POINTS
Preface 1
1. System of Co - ordinates 2
1.1 Cartesian Co-ordinates
1.2 Polar Co-ordinates
2. Distance Formula 2
3. Applications of Distance Formula 3
3.1 Position of three points
3.2 Position of four points
4. Section Formula 4
5. Co-ordinate of some particular points 4
5.1 Centroid
5.2 Incentre
5.3 Circumcentre
5.4 Ortho Centre
6. Area of triangle and quadrilateral 6
6.1 Area of Triangle
6.2 Area of Quadrilateral
7. Transformation of axes 7
7.1 Parallel Transformations
7.2 Rotational Transformation
7.3 Reflection of a Point
8. Locus 8
9. Some important points 9
10. Solved examples 10
STRAIGHT LINE
Preface 13
1. Equation of Straight line 14
2. Equation of Straight line parallel to axes 14
3. Slope of a line 14
4. Different forms of the equation of straight line 14
4.1 Slope – Intercept Form
4.2 Slope Point Form
4.3 Two Point Form
4.4 Intercept Form
4.5 Normal Form
4.6 Parametric Form
5. Reduction of General form of Equations into Standard Forms 15
6. Position of a point relative to a line 17
7. Angle between two straight lines 17
7.1 Parallel lines
7.2 Perpendicular lines
7.3 Coincident lines
8. Equation of parallel & perpendicular lines 18
9. Equation of Straight lines through (x , y ) making an angle α with y = mx + c 18
10. Length of perpendicular 19
10.1 Distance between two Parallel Lines
11. Condition of concurrency 19
12. Bisectors of angles between two lines 20
13. Line passing through the point of intersection of two lines 20
14. Homogeneous equation 20
15. General Equation of Second degree 21
16. Equation of lines joining the intersection points of a line & a curve to the origin 21
17. Some important points 22
18. Solved examples 23
CIRCLE
Preface 29
1. Definition 30
2. Standard Form of equation of a circle 31
2.1 General Form
2.2 Central Form
2.3 Diametral Form
2.4 Parametric Form
3. Equation of a circle in some special cases 31
4. Position of a point with respect to a circle 32
4.1 The least and the greatest distance of a point from a circle
5. Line and circle 33
5.1 Condition of Tangency
5.2 Intercepts made on coordinate axes by the circle
6. Equation of tangent and normal 34
6.1 Equation of Tangent
6.2 Equation of Normal
6.3 Length of Tangent
6.4 Pair of Tangents
7. Chord of contact 35
8. Director circle 35
9. Position of two circles 36
10. Equation of a chord whose middle point is given 38
11. Circle through the point of intersection 38
12. Common chord of two circles 39
13. Angle of intersection of two circles 39
14.1 Condition of Orthogonality
14. Some important points 40
15. Pole and Polar 40
9.1 Equation of Polar
9.2 Co-ordinates of Pole
9.3 Conjugate Points and Conjugate Lines
16. Radical axis and radical centre 41
16.1 Radical Axis
16.2 Radical Centre
17. Solved examples 42
PARABOLA
Preface 50
1. Definition 51
2. Terms related to parabola 51
3. Standard form of equation of parabola 51
3.1 Parameters of the Parabola y2 = 4ax
3.2 Other Standard Parabola
4. Reduction to Standard Equation 53
.
5. General equat
Basic ForTran Programming - for Beginners - An Introduction by Arun Umraossuserd6b1fd
ForTran Programming Notes for Beginners. With complete example. Covers full syllabus. Well structured and commented for easy - self understanding. Good for Senior Secondary Students.
ATOMIC STRUCTURE
1. ATOM & MOLECULES
(a) The smallest particle of a matter that takes part in a chemical reaction is called an atom. The atom of all gases except those of noble gases, cannot exist in free state. These exist in molecular form. The molecules of hydrogen, nitrogen, oxygen and halogens are diatomic (H2, N2). Phosphorus molecule is tetratomic and that of
sulphur is octa atomic.
(b) The smallest particle of a matter that can exist in free state in nature, is known as a molecule.
(c) Some molecules are composed of homoatomic atom, e.g., H2, O2, N2, Cl2, O3 etc., while the molecules of compounds are made up of two or more heteroatomic atoms e.g., HCl, NaOH, HNO3, CaCO3, etc.
2. DALTON’S ATOMIC THEORY
The concepts put forward by John Dalton regarding the composition of matter are known as Dalton’s atomic theory. Its important points are as follows.
(a) Every matter is composed of very minute particles, called atoms that take part in chemical reactions.
(b) Atoms cannot be further subdivided.
(c) The atoms of different elements differ from each other in their properties and masses, while the atoms of the same element are identical in all respects.
(d) The atoms of different elements can combine in simple ratio to form compounds. The masses of combining elements represent the masses of combining atoms.
(e) Atom can neither be created nor destroyed.
2.1 Modern Concept :
Many of the concepts of Dalton’s atomic theory cannot be explained. Therefore, foundation of modern atomic theory was laid down by the end of nineteenth century. The modern theory is substantiated by the existence of isotopes, radioactive disintegration, etc. The important points of the modern atomic theory are as follows.
(a) Prof. Henri Bacquerel discovered the phenomenon of radioactivity and found that an atom is divisible.
(b) An atom is mainly composed of three fundamental particles, viz. electron, proton and neutron.
(c) Apart from the aforesaid three fundamental particles, many others have also been identified, viz. positron, meson, neutrino, antiproton, etc.
(d) Soddy discovered the existence of isotopes, which were atom of the same element having different masses. For example, protium, deuterium and tritium are atoms of hydrogen having atomic masses 1, 2 and 3 a.m.u. respectively.
(e) Atoms having same mass may have different atomic numbers. These are known as isobars. For example,
40 Ar and 40 Ca .
18 20
(f) Atoms of elements combines to form molecules.
(g) It is not necessary that the atoms should combine in simple ratio for the formation of compounds. The atoms in non-stoichiometric compounds are not present in simple ratio. For example, in ferrous sulphide crystals, iron and sulphur atoms are present in the ratio of 0.86 : 1.00.
(h) Atoms participate in chemical reactions.
3. CATHODE RAYS (DISCOVERY OF ELECTRON)
Dry gases are normally bad conductors of electricity. But under low pressure, i.e., 0.1 mm of mercury or lower, electric current can pass thro
The process of finding area of some plane region is called Quadrature. In this chapter we shall find the area bounded by some simple plane curves with the help of definite integral. For solving
(i) The area bounded by a cartesian curve y = f(x), x-axis and ordinates x = a and x = b is given by,
the problems on quadrature easily, if possible first draw the rough sketch of the required area.
b
Area = y dx
a
b
= f(x) dx
a
In chapter function, we have seen graphs of some simple elementary curves. Here we introduce some essential steps for curve tracing which will enable us to determine the required area.
(i) Symmetry
The curve f(x, y) = 0 is symmetrical
about x-axis if all terms of y contain even powers.
about y-axis if all terms of x contain even
powers.
about the origin if f (– x, – y) = f (x, y).
Examples
based on
Area Bounded by A Curve
For example, y2 = 4ax is symmetrical about x-axis, and x2 = 4ay is symmetrical about y- axis and the curve y = x3 is symmetrical about the origin.
(ii) Origin
Ex.1 Find the area bounded by the curve y = x3, x-axis and ordinates x = 1 and x = 2.
2
Sol. Required Area = ydx
1
If the equation of the curve contains no constant
2
= x3 dx =
Lx4 O2 15
MN PQ=
Ans.
term then it passes through the origin.
For example x2 + y2 + 2ax = 0 passes through origin.
(iii) Points of intersection with the axes
If we get real values of x on putting y = 0 in the equation of the curve, then real values of x and y = 0 give those points where the curve cuts the x-axis. Similarly by putting x = 0, we can get the points of intersection of the curve and y-axis.
1 4 1 4
Ex.2 Find the area bounded by the curve y = sec2x,
x-axis and the line x = 4
/4
Sol. Required Area = ydx
x0
For example, the curve x2/a2 + y2 /b2 = 1 intersects the axes at points (± a, 0) and (0, ± b) .
(iv) Region
/4
= sec2
0
xdx = tan x /4 = 1 Ans.
Write the given equation as y = f(x) , and find minimum and maximum values of x which determine the region of the curve.
For example for the curve xy2 = a2 (a – x)
Ex.3 Find the area bounded by the curve y = mx, x-axis and ordinates x = 1 and x = 2
2
Sol. Required area = y dx .
1
y = a
a x
x
2
= mxdx =
LMmx2 2
Now y is real, if 0 < x a , so its region lies between the lines x = 0 and x = a.
1 MN2
= m ( 4 – 1) =
2
PQ1
FGH3 JKm Ans.
Ex.4 Find the area bounded by the curve y = x (1– x)2 and x-axis.
Sol. Clearly the given curve meets the x-axis at (0,0) and (1,0) and for x = 0 to 1, y is positive
Ex.7 Find the area bounded between the curve y2 = 2y – x and y-axis.
Sol. The area between the given curve x = 2y – y2 and y-axis will be as shown in diagram.
so required area- Y
= xb1 xg2 dx
0
1
= ex 2x2 x3 jdx
0
O (0,0) (1,0) X
Lx2
2x3
x4 O1
1 2 1 1
= M P=
– + =
MN2 3 4 PQ0
2 3 4 12
Ans.
Required Area =
e2y y2 jdy
(i
Basic ForTran Programming - for Beginners - An Introduction by Arun Umraossuserd6b1fd
ForTran Programming Notes for Beginners. With complete example. Covers full syllabus. Well structured and commented for easy - self understanding. Good for Senior Secondary Students.
ATOMIC STRUCTURE
1. ATOM & MOLECULES
(a) The smallest particle of a matter that takes part in a chemical reaction is called an atom. The atom of all gases except those of noble gases, cannot exist in free state. These exist in molecular form. The molecules of hydrogen, nitrogen, oxygen and halogens are diatomic (H2, N2). Phosphorus molecule is tetratomic and that of
sulphur is octa atomic.
(b) The smallest particle of a matter that can exist in free state in nature, is known as a molecule.
(c) Some molecules are composed of homoatomic atom, e.g., H2, O2, N2, Cl2, O3 etc., while the molecules of compounds are made up of two or more heteroatomic atoms e.g., HCl, NaOH, HNO3, CaCO3, etc.
2. DALTON’S ATOMIC THEORY
The concepts put forward by John Dalton regarding the composition of matter are known as Dalton’s atomic theory. Its important points are as follows.
(a) Every matter is composed of very minute particles, called atoms that take part in chemical reactions.
(b) Atoms cannot be further subdivided.
(c) The atoms of different elements differ from each other in their properties and masses, while the atoms of the same element are identical in all respects.
(d) The atoms of different elements can combine in simple ratio to form compounds. The masses of combining elements represent the masses of combining atoms.
(e) Atom can neither be created nor destroyed.
2.1 Modern Concept :
Many of the concepts of Dalton’s atomic theory cannot be explained. Therefore, foundation of modern atomic theory was laid down by the end of nineteenth century. The modern theory is substantiated by the existence of isotopes, radioactive disintegration, etc. The important points of the modern atomic theory are as follows.
(a) Prof. Henri Bacquerel discovered the phenomenon of radioactivity and found that an atom is divisible.
(b) An atom is mainly composed of three fundamental particles, viz. electron, proton and neutron.
(c) Apart from the aforesaid three fundamental particles, many others have also been identified, viz. positron, meson, neutrino, antiproton, etc.
(d) Soddy discovered the existence of isotopes, which were atom of the same element having different masses. For example, protium, deuterium and tritium are atoms of hydrogen having atomic masses 1, 2 and 3 a.m.u. respectively.
(e) Atoms having same mass may have different atomic numbers. These are known as isobars. For example,
40 Ar and 40 Ca .
18 20
(f) Atoms of elements combines to form molecules.
(g) It is not necessary that the atoms should combine in simple ratio for the formation of compounds. The atoms in non-stoichiometric compounds are not present in simple ratio. For example, in ferrous sulphide crystals, iron and sulphur atoms are present in the ratio of 0.86 : 1.00.
(h) Atoms participate in chemical reactions.
3. CATHODE RAYS (DISCOVERY OF ELECTRON)
Dry gases are normally bad conductors of electricity. But under low pressure, i.e., 0.1 mm of mercury or lower, electric current can pass thro
The process of finding area of some plane region is called Quadrature. In this chapter we shall find the area bounded by some simple plane curves with the help of definite integral. For solving
(i) The area bounded by a cartesian curve y = f(x), x-axis and ordinates x = a and x = b is given by,
the problems on quadrature easily, if possible first draw the rough sketch of the required area.
b
Area = y dx
a
b
= f(x) dx
a
In chapter function, we have seen graphs of some simple elementary curves. Here we introduce some essential steps for curve tracing which will enable us to determine the required area.
(i) Symmetry
The curve f(x, y) = 0 is symmetrical
about x-axis if all terms of y contain even powers.
about y-axis if all terms of x contain even
powers.
about the origin if f (– x, – y) = f (x, y).
Examples
based on
Area Bounded by A Curve
For example, y2 = 4ax is symmetrical about x-axis, and x2 = 4ay is symmetrical about y- axis and the curve y = x3 is symmetrical about the origin.
(ii) Origin
Ex.1 Find the area bounded by the curve y = x3, x-axis and ordinates x = 1 and x = 2.
2
Sol. Required Area = ydx
1
If the equation of the curve contains no constant
2
= x3 dx =
Lx4 O2 15
MN PQ=
Ans.
term then it passes through the origin.
For example x2 + y2 + 2ax = 0 passes through origin.
(iii) Points of intersection with the axes
If we get real values of x on putting y = 0 in the equation of the curve, then real values of x and y = 0 give those points where the curve cuts the x-axis. Similarly by putting x = 0, we can get the points of intersection of the curve and y-axis.
1 4 1 4
Ex.2 Find the area bounded by the curve y = sec2x,
x-axis and the line x = 4
/4
Sol. Required Area = ydx
x0
For example, the curve x2/a2 + y2 /b2 = 1 intersects the axes at points (± a, 0) and (0, ± b) .
(iv) Region
/4
= sec2
0
xdx = tan x /4 = 1 Ans.
Write the given equation as y = f(x) , and find minimum and maximum values of x which determine the region of the curve.
For example for the curve xy2 = a2 (a – x)
Ex.3 Find the area bounded by the curve y = mx, x-axis and ordinates x = 1 and x = 2
2
Sol. Required area = y dx .
1
y = a
a x
x
2
= mxdx =
LMmx2 2
Now y is real, if 0 < x a , so its region lies between the lines x = 0 and x = a.
1 MN2
= m ( 4 – 1) =
2
PQ1
FGH3 JKm Ans.
Ex.4 Find the area bounded by the curve y = x (1– x)2 and x-axis.
Sol. Clearly the given curve meets the x-axis at (0,0) and (1,0) and for x = 0 to 1, y is positive
Ex.7 Find the area bounded between the curve y2 = 2y – x and y-axis.
Sol. The area between the given curve x = 2y – y2 and y-axis will be as shown in diagram.
so required area- Y
= xb1 xg2 dx
0
1
= ex 2x2 x3 jdx
0
O (0,0) (1,0) X
Lx2
2x3
x4 O1
1 2 1 1
= M P=
– + =
MN2 3 4 PQ0
2 3 4 12
Ans.
Required Area =
e2y y2 jdy
(i
/2
Ex.5 Evaluate : sin2
x dx .
d
If dx
[f(x)] =
(x) and a and b, are two values
0
/2
Sol. sin2 x dx
independent of variable x, then 0
b /2 FG1 cos 2xIJ
(x) dx = f(x) a = f(b) – f(a)
a
= zH 2 dx
1 LM sin 2x O /2
is called Definite Integral of (x) within limits
= x
2
2 PQ
a and b. Here a is called the lower limit and b is called the upper limit of the integral. The interval [a,b] is known as range of integration. It should be noted that every definite integral has
a unique value.
= 1 LM 0
0
= / 4 Ans.
1
x2
Ex.6 Evaluate : xe dx.
0
1
2 x2
Ex.1 Evaluate : x4 dx.
1
Sol. xe dx
0
2 Lx5 O2
1 ex2 1
Sol.
zx4 dx = MP= 32 – 1 = 31
=
Ans. 2 0
MN5 PQ1 5 5 5
/4
= 1 (e –1) Ans.
2
Ex.2 Evaluate : sec2 x dx.
0
1 x3
/4
Sol. sec2 x .dx = tan x /4 = tan / 4 – tan 0 = 1
0
Ex.7 Find the value of
0
1 x8
dx.
0
Ans.
2 1
Sol. Let x4 = t, then 4x3 dx = dt
Ex.3 Evaluate :
1
4 x2
dx.
I =
1 1 dt
4 =
1 [ sin–1 t] 1 =
4 8
2 1 L 2 0
Sol. z dx = Msin GJP
Ans.
1 4 x2
N H2KQ1
= sin–1 (1) – sin–1 (1/2)
z/3 cos x
= – =
Ans.
Ex.8 Evaluate :
0
3 4 sin x dx.
2 6 3
Sol. I =
/3 cos x 3 4 sin x dx.
Ex.4 Evaluate :
z2 1
2 dx
0 4 x2
0
Let 3 + 4 sin x = t 4 cos x. dx = dt
cos x dx = dt/4
Now in the given integral x lies between the
Sol.
4 x2 dx
limit x = 0 to x = / 3 . Now we will decide the limit of t.
1
= tan
2
1 x OP2
0
In 3 + 4 sin x = t, by putting lower limit of x as x = 0; and upper limit as x = / 3 . We
= 1 tan11 0 = / 8 Ans.
2
get lower and upper limit of t respectively.
Putting x = 0 3 + 4 sin 0 = t t = 3
z3 z2
z3 (x) dx
/3 cos x
dx =
zt3 2 3 1 dt
= 2 x2dx +
0
z3b3x 4gdx
0 3 4 sin x
t3 t 4
Fx3 I2 F3x2 I3
= 1 zt3 2 3 1 dt
= GJ+ G
4xJ
4 t3 t
H3 K H2
= 1 log t 32
4
= 8 +
3
27 – 12 – 6 + 8
2
= 1 [ log (3 + 2
4
) – log 3] Ans.
= 37/6 Ans.
Ex.9
sin(tan1 x)
2
dx equals-
Ex.11 Evaluate : |1 x|dx.
0
0 1 x
Sol. Put tan
x = t, then 1 dx = dt
Sol. |1 x| =
RST1 x, when
0 x 1
–1
(1 x2 )
I =
x 1, when 1 x 2
1b1 xgdx + 2 bx 1gdx
/ 2
I = sin t dt [– cos t] /2 = 1 Ans.
0 1
L x2 O1 Lx2 O2
0 Mx
P M xP
0 = MN
2 PQ+
MN2
PQ1
= b1/ 2 0 + b0 1/ 2 = 1 Ans.
z z
i.e. the value of a definite integral remains unchanged if its variable is placed by any other symbol.
[P-4] f(x) dx = f(a x) dx .
0 0
Note :
[P-2]
b
f(x) dx
a
a
= – f(x) dx
b
This property can be used only when lower limit is zero. It is generally used for those complicated
i.e. the interchange of limits of a definite integral
changes only its sign.
zb zc zb
integrals whose denominators are unchanged when x is replaced by a– x. With the hel
Integration is a reverse process of differentiation. The integral or primitive of a function f(x) with
respect to x is that function (x) whose derivative with respect to x is the given function f(x). It is
i. 0. dx = c
ii. 1.dx = x + c
iii. k.dx = kx + c (k R)
xn1
expressed symbolically as -
zf (x) dx (x)
iv. xn dx =
n 1
+ c (n –1)
v. z1 dx = log
x + c
Thus x e
vi. ex dx = ex + c
ax
The process of finding the integral of a function is called Integration and the given function is
vii. ax dx =
loge
a + c = ax loga e + c
called Integrand. Now, it is obvious that the operation of integration is inverse operation of differentiation. Hence integral of a function is also named as anti-derivative of that function.
Further we observe that-
viii. sin x dx = – cos x + c
ix. cos x dx = sin x + c
x. tan x dx = log sec x + c = – log cos x + c
d (x2 )
dx
2 x
xi. cot x dx = log sin x + c
d (x2 2) 2xV| 2xdx x2 constant
xii. sec x dx = log(secx + tanx) + c
dx = – log (sec x –tan x) + c
d 2
dx (x k) 2x|
= log tan
FGH xIJ+ c
So we always add a constant to the integral of function, which is called the constant of
xiii. cosec x dx = – log (cosec x + cot x) + c
Integration. It is generally denoted by c. Due to presence of this constant such an integral is called an Indefinite integral.
= log (cosec x – cot x) + c = log tan
xiv. sec x tan x dx = sec x + c
FGHxIJK+ c
If f(x), g(x) are two functions of a variable x and k is a constant, then-
(i) k f(x) dx = k f(x) dx.
(ii) [f(x) g(x)] dx = f(x)dx ± g(x) dx
(iii) d/dx ( f(x) dx) = f(x)
(iv) f(x)KJdx = f(x)
The following integrals are directly obtained from the derivatives of standard functions.
xv. cosec x cot x dx = – cosec x + c
xvi. sec2 x dx = tan x + c
xvii. cosec2 x dx = – cot x + c xviii. sinh x dx = cosh x + c
xix. cosh x dx = sinh x + c
xx. sech2 x dx = tanh x + c
xxi. cosech2 x dx = – coth x + c
xxii. sech x tanh x dx = – sech x + c
xxiii. cosech x coth x = – cosech x + c
1 1
FxI
eax
R 1FbI
xxiv. xxiv.
x2 + a2 dx =
a tan–1
GHa + c
= a2 b2
sin
STbx tan
GHaJK+ c
xxv. z 1
1
dx = log
FGx a + c
xxxv. zeax cos bx dx
x2 a2
2 a Hx aK
eax
xxvi. z 1
dx = 1 log FGa xIJ + c
= a2 b2
(a cos bx + b sin bx) + c
a2 x2
1
2 a Ha xK
FxI
= cos
STbx tan
1 b V+ c
xxvii. za2 x2 dx = sin–1
GHaJK+ c
FxI
Examples Integration of Function
xxviii. xxviii.
= – cos–1
1
dx = sinh–1
x2 a2
GHaJK+ c
FGxIJ+ c
Ex.1 Evaluate : zx–55 dx
Sol. x–55 dx
x54
= log (x +
) + c
= 54
+ c Ans.
xxix. z 1
dx = cosh–1
FGxIJ+ c
Ex.2 Evaluate :
zex2 1j2
x2 a2
= log (x +
HaK
) + c
Sol.
x
x4 2 x2 1
dx
x
xxx. xxx.
2 2 dx
= zx3 2x 1IJdx
za x
H xK
x4
= x +
2
a . sin–1
2
x + c
a
INDEFINITE INTEGRATION
Preface 1
1. Integration of a Function 2
2. Basic Theorems on Integration 2
3. Standard Integrals 2
4. Methods of Integration 4
4.1 Integration by Substitution
4.2 Integration by Parts
4.3 Integration of Rational Function
4.4 Integration of Irrational Function
4.5 Integration of Trigonometric Function
5. Some Integrates of Different Expression of ex 12
6. Solved Examples 14
DEFINITE INTEGRATION
Preface 24
1. Defination 25
2. Properties Of Definite Integral 26
3. Some Important Formulae 29
4. Summation of Series by Integration 30
5. Solved Examples 32
QUADRATURE
Preface 41
1. Introduction 42
2. Curve Tracing 42
3. Area Bounded by A Curve 42
4. Symmetrical Area 43
5. Positive and Negative Area 44
6. Area Between Two Curves 45
7. Solved Examples 47
DIFFERENTIAL EQUATION
Preface 52
1. Introduction 53
2. Differential Equation 53
2.1 Order of Differential Equation
2.2 Degree of Differential Equation
3. Linear And Non- Linear Differential Equation 53
4. Formation of Differential Equation 53
5. Solution of Differential Equation 54
6. Methods of Solving A first Order & A first Degree Differential EqN 55
6.1 Differential Equation of the form dy/dx = f(x)
6.2 Differential Equation of the form dy/dx = f(x) g(y)
6.3 Differential Equation of the form dy/dx = f(ax+by+ c)
6.4 Differential Equation of homogeneous type
6.5 Differential Equation reducible to homogeneous form
6.6 Linear Differential Equation
6.7 Equation reducible to linear form
6.8 Differential Equation of the form of d2y / dx2 = f(x)
7. Solved Examples 62
All Rights Reserved with CAREER POINT Revised Edition
LEVEL # 1
Order and degree of differential equation
Q.7 The degree of the differential equation
d2y
Q.1 A differential equation of first order and first degree is-
dy 2
dx2 +
(A) 1
= 0 is-
(B) 2 (C) 3 (D) 6
(A) x
d2y
– x + a = 0
Q.8 The order of the differential equation whose solution is y = a cos x + b sin x + c e–x is-
(A) 3 (B) 2
(B) (B)
dx2 + xy = 0
(C) 1 (D) None of these
(C) dy + dx = 0
(D) None of these
Q.2 The order and degree of differential equation
Q.9 The differential equation of all circles of radius a is of order-
(A) 2 (B) 3
(C) 4 (D) None of these
dx + y
dy = 0 are respectively-
Q.10 The order of the differential equation of all
(A) 1,2 (B) 1,1
(C) 2,1 (D) 2,2
Q.3 The order and degree of the differential
circles of radius r, having centre on y-axis and passing through the origin is-
(A) 1 (B) 2 (C) 3 (D) 4
Q.11 The degree of the differential equation
dy d2 y
dy 2
d2 y
equation y = x dx +
is -
+ 3 = x2 log
2 is-
dx2
dx
dx
(A) 1,2 (B) 2,1
(C) 1,1 (D) 2, 2
Q.4 The order and degree of the differential
(A) 1 (B) 2
(C) 3 (D) None of these
Q.12 The differential equation
dy 2 2 / 3 2
equation
4
d y
= are-
d2 y 2
dy 4
dx
dx2
x +
+ y = x2 is of -
dx2
dx
(A) 2, 2 (B) 3, 3
(C) 2, 3 (D) 3, 2
Q.5 The order and the degree of differential
(A) Degree 2 and order 2
(B) Degree 1 and order 1
(C) Degree 4 and order 3
d4 y d3 y
d2y dy
(D) Degree 4 and order 4
equation
dx 4 – 4 dx3 + 8
dx2 – 8 dx
+ 4y = 0 are respectively-
(A) 4,1 (B) 1,4
(C) 1,1 (D) None of these
Q.13
Linear and non linear differential equation
Which of the following equation is linear-
Q.6 The order and degree of differential equation (xy2 + x) dx + (y – x2 y) dy = 0 are-
(A) 1, 2 (B) 2,1
(C) 1,1 (D) 2, 2
(A)
(C)
dy + xy2 = 1 (B) x2
dx
dy + 3y = xy2 (D) x
dx
dy + y = ex
dx
dy + y2 = sinx
dx
Q.14 Which of the following equation is non- linear-
dy
(A) dx = cos x
Q.18 The differential equation of the family of curves y2 = 4a (x + a) , where a is an arbitrary constant, is-
dy 2 dy
d2y
(A) y
1
dx
= 2x dx
(B)
dx2 + y = 0
(C) dx + dy = 0
dy 2 dy
dy 3
(B) y
1
dx
= 2x dx
(D) x
dx +
dy = y2
dx
Q.15 Which of the following equation is linear-
(C)
d2y dx2
dy
+ 2 dx = 0
d2 y 2
dy 2
dy 3 dy
(A) 2 + x2
= 0
(D) + 3
+ y = 0
dx
dx
dx dx
dy
(B) y = dx +
Q.19 The differential equation of all the lines in the xy- plane is-
dy y
dy d2y dy
(C) dx + x = log x
dy
(A) dx – x = 0 (B) dx2 – x dx = 0
d2y d2y
(D) y dx – 4 = x
(C) = 0 (D) + x = 0
dx2 dx2
Formation of differential equation
Q.20 The differential equation of the family
LEVEL # 1
Area bounded by a curve
Q.1 The area between the curves y = 6 – x – x2
and x-axis is -
(A) 125/6 (B) 125/2
(C) 25/6 (D) 25/2
Q.2 The area between the curve y =ex and x-axis which lies between x = – 1 and x = 1 is-
(A) e2 – 1 (B) (e2 –1)/e
(C) (1–e)/e (D) ( e– 1)/e2
Q.3 The area bounded by the curve y = sin 2x,
Q.9 The area bounded by the curve y = 1 + 8/x2, x-axis, x = 2 and x = 4 is-
(A) 2 (B) 3 (C) 4 (D) 5
Q.10 The area between the curve y = log x and x-axis which lies between x = a and x = b (a > 1, b > 1) is-
(A) b log (b/e) – a log (a/e)
(B) b log (b/e) + a log (a/e)
(C) log ab
(D) log (b/a)
Q.11 Area bounded by the curve y = xex2 , x- axis and the ordinates x = 0, x = is-
x- axis and the ordinate x = /4 is- (A) /4 (B) /2 (C) 1 (D) 1/2
(A)
e2 1
2
sq. units (B)
e2 1
2
sq.units
Q.4 The area between the curve xy = a2, x-axis, x = a and x = 2a is-
(A) a log 2 (B) a2 log 2
(C)
e 2
1 sq. units (D)
e 2
1sq.units
(C) 2a log 2 (D) None of these
Q.5 Area under the curve y = sin 2x + cos 2x
between x = 0 and x = 4 , is-
(A) 2 sq. units (B) 1 sq. units
(C) 3 sq. units (D) 4 sq. units
Q.6 The area bounded by the curve y = 4x2 ; x = 0, y = 1 and y = 4 in the first quadrant is-
Q.12 The area bounded between the curve y = 2x2 + 5, x-axis and ordinates x = – 2 and x = 1 is-
(A) 21 (B) 29/5 (C) 23 (D) 24
Q.13 Area bounded by curve xy = c, x-axis between x = 1 and x = 4, is-
(A) c log 3 sq. units
(B) 2 log c sq. units
(C) 2c log 2 sq. units
(D) 2c log 5 sq. units
2 1
(A) 2 3 (B) 3 3
Q.14 The area bounded by the curve y = x sin x2,
(C) 2 1
3
(D) 3 1
2
x-axis and x = 0 and x =
is-
Q.7 The area between the curve y = sec x and y-axis when 1 y 2 is-
(A) 1/2 (B) 1/
(C) 1/4 (D) /2
(A)
2 – log ( 2 + )
3
Q.15 The area bounded between the curve
x – y + 1 = 0, x = – 2, x = 3 and x-axis is-
2 4 2
(B) 3 + log ( 2 + )
(C) – 1 log (2 + )
(A) 45/4 (B) 45/2
(C) 15 (D) 25/2
Q.16 The area bounded by curves y = tan x,
3 2
(D) None of these
Q.8 The area bounded by the lines y = x, y = 0
x- axis and x =
3 is-
and x = 2 is-
(A) 2 log 2 (B) log 2
(A) 1 (B) 2
(C) log
FG2
J (D) 0
(C) 4 (D) None of these
H3 K
Q.17 The area between the curve x2 = 4ay, x-axis, and ordinate x = d is-
(A) d3 /12a (B) d3/a
(C) d3/2a (D) d3/6a
Q.18 Area bounded by the curve y = x (x – 1)2 and x-axis is-
(A) 4 (B) 1/3 (C) 1/12 (D) 1/2
Q.19 The area bounded by the curve y = loge x, x-axis and ordinate x = e is-
(A) loge2 (B) 1/2 unit
(C) 1 unit (D) e unit
1
Q.20 The area bounded by the curve y = cos 2 x ,
Q.28 The area of a loop bounded by the curve y = a sin x and x-axis is-
(A) a (B) 2a2 (C) 0 (D) 2a
Q.29 The area between the curves x = 2 – y – y2 and y-axis is-
(A) 9 (B) 9/2 (C) 9/4 (D) 3
Q.30 The area bounded by y = 4x – x2 and the x-axis is-
(A) 30/7 (B) 31/7 (C
LEVEL # 1
Q.1
Integration of function
1 sin2x dx equals-
dx
Q.7 4x2 9
1
dx is equal to -
3x 1
2x
(A) sin x + cos x + c
(B) sin x – cos x + c
(A) tan–1 + c (B) tan–1 + c
(C) cos x – sin x + c
(D) None of these
(C)
2x
tan–1 3 + c (D)
2x
tan–1 3 + c
2 2
4 5 sin x
Q.2 cos2 x
dx equals-
Q.8 cos2x sin4x dx is equal to -
(A) 4 tan x – sec x + c
(B) 4 tan x + 5 sec x + c
(A)
1 (cos 6x + 3 cos 2x ) + c
12
1
(C) 9 tan x + c
(D) None of these
(B) 6 (cos 6x + 3 cos 2x) + c
1
(C) – 12 (cos 6x + 3 cos 2x) + c
Q.3 (tan x + cot x) dx equals-
(A) log (c tan x)
(B) log (sin x + cosx) + c
(C) log (cx)
(D) None of these
(D) None of these
2dx
Q.9 x2 – 1 equals -
1 x 1
1 x – 1
(A) 2 log x – 1 + c (B)
log x 1 + c
Q.4
e5 loge x e4 loge x
e3 loge x e2 loge x dx equals-
x 1
2
x – 1
(C) log
x – 1
+ c (D) log
x 1
(A)
x + c (B)
2
4
x + c
3
Q.10
(ax bx )2 axbx
dx equals-
(C) x
4
+ c (D) None of these
(A) (a/b)x + 2x + c (B) (b/a)x + 2x + c
(C) (a/b)x – 2x + c (D) None of these
Q.5 1 cos 2x 1 cos 2x
dx equals-
Q.11
dx
sin2 xcos2 x
equals-
(A) tan x + x + c (B) tan x – x + c
(C) sin x – x + c (D) sin x + x + c
(A) tan x – cot x + c (B) tan x + cot x + c
(C) cot x – tan x + c (D) None of these
Q.6 The value of (1 x)
(A) x – x2 – x3 + c
(B) x + x2 – x3 + c
(C) x + x2 + x3 + c
(D) None of these
(1 + 3x) dx equal to -
sin x
Q.12 dx equals-
1 cos x
(A) 2 cos (x/2) + c
(B) 2 sin(x/2) + c (C)2 2 cos (x/2) + c
(D) –2 cos(x/2)+c
Q.13 sec x (tan x + sec x) dx equals-
Q.20
2x 3 x
5x
dx equals-
(A) tan x – sec x + c
(B) sec x – tan x+ c
(C) tan x + sec x + c
(A)
b2 / 5gx log 2 / 5 +
b3 / 5gx
log 3 / 5 + c
(D) None of these
Q.14 The value of sin x cos x
1 sin 2x
dx is-
(B) loge (2x/5) + loge (3x/5) + c
(C) x + c
(D) None of these
sin2 x
(A) sin x + c (B) x + c
Q.21 The value of
dx is-
1 cos x
(C) cos x + c (D) 1 (sin x + cos x)
2
1
(A) x – sin x + c
(B) x + sin x + c
(C) – x – sin x + c
Q.15 The value of
1 cos x dx is-
(D) None of these
(A) 1 2
cot (x/2) + c
1
(B) – 2 cot (x/2) + c
Q.22 e2x+3 dx equals-
(C) – cot (x/2) + c (D) – tan (x/2) + c
(A) 1 e2x+3 + c (B) 1 e2x+5 + c
cos 2x 2 sin2 x 2 2
Q.16
cos2 x
dx equals-
(C) 1 e2x+3 + c (D) 1 e2x+4 – c
(A) cot x + c (B) sec x + c
(C) tan x + c (D) cosec x + c
Q.23
3 2
1 tan x
1 tan x dx equals-
Q.17 cosx FG1
sin x IJ
dx equals-
(A) log (cos x + sin x) + c
Hsin2 x
cos3 xK
(B) log (cos x – sin x) + c
(A) sec x – cosec x + c
(B) cosec x – sec x + c
(C) sec x + cosec x + c
(D) None of these
(C) log (sin x – cos x) + c
(D) None of these
sin4 x cos4 x
Q.24
sin2 xcos2 x
dx equals-
Q.18
sin3 x cos3 x
In this chapter, we shall study the nature of a
function which is governed by the sign of its derivative. If the graph of a function is in upward going direction or in downward coming direction then it is called as monotonic function, and this property of the function is called Monotonicity. If a function is defined in any interval, and if in some part of the interval, graph moves upwards
A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain.
NOTE : If x < x f(x ) < f(x ) x , x D, then
and in the remaining part moves downward then
1 2 1
2 1 2
function is not monotonic in that interval.
f(x) is called strictly increasing in domain D.
These are of two types –
2.1 Monotonic Increasing :
A function f(x) defined in a domain D is said to be monotonic increasing function if the value of f(x) does not decrease (increase) by increasing (decreasing) the value of x or
We can say that the value of f(x) should increase (decrease) or remain equal by increasing
(Decreasing) the value of x.
Similarly if x1 < x2 f(x1) > f(x2), x1, x2 D then it is called strictly decreasing in domain D.
R
If Tor x1 x2 f(x1) f(x2 )
, x1 , x2 D
or SR x1 x2 f(x1) f(x2 ) , x1 , x2 D
2.2 Monotonic Decreasing :
A function f(x) defined in a domain D is said to be monotonic decreasing function if the value of f(x) does not increase (decrease) by increasing (decreasing) the value of x or
We can say that the value of f(x) should decrease (increase) or remain equal by increasing
(Decreasing) the value of x.
For Example
(i) f(x) = ex is a monotonic increasing function where as g(x) = 1/x is monotonic decreasing function.
(ii) f(x) = x2 and g(x) = | x | are monotonic increasing for x > 0 and monotonic decreasing for x < 0. In general they are not monotonic functions.
(iii) Sin x, cos x are not monotonic function whereas tan x, cot x are monotonic.
(i) At a Point : A function f(x) is said to be monotonic increasing (decreasing) at a point x = a of its domain if it is monotonic increasing
R
If Tor x1 x2 f(x1) f(x2 )
, x1 , x2 D
(decreasing) in the interval (a – h, a + h) where h is a small positive number. Hence we may
or SR x1 x2 f(x1) f(x2 ) , x1 , x2 D
observe that if f(x) is monotonic increasing at x = a, then at this point tangent to its graph will make an acute angle with the x–axis where as if the function is monotonic decreasing these tangent will make
an obtuse angle with x–axis. Consequently f' (a) will be positive or negative according as f(x) is monotonic increasing or decreasing at x = a.
Ex. Function f(x) = x2 + 1 is monotonically decreasing in [ –1, 0] because
f' (x) = 2x < 0, x (–1, 0)
Ex. Function f(x) = x2 is not a monotonic function in the interval [–1, 1] because
f' (x) > 0, when x = 1/2
Ex. Function f(x) = sin2x + cos2x is constant
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
CONTINUITY AND DIFFERENTIABILITY
Total No. of questions in Continuity and Differentiability are-
In Chapter Examples 14
Solved Examples 17
Total No. of questions 31
Ex.
The word 'Continuous' means without any break or gap. If the graph of a function has no break or gap or jump, then it is said to be continuous.
A function which is not continuous is called a
discontinuous function. In other words,
If there is slight (finite) change in the value of a function by slightly changing the value of x then function is continuous, otherwise discontinuous, while studying graphs of functions, we see that graphs of functions sin x, x, cos x, ex etc. are continuous but greatest integer function [x] has break at every integral point, so it is not continu- ous. Similarly tan x, cot x, secx, 1/x etc. are also discontinuous function.
(Discontinuous)
For examining continuity of a function at a point, we find its limit and value at that point, If these two exist and are equal, then function is continu- ous at that point.
A function f(x) is said to be continuous at a point x = a if
(i) f (a) exists
(ii) x Lim
f(x) exists and finite
Lim Lim
so x a f(x) = x a f(x)
(iii) x Lim
f(x) = f(a) .
or function f(x) is continuous at x = a.
Lim Lim
(Continuous function)
If x a f(x) = x a f(x) = f(a).
i.e. If right hand limit at 'a' = left hand limit at 'a'= value of the function at 'a'.
If x Lim
f(x) does not exist or
Lim
x a
f(x) f(a),
then f(x) is said to be discontinuous at x= a.
Ex.1 Examine the continuity of the function
X R|x2 9
f (x) =
S| x 3 , when x 3
at x = 3.
T6, when x 3
Sol. f (3) = 6 ( given)
bx 3gbx 3
Lim
x 3
f(x) =
Lim
x 3
bx 3g = 6
(Discontinuous at x = 0)
Lim
x 3
f(x) = f(3)
f (x) is continuous at x = 3.
Ex.2 If f(x) =
log (1 2ax) log 1 bx
x 0
x
Thus a function f(x) is continuous at a point x = a if it is left continuous as well as right continuous at x = a.
Tk ,
x 0
If function is continuous at x = 0 then the value of k is –
(A) a + b (B) 2a +b
Ex.4 Examine the continuity of the function
(C) a – b (D) 0
x2 1, when x 2
f(x) =
|T2x, when x 2
Sol.
Lim
x 0
logFGH1 2axIJ
x
at the point x = 2.
Sol. f(2) = 22 + 1 = 5
= x Lim
FG1 bx J.
b1 bxgb2ag b1 2axgbb
2
f(2– 0) =
Lim
h 0
Lim
b2 hg2 1 = 5
0 H1 2ax
2a b
0 b1 bxgb1 2 axg
b1 bxg
b2a b
b1gb1g
f (2+ 0) = h 0 2( 2+ h) = 4
f(2– 0) f(2+ 0) f(2)
x Lim
= = 2a + b
Ans.[B]
f(x) is not continuous at x = 2.
Ex.5 Check the continuity of the function
1 cos 4x, x 0
R|x 2
x 3
Ex.3 If f(x) = S| x2
is continuous then
f(x) = 5 x 3
at x = 3.
Ta,
x 0
T8 x x 3
the value of a is equal to –
(A) 0 (B) 1
(C) 4 (D) 8
Sol. Since the given function is continuous at x= 0
Sol.
02Application of Derivative # 1 (Tangent & Normal)~1 Module-4.pdfRajuSingh806014
If y = f(x) be a given function, then the differential coefficient f' (x) or dy at the point P (x , y ) is
(i) If the tangent at P (x1,y1) of the curve y = f(x) is parallel to the x- axis (or perpendicular to y- axis) then = 0 i.e. its slope will be
zero.
FGdy J
dx 1 1
m = H K = 0
the trigonometrical tangent of the angle (say) which the positive direction of the tangent to the curve at P makes with the positive direction of x- axis Gdy J, therefore represents the slope of the
tangent. Thus
dx (x1,y1)
The converse is also true. Hence the tangent at (x1,y1) is parallel to x- axis.
GJ = 0
(x1,y1)
(ii) If the tangent at P (x , y ) of the curve y =
1 1
f (x) is parallel to y - axis (or perpendicular
to x-axis) then = / 2 , and its slope will be infinity i.e.
dy
m = =
dx (x ,y )
The converse is also true. Hence the tangent at (x1, y1) is parallel to y- axis
Fdy
(x1,y1)
Thus
(i) The inclination of tangent with x- axis.
dy
(iii) If at any point P (x1, y1) of the curve y = f(x), the tangent makes equal angles with the
axes, then at the point P, = / 4 or 3 / 4 ,
= tan–1
GHdxJK
Hence at P, tan = dy/dx = 1. The
(ii) Slope of tangent = dy
dx
(iii) Slope of the normal = – dx/dy
Ex.1 Find the following for the curve y2 = 4x at point (2,–2)
(i) Inclination of the tangent
(ii) Slope of the tangent
(iii) Slope of the normal
Sol. Differentiating the given equation of curve, we get dy/dx = 2/y = –1 at (2,–2)
so at the given point.
(i) Inclination of the tangent = tan–1(–1) = 135º
(ii) Slope of the tangent = –1
(iii) Slope of the normal = 1
converse of the result is also true. thus at
(x1,y1) the tangent line makes equal angles with the axes.
GJ = 1
(x1,y1)
Ex.2 The equation of tangent to the curve y2 = 6x at (2, – 3).
(A) x + y – 1 = 0 (B) x + y + 1 = 0 (C) x – y + 1 = 0 (D) x + y + 2 = 0
Sol. Differentiating equation of the curve with respect to x
(a) Equation of tangent to the curve y = f(x) at A (x1,y1) is
2y dy = 6
dx
FGdy J
dx (2,3)
= 3 = –1
3
y – y1 =
FGdy J
(x1,y1)
(x–x1)
Therefore equation of tangent is y + 3 = – (x – 2)
x + y + 1 = 0 Ans. [B]
Ex.3 The equation of tangent at any of the curve x = at2, y = 2at is -
(A) x = ty + at2 (B) ty + x + at2 = 0
(C) ty = x + at2 (D) ty = x + at3
2 a 1
Sol. dy/dx = (dy/dt)/(dx/dt) = 2 at = t
equation of the tangent at (x,y) point is
(y – 2 at) = 1 (x – at2)
t
ty = x + at2 Ans.[C]
Ex.4 The equation of the tangent to the curve x2 (x – y) + a2 (x + y) = 0 at origin is-
(A) x + y + 1 = 0 (B) x + y + 2 = 0 (C) x + y = 0 (D) 2x – y = 0
Sol. Differentiating equation of the curve w.r.t. x
dy/dx = – y x
(i) If tangent line is parallel to x - axis, then dy/dx = 0 y = 0 and x = a
Thus the point is (a,0)
(ii) If tangent is parallel to y – axis , then dy/dx = x = 0 and y = a
Thus the point is (0,a)
(iii) If tangent line makes equal angles with both axis , then dy/dx = 1
y =
(a) Natural Numbers : N = {1,2,3,4,...}
(b) Whole Numbers : W = {0,1,2,3,4, }
(c) Integer Numbers :
or Z = {...–3,–2,–1, 0,1,2,3, },
Z+ = {1,2,3,....}, Z– = {–1,–2,–3, }
Z0 = {± 1, ± 2, ± 3, }
(d) Rational Numbers :
p
Q = { q ; p, q z, q 0 }
(i) R0 : all real numbers except 0 (Zero).
(j) Imaginary Numbers : C = {i,, }
(k) Prime Numbers :
These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers.
Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,...
(l) Even Numbers : E = {0,2,4,6, }
(m) Odd Numbers : O = {1,3,5,7, }
Ex. {1,
Note :
5
, –10, 105,
3
22 20
7 , 3
, 0 ....}
The set of the numbers between any two real numbers is called interval.
(a) Close Interval :
(i) In rational numbers the digits are repeated after decimal.
(ii) 0 (zero) is a rational number.
(e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers
Ex. { , ,21/3, 51/4, ,e, }
Note:
(i) In irrational numbers, digits are not repeated after decimal.
(ii) and e are called special irrational quantities.
(iii) is neither a rational number nor a irrational number.
(f) Real Numbers : {x, where x is rational and irrational number}
20
[a,b] = { x, a x b }
(b) Open Interval:
(a, b) or ]a, b[ = { x, a < x < b }
(c) Semi open or semi close interval:
[a,b[ or [a,b) = {x; a x < b}
]a,b] or (a,b] = {x ; a < x b}
Let A and B be two given sets and if each element a A is associated with a unique element b B under a rule f , then this relation is called function.
Here b, is called the image of a and a is called the pre- image of b under f.
Note :
(i) Every element of A should be associated with
Ex. R = { 1,1000, 20/6, ,
, –10, –
,.....}
3
B but vice-versa is not essential.
(g) Positive Real Numbers: R+ = (0,)
(h) Negative Real Numbers : R– = (– ,0)
(ii) Every element of A should be associated with a unique (one and only one) element of but
any element of B can have two or more rela- tions in A.
3.1 Representation of Function :
It can be done by three methods :
(a) By Mapping
(b) By Algebraic Method
(c) In the form of Ordered pairs
(A) Mapping :
It shows the graphical aspect of the relation of the elements of A with the elements of B .
Ex. f1:
f2 :
f3 :
f4 :
In the above given mappings rule f1 and f2
shows a function because each element of A is
associated with a unique element of B. Whereas
f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element
of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B.
(B) Algebraic Method :
It shows the relation between the elem
01. Differentiation-Theory & solved example Module-3.pdfRajuSingh806014
Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
Level # 1 ........................................ 48
Level # 2 ........................................ 16
Level # 3 ........................................ 12
Level # 4 ........................................ 16
LEVEL # 1
Q.1 When x < 0, function f(x) = x2 is
(A) Decreasing
(B) Increasing
(C) Constant
(D) Not monotonic
Q.2 When x > 1, function f(x) = x3 is
(A) Increasing (B) Decreasing
(C) Constant (D) not monotonic
Q.3 In the interval (0, 1), f(x) = x2 – x + 1 is
(A) Monotonic (B) Not monotonic
(C) Decreasing (D) Increasing
Q.4 f(x) = x + 1/x, x 0 is increasing when
(A) | x | < 1 (B) | x | > 1
(C) | x | < 2 (D) | x | > 2
|x|
Q.11 For which value of x, the function f(x) = x2 –2x is decreasing
(A) x > 1 (B) x > 2
(C) x < 1 (D) x < 2
x 2
Q.12 Function f(x) = x 1 , x –1 is
(A) Increasing
(B) Decreasing
(C) Not monotonic
(D) None of these
Q.13 Function f(x) = x3 is
(A) Increasing in (0, ) and decreasing in (–, 0)
(B) Decreasing in (0, ) and increasing in (–, 0)
(C) Decreasing throughout
(D) Increasing throughout
Q.5 The function f(x) =
(A) Decreasing
(B) Increasing
x (x 0), x > 0 is
Q.14 Function f(x) = x | x | is
(A) Monotonic increasing
(B) Monotonic decreasing
(C) Constant function
(D) None of these
Q.6 When x (0, 1), function f(x) = 1 / is
(A) Increasing
(B) Decreasing
(C) Neither increasing nor decreasing
(D) Constant
Q.7 Function f(x) = 3x4 + 7x2 + 3 is
(A) Monotonically increasing
(B) Monotonically decreasing
(C) Not monotonic
(D) Odd function
Q.8 For what values of x, the function
(C) Not monotonic
(D) None of these
Q.15 If f and g are two decreasing functions such that fog is defined then fog is
(A) Decreasing (B) Increasing
(C) Can't say (D) None of these
Q.16 For the function f(x) = | x |, x > 0 is
(A) Decreasing
(B) Increasing
(C) Constant function
(D) None of these
Q.17 In the following , monotonic increasing
4
f(x) = x + x2
is monotonically decreasing
fucntion is
(A) x + | x | (B) x – | x |
(A) x < 0 (B) x > 2
(C) x < 2 (D) 0 < x < 2
(C) | x | (D) x | x |
Q.9 If f(x) = x 2
for –7 x 7, then f(x) is
x 1
Q.18 At x = 0, f(x) = is
2 x x 2
increasing function of x in the interval (A) [7, 0] (B) (2, 7]
(C) [–2, 2] (D) [0, 7]
(A) Increasing (B) Decreasing
(C) Not monotonic (D) Constant
x
Q.10 The function y = 1 x2
interval
decreases in the
Q.19 If f(x) = 2x3 – 9x2 + 12x – 6, then in which interval f(x) is monotonically increasing
(A) (1, 2) (B) (–, 1)
(A) (–, –) (B) (–1, –1)
(C) (0, ) (D) (–, –1)
(C) (2, ) (D) (–, 1) or (2, )
Q.20 For the function f(x) = x3 – 6x2 – 36x + 7 which of the following statement is false
(A) f(x) is decareasing, if –2 < x < 6
(B) f(x) is increasing, if –3 < x < 5
(C) f(x) is increasing, if x < –2
(D) f(x) is increasing, if x > 6
Q.28 Which of the following function is not monotonic
(A) ex – e–x (B) ex + e–x
(C) e–1/x (D) None of these
Q.29 In the following, decreasing function
Level # 1 ........................................ 92
Level # 2 ........................................ 27
Level # 3 ........................................ 30
Level # 4 ........................................ 26
LEVEL # 1
R|sin1 ax
continuity of a function at a point
Q.7 If f(x) =
S x
Tk,
, x 0
x 0
is continuous at
Q.1 Function f(x) =
R1 x,
T
when x 2
x = 2 is
x = 0, then k is equal to-
(A) 0 (B) 1
5 x, when x 2
continuous at x = 2, if f(2) equals-
(A) 0 (B) 1
(C) 2 (D) 3
Rx cos1/ x, x 0
(C) a (D) None of these
Q.8 What is the value of (cos x)1/x at x = 0 so that it becomes continuous at x = 0-
(A) 0 (B) 1
(C) –1 (D) e
Q.2 If f(x) = k,
x 0
is continuous at
R|k
cos x
x = 0, then
(A) k > 0 (B) k< 0
Q.9 If f(x) =
S| 2x
, x / 2 x / 2
is a continuous
(C) k = 0 (D) k 0
function at x = / 2 , then the value of k is-
Q.3 If function f(x) = S|x2 2, x 1
is continuous
(A) –1 (B) 1
(C) –2 (D) 2
x3 a3
at x = 1, then value of f(x) for x< 1 is-
Q.10 If function f(x) =
x a
, is continuous at
(A) 3 (B) 1–2x
(C) 1–4x (D) None of these
Q.4 Which of the following function is continuous at x= 0-
x= a, then the value of f(a) is -
(A) 2a (B) 2a2
(C) 3a (D) 3a2
Rsin 1/ x, x 0
(A) f(x) =
RSsin 2x / x, x 0
Q.11 If f(x) = k,
x 0
is continuous at
(B) f(x) =
T1,
R|1 x1/ x,
T1,
x 0
x 0
x 0
x = 0, then k is equal to-
(A) 8 (B) 1
(C) –1 (D) None of these
Q.12 Function f(x) =
G1
xIJ1/ x
is continuous at
(C) f(x) =
R| 1/ x
|T1,
x 0
x 0
H aK
x= 0 if f(0) equals-
(D) None of these
R|6 5 x,x 0
(A) ea (B) e–a
(C) 0 (D) e1/a
1 cos 7bx
Q.5 If f(x) =
S|T2a x, x 0
is continuous at
Q.13 If f(x) =
x
, (x )
is continu-
x = 0, then the value of a is -
(A) 1 (B) 2
(C) 3 (D) None of these
ous at x= , then f( ) equals-
(A) 0 (B) 1
(C) –1 (D) 7
x2 ba 2gx a
R tan x
Q.6 If f(x) = S|
x 2
, x 2
is continu-
Q.14 If f(x) =
S|sin
x
x 0
, then f(x) is -
T2,
x 2
(A) Continuous everywhere
ous at x = 2, then a is equal to-
(A) 0 (B) 1
(C) –1 (D) 2
(B) Continuous nowhere
(C) Continuous at x= 0
(D) Continuous only at x = 0
Q.15 If f(x) =
2x tan x x
is continuous at x = 0,
Q.22 Function f(x) = [x] is discontinuous at-
(A) Every real number
then f(0) equals-
(A) 0 (B) 1
(C) 2 (D) 3
1 x 3 1 x
(B) Every natural number
(C) Every integer
(D) No where
Q.23 Function f(x) = 3x2–x is-
(A) Discontinuous at x = 1
(B) Discontinuous at x = 0
Q.16 If f(x) =
x , (x 0) is continuous
(C) Continuous only at x = 0
at x= 0, then the value of f(0) is- (A) 1/6 (B) 1/4
(C) 2 (D) 1/3
(D) Continuous at x = 0
R|x2, when x 0
Q.24 If f(x) = S|1, when 0 x 1 , then f(x) is-
Q.17 If f (x) =
Rax2 b when 0 x 1 S2 when x 1 is x 1 when 1 x 2
1 / x, when x 1
(A
LEVEL # 1
Questions
based on
inequation
Q.8 If x2 – 1 0 and x2 – x – 2 0, then x line in the interval/set
(A) (–1, 2) (B) (–1, 1)
Q.1 The inequality
2 < 3 is true, when x belongs to-
x
(C) (1, 2) (D) {– 1}
2
2
Questions Definition of function
(A) 3 ,
(B) 3
based on
2 ,
Q.9 Which of the following relation is a function ?
(C)
x 4
(–, 0) (D) none of these
(A) {(1,4), (2,6), (1,5), (3,9)}
(B) {(3,3), (2,1), (1,2), (2,3)}
(C) {(1,2), (2,2,), (3,2), (4,2)}
(D) {(3,1), (3,2), (3,3), (3,4)}
Q.2
x 3 < 2 is satisfied when x satisfies-
(A) (–, 3) (10, ) (B) (3, 10)
(C) (–, 3) [10, ) (D) none of these
Q.10 If x, y R, then which of the following rules is not a function-
(A) y = 9 –x2 (B) y = 2x2
x 7
(C) y = – |x| (D) y = x2 + 1
Q.3 Solution of x 3 > 2 is-
Questions Even and odd function
(A) (–3, ) (B) (–, –13)
(C) (–13, –3) (D) none of these
2x 3
based on
Q.11 Which one of the following is not an odd function -
Q.4 Solution of
3x 5
3 is-
(A) sin x (B) tan x
(C) tanh x (D) None of these
12 5 12
(A) 1, 7
(B) , 4 4
3 7
Q.12 The function f(x) = sin x cos x
is -
, 5
12 ,
x tanx
(C)
3
(D) 7
(A) odd
(B) Even
Q.5 Solution of (x – 1)2 (x + 4) < 0 is-
(A) (–, 1) (B) (–, –4)
(C) (–1, 4) (D) (1, 4)
Q.6 Solution of (2x + 1) (x – 3) (x + 7) < 0 is-
(C) neither even nor odd
(D) odd and periodic
Q.13 A function is called even function if its graph is symmetrical w.r.t.-
(A) origin (B) x = 0
(C) y = 0 (D) line y = x
1 ,3
1 ,3
(A) (– , –7)
2
(B) (– , – 7)
Q.14 A function is called odd function if its graph is symmetrical w.r.t.-
(C) (–, 7) 1 ,3
2
(D) (–, –7) (3, )
(A) Origin (B) x = 0
(C) y = 0 (D) line y = x
Q.15 The even function is-
Q.7 If x2 + 6x – 27 > 0 and x2 – 3x – 4 < 0, then-
(A) x > 3 (B) x < 4
(A) f(x) = x2 (x2 +1) (B) f(x) = sin3 x + 2
(C) f(x) = x (x +1) (D) f(x) = tan x + c
(C) 3 < x < 4 (D) x = 7 2
Q.16 A function whose graph is symmetrical about the y-axis is given by-
Q.25 In the following which function is not periodic-
(A) f(x) = loge
(x + )
(A) tan 4x (B) cos 2x
(C) cos x2 (D) cos2x
(B) f(x + y) = f(x) + f(y) for all x, y R
(C) f(x) = cos x + sin x
(D) None of these
Q.17 Which of the following is an even function ?
ax 1
1
Q.26 Domain of the function f(x) = x 2
is-
(A) x
ax 1
(B) tan x
(A) R (B) (–2, )
(C) [2, ] (D) [0, ]
(C) (C)
ax ax (D)
2
ax 1
ax 1
Q.27 The domain where function f(x) = 2x2 – 1 and g(x) = 1 – 3x are equal, is-
Q.18 In the following, odd function is -
(A) cos x2 (B) (ex + 1)/(ex – 1)
(C) x2 – |x| (D) None of these
Q.19 The function f(x) = x2 – |x| is -
(A) an odd function
(B) a rational function
(C) an even func
x2 y2
Standard Equation of hyperbola is a 2 – b2 = 1
(i) Definition hyperbola : A Hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.
The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called conjugate hyperbola.
Note :
(i) If e1 and e2 are the eccentricities of the
(ii) Vertices : The point A and A where the curve meets the line joining the foci S and S
hyperbola and its conjugate then
1 +
e 2 e
1 = 1
2
are called vertices of hyperbola.
(iii) Transverse and Conjugate axes : The straight line joining the vertices A and A is called transverse axes of the hyperbola. Straight line perpendicular to the transverse axes and passes through its centre called conjugate axes.
(iv) Latus Rectum : The chord of the hyperbola which passes through the focus and is perpendicular to its transverse axes is called
2b2
latus rectum. Length of latus rectum = a .
(ii) The focus of hyperbola and its1 conju2gate are concyclic.
Standard Equation and Difinitions
Ex.1 Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1,2) and
eccentricity 3 .
Sol. Let P (x,y) be any point on the hyperbola. Draw PM perpendicular from P on the directrix.
Then by definition SP = e PM
(v) Eccentricity : For the hyperbola
x2 y2
a 2 – b2
= 1,
(SP)2 = e2(PM)2
2x y 12
b2 = a2 (e2 – 1)
(x–1)2 + (y–2)2 = 3
Conjugate axes 2
5(x2 + y2 – 2x – 4y + 5} =
e = =
1
Transverse
axes
3(4x2 + y2 + 1+ 4xy – 2y – 4x)
7x2 – 2y2 + 12xy – 2x + 14y – 22 = 0
(vi) Focal distance : The distance of any point on the hyperbola from the focus is called the focal distance of the point.
Note : The difference of the focal distance of a point on the hyperbola is constant and is equal to the length
of the transverse axes. |SP – SP| = 2a (const.)
which is the required hyperbola.
Ex.2 Find the lengths of transverse axis and conjugate axis, eccentricity and the co- ordinates of foci and vertices; lengths of the latus rectum, equations of the directrices of the hyperbola 16x2 – 9y2 = –144
Sol. The equation 16x2 – 9y2 = – 144 can be
Sol. y= m1(x –a),y= m2(x + a) where m1m2 = k, given
x 2
written as 9
x2
y 2
– 16 = – 1. This is of the form
y2
In order to find the locus of their point of intersection we have to eliminate the unknown
m1 and m2. Multiplying, we get
y2 = m1m2 (x2 – a2) or y2 = k(x2–a2)
a 2 – b2 = – 1
a2 = 9, b2 = 16 a = 3, b = 4
or x – y
1 k
= a2
which represents a hyperbola.
Length of transverse axis :
The length of transverse axis = 2b = 8
Length of conjugate axis :
The length of conjugate axis = 2a = 6
5
Ex.5 T
An ellipse is the locus of a point which moves in such a way that its distance form a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus and fixed line is called the directrix and the constant ratio is called the eccentricity of a ellipse denoted by (e).
In other word, we can say an ellipse is the locus of a point which moves in a plane so that the sum of it distances from fixed points is constant.
2.1 Standard Form of the equation of ellipse
Let the distance between two fixed points S and S' be 2ae and let C be the mid point of SS.
Taking CS as x- axis, C as origin.
Let P(h,k) be the moving point Let SP+ SP = 2a (fixed distance) then
(ii) Major & Minor axis : The straight line AA is called major axis and BB is called minor axis. The major and minor axis taken together are called the principal axes and its length will be given by
Length of major axis 2a Length of minor axis 2b
(iii) Centre : The point which bisect each chord of an ellipse is called centre (0,0) denoted by 'C'.
(iv) Directrix : ZM and Z M are two directrix and their equation are x= a/e and x = – a/e.
(v) Focus : S (ae, 0) and S (–ae,0) are two foci of an ellipse.
(vi) Latus Rectum : Such chord which passes through either focus and perpendicular to the major axis is called its latus rectum.
Length of Latus Rectum :
If L is (ae, 𝑙 ) then 2𝑙 is the length of
SP+S'P=
{(h ae)2 k 2} +
= 2a
Latus Rectum.
Length of Latus rectum is given by
2b2
.
h2(1– e2) + k2 = a2(1– e2)
Hence Locus of P(h, k) is given by. x2(1– e2) + y2 = a2(1– e2)
2
a
(vii) Relation between constant a, b, and e
a 2 b2
b2 = a2(1– e2) e2 =
a 2
x2
a 2 +
y
a 2 (1 e 2 ) = 1
e =
a 2
Result :
Major Axis
(a) Centre C is the point of intersection of the axes of an ellipse. Also C is the mid point of AA.
(b) Another form of standard equation of ellipse
x 2 y2
a 2 b2
1 when a < b.
Directrix Minor Axis Directrix x = -a/e x = a/e
Let us assume that a2(1– e2 )= b2
The standard equation will be given by
x2 y2
a2 b2
2.1.1 Various parameter related with standard ellipse :
In this case major axis is BB= 2b which is along y- axis and minor axis is AA= 2a along x- axis. Focus S(0,be) and S(0,–be) and directrix are y = b/e and y = –b/e.
2.2 General equation of the ellipse
The general equation of an ellipse whose focus is (h,k) and the directrix is the line ax + by + c = 0 and the eccentricity will be e. Then let P(x1,y1) be any point on the ellipse which moves such that SP = ePM
Let the equation of the ellipse x
y2
a > b
(x –h)2 + (y –k)2 =
e 2 (ax1 by1 c) 2
a 2 b2
1 1 a 2 b2
(i) Vertices of an ellipse : The point of which ellipse cut the axis x-axis at (a,0) & (– a, 0) and y- axis at (0, b) & (0, – b) is called the vertices of an ellipse.
Hence the locus of (x1,y1) will be given by (a2 + b
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Instructions for Submissions thorugh G- Classroom.pptx
content Theory.pdf
1. CONTENTS
POINTS
TOPIC PAGE NO.
Preface .................................................................................................................................................... 1
1. System of Co - ordinates ................................................................................................................... 2
1.1 Cartesian Co-ordinates
1.2 Polar Co-ordinates
2. Distance Formula .................................................................................................................................. 2
3. Applications of Distance Formula ....................................................................................................... 3
3.1 Position of three points
3.2 Position of four points
4. Section Formula..................................................................................................................................... 4
5. Co-ordinate of some particular points ....................................................................................... 4
5.1 Centroid
5.2 Incentre
5.3 Circumcentre
5.4 Ortho Centre
6. Area of triangle and quadrilateral ............................................................................................. 6
6.1 Area of Triangle
6.2 Area of Quadrilateral
7. Transformation of axes ......................................................................................................................... 7
7.1 Parallel Transformations
7.2 Rotational Transformation
7.3 Reflection of a Point
8. Locus ........................................................................................................................................................ 8
9. Some important points.......................................................................................................................... 9
10. Solved examples ................................................................................................................................... 10
STRAIGHT LINE
TOPIC PAGE NO.
Preface .................................................................................................................................................... 13
1. Equation of Straight line ...........................................................................................................14
2. Equation of Straight line parallel to axes .................................................................................... 14
3. Slope of a line .............................................................................................................................14
2. CIRCLE
TOPIC PAGE NO.
Preface .................................................................................................................................................... 29
1. Definition ................................................................................................................................................. 30
2. Standard Form of equation of a circle .......................................................................................... 31
2.1 General Form
2.2 Central Form
2.3 Diametral Form
2.4 Parametric Form
3. Equation of a circle in some special cases ....................................................................................... 31
4. Different forms of the equation of straight line.......................................................................... 14
4.1 Slope – Intercept Form
4.2 Slope Point Form
4.3 Two Point Form
4.4 Intercept Form
4.5 Normal Form
4.6 Parametric Form
5. Reduction of General form of Equations into Standard Forms ................................................. 15
6. Position of a point relative to a line........................................................................................... 17
7. Angle between two straight lines ....................................................................................................... 17
7.1 Parallel lines
7.2 Perpendicular lines
7.3 Coincident lines
8. Equation of parallel & perpendicular lines ....................................................................................... 18
9. Equation of Straight lines through (x1
, y1
) making an angle with y = mx + c .......................... 18
10. Length of perpendicular ............................................................................................................. 19
10.1 Distance between two Parallel Lines
11. Condition of concurrency ........................................................................................................... 19
12. Bisectors of angles between two lines ...................................................................................... 20
13. Line passing through the point of intersection of two lines ...................................................... 20
14. Homogeneous equation........................................................................................................................ 20
15. General Equation of Second degree.................................................................................................. 21
16. Equation of lines joining the intersection points of a line & a curve to the origin ................... 21
17. Some important points ............................................................................................................... 22
18. Solved examples ................................................................................................................................... 23
3. 4. Position of a point with respect to a circle ........................................................................................ 32
4.1 The least and the greatest distance of a point from a circle
5. Line and circle ............................................................................................................................ 33
5.1 Condition of Tangency
5.2 Intercepts made on coordinate axes by the circle
6. Equation of tangent and normal .............................................................................................. 34
6.1 Equation of Tangent
6.2 Equation of Normal
6.3 Length of Tangent
6.4 Pair of Tangents
7. Chord of contact ..................................................................................................................................... 35
8. Director circle ............................................................................................................................. 35
9. Position of two circles ................................................................................................................ 36
10. Equation of a chord whose middle point is given ..................................................................... 38
11. Circle through the point of intersection ..................................................................................... 38
12. Common chord of two circles .................................................................................................... 39
13. Angle of intersection of two circles............................................................................................ 39
14.1 Condition of Orthogonality
14. Some important points ............................................................................................................... 40
15. Pole and Polar ............................................................................................................................ 40
9.1 Equation of Polar
9.2 Co-ordinates of Pole
9.3 Conjugate Points and Conjugate Lines
16. Radical axis and radical centre ................................................................................................. 41
16.1 Radical Axis
16.2 Radical Centre
17. Solved examples ................................................................................................................................... 42
PARABOLA
TOPIC PAGE NO.
Preface........................................................................................................................................ 50
1. Definition................................................................................................................................ 51
2. Terms related to parabola.......................................................................................................51
3. Standard form of equation of parabola..................................................................................51
3.1 Parameters of the Parabola y2
= 4ax
3.2 Other Standard Parabola
4. Reduction to Standard Equation..............................................................................................53
.
4. ELLIPSE
TOPIC PAGE NO.
Preface.............................................................................................................................. 68
1. Definition .......................................................................................................................... 69
2. Equation of Ellipse ............................................................................................................ 69
2.1 Standard Form
Various parameters related to ellipse
2.2 General equation of an ellipse
(Condition) for general eq. of the second degree in xy Real ellipse
3. Parametric equation of ellipse........................................................................................... 71
5. General equation of a Parabola.............................................................................................54
6. Equation of Parabola when its vertex and focus are given....................................................54
7. Parametric equation of a Parabola........................................................................................55
8. Chord.................................................................................................................................... 55
8.1 Equation of Chord
8.2 Length of Intercept
9. Position of a point and a line with respect to a Parabola.....................................................56
9.1 Position of a Point with respect to a Parabola
9.2 Line and Parabola
10. Tangent to the Parabola ........................................................................................................56
10.1 Condition of Tangency
10.2 Equation of Tangent
11. Normal to the Parabola...........................................................................................................57
11.1 Equation of Normal
11.2 Properties of Normal
12. Pair of Tangents.....................................................................................................................59
12.1 Locus of point of intersection of tangents
13. Chord of Contact ...................................................................................................................59
14. Equation of the chord with given midpoint ...........................................................................60
.
15. Diameter of the parabola ......................................................................................................60
16. Geometrical properties of the parabola ................................................................................60
17. Solved Examples...................................................................................................................63
5. 4. Point and Ellipse ............................................................................................................... 71
5. Ellipse and Line................................................................................................................. 71
5.1 Equation of tangent
5.2 Equation of the normal
6. Equation of the pair of Tangents ....................................................................................... 73
7. Equation of the chord of contact ....................................................................................... 73
8. Equation of the Chord Joining mid- Point (X1
, Y1
)............................................................... 73
9. Diameter.............................................................................................................................73
10. Solved Examples ...............................................................................................................74
HYPERBOLA
TOPIC PAGE NO.
Preface............................................................................................................................. 79
1. Standard Equation & Definition .............................................................................................80
2. Conjugate Hyperbola ............................................................................................................80
3. Parametric equations of the Hyperbola.................................................................................. 81
4. Position of a Point P (X1
, Y1
) with respect to Hyperbola ........................................................82
5. Line and Hyperbola ..............................................................................................................82
6. Equation of tangent ..............................................................................................................82
7. Equation of Normal ...............................................................................................................83
8. Equation of Pair of Tangents ..................................................................................................83
9. Chord of contact......................................................................................................................84
10. Equation of a chord whose middle point is given .................................................................84
11. Director Circle ........................................................................................................................84
12. Solved Examples ...................................................................................................................85
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