TS inter 1st year: These blueprints were designed by ‘Basics in Maths’ team. These to-do’s help the TS intermediate first-year Maths students fall in love with mathematics and overcome their fear.
These blueprints cover all the topics of the TS I.P.E first-year maths syllabus and help in I.P.E exams.
Ts Inter Maths 1B Concept: designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate first-year Maths students fall in love with mathematics and overcome the fear.
These notes cover all the topics covered in the TS I.P.E first year maths 1B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.
Locus: The set of points that are satisfying a given condition or property is called the locus of the point.
Ex:- If a point P is equidistant from the points A and B, then AP =BP
Ex 2: – set of points that are at a constant distance from a fixed point. 
here the locus of a point is a circle.
• In a right-angled triangle PAB, the right angle at P and P is the locus of the point, then
AB2 = PA2 + PB2
•Area of the triangle formed by the vertices A (x1, y1), B (x2, y2), and C (x3, y3) is
Transformation of axes:
When the origin is shifted to (h, k), without changing the direction of axes then
•To remove the first degree terms of the equation ax2 + 2hxy + by2 +2gx +2fy+ c = 0, origin should be shifted to the point 
•If the equation ax2 + by2 +2gx +2fy+ c = 0, origin should be shifted to the point 
Rotation of axes:
When the axes are rotated through an angle θ then
•To remove the xy term of the equation ax2 + 2hxy + by2 = 0, axes should be rotated through an angle θ is given by
Slope:- A-line makes an angle θ with the positive direction of the X-axis, then tan θ is called the slope of the line.
It is denoted by “m”. 
m= tan θ
• The slope of the x-axis is zero.
• Slope of any line parallel to the x-axis is zero.
• The y-axis slope is undefined.
• The slope of any line parallel to the y-axis is also undefined.
• The slope of the line joining the points A (x1, y1) and B (x2, y2) is
Slope of the line ax + by + c = 0 is 
Slope- intercept form
The equation of the line with slope m and y-intercept c is y = mx + c.
Slope point form:
The equation of the line passing through the point (x1, y1) with slope m is
y – y1 = m (x – x1)
Two points form:
The equation of the line passing through the points (x1, y1) and (x2, y2) ’ is
Intercept form:
The equation of the line with x-intercept a, y-intercept b is
• The equation of the line ∥ el to ax +by + c = 0 is ax +by + k = 0.
• The equation of the line ⊥ler to ax +by + c = 0 is bx −ay + k = 0.
Note: –
m1 = m2
m1 × m2 = – 1
with the coordinate axes is 
Perpendicular distance (Length of the perpendicular):
The perpendicular distance from a point P (x1, y1) to the line ax + by + c = 0 is
• The perpendicular distance from origin to the line ax + by + c = 0 is 
Distance between two parallel lines:
•The distance between the parallel lines ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0 is
Perpendicular form or Normal form:
The equation of the line which is at a distance of ‘p’ from the origin and α (0≤ α ≤ 3600) is the angle made by the perpendicular with 
the positive direction of the x-axis is x cosα + y sinα = p.
Symmetric form:
The equation of the line passing through point P (x1, y1) and having inclination θ is
Parametric form:
if P (x, y) is any point on the line passing through A (x1, y1) and
making inclination θ, then 
x = x1 + r cos θ, y = y1 + r sin θ
where ‘r’ is the distance from P to A.
• The ratio in which the line L ≡ ax + by + c = 0 divide the line segment joining the points A (x1, y1), B (x2, y2) is – L11: L22.
Where L11 = ax1 + by1 + c and L22 = ax2 + by2 + c.
Note: – the points A (x1, y1), B (x2, y2) lie on the same side or opposite side of line L = 0 according to L11 and L22 have the same sign or opposite sign.
∗ x-axis divides the line segment joining the points A (x1, y1), B (x2, y2) in the ratio – y1: y2.
∗ y-axis divides the line segment joining the points A (x1, y1), B (x2, y2) in the ratio – x1: x2.
Point of intersection of two lines:
the point of intersection of two lines a1x + b1y + c = 0 and a2x + b2y + c = 0 is
Three or more lines are said to be concurrent lines if they have a point in common.
The common point is called the point of concurrence.
∗ The condition that the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 to be concurrent is
a3(b1c2 – b2c1) + b3(c1a2 – c2a1) + c3(a1b2 – a2b1).
∗ The condition that the lines ax + hy +g = 0, hx + by + f = 0 and gx +fy + c = 0 is
abc + 2fgh – af2 – bg2 – ch2 = o.
Note: – if two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 said to be identical (same) if 
Family of straight lines: – A set of straight lines having a common property is called a family of straight lines.
Let L1 ≡ a1x + b1y + c1 = 0 and L2 ≡ a2x + b2y + c2 =0 represent two intersecting lines, theThe equation λ1 L1 + λ2 L2 = 0 represent a family of straight lines passing through the point of intersection of the lines L1 = 0 and L2 = 0.
∗ The equation of the straight line passing through the point of intersection of the lines L1 = 0 and L2 = 0 is L1 + λL2 = 0.
The angle between two lines:
If θ is the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 then
∗ If θ is an acute angle then
∗ If θ is the angle between two lines, then (π – θ) is another angle between two lines.
∗ If θ≠π/2 is angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then
∗ If m1, m2 are the slopes of two lines then
Note: – The lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are
∗ Parallel iff 
∗ Perpendicular iff a1a2 + b1b2 = 0
The foot of the perpendicular:
If Q (h. k) is the foot of the perpendicular from a point P (x1, y1)to the line ax + by +c = 0 then 
Image of the point:
If Q (h. k) is the image of point P (x1, y1) with respect to the line ax + by +c = 0 then 
Collinear Points:
If three points are said to be collinear, then they lie on the same line.
∗ If A, B, and C are collinear, then
Slope of AB = Slope of BC (or) Slope of BC = Slope of AC (or) Slope of AB = Slope of AC
∎ ax2 + 2hxy + by2 = 0 is called the second-degree homogeneous equation in two variable x and y.
This equation always represents a pair of straight lines which are passing through the origin.
∎ If l1x + m1y = 0 and l2x + m2y = 0 are two lines represented by the equation ax2 + 2hxy + by2 = 0, then ax2 + 2hxy + by2 = (l1x + m1y) (l2x + m2y)
⇒ a = l1l2; 2h = l1m2 + l2m1; b = m1m2
∎ If m1, m2 are the slopes of the lines represented by the equation ax2 + 2hxy + by2 = 0, then
m1+ m2 = – 2h/b and m1 m2 = a/b
∎ The lines represented by the equation ax2 + 2hxy + by2 = 0 are 
∎ If h2 = ab, then the lines represented by the equation ax2 + 2hxy + by2 = 0 are coincident.
∎ If two lines represented by the equation ax2 + 2hxy + by2 = 0 are equally inclined to the coordinate axes then h = 0 and ab < 0.
∎ The equation of the pair of lines passing through the point (h, k) and
(i) Parallel to the lines represented by the equation ax2 + 2hxy + by2 = 0 is
a (x – h)2 + 2h (x – h) (y – k) + b (y – k)2 = 0
(ii) Perpendicular to the lines represented by the equation ax2 + 2hxy + by2 = 0 is
b (x – h)2 – 2h (x – h) (y – k) + a (y – k)2 = 0
Angle between the lines:
If θ is the angle between the lines represented by the equation ax2 + 2hxy + by2 = 0, then 
∎ If a + b = 0, then two lines are perpendicular.
Area of the triangle:
The area of the triangle formed by the lines ax2 + 2hxy + by2 = 0 and the line lx + my + n = 0 is
Angular Bisectors:
⇒ the angle between angular bisectors is always 900
L1 = o, L2 = o are two non-parallel lines the locus of the point P such that the perpendicular distance from P to the first lie is equal to the perpendicular distance from P to second line is called the angular bisector of two lines.
⇒ If two lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the angular bisectors are
∎ The equation of the pair of angular bisectors of ax2 + 2hxy + by2 = 0 is
h (x2 – y2) = (a – b =) xy.
∎ If ax2 + 2hxy + by2 + 2gx 2fy + c= 0 represents a pair of straight lines then
(i) abc + 2fgh – af2 – bg2 – ch2 = 0
(ii) h2 ≥ ab, g2 ≥ ac and f2 ≥ bc
⇒ If two lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 are
l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0, then
ax2 + 2hxy + by2 + 2gx 2fy + c = (l1x + m1y + n1) (l2x + m2y + n2)
a = l1l2; 2h = l1m2 + l2m1; b = m1m2 ; 2g = l1n2 + l2n1; 2f = m1n2 + m2n1 and c = n1n2
⇒ The point of intersection of the lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 is 
∎ If two line have same homogeneous path then the lines represented by the first pair is parallel to the lines represented by the second pair.
ax2 + 2hxy + by2 + 2gx 2fy + c= 0 …………… (2)
ax2 + 2hxy + by2 = 0 …………… (1)
equation (1) and equation (2) form a parallelogram, one of the diagonals of parallelogram which is not passing through origin is 2gx + 2fy + c = 0.
∎ If two lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 are parallel then
⇒ Distance between parallel lines is 
• Let X’OX, Y’OY be two mutually perpendicular lines passing through a fixed point ‘O’. These two lines determine the XOY – plane (XY- plane). Draw the line Z’OZ perpendicular to XY – plane and passing through ‘O’.
The fixed point ‘O’ is called origin and three mutually perpendicular lines X’OX, Y’OY, Z’OZ are called Rectangular coordinate axes.
Three coordinate axes taken two at a time determine three planes namely XOY- plane, YOZ-plane, ZOY-plane or XY-plane, YZ-plane, ZX-plane respectively.
For every point P in space, we can associate an ordered triad (x, y, z) of real numbers formed by its coordinates.
The set of points in space is referred to as ‘Three-Dimensional Space’ or R3– Space.
∗ If P (x, y, z) is a point in a space, then
x is called x-coordinate of P
y is called y-coordinate of P
z is called z-coordinate of P
Distance between two points in space:
∗ Distance between the points A (x1, y1, z1) and B (x2, y2, z2) is 
∗ Distance between the point P (x, y, z) to the origin is 
Translation of axes:
When the origin is shifted to the point (h, k, l), then
X = x – h; Y = y – k; Z = z – l and x = X + h; y = Y + k; z = Z + l
∗ The foot of the perpendicular from P (x, y, z) to X-axis is A (x, 0, 0).
The perpendicular distance of P from X-axis is 
Similarly,
The perpendicular distance of P from Y-axis is 
The perpendicular distance of P from Z-axis is 
Collinear points: If three or more points lie on the same line are called collinear points.
Section formula:
The point dividing the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) in the ratio m : n is given by 
The mid-point of the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) is 
The centroid of the triangle whose vertices are A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is 
Tetrahedron:
→ It has 4 vertices and 6 edges.
→ A Tetrahedron is a closed figure formed by four planes not all passing through the same point.
→ Each edge arises as the line of intersection of two of the four planes.
→ The line segment joining the vertices to the centroid of opposite face. The point of concurrence is called centroid of Tetrahedron.
→ Centroid divides the line segment in the ratio 3:1.
→ The centroid of the Tetrahedron whose vertices are A (x1, y1, z1), B (x2, y2, z2), C (x3, y3, z3) and C (x4, y4, z4) is 
The line segment joining the points (x1, y1, z1), (x2, y2, z2) is divided by
XY – plane in the ratio – z1: z2
YZ – plane in the ratio – x1: x2
XZ – plane in the ratio – y1: y2
Consider a ray OP passing through origin ‘O’ and making angles α, β, γ respectively with the positive direction of X, Y, Z axes.
Cos α, Cos β, Cos γ are called Direction Cosines (dc’s) of the ray OP.
Dc’s are denoted by (l, m, n), where l = Cos α, m = Cos β, n = Cos γ
• A line in a space has two directions, it has two sets of dc’s, one for each direction. If (l, m, n) is one set of dc’s, then (-l, -m, -n) is the other set.
• Suppose P (x, y, z) is any point in space such that OP = r. If (l, m, n) are dc’s of a ray OP then x = lr, y= mr, z = nr.
• If OP = r and dc’s of OP are (l, m, n) then the coordinates of P are (lr, mr, nr).
• If P (x, y, z) is a point in the space, then dc’s of OP are 
• If (l, m, n) are dc’s of a line then l2 + m2 + n2 = 1.
⇒ cos2α + cos2β + cos2γ = 1.
Direction Ratios:
Any three real numbers which are proportional to the dc’s of a line are called direction ratios (dr’s) of that line.
• Let (a, b, c) be dr’s of a line whose dc’s are (l, m, n). Then (a, b, c) are proportional to (l, m, n) 
and a2 + b2 + c2 ≠ 1.
• Dr’s of the line joining the points (x1, y1, z1), (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1)
• If (a, b, c) are dr’s of a line then its dc’s are 
• If (l1, m1, n1), (l2, m2, n2) are dc’s of two lines and θ is angle between them then Cos θ = l1l2 + m1m2 + n1n2
If two lines perpendicular then l1l2 + m1m2 + n1n2 = 0.
• If (a1, b1, c1), (a2, b2, c2) are dr’s of two lines and θ is the angle between them then
If two line are perpendicular then a1a2 + b1b2 + c1c2 = 0.
Plane: A plane is a proper subset of R3 which has at least three non-collinear points and any two points in it.
∎ Equation of the plane passing through a given point A (x1, y1, z1), and perpendicular to the line whose dr’s (a, b, c) is a(x – x1) + a(y – y1) + a(z – z1) = 0.
∎ The equation of the plane hose dc’s of the normal to the plane (l, m, n) and perpendicular distance from the origin to the pane p is lx + my + nz = p
∎ The equation of the plane passing through three non-collinear points A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is
∎ The general equation of the plane is ax + by + cz + d = 0, where (a, b, c) are Dr’s of the normal to the plane.
Normal form:
The equation of the plane ax + by + cz + d = 0 in the normal form is
Perpendicular distance:
The perpendicular distance from (x1, y1, z1) to the plane ax + by + cz + d = 0 is 
The perpendicular distance from the origin to the plane ax + by + cz + d = 0 is
Intercepts:
X- intercept = aIf a plane cuts X –axis at (a, 0, 0), Y-axis at (0, b, 0) and Z-axis at (0, 0, c) then
Y-intercept = b
Z-intercept = c
The equation of the plane in the intercept form is 
The intercepts of the plane ax + by + cz + d =0 is -d/a, -d/b, -d/c
∎ The equation of the plane parallel to ax + by + cz + d = 0 is ax + by + cz + k = 0.
∎ The equation of XY – plane is z = 0.
∎ The equation of YZ – plane is x = 0.
∎ The equation of XZ – plane is y = 0.
∎ Distance between the two parallel planes ax + by + cz + d1 =0 and ax + by + cz + d2 =0 is 
The angle between two planes:
The angle between the normal to two planes is called the angle between the planes.
If θ is the angle between the planes a1 x + b1 y + c1 z + d1 =0 and a2 x + b2 y + c2 z + d2 =0 then
If two line are perpendicular then a1a2 + b1b2 + c1c2 = 0.
∎ The distance of the point P (x, y, z) from
Intervals:
Let (a, b) ∈ R such that a ≤ b, then the set
If f(x) is a function of x such that if x approaches to a constant value ‘a’, then the value of f(x) also approaches to ‘l’. Then the constant ‘I’ is called a limit of f(x) at x = a
⟹ 
Or
A real number l is called the limit of the function f, if for all ϵ> 0 there exist δ > 0 such that 
whenever
⟹
Sand witch theorem:( Squeez Principle):
f, g, and h are functions such that f(x) ≤ g(x) ≤ h(x), then 
and if 
If x < a, then is 
called left-hand limit
If x > a, then 
is called right-hand limit
Note:
if a function f(x) any of the following forms at x = a:
Then f(x) is said to be indeterminate at x = a.
Continuity:
Condition 1: If the condition is like x = a and x ≠ a, then we use following property.
If 
then f(x) is continuous at x = a, otherwise f(x) is not continuous.
Condition 2: If the condition is like x ≤ a and x >a, or x < a and x ≥a then we use following property.
If 
then f(x) is continuous at x = a, otherwise f(x) is not continuous.
Let f be a function defined on a neighbourhood of a real number ‘a’ if 
exist then we say that f is differentiable at x a and it is denoted by f'(a).
∴ f’(a) =
∎ If right hand derivative = left hand derivative, then f is differentiable at ‘a’.
i.e., 
The first principle of the derivative of f at any real number ‘x’ is f’(x) = 
∎ The differentiation of f(x) is denoted by 
means differentiation of ‘y’ with respect to ‘x’
∎ The derivative of constant function is zero i.e., f’(a) = 0 where ‘a’ is any constant.
∎ Let I be an interval in R u and v are real valued functions on I and x ∈ I. Suppose that u and v are differentiable at ‘x’, then
is also differentiable at ‘x’ and 
∎ (f o g)’ (x) = f’(g(x)). g’(x).
∎
If x = f(t) and y = g(t) then the procedure of finding in terms of the parameter ‘t’ is called parametric equations. 
Implicitly differentiation:
An equation involving two or more variables is called an implicit equation.
ax2 + 2hxy + b y2 = 0 is an implicit equation in terms of x and y.
The process of finding from an implicit equation is called implicitly differentiation.
Derivative of one function w.r.t. another function:
The derivative of f(x) w.r.t g(x) is 
Second order derivative:
Let y = f(x) be a function, if y is differentiable then the derivative of f is f’(x). If ‘(x) is again differentiable then the derivative of f’(x) is called second order derivative. And it is denoted by f” (x) or 
Approximations:
Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then ∆y be the corresponding change in y thus ∆y = f (x + ∆x) – f (x).
Let 
⇒ 
where ϵ is very small
⇒
For ‘ϵ.∆x’ is very small and hence, 
Approximate value is f (x + ∆x) = f(x) + f’(x). ∆x
Differential:
Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then
called differential of y = f(x) and it is denoted by df.
∴ dy = f’(x). ∆x
Errors:
Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then ∆y be the corresponding change in y.
CIRCLE: 
If ‘r’ is radius, ‘d’ is diameter ‘P’ is the perimeter or circumference and A is area of the circle then
d= 2r, P = 2πr = πd and A = πr2sq.u
SECTOR:
If ‘r’ is the radius, ‘l’ is the length of arc and θ is of the sector then
Area = ½ l r = ½ r2θsq.u.
Perimeter = l + 2r = r (θ + 2) u.
CYLINDER:
Length of the Arc ‘l’ = rθ (θ must be in radians).
If ‘r is the radius of the base of cylinder and ‘h’ is the height of the cylinder, then
Area of base = πr2 sq.units.
Lateral surface area = 2πrh units.
Total surface area = 2πr (h + r) units.
Volume = πr2 h cubic units.
CONE: 
If ‘r’ is the radius of base, ‘h’ is the height of cone and ‘l’ is slant height then
l 2 + r2 = h2
Lateral surface area = πrl units.
Total surface area = πr (l + r) sq. units.
Volume = 
cubic units.
SPHERE: 
If ‘r’ is the radius of the Sphere then
Surface area = πr2 sq. units.
Volume = 
πr3 cubic units.
Tangent of a Curve:
If the secant line PQ approaches to the same position as Q moves along the curve and approaches to either side then limiting position is called a ‘Tangent line’ to the curve at P. The point P is called point of contact
Let y = f(x) be a curve, P a point on the curve. If Q(≠P) is another point on the curve then the line PD is called secant line.
Gradient of a curve:
Let y = f(x) be a curve and P (x, y) be a point on the curve. The slope of the tangent to the curve y = f(x) at P is called gradient of the curve.
Slope of the tangent to the curve y = f(x) at P (x, y) is m = 
∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = m (x – x1) where m =
Normal of a curve: 
Let y = f(x) be a curve and P (x, y) be a point on the curve. The line passing through P and perpendicular to the tangent of the curve y = f(x) at P is called Normal of the curve.
∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = -1/m (x – x1).
Slope of the normal is -1/m. where m =
Lengths of tangent, normal, subtangent and subnormal: 
PT → Normal; QN → subnormal
PN → Tangent; QT → subtangent
∎ if m = then
Angle between two curves:
If two curves intersect at a point P., then the angle between the tangents of the curves at P is called the angle between the curves at P.
∎ If m1, m2 are the slopes of two tangents of the two curves and θ is the angle between the curves then
Tanθ = 
Note:
Average rate of change:
if y = f(x) then the average rate of change in y between x = x1 and x = x2 is defined as 
Instantaneous rate of change:
if y = f(x), then the instantaneous rate of change of a unction f at x = x0 is defined as 
Rectilinear Motion:
A motion of a particle in a line is called Rectilinear motion. The rectilinear motion is denoted by s = f(t) where f(t) is the rule connecting ‘s’ and ‘t’.
Velocity, Acceleration:
A particle starts from a fixed point and moves a distance ‘S’ along a straight-line during time ‘t’ then
Velocity = 
Acceleration = 
Note:
(i) If v> 0, then the particle s moving away from the straight point.
(ii) If v < 0, then particle s moving away towards the straight point.
(iii) If v = 0, then the particle comes rest.
Rolle’s Theorem:
Suppose a, b (a < b) are two real numbers. Let f: [a, b] → R be a function satisfying the following conditions:
(i) f is continuous on [a, b]
(ii) f is differentiable on (a, b) and
(iii) f(a) = f(b)
then there exists at least one c ∈ (a, b) such that f’(c)= 0.
Lagrange’s Theorem:
Suppose a, b (a < b) are two real numbers. Let f: [a, b] → R be a function satisfying the following conditions:
(i) f is continuous on [a, b]
(ii) f is differentiable on (a, b) and
then there exists at least one c ∈ (a, b) such that f’(c)= 
⨂ Let f be a real function on an interval I then f is said to be
(i) an increasing function on I if
x1 < x2 ⇒ f (x1) ≤ f (x2) ∀ x1, x2 ∈ I
(ii) decreasing function on I if
x1 < x2 ⇒ f (x1) ≥ f (x2) ∀ x1, x2 ∈ I
⨂ Let f be a real function on an interval I then f is said to be
(i) strictly increasing function on I if
x1 < x2 ⇒ f (x1) < f (x2) ∀ x1, x2 ∈ I
(ii) strictly decreasing function on I if
x1 < x2 ⇒ f (x1) > f (x2) ∀ x1, x2 ∈ I
⨂ Let f(x) be a real valued function defined on I = (a, b) or [a, b) or (a, b] or [a, b]. Suppose f is continuous on I and differentiable in (a, b). If
(i) f’ (c) > 0 ∀ c ∈ (a, b), then f is strictly increasing on I
(ii) f’ (c) < 0 ∀ c ∈ (a, b), then f is strictly decreasing on I
(iii) f’ (c) ≥ 0 ∀ c ∈ (a, b), then f is increasing on I
(iv) f’ (c) ≤ 0 ∀ c ∈ (a, b), then f is decreasing on I
Critical point:
A point x = c in the domain of the function said to be ‘critical point’ of the function f if either f’ (c) = 0 or f’ (c) does not exists.
Stationary point:
A point x = c in the domain of the function said to be ‘stationary point’ of the function f if f’ (c) = 0.
MAXIMA & MINIMA
Global maxima – Global minima:
Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be
(i) a global maximum on D if f(c) ≥ f(x)
(ii) a global minimum on D if f(c) ≤ f(x)
Relative maximum:
Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be relative maximum at c if there exist δ > 0 such that f(c) ≥ f(x) ∀ x ∈ (c – δ, c + δ).
Here, f (c) is called relative maximum value of f(x) at x = c and the point x = c is called point of relative maximum.
Relative minimum:
Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be relative maximum at c if there exist δ > 0 such that f(c) ≤ f(x) ∀ x ∈ (c – δ, c + δ).
Here, f (c) is called relative maximum value of f(x) at x = c and the point x = c is called point of relative minimum.
⨂ The relative maximum and minimum value of f are called extreme values.
⨂ If f is either minima or maxima f’ (α) = 0.
⨂ Let f be a continuous function om [a, b] and α ∈ (a, b)
(i) If f’ (α) = 0 and f’’ (α) >0, then f(α) is relative minimum.
(ii) if f’ (α) = 0 and f’’ (α) <0, then f(α) is relative maximum
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Set: A collection of well-defined objects is called a set.
Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).
Cartesian product: Let A and B are two non-empty sets. The Cartesian product of A and B is denoted by A × B and is defined as a set of all ordered pairs (a, b) where a ϵ A and b ϵB
Relation: Let A and B are two non-empty sets the relation R from A to B is a subset of A×B.
⇒ R: A→B is a relation if R⊂ A × B
A relation f: A → B is said to be a function if ∀ aϵ A there exists a unique element b such that (a, b) ϵ f. (Or)
A relation f: A → B is said to be a function if
(i) x ϵ A ⇒ f(x) ϵ B
(ii) x1 , x2 ϵ A , x1 = x2 in A ⇒ f(x1) = f(x2) in B.
Note: If A, B are two finite sets then the no. of functions that can be defined from A to B is n(B)n(A)
VARIOUS TYPES OF FUNCTIONS
One– one Function (Injective):- A function f: A→ B is said to be a one-one function or injective if different elements in A have different images in B.
(Or)
A function f: A→ B is said to be one-one function if f(x1) = f(x2) in B ⇒ x1 = x2 in A.
Note: No. of one-one functions that can be defined from A into B is n(B) p n(A) if n(A) ≤ n(B)
On to Function (Surjection): – A function f: A→ B is said to be onto function or surjection if for each yϵ B ∃ x ϵ A such that f(x) =y
Note: if n(A) = m and n(B) = 2 then no. of onto functions = 2m – 2
Bijection: – A function f: A→ B is said to be Bijection if it is both ‘one-one and ‘onto’.
Constant function: A function f: A→ B is said to be constant function if f(x) = k ∀ xϵA
Identity function: Let A be a non-empty set, then the function defined by IA : A → A, I(x)=x is called identity function on A.
Equal function: Two functions f and g are said to be equal if
(i) They have same domain (D)
(ii) f(x) = g(x) ∀ xϵ D
Even function: A function f: A→ B is said to be even function if f (- x) = f(x) ∀ xϵ A
Odd function: A function f: A→ B is said to be odd function if f (- x) = – f(x) ∀ xϵ A
Composite function: If f: A→B, g: B→C are two functions then the composite relation is a function from A to C.
gof: A→C is a composite function and is defined by gof(x) = g(f(x)).
Step function: A number x = I + F
I → integral part = [x]
F → fractional part = {x}
∴ x = [x] + {x}
If y = [x] then domain = R and
Range = Z
0 ≤ x ≤ 1, [x] = 0
1≤ x ≤ 2, [x] = 1
-1 ≤ x ≤ 0, [x] = -1
If k is any integer [ x + k] = k + [x]
The value of [x] is lies in x – 1 < [x] ≤ 1.
Inverse function: If f: A → B is bijection then f -1 is exists
f-1: B → A is an inverse function of f.
of subsets of a set of n elements is 2n
of proper subsets of a set of n elements is 2n – 1
Let A and B are two non-empty finite sets and f: A → B is a function. This function will
One-one if n(A) ≤ n(B)
On to if n(A) ≥ n(B)
Bijection if n(A) = n(B).
Matrix: An ordered rectangular array of elements is called a matrix
Oder of Matrix: A matrix having rows and ‘n’ columns is said to be of order m x n. Read as m by n.
Principal diagonal ( diagonal) Matrix: If A = [aij] is a square matrix of order ‘n’ the elements a11 , a22 , a33 , ………. ann is said to constitute its principal diagonal.
Trace Matrix: The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A).
Ex:-
Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.
Scalar Matrix: If each non-diagonal elements of a square matrix are ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix.
Identity Matrix or Unit Matrix: If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called a unit matrix.
Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix.
Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix.
Triangular matrices:
A square matrix A = [aij] is said to be upper triangular if aij = 0 ∀ i > j
A square matrix A = [aij] is said to be lower triangular matrix aij = 0 ∀ i < j
Equality of matrices: matrices A and B are said to be equal if A and B of the same order and the corresponding elements of A and B are equal.
Let A = [aik]mxn and B = [bkj]nxp be two matrices ,then the matrix C = [cij]mxp where
Note: Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix.
Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by AI or AT.
Note: (i) (AI)I = A (ii) (k AI) = k . AI (iii) (A + B )T = AT + BT (iv) (AB)T = BTAT
Symmetric Matrix: A square matrix A is said to be symmetric if AT =A
If A is a symmetric matrix, then A + AT is symmetric.
Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A
If A is a skew-symmetric matrix, then A – AT is skew-symmetric
Minor of an element: Consider a square matrix
the minor an element in this matrix is defined as the determinant of the 2×2 matrix obtained after deleting the rows and the columns in which the element is present.
Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 matrix is defined as it’s minor multiplied by (- 1 ) i+j .
Singular and non-singular matrices: A Square matrix is said to be singular if its determinant is zero, otherwise it is said to be the non-singular matrix.
Ad joint of a matrix: The transpose of the matrix formed by replacing the elements of a square matrix A with the corresponding cofactors is called the adjoint of A.
Invertible matrix: Let A be a square matrix, we say that A is invertible if there exists a matrix B such that AB =BA = I, where I is the unit matrix of the same order as A and B.
Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (D) is called the augmented matrix. It is denoted by [AD].
Sub matrix: A matrix obtained by deleting some rows and columns (or both) of a matrix is called the submatrix of the given matrix.
Let A be a non-zero matrix. The rank of A is defined as the maximum of the order of the non-singular submatrices of A.
(iii) 3, if A is non – singular.
Consistent and Inconsistent: The system of linear equations is consistent if it has a solution, in-consistent if it has no solution.
The system of equations AX = 0 has
Directed line: If A and B are two distinct points in the space, the ordered pair (A, B) denoted by AB is called a directed line segment with initial point A and terminal point B.
⇒ A directed line passes through three characteristics: (i) length (ii) support (iii) direction
Scalar: A quantity having magnitude only is called a scalar. We identify real numbers as a scalar.
Ex: – mass, length, temperature, etc.
Vector: A quantity having length and direction is called a vector.
Ex: – velocity, acceleration, force, etc.
⇒ If 
is a vector then its length is denoted by 
Position of vector: If P (x, y, z) is any point in the space, then 
is called the position vector of the point P with respect to origin (O). This is denoted by 
Like and unlike vectors: If two vectors are parallel and having the same direction then they are called like vectors.
If two vectors are parallel and having opposite direction then they are called, unlike vectors.
Coplanar vectors: Vectors whose supports are in the same plane or parallel to the same plane are called coplanar vectors.
Triangle law: If 
are two vectors, there exist three points A, B, and C in a space such that
defined by 
Parallelogram law: If two vectors
and 
represented by two adjacent sides of a parallelogram in magnitude and direction then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point.
Scalar multiplication: Let be a vector and λ be a scalar then we define vector λ to be the vector
if either is zero vector or λ is the scalar zero; otherwise λ is the vector in the direction of with the magnitude 
if λ>0 and λ = (−λ)(− ) if λ<0.
The angle between two non-zero vectors: Let be two non-zero vectors, let 
then ∠AOB has two values. The value of ∠AOB, which does not exceed 1800 is called the angle between the vectors and , it is denoted by ( ).
Section formula: Let be two position vectors of the points A and B with respect to the origin if a point P divides the line segment AB in the ratio m:n then
Linear combination of vectors: let 
be vectors x1, x2, x3…. xn be scalars, then the vector
is called the linear combination of vectors.
Components: Consider the ordered triad (a, b, c) of non-coplanar vectors
If r is any vector then there exist a unique triad (x, y, z) of scalars such that 
. These scalars x, y, z are called the components of with respect to the ordered triad (a, b, c).
= r = x i + y j +z k and
Regular polygon: A polygon is said to be regular if all the sides, as well as all the interior angles, are equal.
⇒The vector equation of the line passing through point A () and ∥el to the vector is
Proof:-
Then AP, are collinear vector proof: let P ( ) be any point on the line a
the equation of the line passing through origin and parallel to the vectoris
) and B( ) is 
) and parallel to the vectors and
is 
), B( ) and parallel to the vector is 
), B( ) and C( ) is
Dot product (Scalar product): Let are two vectors. The dot product or direct product of and 
is denoted by
and is defined as
= 0, = 0 ⟹ = 0.≠0, ≠ 0 then
are two vectors, then 
then. ⟹ 
⟹ If > 0, then θ is an acute angle
⟹ If < 0, then θ is obtuse angle 0
⟹ If = 0, then is perpendicular to
is any vector then 
Component and Orthogonal Projection:
Let=
,=
be two non-zero vectors. Let the plane passing through B ( ) and perpendicular to intersects
In M, then 
is called the component of on
on is 
perpendicular to = 
If 
,
, 
form a right-handed system of an orthonormal triad, then
then = a1b1 + a2b2 + a3b3 then 
Parallelogram law:
In a parallelogram, the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of its sides.
In ∆ABC, the length of the median through vertex A is 
Vector equation of a plane:
The vector equation of the plane whose perpendicular distance from the origin is p and unit normal drawn from the origin towards the plane is,
•The vector equation of the plane passing through point A ( ) and perpendicular to the
is
•If θ is the angle between the planes 
then 
Cross product (vector product): Let and be two non-zero collinear vectors. The cross product of and is denoted by × (read as a cross ) and is defined as 
•
are orthogonal triad then
• The vector × is perpendicular to both and and also perpendicular to the plane containing them
• The unit vector perpendicular to both and is
• Let
then 
• If and are two sides of a triangle then the area of the triangle =
• If A ( ), B ()and C ( )are the vertices of a ∆ABC, then its area
• The area of the parallelogram whose adjacent sides and is 
• The area of the parallelogram whose diagonals and is
• If A ( ), B ( )and C ( )are three points then the perpendicular distance from A to the line passing through B, C is
Let,and be three vectors, then (
) . is called the scalar triple product of,andand it is denoted by
If
then
•In determinant rows(columns) are equal then the det. Value is zero.
•In a determinant, if we interchange any two rows or columns, then the sign of det. Is change.
•Four distinct points A, B, C, and D are said to be coplanar iff 
The volume of parallelepiped:
If ,andare edges of a parallelepiped then its volume is 
The volume of parallelepiped:
The volume of Tetrahedron with, and are coterminous edges is
The volume of Tetrahedron whose vertices are A, B, C and D is 
Vector equation of a plane:
The vector equation of the plane passing through point A () and parallel to the vectors and is 
The vector equation of the plane passing through the points A ( ) and B( ) and parallel to the vector is
The vector equation of the plane passing through the points A (), B( ) and C( ) is 
Skew lines:
The lines which are neither intersecting nor parallel are called Skew lines
The shortest distance between the Skew lines:
If 
are two skew lines, then the shortest distance between them is 
If A, B, C and D are four points, then the shortest distance between the line joining the points AB and CD is 
•The plane passing through the intersection of the planes
is 
the perpendicular distance from point A (a ̅) to the plane 
is 
Let ,and be three vectors, then
is called the vector triple product of, and.
Scalar product of four vectors:
Vector product of four vectors:
The word ’trigonometry’ derived from the Greek words ‘trigonon’ and ‘metron’. The word ‘trigonon’ means a triangle and the word ‘metron’ means a measure.
Angle: An angle is a union of two rays having a common endpoint in a plane.
There are three systems of measurement of the angles.
Sexagesimal system: – In this system, a circle can be divided into 360 equal parts. Each part is called one degree (0). One circle = 3600
Further, each degree can be divided into 60 equal parts. Each part is called one minute (‘).
and each minute can be divided into 60 equal parts. Each part is called one second (“)
Sexagesimal system: – In this system, a circle can be divided into 400 equal parts. Each part is called one grade (g). One circle = 400g
Further, each grade can be divided into 100 equal parts. Each part is called one minute (‘).
and each minute can be divided into 100 equal parts. Each part is called one second (“)
Circular measure: Radian is defined as the amount of the angle subtended by an arc of length ’r’ of a circle of radius ‘r’.
One radian is denoted by 1c. One circle = 2πc
Relation between the three measures:
3600 = 400g = 2 πc
1800 = 200g = πc
Trigonometric Ratios:
Trigonometric identities: –
∗ sin2θ + cos2θ = 1
1 – cos2θ = sin2θ
1 – sin2θ = cos2θ
∗ sec2θ − tan2θ = 1
sec2θ = 1 + tan2θ
tan2θ = sec2θ – 1
(secθ − tanθ) (secθ + tanθ) = 1
∗ cosec2θ − cot2θ = 1
co sec2θ = 1 + cot2θ
cot2θ = cosec2θ – 1
(cosec θ – cot θ) (cosec θ + cot θ) = 1
• sin θ. cosec θ = 1
sec θ. cos θ = 1
tan θ. cot θ = 1
All Silver Tea Cups Rule:
Note: If 900 ±θ or 2700 ±θ then
‘sin’ changes to ‘cos’; ‘tan’ changes to ‘cot’; ‘sec’ changes to ‘cosec’
‘cos’ changes to ‘sin’; ‘cot’ changes to ‘tan’; ‘cosec’ changes to ‘sec’.
If 1800 ±θ or 3600 ±θ then, no change in ratios.
Values of Trigonometric Ratios:
Complementary angles: Two angles A and B are said to be complementary angles, if A + B = 900.
supplementary angles: Two angles A and B are said to be supplementary angles, if A + B = 1800.
Let E ⊆ R and f: E → R be a function, then f is called periodic function if there exists a positive real number ‘p’ such that
If such a positive real number ‘p’ exists, then it is called a period of f.
The algebraic sum of two or more angles is called a ‘compound angle’.
For any two real numbers A and B
⋇ sin (A + B) = sin A cos B + cos A Cos B
⋇ sin (A − B) = sin A cos B − cos A Cos B
⋇ cos (A + B) = cos A cos B − sin A sin B
⋇ cos (A − B) = cos A cos B + sin A sin B
⋇ tan (A + B) =
⋇ tan (A − B) =
⋇ cot (A + B) =
⋇ cot (A − B) = 
⋇ sin (A + B + C) = ∑sin A cos B cos C − sin A sin B sin C
⋇ cos (A + B + C) = cos A cos B cos C− ∑cos A sin B sin C
⋇ tan (A + B + C) =
⋇ cot (A + B + C) =
⋇ sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A
⋇ cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A
Extreme values of trigonometric functions:
If a, b, c ∈ R such that a2 + b2 ≠ 0, then
Maximum value = 
Minimum value =
If A is an angle, then its integral multiples 2A, 3A, 4A, … are called ‘multiple angles ‘of A and the multiple of A by fraction like
are called ‘submultiple angles.
⋇ sin 2A = 2 sin A cos A =
⋇ cos 2A = cos2 A – sin2 A
= 2 cos2 A – 1
= 1 – 2sin2 A
=
⋇ tan 2A =
⋇ cot 2A =
∎ If 
is not an add multiple of 
⋇ sin A = 2 sin cos =
⋇ cos A = cos2 – sin2
= 2 cos2 – 1
= 1 – 2sin2
=
⋇ tan A =
⋇ cot A =
⋇ sin3A = 3 sin A −4 sin3 A
⋇ cos 3A = 4 cos3 A – 3 cos A
⋇ tan 3A =
⋇ cot 3A =
⋇ tan A + cot A = 2 cosec 2A
⋇ cot A – tan A = 2 cot 2A
For A, B∈ R
⋇ sin (A + B) + sin (A – B) = 2sin A cos B
⋇ sin (A + B) −sin (A – B) = 2cos A sin B
⋇ cos (A + B) + cos (A – B) = 2 cos A cos B
⋇ cos (A + B) − cos (A – B) = − 2sin A sin B
For any two real numbers C and D
⋇ sin C + sin D = 2sin
cos
⋇ sin C −sin D= 2cos sin
⋇ cos C + cos D = 2 coscos
⋇ cos C − cos D = − 2sin sin
If A + B + C = π or 1800, then
⋇ sin (A + B) = sin C; sin (B + C) = sin A; sin (A + C) = sin B
⋇ cos (A + B) = − cos C; cos (B + C) = −cos A; cos (A + C) = − cos B
If A + B + C = 900 or then
⋇ sin 
= cos
; sin
= cos
; sin
= cos
⋇ cos = sin ; cos = sin ; cos = sin
If 
then
⋇ sin (A + B) = cos C; sin (B + C) = cos A; sin (A + C) = cos B
⋇ cos (A + B) = sin C; cos (B + C) = sin A; cos (A + C) = sin B
Trigonometric equation: An equation consisting of the trigonometric functions of a variable angle θ ∈ R is called a ‘trigonometric equation’.
The solution of the equation: The values of the variable angle θ, satisfying the given trigonometric equation is called a ‘solution’ of the equation.
The set of all solutions of the trigonometric equation is called the solution set’ of the equation. A ‘general solution’ is an expression of the form θ0 + f(n) where θ0 is a particular solution and f(n) is a function of n ∈ Z involving π.
⨂ If k ∈ [− 1, 1] then the principle solution of θ of sin x = k lies in 
General solution of sin x = sin θ is x = nπ + (−1) n θ, n ∈ Z
⨂ If k ∈ [− 1, 1] then the principle solution of θ of cos x = k lies in 
General solution of cos x = cos θ is x = 2nπ ± θ, n ∈ Z
⨂ If k ∈R then the principle solution of θ of tan x = k lies in 
General solution of tan x = tan θ is x = nπ + θ n ∈ Z
⨂ If sin θ = 0, then the general solution is θ = nπ, n ∈ Z
⨂ If tan θ = 0, then the general solution is θ = nπ, n ∈ Z
⨂ If cos θ = 0, then the general solution is θ = (2n + 1) , n ∈ Z
⨂ If sin2 θ = sin2 𝛂, cos2 θ = cos2 𝛂 or tan2 θ = tann2 𝛂 then the general solution is 𝛉 = nπ ± θ, n ∈ Z
If A, B are two sets and f: A→ B is a bijection, then f-1 is existing and f-1: B → A is an inverse function.
⨂ The function Sin-1: [−1, 1] →
is defined by Sin-1 x = θ ⇔ θ∈ and sin θ = x
⨂ The function Cos-1: [−1, 1] → [0, π] is defined by Sin-1 x = θ ⇔ θ∈ [0, π] and cos θ = x
⨂ The function Tan-1: R →
is defined by Tan-1 x = θ ⇔ θ∈ and tan θ = x
⨂ The function Sec-1: [−∞, −1] ∪ [1, ∞] →
is defined by Sin-1 x = θ ⇔ θ∈ and sec θ= x
⨂ The function Cosec-1: [−∞, −1] ∪ [1, ∞] →
is defined by cosec-1 x = θ ⇔ θ∈ and Cosec θ= x
⨂ The function Cot-1: R → (0, π) is defined by Cot-1 x = θ ⇔ θ ∈ (0, π) and cot θ = x
Properties of Inverse Trigonometric functions:
⨂ Sin-1 x = Cosec-1(1/x) ∀ x ∈ [−1, 1] – {0}
⨂ Cos-1x = Sec-1(1/x) ∀ x ∈ [−1, 1] – {0}
⨂ Tan-1 x = Cot-1(1/x), if x > 0
⨂ Tan-1 x = Cot-1(1/x) −π, if x < 0
⨂ Sin-1 (−x) = − Sin-1(x) ∀ x ∈ [−1, 1]
⨂ Cos-1 (−x) = π − Cos-1(x) ∀ x ∈ [−1, 1]
⨂ Tan-1 (−x) = − Tan-1(x) ∀ x ∈ R
⨂ Cosec-1 (−x) = − Cose-1(x) ∀ x ∈ (− ∞, − 1] ∪ [1, ∞)
⨂ Sec-1 (−x) = π − Sec-1(x) ∀ x ∈ (− ∞, − 1] ∪ [1, ∞)
⨂ Cot-1 (−x) =π − Cot-1(x) ∀ x ∈ R
⨂
(i) If θ∈, then Sin−1(sin θ) = θ and if x ∈ [−1, 1], then sin (Sin−1x) = x
(ii) If θ∈ [0, π], then Cos−1(cos θ) = θ and if x ∈ [−1, 1], then cos (Cos−1x) = x
(iii) If θ∈ , then tan−1(tann θ) = θ and if x ∈ R, then tan (Tan−1x) = x
(iv) If θ∈ (0, π), then Cot−1(cot θ) = θ and if x ∈ R, then cot (Cot−1x) = x
(v) If θ∈ [0, 
) ∪ ( , π], then Sec−1(sec θ) = θ and
if x ∈ (− ∞, − 1] ∪ [1, ∞), then sec (Sec−1x) = x
(vi) If θ∈ , then Cosec−1(cosec θ) = θ and
if x ∈ (− ∞, − 1] ∪ [1, ∞), then cosec (Cosec−1x) = x
⨂
(i) If θ∈ , then Cos−1(sin θ) = 
(ii) If θ∈ [0, π], then Sin−1(cos θ) =
(iii) If θ∈ , then Cot−1(tan θ) =
(iv) If θ∈ (0, π), then Tan−1(cot θ) =
(v) If θ∈ , then Cosec−1(sec θ) =
(vi) If θ∈ , then Sec−1(cosec θ) =
⨂
)if 0 ≤ x ≤ 1 and Sin−1x =− Cos−1 ( ) if −1 ≤ x ≤ 0
if x ∈ (−1, 1)
) if x ∈ [0, 1] and Cos−1x = π − Sin−1 () if x ∈ [−1, 0]
= Cos−1 
or x > 0⨂ Cos−1 x + Sin−1x = ∀ x ∈ [−1, 1]
⨂ Tan−1 x + Cot−1x = ∀ x ∈ R
⨂ Sec−1 x + Cosec−1x = ∀ x ∈ (−∞, −1] ∪ [1, ∞)
⨂ Sin−1 x + Sin−1y = Sin−1(x 
+ y ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 ≤ 1
=π− Sin−1(x + y ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 > 1
⨂ Cos−1 x + Cos−1y = Cos−1(x y − ) if 0 ≤x, y ≤ 1and x2 + y2 ≥ 1
=π− Cos−1(x y − ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 < 1
⨂ Tan−1 x + Tan−1y = Tan−1
if x > 0, y> 0 and xy < 1
=π + Tan−1 if x > 0, y> 0 and xy > 1
= Tan−1 if x < 0, y< 0 and xy > 1
= −π + Tan−1 if x < 0, y< 0 and xy < 1
⨂ Tan−1 x − Tan−1y = Tan−1 
if x > 0, y> 0 or x < 0, y< 0
⨂ 2 Sin−1 x = Sin−1 (2x ) if x≤
= π− Sin−1 (2x ) if x >
⨂ 2 Cos−1 x = Cos−1(2x2 – 1) if x ≥
=Cos−1(1–2x2) if x <
⨂ 2 Tan−1 x = Tan−1 
if
< 1
= π + Tan−1 if ≥ 1
= Sin−1 
if x ≥ 0
= Cos−1 
if x ≥ 0
⨂ 3Sin−1x = Sin−1(3x – 4x3)
⨂ 3Cos−1x = Cos−1(4x3 – 3x)
⨂ 3Tan−1x = tan−1
⨂
⨂ The function f: R→R defined by f(x) = ∀ x ∈ R is called the ‘hyperbolic sin’ function. It is denoted by sinh x.
∴ sinh x =
Similarly,
⨂ cosh x = 
∀ x ∈ R
⨂ tanh x = 
∀ x ∈ R
⨂ coth x =
∀ x ∈ R
⨂ sech x = 
∀ x ∈ R
⨂ cosech x = 
∀ x ∈ R
Identities:
⨂ cosh2x – sinh2 x = 1
cosh2x = 1 + sinh2 x
sinh2 x = cosh2 x – 1
⨂ sech2 x = 1 – tanh2 x
tanh2 x = 1 – sech2 x
⨂ cosech2 x = coth2 x – 1
coth2 x = 1 + coth2 x
Addition formulas of hyperbolic functions:
⨂ sinh (x + y) = sinh x cosh y + cosh x sinh y
⨂ sinh (x − y) = sinh x cosh y − cosh x sinh y
⨂ cosh (x + y) = cosh x cosh y + sinh x sinh y
⨂ cosh (x − y) = cosh x cosh y − sinh x sinh y
⨂ tanh (x + y) = 
⨂ tanh (x − y) = 
⨂ coth (x + y) =
⨂ sinh 2x = 2 sinh x cosh 2x = 
⨂ cosh 2x = cosh2x + sinh2 x = 2 cosh2x – 1 = 1 + 2 sinh2x =
⨂ tanh 2x =
⨂ sinh 3x = 3 sinh x + 4 sinh3x
⨂ cosh 3x = 4 cosh3 x – 3 cosh x
⨂ tanh 3x = 
Inverse hyperbolic functions:
⨂ Sinh−1x =
∀ x ∈ R
⨂ Cosh−1x = 
∀ x ∈ (1, ∞)
⨂ Tanh−1x = 
∀ < 1
⨂ Coth−1x = 
∀ > 1
⨂ Sech−1x = 
∀ x ∈ (0, 1]
⨂ Cosech−1x = 
if x < 0 and x ∈ (−∞, 0)
= if x > 0
In ∆ABC,
Lengths AB = c; BC = a; AC =b
Area of the tringle is denoted by ∆.
Perimeter of the triangle = 2s = a + b + c
A = ∠CAB; B = ∠ABC; C = ∠BCA.
R is circumradius.
Sine rule:
In ∆ABC,
⟹ a = 2R sin A; b = 2R sinB; c = 2R sin C
Where R is the circumradius and a, b, c, are lengths of the sides of ∆ABC.
Cosine rule:
In ∆ABC,
a2 = b2 + c2 – 2bc cos A ⟹cos A = 
b2 = a2 + c2 – 2ac cos B ⟹ cos B = 
c2 = a2 + b2 – 2ab cos C ⟹ cos A = 
projection rule:
In ∆ABC,
a = b cos C + c cos B
b = a cos C + c cos A
c = a cos B + b cos A
Tangent rule (Napier’s analogy):
In ∆ABC,
Half angle formulae and Area of the triangle:
In ∆ABC, a, b, and c are sides
and area of the triangle 
1.Half angle formulae: –
2.Formulae for ∆: –
∆ = ½ ab sinC= ½ bc sin A=½ ac sin B
where
= 2R2sin A sin B sinC
= r.s
=
=
In circle and Excircles of a triangle:
⋇The circle that touches the three sides of an ∆ABC internally is called ‘incircle’. The centre of the incircle is ‘I’ and the radius is ‘r’.
Formulae for ‘r’: –
r = 
= (s – a) tan
= (s – b) tan
= (s – c) tan
= 4R sinsin sin
=
⋇ The circle that touches the side BC internally and the other two sides AB and AC externally is called the ‘Excircle’ opposite to the angle A. Its centre is I1 and the radius is r1. A 
triangle has three ex circles. The remaining circles centre and radius are respectively I2, r2 and I3, r3.
Formulae for ‘r1’: –
r1 = 
= s tan
= (s – b) cot = (s – c) cot
= 4R sin cos cos
=
Formulae for ‘r2’: –
r2 = 
= s tan
= (s – c) cot = (s – a) cot
= 4R cos sin cos
=
Formulae for ‘r3’: –
r3 = 
= s tan
= (s – a) cot = (s – b) cot
= 4R cos cos sin
=
