b+ab for all a , b € G.
9. Show the set {[1], [3], [4], [5] , [9] } forms an abelian group under multiplication modulo 11.
10. Show that the set of all matrices of the form (𝑎 00 0
) , a € R-{0} forms an abelian group
under matrix multiplication.
Prepared by,
B.MOHANKUMAR M.Sc., B.Ed
Maths academy, Madurantakam
Cell:9787754734
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10. PROBABILITY DISTRIBUTION
6 Marks:
1. A continuous random variable X follows the probability law,
f(x) = {𝑘𝑥(1 − 𝑥)10, 0 < 𝑥 < 1
0 , 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒. Find k.
2. If F(x)= 1
𝜋 (
𝜋
2 + tan-1 x)-∞<x<∞is a distribution function of continuous variable X, find P(0≤ x≤
1).
3. For the p.d.f(x)={𝑐𝑥(1 − 𝑥)3, 0 < 𝑥 < 1
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Find (i) the constant c (ii) P(𝑥 < 1/2) 4. A continuous random variable x has the p.d.f defined by
F(x)={𝑐𝑒−𝑎𝑥, 0 < 𝑥 < ∞0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
. Find the value of c if a >0.
5. A discrete random variable X has the following probability distribution
X 0 1 2 3 4 5 6 7 8
P(X=x) a 3a 5a 7a 9a 11a 13a 15a 17a
(i) Find the value of a (ii)Find(x<3) (iii)Find P(3<x<7). 6. An urn contains 4 white and 3 red balls. Find the probability distribution of the number of red balls in three draws when a ball is drawn at random with replacement. Also find the mean and variance.
7. Find the mean and variance of the distribution f(x)={3e−3x, 0 < 𝑥 < ∞
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
8. In a gambling game a man wins Rs.10 if he gets all heads or all tails and loses Rs.5 if he gets 1 or 2 head when three coins are tossed once. Find his expectation of gain. 9. Find the Mean and variance for the following probability density functions
(i) f(x)={1
24, −12 ≤ 𝑥 ≤ 12
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (ii) f(x)={
𝛼e−αx, 𝑖𝑓 𝑥 > 00, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(iii)f(x)={𝑥𝑒−𝑥, 𝑖𝑓 𝑥 > 00 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
10. The difference between the mean and the variance of a Binomial distribution is 1 and the difference between their squares is 11. Find n.
11. If on an average 1 ship out of 10 do not arrive safely to ports. Find the mean and the
standard deviation of ships returning safely out of a total of 500 ships.
12. Four coins are tossed simultaneously. What is the probability of getting (a) exactly 2 heads
(b) at least two heads. (c) at most 2 heads.
13. Suppose that the probability of suffering a side effect from a certain vaccine is 0.005.
If 1000 persons are inoculated. Find approximately the probability that (i) at most 1 person
suffer. (ii) 4,5,or 6 persons suffer.
Prepared by,
B.MOHANKUMAR M.Sc., B.Ed
Maths academy, Madurantakam
Cell:9787754734
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14. In a Poisson distribution if P(X=2)=P(X=3) find P(X=5) [given e-3=0.050].
15. If the probability of a defective fuse from a manufacturing unit is 2% in a box of 200 fuses
find the probability that (i) exactly 4 fuses are defective. (ii) more than 3 fuses are defective.
[e-4 =0.0183].
16. The life of army shoes is normally distributed with mean 8 months and standard deviation 2
months. If 5000 pairs are issued, how many pairs would be expected to need replacement within
12 months.
17. The mean weight of 500 male students in a certain college in 151 pounds and the standard
deviation is 15 pounds. Assuming the weights are normally distributed, find how many students
weigh (i) between 120 and 155 pounds (ii) more than 185 pounds.
18. If the heights of 300 students are normally distributed with mean 64.5 inches and standard
deviation 3.3 inches, find the height below which 99% of the student lie.
19. Mark in an aptitude test given to 800 students of a school was found to be normally
distributed. 10 % of the students scored below 40 marks and 10% of the students scored above
90 marks. Find the number of students scored between 40 and 90.
20. Let Z be a standard normal variate. Calculate the following probabilities.
(i) P(0≤ Z≤ 1.2) (ii) P(-1.2≤Z≤0)
(iii) Area to the right of Z=1.3 (iv) Area to the left of Z =1.5(v) P(-1.2≤Z≤2.5)
(v) P(-1.2≤Z≤-0.5) (vii) P(1.5≤Z≤2.5).
Prepared by,
B.MOHANKUMAR M.Sc., B.Ed
Maths academy, Madurantakam
Cell:9787754734
www.Padasalai.Net
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10. PROBABILITY DISTRIBUTION
Ten marks:
1. A random variable X has the following probability mass function
x 0 1 2 3 4 5 6
P(X=x) k 3k 5k 7k 9k 11k 13k
i) Find k ii) Evaluate P(X< 4), 𝑃(𝑋 ≥ 5) 𝑎𝑛𝑑 𝑃(3 < 𝑋 ≤ 6)
iii) What is the smallest value of x for which P(X≤ 𝑥)> 1
2
2. An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls in three draws one by one from the urn. i) with replacement ii) without replacement
3. If the probability density function of a random variable is given by f(x) ={𝑘(1 − 𝑥2), 0 < 𝑥 < 1
0 , 𝑒𝑙𝑠𝑒 𝑤ℎ𝑒𝑟𝑒
Find (i) k (ii) the distribution function of the random variable. 4. The probability density function of a random variable x is
f(x)= {𝑘 𝑥𝛼−1𝑒−𝛽𝑥𝛼
, 𝑥, 𝛼, 𝛽 > 00 , 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒.
Find i) k ii) P(X>10)
5. In a hurdle race a player has to cross 10 hurdles . The probability that he will clear ecach hurdle is 5/6 What is the probability that he will knock down less than 2 hurdles. 6. Alpha particles are emitted by a radioactive source at an average rate of 5 in a 20 minutes interval . Using poisson distribution find the probability that there will be (i) 2 emission (ii) at least 2 emission
in a particular 20 minutes interval . [𝑒−5 = 0.0067] 7. The number of accidents in a year involving taxi drivers in a city follows a poisson in a distribution with mean equal to 3 out of 1000 taxi drivers, find approximately the number of drivers with (i) no accident in a year (ii) more than three accident in a year . [𝑒−3=0.0498 ] 8. Find c, 𝜇 and 𝜎2 of the normal distribution whose probability function is given by f(x) = 𝑒−𝑥2+3𝑥 ,
-∞<x<∞. 9. Obtain K, 𝜇 and 𝜎2 of the normal distribution whose probability distribution function is given by
f(x) = k𝑒−2𝑥2+4𝑥 , -∞<x<∞. 10. The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value of 31 Psi and standard deviation 0.2Psi (i) what is the probability that the pressure for a randomly selected tyre (a)between 30.5 and 31.5 Psi (b) between 30 and 32 Psi (ii) what is the probability that the pressure for a randomly selected tyre exceeds 30.5 Psi ? [ 2.5 = 0.4938]
Prepared by,
B.MOHANKUMAR M.Sc., B.Ed
Maths academy, Madurantakam
Cell:9787754734
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