The document contains questions related to trigonometric functions, sets, relations and functions, complex numbers, and sequences and series. Some questions ask students to prove trigonometric identities, find sets operations, determine if relations are functions, solve complex equations, and evaluate infinite geometric series. The document provides hints for many questions and includes the answers for some questions.

1. ASSIGNMENT FOR --XI
TRIGONOMETRIC FUNCTIONS
Some valuable points to remember:
Law of sine :
Law of cosine :
Q .1 Find in degrees and radians the angle subtended b/w the hour
hand and the minute hand Of a clock at half past three. [answer
is 750
, 5𝝅/12]
Q.2 Prove that: tan300
+ tan150
+ tan300
. tan150
=1.
[hint: take tan450
= tan(300
+150
)

2. Q.3 Prove that (1+cos
𝝅
𝟖
) (1+ cos
𝟑𝝅
𝟖
) (1+ cos
𝟓𝝅
𝟖
) (1+ cos
𝟕𝝅
𝟖
) =
𝟏
𝟖
.
[Hint : cos
𝟕𝝅
𝟖
= cos( π -
𝝅
𝟖
) = - cos
𝝅
𝟖
, cos
𝟓𝝅
𝟖
= cos ( π -
𝟑𝝅
𝟖
) = -
cos
𝟑𝝅
𝟖
]
Q.4 (i) Prove that sin200
sin400
sin600
sin800
=
𝟑
𝟏𝟔
[Hint : L.H.S. sin200
sin400
sin600
sin800
(√ 𝟑/𝟐)
⇨
√𝟑
𝟐×𝟐
(2sin200
sin400
sin800
) ⇨
√𝟑
𝟒
[(cos200
– cos600
) sin800
(ii) Prove that: cos
𝝅
𝟕
cos
𝟐𝝅
𝟕
cos
𝟒𝝅
𝟕
= -
𝟏
𝟖
.
[Hint: let x =
𝝅
𝟕
, then
𝟏
𝟐𝒔𝒊𝒏𝒙
(2sinx cosx cos2x cos4x)]
Sin2x
(iii) Prove that: tan200
tan400
tan800
= tan600
.
[hint: L.H.S.
𝒔𝒊𝒏𝟐𝟎 𝟎 𝒔𝒊𝒏𝟒𝟎 𝟎 𝒔𝒊𝒏𝟖𝟎 𝟎
𝒄𝒐𝒔𝟐𝟎 𝟎 𝒄𝒐𝒔𝟒𝟎 𝟎 𝒄𝒐𝒔𝟖𝟎 𝟎 solve as above method.]
Q.5 If cos(A+B) sin(C-D) = cos(A-B) sin(C+D) , then show that tanA
tanB tanC + tanD = 0
[Hint: We can write above given result as
𝐜𝐨𝐬(𝐀+𝐁)
𝐜𝐨𝐬(𝐀−𝐁)
=
𝐬𝐢𝐧(𝐂+𝐃)
𝐬𝐢𝐧(𝐂−𝐃)
By
C & D
𝐜𝐨𝐬( 𝐀+𝐁)+𝐜𝐨𝐬(𝐀−𝐁)
𝐜𝐨𝐬( 𝐀+𝐁)−𝐜𝐨𝐬(𝐀−𝐁)
=
𝐬𝐢𝐧( 𝐂+𝐃)+𝐬𝐢𝐧(𝐂−𝐃)
𝐬𝐢𝐧( 𝐂+𝐃)−𝐬𝐢𝐧(𝐂−𝐃)
]
Q.6 If 𝛂, are the acute angles and cos2𝛂 =
𝟑𝒄𝒐𝒔𝟐𝜷−𝟏
𝟑−𝒄𝒐𝒔𝟐𝜷
, show that
tan 𝛂 = √𝟐 tan𝛃.

3. [Hint: According to required result , we have to convert given
part into tangent function By using cos2𝛂 =
𝟏−𝒕𝒂𝒏²𝜶
𝟏+𝒕𝒂𝒏²𝜶
∴ we will get
𝟏−𝒕𝒂𝒏²𝜶
𝟏+𝒕𝒂𝒏²𝜶
=
𝟑(
𝟏−𝒕𝒂𝒏²𝜷
𝟏+𝒕𝒂𝒏²𝜷
)−𝟏
𝟑−
𝟏−𝒕𝒂𝒏²𝜷
𝟏+𝒕𝒂𝒏²𝜷
=
𝟑−𝟑𝒕𝒂𝒏²𝜷−𝟏−𝒕𝒂𝒏²𝜷
𝟑+𝟑𝒕𝒂𝒏²𝜷−𝟏+𝒕𝒂𝒏²𝜷
[
Q.7 Solve the equation: sin3θ + cos2θ = 0
{ Answer: -θ = nπ+
𝝅
𝟐
and 5θ = nπ-
𝝅
𝟐
, ∀ nєz. }
Q.8 Solve: 3cos2
x - 2 √ 𝟑 sinx .cosx – 3sin2
x = 0. [ans. X=
n𝝅+𝝅/𝟔 or X= n𝝅-𝝅/𝟑 ∀ nєz. ]
Q.9 In △ ABC, if acosA= bcosB , show that the triangle is either isosceles
or right angled.
[hint: ksinAcosA= k sinBcosB ⇨ sin2A= sin2B ]
Q.10 Prove that
𝒄−𝒃𝒄𝒐𝒔𝑨
𝒃−𝒄𝒄𝒐𝒔𝑨
=
𝒄𝒐𝒔𝑩
𝒄𝒐𝒔𝑪
[use cos A =
𝒃 𝟐+𝒄 𝟐−𝒂 𝟐
𝟐𝒃𝒄
on L.H.S]

4. SETS
Some valuable points to remember:
1. If A and B are finite sets, and A B = then number
of elements
in the union of two sets
n(AUB)= n(A) + n(B)
2. If A and B are finite sets, A B ≠ then
n(AU B ) = n(A) + n(B) - n(A ∩B)
3. n(A B) = n(A – B) + n(B – A) + n(A B)
4. n(A B  C) = n(A) + n(B) + n(C) – n(B∩C) – n(A∩B) –
n(A∩C) +n(A∩B∩C)
(A) Number of elements in the power set of a set with n
elements =2n
.
(B)Number of Proper subsets in the power set = 2n
-2
Venn diagram of only A or A – B= A∩B’
A ⊆B(subset), Set A is a subset of set B because all
members of set A are in set B. If they are not, we have a
proper subset.
A ⊂B(proper subset)
Questionbank for recapitulation:

5. Q. 1 If U = {1,2,3......,10} , A = {1,2,3,5}, B = {2,4,6,7}, then
find (A-B)’.
Q.2 In a survey it was found that 21 people liked product A,
26 liked product B and 29 liked product C. If 14 people liked
products A and B, 12 people liked products C and A, 14
people liked products B and C and 8 liked all the three
products. Find how many liked product C only, ProductA
and C but not product B , atleast one of three products.
[Answer n(A’∩B’∩C’)=11, n(A∩B’∩C)=4 , n(AUBUC)=44.]
Question: 3 If U = {x :x ≤ 10, x∈ N}, A = {x :x ∈ N, x is prime},
B = {x : x∈ N, x is even}. Write A ∩B’ in roster form.
Question: 4 In a survey of 5000 people in a town, 2250
were listed as reading English Newspaper, 1750 as reading
Hindi Newspaper and 875 were listed as reading both Hindi
as well as English. Find how many People do not read Hindi
or English Newspaper.Find how many people read only
English Newspaper. [Answer:People read only English
Newspaper n(E’∩H) = n(E) – n(E∩ H) = 1375. ]
Q.5 A and B are two sets such that n(A-B) = 14 + x, n(B-A) =
3x and n(A ∩B) =x. If n(A) = n(B) then find the value of
x.[Answer n(A-B) = 14 + x= n(A ∩B’) = n(A) - n(A ∩B)⇨ x=7]
Q.6 (i) Write roster form of {x:
𝒏
𝒏²+𝟏
and 1≤ n ≤3 , n∈ N}
(ii) Write set-builder form of {-4,-3,-2,-1,0,1,2,3,4}
Q.7 If A ={1}, find number of elements in P[P{P(A)}].

6. RELATIONS & FUNCTIONS
Some valuable points to remember:
1. AXB = { (a,b): a ε A, b ε B} for non-empty sets A,B otherwise
AXB=φ, this is the set of all ordered pairs of elements from A and
B. n(AXB)=n(A)Xn(B)
2. AXB,BXA are not equal (not commutative).
3. If R is relation on a finite set having n elements, then the number
of relations on A is 2nxn .
4. Is the relation a function?{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}
Since x = 2 gives me two possible destinations, then this relation is
not a function.
5. Inverse Function is a relation from B to A defined by
{ (b,a) :(a,b)εR}
for modulas function(|x|).
6.Graph of f : R → R such that f(x) = |x+1|
Questionbank for recapitulation:
Q.1 Find the domain and range of f(x) =
|𝒙−𝟒|
𝒙−𝟒
. [ans. Domain of f =
R – {4}, Range of f = {-1, 1}]
Q.2 Let f(x) = , find f(-1) , f(3).

7. Q.3 If Y= f(x)= , then find x=f(y).
Q.4 Let R be the relation on the set N of natural numbers defined
by R = {(a, b): a+3b = 12, a, bЄ N }.Find R, domain of R and
range of R. {ans. (9,1),(6,2),3,3)}
Q.5 Write the domain & range of f(x) = 1/ (5x – 7).
Q.6 If f : R → R Be defined by f(x) = x2
+2x+1 then find
(i) f(-1) x f(1) ,is f(-1)+f(1)=f(0)
(ii) f(2) x f(3) , is f(2) x f(3)=f(6)
Q.7 If f(x)= x2
–
𝟏
𝑿 𝟐, then find the value of : f(x) +f(1/x) . [ans.
0]
Q.8 Find the domain and Range of the function
f(x) =
𝟏
𝟐−𝒔𝒊𝒏𝟑𝒙
. {Domain = R , Range = [1/ 3 , 1] }
Q.9 Draw a graph of f : R → R defined by f(x) = |x-2|
Q.10 If R is a relation from set A={11,12,13} to set
B={8,10,12} defined by y=x-3,then write R-1
.
{Ans.(8,11),(10.13)}

8. COMPLEX NUMBERS & QUADRATIC EQUATIONS
Some valuable points to remember:
A complex number z is a number of the form z = x + yi.
Its conjugate is a number of the form = x- yi.
z = (x+ yi)(x- yi) = x2
+ y2
= |z|2
Triangle Inequality:
1. |z1 + z2| |z1| + |z2|
2. |z1 + z2| |z1| - |z2|
3. |z1 - z2| |z1| - |z2|
4. |z1 + z2 + z3| |z1| + |z2| + |z3|
5. |z1z2| = |z1||z2|
Evaluate i203
. We have to replace 203 with its remainder on division
by 4. 203 = 4X50 + 3; i203
= i3
= – i
ϴ, x>0, y>0
π - ϴ, x<0, y>0
arg (z) = ϴ - π , x<0, y<0
- ϴ, x>0, y<0
Can be used in finding principal argument of complex numbers.
Questionbank for recapitulation:

9. Q.1 Solve X2
– (3√2 – 2i) x - 6√2i = 0 (3√2, -2i)
Q.2 If z =
𝟏
( 𝟏−𝒊)(𝟐+𝟑𝒊)
, then |z| is [ans. 1/√(26) ]
Q.3 If z = (
𝟏+𝒊
𝟏−𝒊
), then z4 is [ans. 1]
Q.4 If z =
𝟏
𝟏−𝒄𝒐𝒔𝝋−𝒊𝒔𝒊𝒏𝝋
, then Re(z) is [ans. ½]
Q.5 If (1 - i) (1 - 2i)(1 - 3i)...........(1 - ni) = (x - yi) , show that
2.5.10...........(1+n2) = x2+y2
Q.6Express in polar form:
−𝟐
𝟏+𝒊√𝟑
. [ ans. cos2π/3 +isin2 π/3]
Q.7 If x = - 5 +2√(-4) , find the value of x4
+9x3
+35x2
– x+4.
[Hint: Divide given poly. By x2 +10x +41=0 as (x+5)2= (4i)2
Remainder is -160 ]
Q.8 Q.10 Find the values of x and y if x2
– 7x +9yi and y2
i+20i –
12 are equal.
[ans. x =4, 3 and y =5, 4.]

10. SEQUENCE & SERIES
Some valuablepoints to remember:
S∞ =
𝒂
𝟏−𝒓
, a is firstterm, r isthecommonratio. Asn→∞ rn
→ 0 for |r|<1.
Geometric mean betweentwo numbers a & b is √ 𝒂𝒃 i.e., G2
= ab or G =
√ 𝒂𝒃 , Harmonic mean (H.M.)=
𝟐𝒂𝒃
𝒂+𝒃
, If a , b, c are in G.P. then b/a = c/b ⇨
b2
= ac. A – G =
𝒂+𝒃
𝟐
- √ 𝒂𝒃 ≥ 0 ⇨ A ≥ G≥H, R = (
𝒃
𝒂
)
𝟏
𝒏+𝟏 , if a,b,c are in
A.P. then 2b = a+c , d =
𝒃−𝒂
𝒏+𝟏
and nth
term = Tn = Sn – Sn-1 and last term in
G.P. is arn-1 , A.M. b/w two numbers a & b is (a+b)/2 .
Questionbank for recapitulation:
Q. 1 Find k so that 2/3, k, 5k/8 are in A.P.
Q. 2 There are n A.M.’s between 7 and 85 such that (n-3)th mean :
nth mean is 11 : 24.Find n. [Hint: 7, a2, a3, a4,…………..an+1, 85 , n=5]
Q. 3 Find the sum of (1+
𝟏
𝟐 𝟐) + (
𝟏
𝟐
+
𝟏
𝟐 𝟒) + (
𝟏
𝟐 𝟐 +
𝟏
𝟐 𝟔) +…….to ∞
[answer is 7/3 ]
Q.4 If a, b, c, d are in G.P. prove that a2
-b2
, b2
-c2
, c2
-d2
are also in
G.P. [HINT: take a, b= ar, c=ar2
, d = ar3
]
Q.5 If one geometric mean G and two arithmetic mean p and q be
inserted between two quantities Show that G2
= (2p-q)(2q-p).
[Hint: G2 = ab, a, p, q, b are in A.P. d= (b-a)/3 ,p = a+d=
𝟐𝒂+𝒃
𝟑
, q =
a+2d =
𝒂+𝟐𝒃
𝟑
]
Q. 6 In an increasing G.P., the sum of first and last term is 66, and
product of the second and last but one term is 128. If the sum of
the series is 126, find the number of terms in the series.
[Ans. r=2 and n = 6]

11. Permutation & combination
Some valuablepoints to remember:
Permutation:
An arrangement is called a Permutation. It is the
rearrangement of objects or symbols into distinguishable
sequences. When we set things in order, we say we have made an
arrangement. When we change the order, we say we have changed
the arrangement. So each of the arrangement that can be made by
taking some or all of a number of things is known as Permutation.
Combination:
A Combination is a selection of some or all of a number of
different objects. It is an un-ordered collection of unique sizes.In a
permutation the order of occurence of the objects or the
arrangement is important but in combination the order of
occurence of the objects is not important.
Formula:
Permutation = n
Pr = n! / (n-r)!
Combination = n
Cr = n
Pr / r!
where,
n, r are non negative integers and 0≤ r≤n.
r is the size of each permutation.
n is the size of the set from which elements are permuted.
! is the factorial operator.
Questionbank for recapitulation:
Q.1 A child has 3 pocket and 4 coins. In how many ways can he put
the coins in his pocket. [Ans. Use 𝒏 𝒓
, 81]
Q.2 The principal wants to arrange 5 students on the platform such
that the boy SALIM occupies the second position and such

12. that the girl SITA is always adjacent to the girl RITA . How many
such arrangements are possible? [answer is 8]
Q. 3 When a group- photograph is taken, all the seven teachers
should be in the first row and all the twenty students should be in
the second row. If the two corners of the second row are reserved
for the two tallest students, interchangeable only b/w them, and if
the middle seat of the front row is reserved for the principal, how
many such arrangements are possible? [total no. Of ways =
6!.(18)!.2! ]
Q.4 A team of 8 students goes on an excursion, in two cars, of
which one can seat 5 and the other only 4. In how many ways can
they travel? [ Hint: 8
C3 + 8
C4 = 56 + 70 = 126.]
Q.5 If there are six periods in each working day of a school, in how
many ways can one arrange 5 subjects such that each subject is
allotted at least one period? [ P(6,5)X5 ans. 3600.]
Q. 6 Out of 7 consonants and 4 vowels, how many words of 3
consonants and 2 vowels can be formed? [ans. Required number
of ways = (210 x 120) = 25200]
Q. 7 How many numbers greater than 4,00,000 can be formed by
using the digits 0, 2, 2, 4, 4, 5? [ans. total no. Of ways 90.]

13. BINOMIAL EXPANSION
Some valuablepoints to remember:
Binomial theorem formula
general term = (r+1)th
term= Tr+1 = n
Cr xn-r
. ar
(i) n
Cr + n
Cr-1 = n+1
Cr (ii) n
Cx = n
Cy x = y or x + y = n
Questionbank for recapitulation:
Q. 1 If the co-efficient of x in (x2
+
𝒌
𝒙
)5
is 270 , then find k.
Q.2 Show that 2(4n+4)
– 15n – 16 is divisible by 225 ∀ n∊N.
Q.3 Find the middle term in the expansion of (
𝟐𝒙 𝟐
𝟑
+
𝟑
𝟐𝒙 𝟐 )10
Q.4 Let ‘n’ be a positive integer. If the coefficients of second, third
and fourth terms in (1+x)n
are in arithmetic progression, then find
the value of n.
Q.5 If the coefficients of (2r + 1)th term and (r + 2)th term in the
expansion of (1 + x)43
are equal, find ‘r’.
Q.6 Find the 4th
term from the end in the expansion of [
𝒙 𝟑
𝟐
-
𝟐
𝒙 𝟒 ] 7
[ans. 1. k =3 ,3. 252, 4. n= 7, 5. 2r + r + 1 = 43, 6. 70x]

14. STRAIGHT LINES
Some valuablepoints to remember:
1.Slopes of parallel lines: Slopes are equal or
where m1 and m2 are the slopes of the lines L, and L2,
respectively.
2. Slopes of perpendicular lines: Slopes are negative reciprocals or
3.Point-slope form of a straight line:
4.Slope-intercept form of a straight line: y = mx + b
where b is the y- intercept.
5.Normal form of a straight line:
where p is the line's perpendicular distance from the origin and Ѳ
is the angle between the perpendicular and the X- axis
6.Distance from a point to a line:
Questionbank for recapitulation:

15. Q. 1 Determine the ratio in which the line 3x+y – 9 = 0 divides the
segment joining the points (1,3) and (2,7).
[ Hint: use section formula , k=3/4]
Q.2 (i) Find the co-ordinates of the orthocenter of the ∆ whose
angular points are (1,2), (2,3), (4,3) (ii) Find the co-ordinates of the
circumcenter of the ∆ whose angular points are (1,2), (3,-4), (5,-6).
[ answer (i) (1,6) (ii) (11,2)]
Q. 3 Find the equation of the line through the intersection of the
lines x -3y+1=0 and 2x+5y -9=0 and whose distant from the origin is
√𝟓 . [ Hint: (x -3y+1) +k(2x+5y -9)=0 -------(1) , then find distant
from (0,0) on the line (1) ,Answer is 2x+y – 5=0]
Q. 4 The points (1,3) and (5,1) are the opposite vertices of a
rectangle.The other two vertices lie on the line y = 2x+c. Find c and
the remaining vertices.
[Hint: D(𝜶, 𝜷) C(5,1)
y=2x+c
M (3,2)
A(1,3) B(X,2X-4)
c = -4 , use pythagoras B(2,0) then D(4,4)]
Q.5 The extremities of the base of an isosceles ∆ are the points
(2a,0) & (0,a). The equation of the one of the sides is x=2a. Find the
equation of the other two sides and the area of the ∆. [ ans. Given
points A&B then C (2a,5a/2), x+2y-2a=0 & 3x – 4y+4a=0 & area of
∆ACB is 5a2/2 sq.units.]

16. CONIC SECTION
Some valuablepoints to remember: A conic section is the
locus of all points in a plane whose distance from a fixed point is a
constant ratio to its distance from a fixed line. The fixed point is the
focus, and the fixed line is the directrix.The ratio referred to in the
definition is called the eccentricity (e)
Questionbank for recapitulation:
Q. 1 A circle has radius 3 & its centre lies on the y = x-1. Find the
eqn. of the circle if it passes through (7,3).[hint: h=4,7 k = 3,6]
Q.2 Find the eqn. of circle of radius 5 which lies within the circle
x2
+y2
+14x+10y – 26 = 0 and which touches the given circle at the
point (-1,3). [ans. (x+4)2
+ (y+1)2
= 52
]
Q.3 Find the eqn. of circle circumscribing the ∆ formed by the lines
x + y = 6, 2x+y = 4 & x+2y = 5. [ans. (7,-1), (-2,8) & (1,2) is x2
+y2
-3x-
2y -21=0]
Q.4 Reduce the equation to standard form,find it’s vertex
. [eqn. of the parabola with its vertex at (-1,3).]
Q.5 Find the eqn. of ellipse whose centre is at origin , foci are (1,0)
& (-1,0) and e=1/2. [hint: PF+PF’ =2a , we get 3x2
+4y2
-12=0]
Q.6 Find the eqn. of hyp. Whose conjugate axis is 5 and the
distance b/w the foci is 13.[ ans. 25x2
– 144y2
= 900]

17. LIMITS
Some valuable points to remember:
, 𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒙−𝟏
𝒙
=1 , 𝐥𝐢𝐦
𝒙→𝟎
𝐥𝐨𝐠(𝟏+𝒙)
𝒙
=1, 𝐥𝐢𝐦
𝒙→𝒂
𝒙 𝒏−𝒂 𝒏
𝒙−𝒂
= nan-1
e-∞ =0, e∞ =∞, 𝐥𝐢𝐦
𝒙→𝟎
𝟏
𝒙
=∞, 𝐥𝐢𝐦
𝒙→∞
𝟏
𝒙
=0 , {0/0 form 𝐥𝐢𝐦
𝑿→𝟏
𝑿²−𝟏
𝑿−𝟏
)
c∞ = ∞ if c > 1
= 0 , 0 ≤ c ≤ 1
= 1 , c = 1.
Question bank for recapitulation:
Q.1 𝐥𝐢𝐦
𝒙→𝟑+
𝒙
[𝒙]
and 𝐥𝐢𝐦
𝒙→𝟑−
𝒙
[𝒙]
where [x] denotes the integral part of x. Are they
equal?
Q.2 Is 𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒙−𝟏
√ 𝟏−𝒄𝒐𝒔𝒙
exist?
Q. 3 Evaluate:(i) 𝐥𝐢𝐦
𝒙→𝝅
𝒔𝒊𝒏𝟑𝒙−𝟑𝒔𝒊𝒏𝒙
(𝝅−𝒙)³
[answer is -4]
(ii) 𝐥𝐢𝐦
𝒙→𝝅/𝟐
𝒄𝒐𝒕𝒙−𝒄𝒐𝒔𝒙
𝒄𝒐𝒔³𝒙
[ answer is ½]
(iii) 𝐥𝐢𝐦
𝒙→𝟎
𝒆 𝒔𝒊𝒏𝟑𝒙−𝟏
𝐥𝐨𝐠(𝟏+𝒕𝒂𝒏𝟐𝒙)
[ans. 3/2]
Q.4 Let f(x) = {
𝟑 − 𝒙² , 𝒙 ≤ −𝟐
𝒂𝒙 + 𝒃 , − 𝟐 < 𝑥 < 2
𝒙 𝟐
𝟐
, 𝒙 ≥ 𝟐
Finda,b sothat 𝐥𝐢𝐦
𝒙→𝟐
𝒇(𝒙) and 𝐥𝐢𝐦
𝒙→−𝟐
𝒇(𝒙)exist. [ans. a=3/4,b=1/2]
Q.5 Find k such that following functionis continuous at indicatedpoint

18. f(x) ={
𝟏−𝒄𝒐𝒔𝟒𝒙
𝟖𝒙²
, 𝒙 ≠ 𝟎
𝒌 , 𝒙 = 𝟎
[hint: if L.H.Lt=R.H.Lt=f(a)→f is cts. at x=a,k=1]
DERIVATIVES
Some valuable points to remember:
First principle(ab-initio)
Leibnitz product rule, Quotient rule
Apply the chain rule in composition of functions
Question bank for recapitulation:
Q.1 Find the derivative of the following functions
from first principle:
(i) sin (x + 1)
(ii)
(iii) √ 𝒔𝒊𝒏𝒙
Q.2 Find the derivative of
(i)
(ii)
𝒔𝒊𝒏𝒙+𝒙𝒄𝒐𝒔𝒙
𝒙𝒔𝒊𝒏𝒙−𝒄𝒐𝒔𝒙
(iii) 𝒄𝒐𝒔 𝟑
(√ 𝒙 𝟐 + 𝟐)
Q.3 If y =
𝒙
𝒙+𝒂
, prove that x
𝒅𝒚
𝒅𝒙
= y(1-y).

19. Q.4 Write the value of derivative of
𝟏+𝒕𝒂𝒏𝒙
𝟏−𝒕𝒂𝒏𝒙
at x = 0 [ans. Is 2]
PROBABILITY
Some valuable points to remember:
1. P(A∩B’) = P(A-B) = P(A) – P(A∩ B) = P( Only A)
2. P(A’∩ B’) = P(AUB)’ = 1 – P(AUB) and P(A’UB’) = 1 – P(A∩B)
3. P(at least one) = 1 – P(None) = 1 – P(0)
4. For any two events A and B,
P(A∩B) ≤ P(A)≤P(AUB)≤P(A)+P(B)
Question bank for recapitulation:
Q.1 Three squares of chess board are selected at random. Find the
porb. Of getting 2 squares of one Colour and other of a different
colour. [ans. Is 16/21]
Q.2 A box contains 100 bolts and 50 nuts, It is given that 50% bolt
and 50% nuts are rusted. Two Objects are selected from the box at
random. Find the probability that both are bolts or both are rusted.
[ans. Is 260/447.]
Q.3 Fine the probability that in a random arrangement of the
letters of the word “UNIVERSITY” the two I’s come together.
[ans. Is. 1/5 ]
Q.4 A five digit number is formed by the digits 1, 2, 3, 4, 5 without
repetition. Find the probability that the number is divisible by 4.
[ans. Is. 1/5 ]
Q.5 A pair of dice is rolled. Find the probability of getting a doublet
or sum of number to be at least 10.
[ans. is P(AUB) = 5/18.]
Q.6 Two unbiased dice are thrown. Find the prob. That neither a
doublet nor a total of 10 will appear. [ Answer: is 7/9. ]

20. Q.7 The prob. Of occurrence of atleast one of the events A & B
Is 0.6.If A & B occur simultaneously with a prob. Of 0.2. find
P(A’)+P(B’) [ P(A∩B)= 0.2 ans. 1.2]