Course

# 11th Standard - Maths Tutorial - NCERT - CBSE Pattern

We will cover below chapters.
• 1 - SETS
• 2 - RELATIONS AND FUNCTIONS
• 3 - TRIGONOMETRIC FUNCTIONS
• 4 - COMPLEX NUMBERS AND QUADRATIC EQUATIONS
• 5 - LINEAR INEQUALITIES
• 6 - PERMUTATIONS AND COMBINATIONS
• 7 - BINOMIAL THEOREM
• 8 - SEQUENCES AND SERIES
• 9 - STRAIGHT LINES
• 10 - CONIC SECTIONS
• 11 - INTRODUCTION TO THREE DIMENSIONAL GEOMETRY
• 12 - LIMITS AND DERIVATIVES
• 13 - STATISTICS
• 14 - PROBABILITY

# 1 - SETS

A set is a well-defined collection of objects.
• A set which does not contain any element is called empty set.
• A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.
• Two sets A and B are said to be equal if they have exactly the same elements.
• A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R.
• The union of two sets A and B is the set of all those elements which are either in A or in B.
• The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.
• The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.
For any two sets A and B,
$(A ∪ B)' = A' ∩ B' and ( A ∩ B )' = A' ∪ B'$

# 2 - RELATIONS AND FUNCTIONS

• Ordered pair A pair of elements grouped together in a particular order.
• Cartesian product A × B of two sets A and B is given by$A × B = {(a, b): a ∈ A, b ∈ B}$
In particular $R × R = {(x, y): x, y ∈ R}$  and $R × R × R = {(x, y, z): x, y, z ∈ R}$
• If (a, b) = (x, y), then a = x and b = y.
• If n(A) = p and n(B) = q, then n(A × B) = pq.
• A × φ = φ
• In general, A × B ≠ B × A.
• Relation A relation R from a set A to a set B is a subset of the cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B.
• The image of an element x under a relation R is given by y, where (x, y) ∈ R,
• The domain of R is the set of all first elements of the ordered pairs in a relation R.
• The range of the relation R is the set of all second elements of the ordered pairs in a relation R.
• Function A function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. We write f: A→B, where f(x) = y.
• A is the domain and B is the codomain of f.
• The range of the function is the set of images.
• A real function has the set of real numbers or one of its subsets both as its domain and as its range.
• Algebra of functions For functions f : X → R and g : X → R, we have
(f + g) (x) = f (x) + g(x), x ∈ X
(f – g) (x) = f (x) – g(x), x ∈ X
(f.g) (x) = f (x) .g (x), x ∈ X
(kf) (x) = k ( f (x) ), x ∈ X, where k is a real number.

# 3 - TRIGONOMETRIC FUNCTIONS

• If in a circle of radius r, an arc of length l subtends an angle of θ radians, then l = r θ
• $Radian = \frac {\pi} {180} \times Degree$
• $Degree = \frac {180} {\pi} \times Radian$
• $cos^2x + sin^2x = 1$
• $1 + tan^2x = sec^2x$
• $1 + cot^2x = cosec^2x$
• $cos(2n\pi + x) = cos x$
• $sin(2n\pi + x) = sin x$
• $sin(-x) = - sin (x)$
• $cos(-x) = cos (x)$
• $cos(x + y) = cos (x) cos (y) - sin (x) sin(y)$
• $cos(x - y) = cos (x) cos (y) + sin (x) sin(y)$
• $cos( \frac{\pi}{2} -x) = sin (x)$
• $sin( \frac{\pi}{2} -x) = cos (x)$
• $sin(x+y) = sin (x) cos(y) + cos(x) sin(y)$
• $sin(x-y) = sin (x) cos(y) - cos(x) sin(y)$
• $cos( \frac{\pi}{2} +x) = -sin (x)$

# 4 - COMPLEX NUMBERS AND QUADRATIC EQUATIONS

A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number. Let z1 = a + ib and z2 = c + id. Then
(i) z1 + z2 = (a + c) + i (b + d)
(ii) z1 z2 = (ac – bd) + i (ad + bc)
For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists the complex number
$\frac{a}{a^2 + b^2} + i \frac{-b}{a^2 + b^2} ) = -sin (x)$
denoted by $z^-1$ called the implicative inverse of z such that
$(a + ib) \frac{a}{a^2 + b^2} + i \frac{-b}{a^2 + b^2} ) = 1 + i0 = 1$
For any integer k,
$i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i$
The conjugate of the complex number z = a + ib, denoted by $\overline z$ , is given by   $\overline z = a - ib$ .

# 5 - LINEAR INEQUALITIES

• Two real numbers or two algebraic expressions related by the symbols $<, >, ≤ or ≥$ form an inequality.
• Equal numbers may be added to (or subtracted from ) both sides of an inequality.
• Both sides of an inequality can be multiplied (or divided ) by the same positive number. But when both sides are multiplied (or divided) by a negative number, then the inequality is reversed.
• The values of x, which make an inequality a true statement, are called solutions of the inequality.
• To represent $x < a (or \text { } x > a)$ on a number line, put a circle on the number a and dark line to the left (or right) of the number a.
• To represent x ≤ a (or x ≥ a) on a number line, put a dark circle on the number a and dark the line to the left (or right) of the number x.

# 6 - PERMUTATIONS AND COMBINATIONS

Fundamental principle of counting If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m × n. The number of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by  $^nP_r$ is given by $^nP_r = \frac {n!}{(n-r)!}$where $0 <= r <= n$The number of permutations of n different things, taken r at a time, where repeatition is allowed, is $n^r$. The number of permutations of n objects taken all at a time where $p_1$ objects are of first kind, $p_2$ objects are of the second kind, ..., $p_k$objects are of the $k^{th}$ kind and rest, if any, are all different is
$\frac {n!} {p_1! p_2! ...p_k!}$
The number of combinations of n different things taken r at a time, denoted by$^nC_r$ is given by
$^nC_r = \frac {n!} {r! (n-r)!} , 0<=r<=n$

# 7 - BINOMIAL THEOREM

The expansion of a binomial for any positive integral n is given by Binomial Theorem, which is
$(a + b)^n=^nC_0a^n + ^nC_1 a^{n-1}b + ^nC_2 a^{n-2}b^2 + ...... + ^nC_n b^n$
The coefficients of the expansions are arranged in an array. This array is called Pascal's triangle.

# 8 - SEQUENCES AND SERIES

By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type  ${1, 2, 3, ....k}$ . A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence. Let a1, a2, a3, ... be the sequence, then the sum expressed as a1 + a2 + a3 + ... is called series. A series is called finite series if it has got finite number of terms. A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the $n^{th}$ term of GP is given by
$a_n = ar^{n-1}$
. The sum S_n of the first n terms of GP is given by
$S_n = \frac {a(a^n-1)} {r-1} or \frac {a(1 - r^n)} {1-r} , \text { if } r \neq 1$
. The geometric mean (G.M.) of any two positive numbers a and b is given by
$\sqrt {ab}$
. i.e., the sequence a, G, b is G.P.

# 9 - STRAIGHT LINES

Slope (m) of a non-vertical line passing through the points (x1, y1) and (x2, y2) is given by
$m = \frac {y2-y1} {x2-x1} = \frac {y1-y2} {x1-x2} , x1 \neq x2$
If a line makes an angle α with the positive direction of x-axis, then the slope of the line is given by
$m = tan α, α ≠ 90°$
Slope of horizontal line is zero and slope of vertical line is undefined. An acute angle (say θ) between lines L1 and L2 with slopes m1 and m2 is given by
$tan \theta = \lvert \frac {m_2-m_1}{1 + m_1m_2} \rvert , 1 + m_1m_2 \neq 0$
• Two lines are parallel if and only if their slopes are equal.
• Two lines are perpendicular if and only if product of their slopes is -1.
• Three points A, B and C are collinear, if and only if slope of AB = slope of BC.
Equation of the horizontal line having distance a from the x-axis is
$\text {either } y = a \text{ or } y = - a$
Equation of the vertical line having distance b from the y-axis is
$\text {either } x = b \text{ or } x = - b$
The point (x, y) lies on the line with slope m and through the fixed point$(x_0, y_0)$ , if and only if its coordinates satisfy the equation
$y - y_0 = m (x - x_0).$
Equation of the line passing through the points (x1, y1) and (x2, y2) is given by
$y - y_1 = \frac {y_2-y_1} {x_2-x_1} (x-x_1)$
The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c.
If a line with slope m makes x-intercept d. Then equation of the line is y = m (x - d).
Equation of a line making intercepts a and b on the x-and y-axis, respectively, is
$\frac {x} {a} + \frac {y} {b} = 1$
Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.
The perpendicular distance (d) of a line Ax + By+ C = 0 from a point (x1, y1) is given by
$d = \frac { \lvert Ax_1 + By_1 + C \lvert } {\sqrt {A^2 + B^2}}$
Distance between the parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0, is given by
$d = \frac { \lvert C_1 - C_2 \lvert } {\sqrt {A^2 + B^2}}$

# 10 - CONIC SECTIONS

A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. The equation of a circle with centre (h, k) and the radius r is
${ (x-h)^2 } + (y-k)^2 = r^2$
A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. The equation of the parabola with focus at $(a, 0) a > 0$and directrix x = - a is
$y^2 = 4ax$
Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.

Length of the latus rectum of the parabola  $y^2 = 4ax$   is 4a. An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
The equation of an ellipse with foci on the x-axis is
$\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$
Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.

Length of the latus rectum of the ellipse
$\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$
is
$\frac {2 b^2} {a}$
The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

The equation of a hyperbola with foci on the x-axis is
$\frac {x^2} {a^2} - \frac {y^2} {b^2} = 1$
Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.

Length of the latus rectum of the hyperbola
$\frac {x^2} {a^2} - \frac {y^2} {b^2} = 1$
is
$\frac {2 b^2} {a}$
The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola.

# 11 - INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

In three dimensions, the coordinate axes of a rectangular Cartesian coordinate system are three mutually perpendicular lines. The axes are called the x, y and z-axes. ®The three planes determined by the pair of axes are the coordinate planes, called XY, YZ and ZX-planes. ®The three coordinate planes divide the space into eight parts known as octants. ®The coordinates of a point P in three dimensional geometry is always written in the form of triplet like (x, y, z). Here x, y and z are the distances from the YZ, ZX and XY-planes.
• Any point on x-axis is of the form (x, 0, 0)
• Any point on y-axis is of the form (0, y, 0)
• Any point on z-axis is of the form (0, 0, z)
Distance between two points P(x1, y1, z1) and Q (x2, y2, z2) is given by
$\sqrt {(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 }$

# 12 - LIMITS AND DERIVATIVES

Limits and derivatives fall under calculus. So let us try to understand calculus first. The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. Limit of a function at a point is the common value of the left and right hand limits, if they coincide. For a function f and a real number a, $\lim\limits_{x \to a} f(x)$  and $f(a)$ may not be same (In fact, one may be defined and not the other one). For functions f and g the following holds:
$\lim\limits_{x \to a} [f(x) \pm g(x)] = \lim\limits_{x \to a} f(x) \pm \lim\limits_{x \to a} g(x)$
$\lim\limits_{x \to a} [f(x) \cdot g(x)] = \lim\limits_{x \to a} f(x) \cdot \lim\limits_{x \to a} g(x)$
$\lim\limits_{x \to a} [ \frac {f(x) } { g(x)}] = \frac {\lim\limits_{x \to a} f(x)} {\lim\limits_{x \to a} g(x)}$
Following are some of the standard limits
$\lim\limits_{x \to a} [ \frac {x^n - a^n } { x -a }] = na^{n-1}$
$\lim\limits_{x \to 0} [ \frac {sin(x) } { x }] = 1$
$\lim\limits_{x \to 0} [ \frac {1 - cos(x) } { x }] = 0$
The derivative of a function f at a is defined by
$f'(a) = \lim\limits_{h \to 0} [ \frac { f(a+h) - f(a) } { h }]$
Derivative of a function f at any point x is defined by
$f'(x) = \frac {df(x)} {dx} = \lim\limits_{h \to 0} [ \frac { f(x+h) - f(x) } { h }]$
For functions u and v the following holds
$(u \pm v)' = u' \pm v'$
$(u v)' = u'v + uv'$
$( \frac{u} {v} )' = \frac {u'v - uv'} { v^2}$
provided all are provided

Some standard derivatives -
$\frac {d} {dx} (x^n) = nx^{n-1}$
$\frac {d} {dx} (sin (x)) = cos(x)$
$\frac {d} {dx} (cos(x)) = -sin (x)$

# 13 - STATISTICS

Measures of dispersion Range, Quartile deviation, mean deviation, variance, standard deviation are measures of dispersion.

Range = Maximum Value – Minimum Value

Mean deviation for ungrouped data
$M.D. ( \overline x) = \frac {\sum \lvert x_i - \overline x \rvert} {n}$
$M.D. ( M) = \frac {\sum \lvert x_i - M \rvert} {n}$
Mean deviation for grouped data
$M.D. ( \overline x) = \frac {\sum f_i \lvert x_i \overline x \rvert} {N}$
$M.D. ( M) = \frac {\sum f_i \lvert x_i M \rvert} {N}$
where$N = \sum f_i$

Variance and standard deviation for ungrouped data
$\sigma ^2 = \frac {1}{n} {\sum (x_i - \overline x)^2}$
,
$\sigma = \sqrt { \frac {1}{n} {\sum (x_i - \overline x)^2} }$
Variance and standard deviation of a discrete frequency distribution
$\sigma ^2 = \frac {1}{N} {\sum f_i (x_i - \overline x)^2}$
,
$\sigma = \sqrt { \frac {1}{N} {\sum f_i (x_i - \overline x)^2} }$
Variance and standard deviation of a continuous frequency distribution
$\sigma ^2 = \frac {1}{N} {\sum f_i (x_i - \overline x)^2}$
,
$\sigma = \frac {1}{N} \sqrt { N {\sum f_i x_i^2 - (\sum f_i x_i )^2} }$
Shortcut method to find variance and standard deviation.
$\sigma ^2 = \frac {h^2}{N^2} [N \sum f_iy_i^2 - (\sum f_i y_i )^2]$
,
$\sigma = \frac {h}{N} \sqrt {[N \sum f_iy_i^2 - (\sum f_i y_i )^2] }$
, where
$y_i = \frac {x_i - A}{h}$

# 14 - PROBABILITY

• Event: A subset of the sample space
• Impossible event : The empty set
• Sure event: The whole sample space
• Complementary event or 'not event' : The set A' or S - A
• Event A or B: The set A ∪ B
• Event A and B: The set A ∩ B
• Event A and not B: The set A – B
Mutually exclusive event: A and B are mutually exclusive if A ∩ B = φ Exhaustive and mutually exclusive events: Events E1, E2,..., En are mutually exclusive and exhaustive if E1 ∪ E2 ∪ ...∪ En = S and Ei ∩ Ej = φ V i ≠ j Probability: Number P ($ω_i$ ) associated with sample point $ω_i$ such that (i) 0 ≤ P ($ω_i$) ≤ 1 (ii) Σ P ($ω_i$) for all $ω_i$ ∈ S = 1 (iii) P(A) = Σ P($ω_i$) for all $ω_i$ ∈ A. The number P$ω_i$ is called probability of the outcome $ω_i$.

Equally likely outcomes: All outcomes with equal probability

Probability of an event: For a finite sample space with equally likely outcomes Probability of an event
$P(A) = \frac {n(A)} {n(S)}$
, where n(A) = number of elements in the set A, n(S) = number of elements in the set S.

If A and B are any two events, then P(A or B) = P(A) + P(B) – P(A and B) equivalently, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive, then P(A or B) = P(A) + P(B)

If A is any event, then P(not A) = 1 – P(A)