MESSAGE FROM DEPUTY COMMISSIONER
-
“Dear Students, this is a dedicated resource crafted to fortify
your academic foundation. Your journey towards success begins
here, supported by our commitment to your growth. Wishing you
determination and achievements ahead.”
- Sh. C. S. Azad, DC, KVS RO Guwahati
MESSAGE FROM ASSISTANT COMMISSIONERS
“Dear students, this booklet is tailored to support your academic
journey. May it be a helpful companion in your pursuit of
excellence. Best wishes”
- Sh. R.K. Panigrahi, AC, KVS RO Guwahati
"Dear , this resource is a dedicated tool to support your academic
efforts. Embrace the opportunity for growth and success in your
learning journey. We hope that you will find it useful and
enjoyable."
- Sh. N. Kumar, AC, KVS RO Guwahati
MESSAGE FROM PRINCIPAL
“This booklet is our sincere effort in empowering students with
targeted support for academic success”
- Sh. Vivek Kumar, Principal, KV IIT Guwahati
CONTRIBUTORS
This remedial booklet is a collaborative effort of team of 30 dedicated and experienced PGT MATHS from
various Kendriya Vidyalayas of Guwahati region. They have contributed their expertise, knowledge, and
creativity to design and develop this booklet for the benefit of the students.
Sr. No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
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30
Name of the participant
Mrs. CHANDRA BISWAS
MR. RAVI BACHHETY
MR. MANISH SHARMA
MR. SHIVESH SRIVASTAVA
MR. GYAN BAHADUR SONAR
MR. HIMANSHU VERMA
MR. GURU PRASAD C.
MR. JEETENDRA KUMAR VERMA
MR. SUNIL GALAWA
MR. MOOL PAL
MR. PANKAJ KUMAR
Dr. (MRS.) AAYUSHI JAIN
MR. VIKAS CHAUDHARY
MR. NAVEEN SINGH
MR. GOPAL SINGH RAWAT
MR. AMIT KUMAR
MR. SHAILENDRA KUMAR SAHARAN
MR. BIJOY BHAKTA
MR. ANIL KUMAR KANAUJIYA
MR. RAHUL JAIN
MR. RANVEER SINGH
MR. VINOD KUMAR
MR. KAPIL KANSARA
MR. NAVEEN KUMAR
MR. SUBHAJIT DAS
MR. BALKAR
MR. SANDEEP CHANDRAVANSHI
MR. ABHISHEK JAIN
MR. GOLOK SONOWAL
MR. PRADEEP KUMAR
Name of the KV
PM SHRI KV CRPF (GC) AMERIGOG
PM SHRI KV AFS DIGARU
KV DIPHU
KV DOOM DOOMA (ARC)
KV GERUKAMUKH
KV GOALPARA
KV HAFLONG (SSB)
KV IIT GUWAHATI
KV JAGIROAD (HPCL)
KV JORHAT (ONGC)
PM SHRI KV KHANAPARA
PM SHRI KV KHANAPARA
KV KOKRAJHAR
PM SHRI KV LOKRA
KV MALIGAON
KV MALIGAON
KV MANGALDAI
PM SHRI KV NAGAON
KV TEZPUR NO. 4
PM SHRI KV NEW BONGAIGAON
KV IOC NOONMATI
PM SHRI KV NORTH LAKHIMPUR
KV PANBARI
KV SIVASAGAR (NAZIRA)
PM SHRI KV TAMULPUR
KV TEZPUR NO. 1
KV TEZPUR NO. 2
KV BARPETA
PM SHRI KV GOLAGHAT
KV LUMDING
EDITORS
Sr. No.
Name of the Editors
MR. SHUBHENDU CHAKRABORTY
1
MR. V N SASIDHAR VALLURI
2
MR. RAM RAJ SHARMA
3
Name of the KV
PM SHRI KV NEW BONGAIGAON
PM SHRI KV NARANGI
PM SHRI KV JORHAT (AFS)
TECHNICAL SUPPORT GIVEN BY
Sr. No.
Name of the Person
Dr. (MRS.) AAYUSHI JAIN
1
MR. RAHUL JAIN
2
Name of the KV
PM SHRI KV KHANAPARA
PM SHRI KV NEW BONGAIGAON
10 DAYS SUGGESTIVE REMEDIAL PLAN
CLASS XII
DAY
TOPIC
DAY-1
MATRIX & DETERMINANTS
DAY-2
LPP
DAY-3
VECTORS AND 3D GEOMETRY
DAY-4
CONTINUITY & DIFFERENTIABILITY
DAY-5
APPLICATION OF DERIVATIVE
DAY-6
DEFINITE & INDEFINITE INTEGRAL
DAY-7
APPLICATION OF INTEGRALS
DAY-8
DIFFERENTIAL EQUATIONS
DAY-9
PROBABILITY
DAY-10
RELATIONS AND FUNCTIONS
SLIP TESTS TO BE CONDUCTED AFTER COMPLETION OF EACH TOPIC
S.N.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Chapter 1: Relations and Functions
MULTIPLE CHOICE QUESTIONS
Let f : R → R be defined by f (x) = 1/x ∀ x ∈ R. Then f is
(a) one-one
(b) onto
(c) bijective
(d) f is not defined
If the set A contains 5 elements and the set B contains 6 elements, then the number of oneone and onto mappings from A to B is
(a) 720
(b) 120
(c) 0
(d) none of these
3
A function f:R→R is defined by f(x)=5x ‒ 8 .The type of function is __
(a) one –one
(b) onto
(c) many-one
(d) both one-one and onto
2
Let 𝑓∶R → R defined by f(x) = 1+ x . Choose the correct answer
(a) both one -one and onto
(b)one-one but not onto
(c) onto but not one-one
(d) Neither one- one nor onto
Let the relation R in the set A = { xϵ Z: 0 ≤ x ≤ 12}, given by R = { (a,b): |a-b| is multiple of
4 }. Then the equivalence class of 1 is
(a) {1, 5, 9}
(b) { 0, 1, 2, 5}
(c) ∅
(d) A
Let A = { 1, 2, 3, ….,n}and B= {a, b}. then the number of surjections from A to B is
(a) nP2
(b) 2n ‒ 2
(c)2n ‒1
(d) none of these
Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}.
Then R is —
(a) reflexive but not symmetric
(b) reflexive but not transitive
(c) symmetric and transitive
(d) neither symmetric, nor transitive
The number of bijective functions from set A to itself when A contains 106 elements is —
(a) 106
(b) (106)2
(c) 106!
(d) 2106
Let us define a relation R in R as aRb if a ≥ b. Then R is
(a) an equivalence relation
(b) reflexive, transitive but not symmetric
(c) symmetric, transitive but not reflexive
(d) neither transitive nor reflexive but symmetric
For real numbers x and y, define xRy if and only if x – y + √2 is an irrational number. Then
the relation R is
(a) reflexive
(b) symmetric
(c) transitive
(d) none of these
Let L denotes the set of all straight lines in a plane. Let a relation R be defined by mRn iff m
is perpendicular to n for all m,nϵ L .Then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) equivalence
If a relation R on the set {1,2,3} be defined by R = {(1,2)} , then R is
(a) reflexive
(b) symmetric (c) transitive
(d) none of these
MARKS
1
1
1
1
1
1
1
1
1
1
1
1
13.
The following figure depicts which type of function?
(a) one-one
(c) not one-one
14.
15.
16.
17.
18.
19.
20.
1
(b) onto
(d) both one-one and onto
Let S be the set of all real numbers. Then, the relation R = {(a, b) : 1 +ab > 0}
on S is —
(a) Reflexive and symmetric but not transitive
(b) Reflexive and transitive but not symmetric
(c) Symmetric, transitive but not reflexive
(d) reflexive, transitive and symmetric
If R is a relation in a set A such that (a, a) ∈ R for every a ∈ A, then the relation R is called
(a) symmetric
(b) reflexive
(c) transitive
(d) symmetric or transitive
Let A = {1, 2, 3} and R={(1, 2), (2, 3)} be a relation in A. Then, the minimum number of
ordered pairs may be added, so that R becomes an equivalence relation, is
(a) 7
(b) 5
(c) 1
(d) 4
Let A={1, 2, 3} and B={a, b, c}, and let f = {(1, a), (2, b), (P, c)} be a function from A to B.
For the function f to be one-one and onto, the value of P =
(a) 1
(b) 2
(c) 3
(d) 4
4
Let f : R → R be defined as f(x) = x , then
(a) f is one-one onto
(b) f is one-one
(c) f is one-one but not onto
(d) f is neither one-one nor onto
The following questions consist of two statements-Assertion (A) and Reason (R).Answer
these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A
(b) Both A and R are true and R is not correct explanation for A
(c) A is true and R is false.
(d) A is false and R is true.
Assertion (A): Let A ={1,2,3} then the relation on A as R={(1,2),(2,1)} R is not transitive
relation
Reason (R) :A relation R defined on a non empty set A is said to be transitive
relation if (a,b), (b,c) 𝜖 R ⇒ (a,c) 𝜖 R
Let 𝑓: 𝑅 → 𝑅 such that 𝑓(𝑥) = 𝑥 3
Assertion (A): f(x) is one - one function.
Reason (R) : f(x) is one - one function if co-domain = range
1
1
1
1
1
1
1
Subjective Questions
Q NO
1
2
𝑥−1
Let A=R-{2},B=R-{1}. Let 𝑓: 𝐴 → 𝐵 be defined by 𝑓(𝑥) = 𝑥−2 ∀ 𝑥 ∈ 𝐴 .Show that 𝑓(𝑥) is
One-one function
A traffic light is indicated according to the range of the function as given below 𝑓(𝑥) =
|𝑥−1|
𝑥−1
; 𝑥 ≠ 1.Then find the range of the function.
MARKS
2
2
3
Show that the relations S in the set of real numbers defined as
(a,b): a,b 𝜖 R and a≤ 𝑏 3 } is neither reflexive nor symmetric nor transitive
4
Let A=R-{ },show that the function f in set A defined by 𝑓(𝑥) =
5
and onto
Show that the function f: R → R, defined as f(x) = x2, is neither one-one nor onto.
6
7
8
9
10
11
12
13
2
4𝑥−3
3
6𝑥−4
S={
∀ 𝑥 ∈ 𝐴, is one-one
Let T be the set of all triangles in a plane with R a relation in the set T given by R={(T1, T2) :
T1≅T2}. Show that R is an equivalence relation.
Are the following set of ordered pairs functions?.
(i) {(x, y): x is a person, y is the mother of x}.
(ii) {(a, b): a is a person, b is an ancestor of a}
If so, examine whether the mapping is one-one, many-one or onto.
Prove that the function f is surjective, where
𝑛+1
, 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑓: ℕ → ℕ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑓(𝑛) = { 𝑛2
, 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
2
Is the function injective? Justify your answer.
Show that the function f: R+ → [4, ∞) given by
f(x) = x2 + 4 is a bijective function.
Prove that the relation in the set A={1,2,3,4,5} given by
R={(a,b):|𝑎 − 𝑏| is an even}
is an equivalence relation
Students of class 12, planned to plant saplings along straight lines, parallel to each other to
one side of the school ground ensuring that they had enough play area.
Let us assume that they planted one of the row of saplings along the line 2𝑥 + 𝑦 = 6 .
Let L be the set of all lines which are parallel on the ground and R be relation on L.
A) Let Relation R be defined by R={(𝐿1 , 𝐿2 ): 𝐿1 ∥ 𝐿2 where 𝐿1 , 𝐿2 ∈ L} what is the
type of Relation R?
B) Check whether the function 𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) =6-2x is bijective or not.
2
2
2
3
3
3
3
3
4
Kendriya Vidyalaya Sangathan conducted cycle race under two different categories- Boys
4
and Girls. There were 32 participants in all. Among all them, finally three from
category -1 and two from category-2 were selected for the final race. Amit form two
sets B and G with these participants form his college project. Let B={𝑏1 , 𝑏2 𝑏3 }, and
G=(𝑔1 , 𝑔2 },where B represents the set of Boys selected and G the set of Girls selected
for the final race.
(A) How many relation from B to G ?
(B) Among all the possible relations from B to G, how many functions can be formed
from B to G?
A function 𝑓: 𝐵 → 𝐺 be defined by 𝑓: 𝐵 → 𝐺 defined by f={(𝑏1 , 𝑔1 ),
(𝑏2 , 𝑔2 ),( 𝑏3 , 𝑔1 )}.Check f is bijective or not ?
CASE STUDY
4
In general election of Lok Sabha in 2019, about 911 million people were eligible to vote and
voter turnout was about 67%, the highest ever. Let A be the set of all citizens of India
14
who were eligible to exercise their voting right in general election held in 2019. A
relation ‘R’ is defined on A as follows:
R = {(V1, V2) ∶V1, V2 ∈A and both use their voting right in general election –
2019}
Read the above passage and answer the following questions.
(I). Mr.’X’ and his wife ‘W’both exercised their voting right in general election 2019, Which of the following is true? (A). (X,W) ∈ R but (W,X) ∉ R
(B). (X,W) ∈ and (W,X) ∈ R
(C). (X,W) ∉ R and (W,X) ∉ R
(D). (W,X) ∈ R but (X,W) ∉ R
(II). Three friends F1, F2 and F3 exercised their voting right in general election2019, then which of the following is true?
(A). (F1,F2 ) ∈R, (F2,F3) ∈ R and (F1,F3) ∈ R
(B). (F1,F2 ) ∈ R, (F2,F3) ∈ R and (F1,F3) ∉ R
(C). (F1,F2 ) ∈ R, (F2,F2) ∈R but (F3,F3) ∉ R
(D). (F1,F2 ) ∉ R, (F2,F3) ∉ R and (F1,F3) ∉ R
(III). Mr. John exercised his voting right in General Election – 2019, then Mr. John is
related to which of the following?
(A). Eligible voters of India
(B). Family members of Mr. John
(C). All citizens of India
(D). All those eligible voters who cast their votes
(IV). The relation R = {(V1, V2) ∶V1, V2 ∈A and both use their voting right in
general election –2019} is ------(A) symmetric but not reflexive
(B) reflexive, symmetric but not transitive
(C) equivalence relation
(D) neither reflexive nor symmetric nor transitive
CASE STUDY
Manikanta and Sharmila are studying in the same KendriyaVidyalaya inVisakhapatnam. The
distance from Manikanta’s house to the school is same as distance from Sharmila’s
house to the school. If the houses are taken as a set of points and KV is taken as
origin, then answer the below questions based on the given information; (M for
Manikanta’s house and S for Sharmila’s house)
i. The relation is given by { ( Distance of point M from origin is
same as distance of point S from origin } is
a) Reflexive, Symmetric and Transitive
b) Reflexive, Symmetric and not Transitive
c) Neither Reflexive nor Symmetric
d) Not an equivalence relation
ii. Suppose Dheeraj’s house is also at the same distance from KV then
a) OM ≠ OS
b) OM ≠ OD
c) OS ≠ OD
d) OM = OS= OD
4
15
16
17
18
19
20
iii. If the distance from Manikanta, Sharmila and Dheeraj houses from KV are same,
then the points form a
a) Rectangle
b) Square
c) Circle
d) Triangle
iv. Let {(0,3),(0,0),(3,0)} , then the point which does not lie on the
circle is
a) (0,3)
b) (0,0)
c) (3,0)
d) None of these
Show that the relation R defined on the set N×N by (a,b) R (c,d) ⇒ 𝑎2 + 𝑑 2 = 𝑏 2 + 𝑐 2
∀ 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑵 is an equivalence relation
If R1 and R2 are two equivalence relations in a set A, show that R1∩R2 is also an
equivalence relation.
Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of
the point P from the origin is same as the distance of the point Q from the origin}, is
an equivalence relation. Further, show that the set of all points related to a point P ≠
(0,0) is the circle passing through P with origin as Centre.
Show that the relation R on the set A = {x ϵ Z : 0 ≤ x ≤ 12}, given by R = {(a,b):│a-b│is a
multiple of 4} is an equivalence relation. Find the set of all elements related to 1 i. e.
Find equivalence class [1]
Let R be the relation in 𝑁 × 𝑁 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 (𝑎, 𝑏) 𝑅(𝑐, 𝑑). If a+d=b+c for (a,b), (c,d) in 𝑁 × 𝑁.
Prove that R is an equivalence relation.
𝑥 + 1, 𝑥 𝑖𝑠 𝑜𝑑𝑑
Show that f : N  N is given by 𝑓(𝑥) = {
is both one-one and onto.
𝑥 − 1, 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛
5
5
5
5
5
Chapter 2: Inverse Trigonometric Functions
1
𝟏
Choose the correct principal values of: 𝒄𝒐𝒔−𝟏 ( − 𝟐)
𝜋
𝜋
𝜋
𝐴)
𝐵)
𝐶)
2
6
3
2𝜋
3
𝐷)
𝜋
3
𝟑𝝅
2
Choose the correct principal values of: 𝒄𝒐𝒔−𝟏 ( 𝒕𝒂𝒏 𝟒 )
𝜋
𝐵) 𝜋
𝐶) 0
𝐴)
2
3
𝐷)
𝟏
Choose the correct principal values of: 𝒄𝒐𝒔−𝟏 ( 𝒔𝒊𝒏 ( 𝒄𝒐𝒔−𝟏 (𝟐)))
𝜋
𝜋
𝐶) 0
𝐵) 6
2
𝟒𝝅
Choose the correct principal values of: 𝒄𝒐𝒔−𝟏 ( 𝒔𝒊𝒏 )
𝟑
𝜋
𝜋
5𝜋
𝐴)
𝐶)
𝐵)
2
3
6
𝐴)
4
5
1
If sec −1 √1−𝑥2 + cot −1
√1−𝑥 2
𝑥
A)
2𝜋
3
C) √1 − 𝑥 2
D) 2𝑥
C) 3𝑥
D) 4𝑥
C) [1,2]
D) [1,1[
π 𝜋
C) [ -2 , 2 ]
D) [0 , 𝜋]
Find the value of cot (𝐜𝐨𝐬 −𝟏 𝒙 )
√1−𝑥2
𝑥
𝑥
𝑥
B) - √1−𝑥 2
C) √1−𝑥 2
1
D) √1−𝑥2
Find the value of cos ( 𝐬𝐢𝐧−𝟏 𝒙 )
10
A) √1 − 𝑥 2
11
A)
𝐷)
= sin−1 𝑘, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘 𝑖𝑠:
A) x√1 − 𝑥 2
B) 2x√1 − 𝑥 2
6 The value of [tan−1(𝑐𝑜𝑡𝑥) − tan−1(𝑐𝑜𝑡2𝑥)] 𝑖𝑠:
B) 2x
A) 𝑥
−𝟏
7 Find the domain of 𝒔𝒊𝒏 √𝒙 − 𝟏
A) [−1,1]
B) [0,1]
−𝟏
8 Find the domain of 𝒔𝒊𝒏 (𝟐𝒙 − 𝟏)
A) [0,1]
B) [ -1,1]
9
𝐷)
1
B) - √1 − 𝑥 2
C ) √1−𝑥 2
Find the principal value of 𝒔𝒆𝒄−𝟏 (𝟐𝐬𝐢𝐧
𝟑𝝅
𝟒
12
B) -
15π
4
B)
𝟒
𝝅
𝝅
D) − 𝟒
C) 𝟒
𝟒
𝟏𝟓𝝅
𝟒
15π
4
13 Find the domain of 𝒔𝒊𝒏−𝟏 𝒙 + 𝐜𝐨𝐬 −𝟏 𝒙
A) [ -1, 1]
B) [-2,2]
14
)
𝟑𝝅
Find the principal value of 𝒕𝒂𝒏−𝟏 (𝐭𝐚𝐧
A) -
𝟑𝝅
1
D) - √1−𝑥 2
)
π
π
C) 4
D) - 4
C)[0,1]
D) [-2 , 2 ]
π 𝜋
√𝟑+𝟏
Find the principal value of 𝒄𝒐𝒔−𝟏 ( 𝟐√𝟐 )
𝜋
A) 8
𝜋
B) 12
𝜋
C) - 12
15 If 𝒄𝒐𝒔−𝟏 𝒙 + 𝒄𝒐𝒔−𝟏 𝒚 + 𝒄𝒐𝒔−𝟏 𝒛 = 𝟎 , 𝒇𝒊𝒏𝒅 𝒙 , 𝒚, 𝒛
A) 1, 1, 1
B) 0,0, 0
C) -1,-1,-1
𝜋
D) 24
D) 1,0,1
𝜋
3
16 If 𝒄𝒐𝒔−𝟏 𝒙 + 𝒄𝒐𝒔−𝟏 𝒚 + 𝒄𝒐𝒔−𝟏 𝒛 = 𝟑𝝅 , 𝒇𝒊𝒏𝒅: 𝐱𝐲 + 𝐲𝐳 + 𝐳𝐱
A) 3
B)2
C) -1
1
𝜋
17
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: tan−1 ( − ) + tan−1( − √3) + tan−1(sin ( − ))
2
√3
𝜋
3𝜋
3𝜋
C) - 4
A) −
B)
4
18
D) 0
𝜋
D) 4
4
What is the simplest form tan−1 (
𝑥
√1−cos 𝑥
),0
√1+cos 𝑥
B) 𝑥
A) 2
<𝑥<𝜋
𝜋
C) 2𝑥
D) 4
In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true
19. Assertion (A): The domain of the function sec −1(2𝑥 + 1)𝑖𝑠 (−∞, −1] ∪ [1, ∞)
Reason (R): 𝒔𝒆𝒄−𝟏 ( −
𝟐
)=
√𝟑
𝟓𝝅
𝟔
20. Assertion (A): The principal value of the function cos −1 (−
Reason (R): 𝒄𝒐𝒔−𝟏 ( −𝒙 ) = 𝝅 − 𝒄𝒐𝒔−𝟏 ( 𝒙 )
√3
)
2
π
= -6
Chapter 3: Matrices
Q NO
QUESTION
2
1 1If for a square matrix 𝐴, 𝐴 − 3𝐴 + 𝐼 = 𝑂 and 𝐴−1 = 𝑥𝐴 + 𝑦𝐼, then the value of 𝑥 + 𝑦 is
(a) −2
(b) 2
(c) 3
(d) 3
2 2If 𝐴 = [3 4] and 2𝐴 + 𝐵 is a null matrix, then 𝐵 ie equal to :
5 2
6 8
−6 −8
(a)
[
]
(b) [
]
10 4
−10 −4
5 8
−5 −8
(c)
[
]
(d) [
]
10 3
−10 −3
3 3If 𝐴 = [ 0 1] and (3𝐼 + 4𝐴)(3𝐼 − 4𝐴) = 𝑥 2 𝐼 , then the value(s) of 𝑥 is are
−1 0
(a)
±√7
(b) 0
(c)
±5
(d) 25
4 4If 𝐴 = [0 1], then 𝐴2023 is equal to
0 0
0 1
0 2023
(a) [
]
(b) [
]
0 0
0
0
0 0
2023
0
(c) [
]
(d) [
]
0 0
0
2023
5 5If [2 0] = 𝑃 + 𝑄 , where 𝑃 is a symmetric and 𝑄 is a skew symmetric matrix, then 𝑄 is equal
5 4
to
5
2
2
(a) [ 5
4
2
(c) [
6
7
8
9
10
0
5
0
(b) [ 5
]
2
5
2
−2 0
2
(d) [ 5
]
2
′
𝜋
(d) 2
1
1
1
1
5
−2
0
]
5
−2
4
]
6If 𝐴 is a square matrix such that 𝐴𝐵 and 𝐴𝐵 both are defined, then order of the matrix B is
(a) 𝑚 × 𝑛
(b) 𝑚 × 𝑚
(c) 𝑛 × 𝑛
(d) 𝑛 × 𝑚
7Number of symmetric matrices of order 3 × 3 with each entry 1 or −1 is
(a)
512
(b) 64
(c)
27
(d) 4
8Number of skew-symmetric matrices of order 3 × 3 with each entry 1 or −1 or 0 is
(a)
512
(b) 64
(c)
8
(d) 4
9𝐴 and 𝐵 are skew-symmetric matrices of same order. 𝐴𝐵 is symmetric, if
(a) 𝐴𝐵 = 0
(b) 𝐴𝐵 = −𝐵𝐴
(c) 𝐴𝐵 = 𝐵𝐴
(d) 𝐵𝐴 = 0
1For what value of 𝑥 ∈ [0, 𝜋] , is 𝐴 + 𝐴′ = √3 𝐼, where 𝐴 = [ cos 𝑥 sin 𝑥 ] ?
2
− sin 𝑥 cos 𝑥
0
𝜋
𝜋
(a) 3
(b) 6
(c) 0
M
1
1
1
1
1
1
11
12
13
14
15
16
17
18
19
20
21
1If 𝐴 is a square matrix and 𝐴2 = 𝐴 then (𝐼 + 𝐴)2 − 3𝐴 is equal to
1
(a) 𝐼
(b) 𝐴
(c) 2𝐴
(d) 3𝐼
1If a matrix 𝐴 = [1 2 3], then the matrix 𝐴𝐴′ (where 𝐴′ is the transpose of 𝐴) is :
1 0 0
2
(a) 14
(b) [0 2 0]
0 0 3
1 2 3
(c) [2 3 1]
(d) [14]
3 1 2
6
1 1 1 1 𝑥
3If [0 1 1] [𝑦] = [3] , then the values of (2𝑥 + 𝑦 − 𝑧) is:
0 0 1 𝑧
2
(a)
1
(b) 2
(c)
3
(d) 5
1If 𝐴 is a 3 × 3 square matrix and 𝐵 is matrix such that 𝐴′ 𝐵 and 𝐴𝐵 ′ are both defined, then the
4order of the matrix 𝐵 is :
(a) 3 × 4
(b) 3 × 3
(c) 4 × 4
(d) 4 × 3
2
1If for a square matrix 𝐴, 𝐴 − 𝐴 + 𝐼 = 𝑂 , then 𝐴−1 equals
5
(a)
𝐴
(b) 𝐴 + 𝐼
(c)
𝐼−𝐴
(d) 𝐴 − 𝐼
1If 𝐴 = [1 0] , 𝐵 = [𝑥 0] and 𝐴 = 𝐵 2 , then 𝑥 equals
2 1
1 1
6
(a) ±1
(b) −1
(c) 1
(d) 2
5 𝑥
1
If 𝐴 = [
] and 𝐴 = 𝐴𝑇 , where 𝐴𝑇 is the transpo1se of a matrix 𝐴, then
𝑦 0
7
(a) 𝑥 = 0, 𝑦 = 5
(b) 𝑥 = 𝑦
(c) 𝑥 + 𝑦 = 5
(d) 𝑥 = 5, 𝑦 = 0
1 𝑤ℎ𝑒𝑛 𝑖 ≠ 𝑗
1
If 𝐴 = [𝑎𝑖𝑗 ] is a square matrix of order 2 such that 𝑎𝑖𝑗 = {
then 𝐴2 is
0
𝑤ℎ𝑒𝑛
𝑖
=
𝑗
8
1 0
1 1
(a) [
]
(b) [
]
1 0
0 0
1 1
1 0
(c) [
]
(d) [
]
1 0
0 1
1If 𝐴 is a square matrix and 𝐴2 = 𝐴 then (𝐼 − 𝐴)3 + 𝐴 is equal to
9
(a) 𝑂
(b) 𝐼
(c) 2𝐴
(d) 𝐼 + 𝐴
2If 𝐴 and 𝐵 are symmetric matrices of same order then (𝐴𝐵 ′ − 𝐵𝐴′ ) is
0
(a)
Null matrix
(b)
Symmetric matrix
(c)
Skew-symmetric matrix
(d)
None of these
2
3  4
T
If
A
=
7 8  ,Show that A  A is a skew symmetric matrix .
2


1
1
1
1
1
1
1
1
1
1
2
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
3Find the value of 𝑥 − 𝑦, if
1
2
2[
𝑦
1 3
]+[
0 𝑥
1
0
5 6
]=[
]
2
1 8
3What is trace of a matrix?
2
2
2
]
Evaluate:[
[
1
2
1
4]
3
2
2Find value of x for which [3 𝑥] = [3 −2].
4 1
4 1
4
2
2
2
2Construct a matrix of 2 x 2 whose elements are given by
2
5
4i  j
a ij 
2
2If A =[3 5] is written as 𝐴 = 𝑃 + 𝑄, where P is a symmetric matrix and Q is skew symmetric 2
7 9
6
matrix, then write the matrix P. ,
1 2
3
2
3
3
Matrix
A
such
that
A=[
]
,
then
show
that
𝐴
−
23𝐴
−
40𝐼
=
𝑂
3
−2
−1
7
4 2
1
2If 𝐀 = [ 𝟏 𝟎 ] , Find k such that A2  8 A  kI  0.
−𝟏 𝟕
1
1 −2 3
2
If
𝐴
=
[
0 −1 4]. Find (𝐴′ )−1
8
−2 2 1
2Find the value of X such that
1 2
−1 4
9
𝑋[
]=[
]
3 4
5 6
3The sum of three numbers is 2. If we subtract the second number from twice the first number,
0we get 3. By adding double the second number and the third number we get 0. Represent it
algebraically and find the numbers using matrix method.
3Determine the product
−4 4
4 1 −1 1
3
[−7 1
3 ] [1 −2 −2]
5 −3 −1 2 1
3
And use it to solve the system of equations
x-y+z=4, x-2y-2z=9, 2x+y+3z=1.
2 3 1
3
If
A=[
1 2 2 ], Find 𝐴−1
4
−3 1 −1
And use it to solve the system of equations
2x+y-3z=13 , 3x+2y+z=4 , x+2y-z=8.
3A shopkeeper has 3 varieties of pens A, B, C . Meenu purchased 1 pen of each varity for a
5total of Rs 21. Jeevan purchased 4 pens of A variety, 3 pens of B variety and 2 pens of C
variety for Rs 60. While Shikha purchased 6 pens of A variety , 2 pens of B variety and 3 pens
of C variety for Rs 70. Using matrix method, find cost of each variety of pen.
3Use the product
7
3
3
3
5
5
5
5
5
37
1 −1 2 −2 0 1
[0 2 −3] [ 9 2 −3] to solve the system of equation
3 −2 4
6 1 −2
𝑥 − 𝑦 + 2𝑧 = 1; 2𝑦 − 3𝑧 = 1 ; 3𝑥 − 2𝑦 + 4𝑧 = 2
3Solve the system of following equation
92 + 3 + 10 = 4; 4 − 6 + 5 = 1;6 + 9 − 20 = 2
𝑥
38
39
40
𝑦
𝑧
𝑥
𝑦
𝑧
𝑥
𝑦
5
𝑧
0 1 2
4
0Obtain the inverse of the matrix A = [1 2 3]
3 1 1
3CASE BASED STUDY-1:
8Read the following passage and answer the questions given
below.
Amit, Biraj and chirag were given the task of creating a square
matrix of order 2.
Below are the matrices created by them. A, B, C are the
matrices created by Amit, Biraj , Chirag respectively.
1 2
4 0
2 0
A=[
],𝐵 = [
],𝐶 = [
]
−1 3
1 5
1 −2
If 𝑎 = 4 and 𝑏 = −2.
i)
Find the sum of the matrices 𝑎𝐴, 𝐵 and 𝑏𝐶.
ii)
Find [(𝑏𝐴)𝐴𝑇 }𝑇
3CASE BASED STUDY 2.
6Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of Rs 160. From the same
shop , Vikram buys 2 pens , 1 bag and 3 instument boxes and pays a sum of Rs 190. Also
Ankur buys 1pen, 2 bags and 4 instrument boxes and pays a sum of Rs 250.
Based on the above information, answer the following questions:
i)
Convert the given situation into a matrix equation of the form AX=B.
ii)
Find |A|
iii)
Find 𝐴−1 .
3
2+2
=4
4
Chapter 4: Determinants
Questions
Q. N.
Marks
Section A (MCQ’s)
1 If 𝐴 is a square matrix of order 3 and |2𝐴| = 𝑘|𝐴|, then the value of 𝑘 is,
(a)4
(b)8
(c)6
(d)2
2 Value of |cos 50° sin 10° | is,
sin 50° cos 10°
1
1
(a)0
(b)1
(c)2
(d)− 2
1
3 If the area of a triangle with vertices (−3, 0), (3, 0) 𝑎𝑛𝑑 (0, 𝑘) is 9 sq units, then value(s) of 𝑘
will,
(a)9
(b)3, -3
(c)-9
(d)6
1
4 Let A be a square matrix of order 3 and |𝐴| = −2, then |𝑎𝑑𝑗 (2𝐴)| is equal to,
(a)−26
(b)4
(c)−28
(d)28
5
1
1
1
2 0 1
The co-factor of a32 in the determinant 5 3 8 is,
3 2 1
(a)11
(b)-11
(c)12
(d)10
6
3 p 6 
If 
is a singular matrix, then the value of ‘𝑝’ is,
2 
1
(a)2
(b)3
(c)0
(d)-1
1
7
2 0 0
If A   0 2 0  , then the value of adj. A is ?
 0 0 2 
(a)24
(b) 26
(c) 23
1
(d) 210
8 If 𝐴 is a skew symmetric matrix of order 3, then the value of A is ?
1
(a)1
(b)3
(c)2
(d)0
9 The system of equations 2 x  y  3z  5; 3x  2 y  2 z  5 ; 5 x  3 y  z  16 is,
1
(a) inconsistent
(b) consistent with a unique solution
(c) consistent with a infinitely many solutions
(d) has its solution lying along x-axis in 3D space
10 If 𝑘 is a natural number and |𝑘 3| = |4 −3|, then value of 𝑘 is,
0 1
4 𝑘
(a)4
(b)-4
(c)4, -4
(d) 16
Section B (Descriptive)
11 Find equation of line joining 𝐴(1, 2) and 𝐵(3, 4) using determinants.
3 2
4 12
𝑥 3
If |
| =|
|−|
|, find value(s) of 𝑥 .
1 𝑥
2 1
−2 1
13 Let 𝐴 be a non singular matrix of order 4 and |𝐴−1 | = 3 , then find the value of |𝑎𝑑𝑗 𝐴|.
12
1
2
2
2
14 Find value(s)of 𝑘, if area of triangle 𝐴𝐵𝐶 is 35 square units and 𝐴(2, −6) and
𝐵(5, 4) 𝑎𝑛𝑑 𝐶(𝑘, 4) .
15 Let 𝐴 be a non singular matrix of order 4 and |𝑎𝑑𝑗 𝐴| = 729 , then find the value of |𝐴−1 |.
2
𝑥
sin 𝜃 cos 𝜃
Prove that, |− sin 𝜃 −𝑥
1 | is independent of 𝜃.
cos 𝜃
1
𝑥
2 −1 3
17
Let 𝐴 = [ 𝑘
0 7] . for which value(s) of 𝑘 , inverse of 𝐴 does not exist.
−1 1 4
1+𝑎
1
1
18
Using expansion, prove that | 1
1+𝑏
1 | = 𝑎𝑏𝑐 + 𝑏𝑐 + 𝑐𝑎 + 𝑎𝑏
1
1
1+𝑐
3
2
16
19 If A is a symmetric matrix and B is skew-symmetric matrix such that
1 2
𝐴−𝐵 =[
]
3 4
then find |2𝐴|
20 Evaluate the determinant ,
log 9
log 3 8
| 4
|
log 4 3 log 3 512
3
3
3
3
Chapter 5: Continuity and Differentiability
MULTIPLE CHOICE QUESTION
Q.No.
1
kx 2 , if x  2
What value of k,the function 
is continuous at x=2.
3
,
if
x

2

(a) ¾
(b) 3
(c) 4/3
(d) 6
2
MARKS
1
1
The relationship between “a” and “b” so that the function ‘f’ defined by:
 ax + 1 if x  3
f(x)= 
is continuous at x=3.
bx
+
3
if
x
>
3

(a)
a= b
(b) a+ b =0
(c) a –b = 2/3
(d) none of these
3
4
5
1
d2y

at  
2
dx
6
(d) 32
If x  a cos3  and y  a sin 3  , then find the value of
(a) 32/27a
(b) 32a/27 (c) 32/27
Choose correct option
1 𝑥
1
1
𝑑𝑦
If 𝑦 = (1 + 𝑥) , then 𝑑𝑥 =
1
1
1
(a) (1 + 𝑥)𝑥 [log(1+𝑥)- 𝑥+1]
6
𝑥
(c)
𝑙𝑜𝑔𝑥
−1
(b) (𝑥𝑙𝑜𝑔𝑥)
Choose correct option
If 𝑓(𝑥) = 𝑡 5 then
(a)
(b)
8
1
(c) 0
(d) 1
Choose correct option
The differential coefficient of 𝑓(𝑙𝑜𝑔𝑥) with respect to 𝑥, where 𝑓(𝑥) = 𝑙𝑜𝑔𝑥 is
(a)
7
1
(b) (1 + 𝑥)𝑥 [log(1+𝑥)]
𝑑𝑦
𝑑𝑥
𝑙𝑜𝑔𝑥
𝑥
(d)
none of these
1
is
5t4
(c) 5t5
𝑡6
(d) none of these
6
1
Choose correct option
If 𝑦 = 𝑥 6 find
(a) 6𝑥
(b) 1
1
5
𝑑𝑦
𝑑𝑡
(c) 0
(d) none of these
9
1
10
1
11
1
12
1
13
1
14
1
15
1
16
1
17
1
18
1
19
ASSERTION - REASON TYPE QUESTIONS Directions : Each of these questions contains
1
two statements, Assertion and Reason. Each of these questions also has four alternative
choices, only one of which is the correct answer. You have to select one of the codes (a),
(b), (c) and (d) given below. (a) Assertion is correct, reason is correct; reason is a correct
explanation for assertion. (b) Assertion is correct, reason is correct; reason is not a
correct explanation for assertion (c) Assertion is correct, reason is incorrect (d)
Assertion is incorrect, reason is correct.
1
20
1
Subjective Problems
𝑑𝑦
1
2
3
Find𝑑𝑥 if x 3 + x 2 y + xy 2 + y 3 = 81.
[2]
Show that the function f(x) = 2x - |x| is continuous at x = 0.
[2]
[2]
𝑘𝑥 + 1 𝑖𝑓 𝑥 ≤ 𝜋
Find the value of k so that function is continuous at the given value. 𝑓(𝑥) = {
at
cos𝑥 𝑖𝑓 𝑥 > 𝜋
𝑥=𝜋
𝑑𝑦
4
If x 𝑦 = y 𝑥 , find 𝑑𝑥 .
5
If x 𝑦 = e 𝑥−𝑦 , prove that 𝑑𝑥 =
[2]
𝑑𝑦
(1+log𝑦)2
[2]
log𝑦
6
7
Differentiate w.r.t x: (sin x)𝑐𝑜𝑠𝑥
8
Find𝑑𝑥 , when y = e 𝑥 log (1 + x 2 )
[2]
9
If y = (sin x)𝑥 + sin −1 √𝑥 , find 𝑑𝑥 .
𝑑𝑦
[2]
𝑥+𝑦
𝑑𝑦
𝑦
Ifsec (𝑥−𝑦) = 𝑎 , prove that 𝑑𝑥 = 𝑥 .
𝑑𝑦
[2]
[2]
10 If y = (sin–1 x) 2 , prove that (1 – x 2 ) 𝑑2 𝑦 − 𝑥 𝑑𝑦 − 2 = 0
𝑑𝑥 2
𝑑𝑥
[3]
11 If x𝑚 y 𝑛 = (x + y) 𝑚+𝑛 , prove that 𝑑𝑦 = 𝑦 .
𝑑𝑥
𝑥
[3]
12 If𝑥 = 𝑎sin2𝑡(1 + cos2𝑡) and 𝑦 = 𝑏cos2𝑡(1 − cos2𝑡) , find 𝑑𝑦 𝑎𝑡𝑡 = 𝜋 .
𝑑𝑥
4
[3]
13 If x = a cos𝜃 + b sin 𝜃 , y = a sin 𝜃 - b cos 𝜃 ,then show that 𝑦 2 𝑑2 𝑦 − 𝑥 𝑑𝑦 + 𝑦 = 0 .
𝑑𝑥 2
𝑑𝑥
[3]
14 If𝑥√1 + 𝑦 + 𝑦√1 + 𝑥 = 0 and x ≠ y, prove that 𝑑𝑦 = − 1 .
(𝑥+1)2
𝑑𝑥
[3]
15 If y = e𝑡𝑎𝑛𝑥 , prove that (cos 2 𝑥) 𝑑2 𝑦 - (1 + sin 2x) 𝑑𝑦 = 0.
𝑑𝑥 2
𝑑𝑥
[3]
16 If x= a(cos t + t sin t) andy = a (sin t - t cos t), then find𝑑2 𝑥 , 𝑑2 𝑦 and 𝑑2 𝑦 .
𝑑𝑡 2 𝑑𝑡 2
𝑑𝑥 2
[3]
17 Differentiate the function with respect to x:tan−1 { 𝑥
√𝑎2
[3]
−𝑥2
18
√1+𝑥 2 +√1−𝑥2
If y =tan−1 (√1+𝑥 2
−√1−𝑥2
} , −𝑎 < 𝑥 < 𝑎 .
[3]
𝑑𝑦
) , x 2 ≤ 1, then find 𝑑𝑥 .
19 Read the text carefully and answer the questions: A potter made a mud vessel, where the shape
of the pot is based on f(x) = |x - 3| + |x - 2|, where f(x) represents the height of the pot.
1.
2.
What is the height in terms of x when x > 4?
Will the slope vary with x value?
3.
What is𝑑𝑥 at x = 3?
𝑑𝑦
4. Will the potter be able to make a pot using the function f(x) = [ x ]? Why or why not?
20 Read the text carefully and answer the questions: Mansi started to read the notes on the topic
’differentiability’ which she has prepared in the class of mathematics. She wanted to solve the
questions based on this topic, which teacher gave as home work. She has written following matter
in her notes:Let f(x) be a real valued function, then its Left Hand Derivative (LHD) is: L𝑓 ′ (𝑎) =
limℎ→0
𝑓(𝑎−ℎ)−𝑓(𝑎)
−ℎ
Right Hand Derivative (RHD) is: R𝑓 ′ (𝑎) = limℎ→0
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
Also, a function
f(x) is said to be differentiable at x = a if its LHD and RHD at x = a exist and areequal. For the
|𝑥 − 3|,
𝑥≥1
2
function, 𝑓(𝑥) = {𝑥
3𝑥
13
− 2 + 4 , 𝑥 <1
4
1.
2.
3.
4.
[4]
Find the value f′(-1).
Find the value of f′(2).
Check the differentiability of function at x = 1.
Check the differentiability of the given function at x = 1.
[4]
Chapter 6: Application of Derivatives
Multiple Choice Questions (MCQs)
S.
No
1 The side of an equilateral triangle is increasing at the rate of 2 𝑐𝑚/𝑠𝑒𝑐. The rate at which area
increases when the side is 10 𝑐𝑚 is –
(a) 10 𝑐𝑚2 /𝑠𝑒𝑐
2
3
4
5
6
7
9
10
1
10
12
(b) 20 𝑟𝑎𝑑/𝑠𝑒𝑐
The function 𝑓(𝑥) =
(c) √3 𝑐𝑚2 /𝑠𝑒𝑐
(d) 10√3 𝑐𝑚2 /𝑠𝑒𝑐
(c) 20 𝑟𝑎𝑑/𝑠𝑒𝑐
(d) 10 𝑟𝑎𝑑/𝑠𝑒𝑐
𝑙𝑜𝑔𝑥
𝑥
(b) (0, 𝑒)
(d) 𝑒 , 2𝑒)
1
(a) Strictly increasing in (1, ∞)
(b) Strictly decreasing in (1, ∞)
(c) Neither increasing nor decreasing in (1, ∞)
(d) None of these
If the rate of change of volume of a sphere is equal to the rate of change of the radius then its
radius is equal to –
2√𝜋
1
𝑢𝑛𝑖𝑡𝑠
(b) √𝜋 𝑢𝑛𝑖𝑡𝑠
(c) 𝜋 𝑢𝑛𝑖𝑡𝑠
27
4
(b) 27
(c) 2
1
1
(d) 𝜋 𝑢𝑛𝑖𝑡𝑠
The value of the function 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 2)2 at its maxima is
5
1
1
(c) (2,2𝑒)
1
1
1
1
is increasing in the interval
For the function 𝑓(𝑥) = 𝑥 cos 𝑥 , 𝑥 ≥ 1, 𝑓(𝑥) is
(a)
14
𝑐𝑚2 /𝑠𝑒𝑐
If the volume of the sphere is increasing at a constant rate, then the rate at which its radius is
increasing is –
(a) A constant
(b) proportional to the radius
(c) inversely proportional to the radius (d) inversely proportional to the surface area
Total revenue in rupees received from sales of 𝑥 units of a product is given by 𝑅(𝑥) = 3𝑥 2 +
36𝑥 + 5. The marginal revenue when 𝑥 = 15 is –
(a) 126
(b) 36
(c) 96
(d) 116
(a)
13
3
1
𝑟𝑎𝑑/𝑠𝑒𝑐
(a) (1,2𝑒)
11
10
1
The point(s) on the curve 𝑦 = 𝑥 2 , at which 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 is changing six times as fast as 𝑥 − 1
𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 is/are –
(a) (6,2)
(b) (2,4)
(c) (3,9)
(d) (3,9) 𝑎𝑛𝑑 (9,3)
1
In which of the following intervals the function 𝑓(𝑥) = 𝑥 2 𝑒 𝑥 is decreasing
(a) (−∞ − 2) ∪ (0, ∞) (b) [-2,0]
(c) −∞, ∞) (d) None of these
1
The interval in which the function 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 3 is increasing is
(a) (1, ∞)
(b)(3, ∞)
(c) (1,2)
(d) None of these
3
1
Function 𝑓(𝑥) = 𝑥 − 27𝑥 + 5 is monotonically increasing, when
(a) 𝑥 < −3
(b) |𝑥| > 3
(c) 𝑥 ≤ −3
(d) |𝑥| ≥ 3
1
Function 𝑓(𝑥) = 𝑙𝑜𝑔𝑥 is increasing on 𝑹, if
(a) 0<x<1
(b) x>1
(c)x<1
(d) x>0
A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the 1
ladder slides at rate of 10 𝑐𝑚/𝑠𝑒𝑐, then the rate at which the angle between the floor and the
ladder is decreasing when the lower end of the ladder is 2 m from the wall is
(a)
8
(b)
Marks
1
(d) 1
The Maximum and Minimum values of the function |sin 4𝑥 + 3| are
(a) 1, 2
(b) 4, 2
(c) 2, 4
(d) 1, 1
1
15
16
17
18
19
The function 𝑥 5 − 5𝑥 4 + 5𝑥 3 − 10 has a maximum, when x = ?
(a) 3
(b) 2
(c) 1
(d) 0
3
2
The maximum value of function 𝑥 − 12𝑥 + 36𝑥 + 17 in the interval [1, 10] is
(a) 177
(b) 17
(c) 77
(d) None of these
3
2
The maximum value of the function 𝑥 + 𝑥 + 𝑥 − 4 is
(a) 127
(b) 4
(c) Does not have a maximum value
(d) None of these
2
The function 𝑥 log 𝑥 in the interval (1, 𝑒) has
(a) A point of maximum
(b) A point of minimum
(c) Points of maximum as well as minimum
(d) Neither a point of maximum nor minimum
1
22
23
24
(c) 4
Local maximum value of the function
(a) e
21
(b) 2
(b) 1
log 𝑥
𝑥
1
(c) 𝑒
1
1
is
(d) 2e
2
2
2
2
𝜋
25
Prove that 𝑦 = 2+𝑐𝑜𝑠𝜃 − 𝜃 is an increasing function of 𝜃 in [0, 2 ]
26
27
Show that the function 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 6𝑥 − 100 is increasing on R.
28
29
30
31
1
(d) 6
Water is dripping out from a conical funnel of semi-vertical angle π/4 at the uniform rate of 2
cm2 /sec in the surface area, through a tiny hole at the vertex of the bottom. When the slant
height of cone is 4 cm, find the rate of decrease of the slant height of water.
The total cost C(x) associated with the production of x units of an item is given by 𝐶(𝑥) =
0.005𝑥 3 − 0.02𝑥 2 + 30𝑥 + 5000. Find the marginal cost when 3 units are produced, where by
marginal cost we mean the instantaneous rate of change of total cost at any level of output.
The volume of a cube is increasing at the rate of 9 𝑐𝑚3 /𝑠. How fast is its surface area
increasing when the length of an edge is 10 cm?
Find the intervals in which the functions f given by f(x)=4x3-6x2-72x+30 is strictly
(a)increasing
(b)decreasing
4𝑠𝑖𝑛𝜃
1
1
The minimum value of |𝑥| + |𝑥 + 2| + |𝑥 − 3| + |𝑥 − 2| is
(a) 0
20
5
1
2
𝜋
Prove that the function f(x) = tan x– 4x is strictly decreasing on (− 3 , 𝜋/3).
Find maximum value of sinx+cosx.
Find maximum and minimum values of |sin4x+3|.
Find the two numbers whose sum is 24 and whose product is as large possible.
2
2
2
2
𝜋
Find the intervals in which the function 𝑓(𝑥) = 𝑠𝑖𝑛3𝑥 , 𝑥 ∈ [0, 2 ] is (a) increasing (b)
3
32
decreasing.
Find the intervals in which the functions 𝑓(𝑥) = −2𝑥 3 − 9𝑥 2 − 12𝑥 + 1 is strictly
increasing or decreasing.
33
Show that y = log(1+x) - 2+𝑥, x>-1 , is an increasing function of x throughout domain.
3
34
Find the absolute maximum and minimum values of a function f given by f (x) =
2x 3 – 15x 2 + 36x +1 on the interval [1, 5].
Fid the maximum and minimum value of 𝑥 + 𝑠𝑖𝑛2𝑥 on [0,2𝜋]
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27
of the volume of the sphere.
3
35
36
37
2𝑥
3
3
3
5
38
39
Show that the right circular cone of least curved surface and given volume has an altitude equal
to √2 time the radius of the base.
Read the following text and answer the following questions on the basis of the same.
The relation between the height of the plant (y in cm) with respect to exposure to sunlight is
5
1+1
+2
1
governed by the following equation y= 4x - 2 x2 ,where x is the number of days exposed to
sunlight.
(i)The rate of growth of the plant with respect to sunlight is ……………
1
(a) 4x - 2 x2
40
(b)4 – x
(c)x – 4
1
(d) x - 2 x2
(ii)What is the number of days it will take for the plant to grow the maximum height?
(a)4
(b)6
(c)7
(d)10
(iii)What is the maximum height of the plant?
(a)12cm
(b)10cm
(c)8cm
(d)6cm.
4
2
The shape of a toy is given as g(x) = 6(2x − x ). To make the toy beautiful 2 sticks which
are perpendicular to each other were placed at a point (2,3), above the toy.
1.Which value is abscissa of critical point?
2.Find the second order derivative of the function at x = 5.
3.At which of the following intervals will f(x) be increasing?
1+1
+2
Q.
NO
1
Chapter 7: Integrals
QUESTION
The value of ∫
𝑒𝑥
𝑥
The value of
(𝑎) 0
3
1
[𝑥(𝑙𝑜𝑔 𝑥)2 + 2 𝑙𝑜𝑔 𝑥]𝑑𝑥is
(𝑎) 2𝑒 𝑥 (𝑙𝑜𝑔 𝑥)2 + 𝐶
(𝑐) − 𝑒 𝑥 (𝑙𝑜𝑔 𝑥)2 + 𝐶
2
(𝑏)x𝑒 𝑥 (𝑙𝑜𝑔 𝑥)2 + 𝐶
(𝑑)𝑒 𝑥 (𝑙𝑜𝑔 𝑥)2 + 𝐶
𝜋
𝑑𝑥
4
𝜋
− 1+𝑐𝑜𝑠 2𝑥
4
1
is
∫
(𝑏)1
(𝑐) − 1
(𝑑) 2
3
2
𝑥3
𝐼𝑓 ∫ √1+𝑥 2 𝑑𝑥 = 𝑎(1 + 𝑥 2 ) + 𝑏√1 + 𝑥 2 + 𝑐, the values of a and b are
1
, 𝑏 = −1
3
2
(𝑐) 𝑎 = , 𝑏 = −1
3
1
1
(𝑏 ) 𝑎 = , 𝑏 = 1
3
1
(𝑑)𝑎 = − , 𝑏 = −1
3
(𝑎 ) 𝑎 =
4
The value of ∫ sec x(sec x - tan x) dx is
(𝑎) 𝑡𝑎𝑛𝑥 − 𝑠𝑒𝑐𝑥 + 𝑐
(𝑏) − 𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥 + 𝑐
(𝑐) 𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 + 𝑐
(𝑑) 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥 + 𝑐
5
𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 ∫02 tan 𝑥+ cot 𝑥 𝑑𝑥
√
√
𝜋
𝜋
𝜋
(𝑎)
(𝑏)
(𝑐)
2
4
8
𝑒 1+log 𝑥
𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 ∫1 ( 𝑥 ) 𝑑𝑥
3
1
(𝑎)
(𝑏)
(𝑐) 𝑒
2
2
6
MARKS
𝜋
𝜋
2
7
1
1
√tan 𝑥
(𝑑)
𝜋
12
1
(𝑑)
1
𝑒
1
9
𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 ∫ 𝑠𝑖𝑛 𝑥 𝑑𝑥
–
(𝑎) 0
8
(𝑏)1
(𝑐) − 1
1
(𝑑) 2
x
) dx
1+x
1
What is the value of ∫ (
0
(𝑎)1 − log 2
(𝑐)1 + log 2
9
𝜋
2
(𝑏) log 2 − 1
(𝑑) log 2
1
10
1
1
11
1
𝐼𝑓 𝐼 = ∫ 𝑆𝑖𝑛2 𝑥 𝐶𝑜𝑠2 𝑥 𝑑𝑥, 𝑡ℎ𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐼 𝑤𝑖𝑙𝑙 𝑏𝑒.
(a) tan x − cos x + c
(c) tan x − cot x + c
12
(b) tan 𝑥 − cosec x + c
(d) tan x − sec x + c
1
What is the value of ∫ 𝑠𝑖𝑛−1 (𝑐𝑜𝑠𝑥) dx
(a)
13
𝜋𝑥
−
2
𝐼𝑓 𝑓(𝑥) =
𝑥2
𝜋
+c (b) + 𝑥 (c) −𝑥 (d) 𝑠𝑖𝑛𝑥
2
2
𝑥
∫0 𝑡𝑠𝑖𝑛𝑡 dx,
then 𝑓 / (𝑥) is
1
(𝑎) 𝑐𝑜𝑠𝑥 + 𝑥𝑠𝑖𝑛𝑥 (b) 𝑥𝑠𝑖𝑛𝑥 (c) 𝑥𝑐𝑜𝑠𝑥 (d) 𝑠𝑖𝑛𝑥 + 𝑥𝑐𝑜𝑠𝑥
1
14
1
What is the value of ∫ 𝑐𝑜𝑠 4 𝑥. 𝑥17 dx
−1
(a) −1 (b) 2 (c) 0 (d) 1
15
𝐼𝑓 𝐼 = ∫
𝑆𝑖𝑛2 𝑥 −𝐶𝑜𝑠2 𝑥
𝑆𝑖𝑛2 𝑥 𝐶𝑜𝑠2 𝑥
(a) tan x + cos x + c
(c) tan x + cot x + c
0 |𝑥|
16
The value of ∫−2
(𝑎) 0
2
∫−2|𝑥| 𝑑𝑥
17
(𝑎) 0
𝑑
18
𝑑𝑥
𝑥
(b) tan 𝑥 + cosec x + c
(d) tan x + sec x + c
1
dx is
(𝑏) − 1
(𝑐)1
(𝑑) − 2
1
𝑖𝑠 𝑒𝑞𝑢𝑎𝑙𝑠 𝑡𝑜,
(𝑏)2
(𝑐) 4
(𝑑) 1
1
∫ 𝑓(𝑥) 𝑑𝑥 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙𝑠 𝑡𝑜,
(𝑎) 𝑓 ′ (𝑥)
(𝑐) 𝑓(𝑥 ′ )
(𝑏) 𝑓(𝑥)
𝑑𝑥
19
1
𝑑𝑥, 𝑡ℎ𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐼 𝑤𝑖𝑙𝑙 𝑏𝑒.
Assertion (A): ∫ 𝑥 2 +2𝑥+3 =
𝑑𝑥
1
1
√2
(𝑑) 𝑓 ′ (𝑥 ′ )
𝑥+1
tan−1 (
2
)+𝑐
1
𝑥
Reason (R): ∫ 𝑥 2 +𝑎2 = 𝑎 tan−1 (𝑎) + 𝑐

20

(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is NOT the correct explanation of A
(c) A is true but R is false
(d) A is false and R is True
Assertion (A): ∫ex[sin x + cos x]dx = ex sin x + c
Reason (R): ∫ex [f(x) +f′(x)]dx = ex f(x) + c
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is NOT the correct explanation of A
(c) A is true but R is false
(d) A is false and R is True
1
21
Evaluate and find the value of ‘a’
𝑎 𝑑𝑥
∫0 4+𝑥 2
22
=
𝜋
8
x3 −x2 +x−1
Evaluate: ∫
x−1
Evaluate: ∫ 5−8x−x2
24
Evaluate: ∫1
26
27
28
29
30
dx
2
2
dx
23
25
2
√3 𝑑𝑥
1+𝑥 2
𝜋
2
𝜋
−
2
2
𝑠𝑖𝑛5 𝑥 𝑑𝑥
Evaluate: ∫
Find the value of ∫ 𝑠𝑖𝑛𝑥 − 𝑙𝑜𝑔𝑐𝑜𝑠𝑥 𝑑𝑥
Find the value of
∫
𝑙𝑜𝑔2 𝑥
𝑥
(𝑥−3)
3
𝑥
dx
1 − 𝑥3
1
Evaluate ∫
dx
cos(𝑥 − 𝑎) cos(𝑥 − 𝑏)
3
Evaluate ∫ √
1
Evaluate ∫ 1+tan 𝑥 𝑑𝑥
32
Evaluate ∫
𝒔𝒊𝒏𝟔 𝒙−𝒄𝒐𝒔𝟔 𝒙
𝒔𝒊𝒏𝟐 𝒙 𝒄𝒐𝒔𝟐 𝒙
𝑑𝑥
34
Evaluate: ∫ (𝑥 2 +1)(𝑥+2) 𝑑𝑥
35
Evaluate ∫02 1+3 𝑠𝑖𝑛2 𝑥dx
36
Evaluate: ∫ √𝑥 2
39
3
𝑥 2 +𝑥+1
5
𝑐𝑜𝑠2 𝑥
5
5𝑥+3
Evaluate:
3
dx
Evaluate ∫ √5𝑥 2
−2𝑥
𝜋
5
𝑑𝑥
+4𝑥+10
𝜋 𝑥.𝑆𝑖𝑛 𝑥
∫0 1+𝐶𝑜𝑠2 𝑥 𝑑𝑥
Evaluate ∫
3
3
33
38
𝑑𝑥
2
2
Evaluate: ∫ (𝑥−1)3 𝑒 𝑥 𝑑𝑥
31
37
2
(x 2
5
2x
dx
+ 1)(x 2 + 2)2
The given integral ∫ f(x) dx can be transformed into another
form by changing the independent variable x to t by substituting
x = g(t),
dx
Consider I = ∫ f(x) , put x = g(t) ⇒
= g ′ (t)
dt
⇒ dx = g ′ (t)dt ⇒ I = ∫ f(x) = ∫ f(g(t)) g ′ (t)dt
This change of variable formula is one of the important tools available to
us in the name of integration by substitution.
Based on the above information, answer the following questions:
1. Find the value of ∫
etan
−1 x
1+x2
dx
5
4
sin−1 x
2. . Find the value of ∫ √1−x2 dx
3. Find the value of ∫
4.Find the value of ∫
40
sin x
dx
(1 + cos x)2
log x
x
dx
There are many practical applications of Definite Integration. Definite integrals can be
4
used to determine the mass of an object if its density function is known. We can also find
work by integrating a force function, and the force exerted on an object submerged in a
liquid. The most important application of Definite Integration is finding the area under the
curve.
Let f be a continuous function defined on the closed interval [a,b] and F be an
antiderivative of f then
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 = [𝐹(𝑥)]𝑏𝑎 = 𝐹(𝑏) − 𝐹(𝑎)
It is very useful because it gives us a method of calculating the definite integral more
easily. There is no need to keep integration constant C because if we consider F(x) + C
instead of F(x).
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 = [𝐹(𝑥) + 𝐶]𝑏𝑎 = 𝐹(𝑏) + 𝐶 − 𝐹(𝑎) − 𝐶
= 𝐹(𝑏) − 𝐹(𝑎)
Based on the above information, answer the following questions:
we get
3
1. Find the value of ∫2 x 2 dx
√3
2. Find the value of ∫1
1
1+x2
1
dx
3. Find the value of ∫−1(x + 1)dx
31
4. Find the value of ∫2 x dx
1
2
Chapter 8: Applications of Integrals
The area of the region bounded by the circle 𝑥 2 + 𝑦 2 = 1 is
(a) 2𝜋 sq. units
(b) 𝜋 sq. units
(c) 3𝜋 sq. units
(d) 4𝜋 sq. Units
The area of the region bounded by the curve 𝑦 = 𝑥 + 1 and the lines 𝑥 = 2 and 𝑥 = 3 is
(a)
(c)
3
7
2
11
2
(b) 2 sq. units
sq. units
(d)
9
(d) 2
7
2
(c)
2
3
32
33
2
(b)
(d)
32
3
16
3
Area lying between the curve 𝑦 2 = 4𝑥 and 𝑦 = 2𝑥 is
(a)
2
3
1
(c) 4
1
(b) 3
3
(d) 4
The area bounded by the parabola 𝑦 2 = 4𝑎𝑥, latus rectum and 𝑥-axis is
(a) 0
2
(c) 3 𝑎2
8
9
(b)
(c) 9
(d) none of these
The area bounded by the parabola 𝑥 = 4 − 𝑦 2 and y-axis, in square units, is
(a)
7
9
(b) 4
The area bounded by 𝑦 = 2 − 𝑥 2 and 𝑥 + 𝑦 = 0 is
(a)
6
sq. units
2
9
(c) 3
5
13
The area of the region bounded by the curve 𝑦 2 = 4𝑥, 𝑦-axis and the line 𝑦 = 3 is
(a) 2
4
9
sq. units
4
(b) 3 𝑎2
(d)
𝑎2
3
𝑥2
𝑦2
The area of the region bounded by the ellipse 𝑎2 + 𝑏2 = 1
9
(a) 𝜋𝑎𝑏
(b) 𝜋𝑎2 𝑏 2
(c) 2𝜋𝑎𝑏
(d) 𝑎𝑏
The area of the region bounded by the circle 𝑥 2 + 𝑦 2 = 𝑎2
(a) 2𝜋𝑎
(b) 𝜋𝑎2
(c) 2𝜋𝑎2
(d) None of these
10
The area of the region bounded by the curve
11
12
13
14
15
𝑥2
4
+
𝑦2
9
=1
(a) 6𝜋
(b) 36𝜋
(c) 18𝜋
(d) None of these
Using integration, find the area of the triangular region whose sides have the equations 𝑦 =
2𝑥 + 1, 𝑦 = 3𝑥 + 1 and 𝑥 = 4.
Find the area bounded by the curve 𝑥 2 = 4𝑦 and the line 𝑥 = 4𝑦 − 2
0
Sketch the graph of 𝑦 = |𝑥 + 3| and hence evaluate ∫−6|𝑥 + 3|𝑑𝑥
Using integration, find the area of the region bounded by the triangle whose vertices are
(−1, 0), (1, 3) and (3, 2).
Using the method of integration find the area of the region bounded by lines:
2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
16
17
18
19
20
21
22
23
24
25
Using integration, find the area of the triangular region whose sides have the equations 𝑦 = 2𝑥 +
1, 𝑦 = 3𝑥 + 1, 𝑥 = 4.
Make a rough sketch of the region given below and find its area {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥 2 + 3, 0 ≤ 𝑦 ≤
2𝑥 + 3, 0 ≤ 𝑥 ≤ 3}
Find the area of the region bounded by the curve
𝑦 = √16 − 𝑥 2 and 𝑥 − 𝑎𝑥𝑖𝑠 .
Find the area of the region bounded by the curve
𝑦 = 𝑥 2 𝑎𝑛𝑑 𝑦 = 16.
Find the area under the curve 𝑦 = 𝑥 2 and the lines
𝑥 = −1, 𝑥 = 2 𝑎𝑛𝑑 𝑥 − 𝑎𝑥𝑖𝑠
Find area of region given by {(x, y) : 𝑥 2 ≤ y ≤ |x|}.
Using integration, find the area of the region bounded by the triangle
whose vertices are (–2, 2) (0, 5) and (3, 2).
Find the area of the region bounded by y = |x – 1| and y = 1
Find the area of region {(x, y) : 𝑦 2 ≤ 4x, 4𝑥 2 + 4𝑦 2 ≤ 9}
Find the area of the region {(x, y) : 𝑥 2 + 𝑦 2 ≤1 ≤ x + y}.
Chapter 9: Differential Equations
Q.1
The sum of order & degree of the differential equation
=(1 + 𝑑𝑥 ) is
𝑑𝑥 3
(a) 3
Q2
(b) 4
Q4
(b)
𝑥2
2
𝑑2 𝑦
1
𝑥2
3
+𝐶
(d) 0
1
𝑥2
(b) 𝑦 = 2𝑙𝑜𝑔𝑥 +
(d) 𝑦 = 2𝑥 +
3
𝑥2
2
+𝐶
+𝐶
1
𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠 (𝑑𝑥 ) = 𝑦
is
(a) 2
(b) 1
(c) 0
(d) None of these
The integrating factor of the differential equation
𝑑𝑦
(1 − 𝑦 2 ) + 𝑦𝑥 = 𝑎𝑦, (−1 < 𝑦 < 1)
𝑑𝑥
(b)
1
(c)
√𝑦 2 −1
The solution of differential equation
1
(a)
𝑥
1
+𝑦 =𝑐
(b)
𝑑𝑥
𝑥
+
𝑑𝑦
𝑦
𝟏
𝑑𝑥 2
1
1
(d) 1−𝑦 2
√𝟏−𝒚𝟐
1
= 0 is
𝑙𝑜𝑔𝑥 − 𝑙𝑜𝑔𝑦 = 𝐶
(c) 𝑥𝑦 = 𝑐
(d) 𝑥 + 𝑦 = 𝑐
What is the product of order and degree of the differential equation
𝑑2 𝑦
1
𝑑𝑦 3
𝑠𝑖𝑛𝑦 + (𝑑𝑥 ) 𝑐𝑜𝑠𝑦 = √𝑦
(a) 2
(b) 1
(c) 0
(d) None of these
𝑑𝑦
The integrating factor of the differential equation 𝑥 𝑑𝑥 − 𝑦 = 2𝑥 2 is
(a) 𝑒 −𝑦
Q10
2
2
1
Q9
(d) √𝑒
3
𝑑𝑦 2
Degree of the differential equation
(a) 𝑦 2 −1
Q8
(c) 1
𝑒
+𝐶
𝑑𝑦
Q7
1
1
(a) 2
(b) 4
(c) 5
The general solution of the differential equation
𝑥 𝑑𝑦 ̶ (1 + 𝑥 2 )𝑑𝑥 = 𝑑𝑥
(c) 𝑦 =
Q6
(d) 8
The degree of differential equation [1 + (𝑑𝑥 ) ] =(𝑑𝑥 2 ) is
(a) 𝑦 = 2𝑥 +
Q5
(c) 5
The integrating factor of differential equation
𝑑𝑦
(𝑥) + 𝑦 = 2𝑙𝑜𝑔𝑥 is
𝑑𝑥
(a) 𝑥
Q3
1
𝑑𝑦 5
𝑑3 𝑦
(b) 𝑒 −𝑥
(c) 𝑥
(d)
1
𝑥
The order and degree (if defined) of the differential equation
𝑑2 𝑦
2
𝑑𝑦 3
1
1
𝑑𝑦
(𝑑𝑥 2 ) + (𝑑𝑥 ) = 𝑥 𝑠𝑖𝑛 (𝑑𝑥 ) respectively are
(a) 2,2
(b) 1,3
(c) 2,3
(d) 2, degree not defined
3
Q11
𝑑𝑦 2 2
𝑑2 𝑦
1
The order of the differential equation [1 + (𝑑𝑥 ) ] = 𝑑𝑥 2 is
(a) 2
Q12
Q13
(b) 4
(c) 5
(d) 8
𝑑𝑦
The integrating factor of the differential equation 𝑥 𝑑𝑥 + 2𝑦 = 𝑥 2 is
(a) 𝑥
(b)3𝑥
(c)𝑥𝑦
(d) 𝑥 2
The particular solution of differential equation
𝑑𝑦
= 𝑦 𝑡𝑎𝑛𝑥 𝑎𝑡 𝑦 = 1, 𝑥 = 0 is
𝑑𝑥
1
1
(a) 𝑦 = 𝑐𝑜𝑠𝑥
(c) 𝑦 = 𝑡𝑎𝑛𝑥
Q14
Q15
(b) 𝑦 = 𝑠𝑒𝑐𝑥
(d) 𝑦 𝑠𝑖𝑛𝑥 = 6
𝑑𝑦 5
𝑥4 −1
(
)
3
Q16
(a) 𝑥 +
Q18
4
𝑥4 −1
(
)
4
1
1
−𝑥4
4
(a) 𝑦 = 𝑒
(b) 𝑦 = 𝑒
(c) 𝑦 = 𝑒 𝑥 (d)𝑦 = 𝑒
Which of the following is not a homogeneous function of 𝑥 𝑎𝑛𝑑 𝑦 ?
2
Q17
𝑑2 𝑦
The degree of the differential equation ( ) +2𝑥 2 ( 2 ) = 0 is
𝑑𝑥
𝑑𝑥
(a) 3
(b) 4
(c) 1
(d) 2
3
Solve
𝑥 𝑦𝑑𝑥 = 𝑑𝑦, 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 1
𝑦
𝑦
2𝑥𝑦 (b) 2𝑥 − 𝑦 (c) 𝑐𝑜𝑠 2 (𝑥 ) + 𝑥
𝑑2 𝑦
𝑑𝑦
the equation 𝑒 𝑥 𝑑𝑥 2 + 𝑠𝑖𝑛 (𝑑𝑥 ) =
1
(d) 𝑠𝑖𝑛𝑥 − 𝑐𝑜𝑠𝑦
The degree of
3 is
(a) 2
(b) 1
(c) 0
(d) None of these
In the following questions, a statement of assertion (A) is followed by a statement of
Reason (R).
1
1
Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true
Q18. The solution of the differential equation
𝑑2 𝑦
𝑑𝑥 2
+ 𝑦 = 0 𝑖𝑠 𝑦 = 𝑓(𝑥) = sin(𝑥 + 𝑏).
Assertion (A) The function 𝑦 = 𝑓(𝑥) is called general solution .
Reason (R) The solution which contain arbitrary constant , is called general solution.
2 marks questions
1. Find the integrating factor of the differential equation.
(
𝑒 −2√𝑥
√𝑥
2. Show that xy = log y + C, is the solution of
3. Solve the differential equation,
dy
dx
dy
−
𝑦
𝑑𝑥
=1
√𝑥 𝑑𝑦
)
y2
= 1−xy (xy ≠ 1)
dx
= √4 − y 2 ; (−2 < 𝑦 < 2)
4. Solve the differential equation, (ex + e−x )dy − (ex − e−x )dx = 0
5. Solve the differential equation,
dy
dx
= (1 + x 2 )(1 + y 2 )
3 marks questions
1. Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2)ex dx = 0, given that y = 1 when
x = 0.
2. Solve the differential equation yex/y dx = (x ex/y + y2)dy ( y ≠ 0).
3. Find a particular solution of the differential equation (dy/dx) + y cot x = 4x cosec x, (x≠ 0), given that y = 0
when x = π/2.
4. Solve the differential equation (1 + x2) dy + 2xy dx = cot x dx (x ≠ 0)
5. Solve:
dy
dx
+ 2y tanx = sinx
Case study
1. A first order first degree differential equation is of the form
dy
dx
= F(x, y)
if F(x, y) can be expressed as a product of g(x)h(y), where g(x) is the function of x and
h(y) is the function of y.
The solution of differential equation by this method is called "variable separable".
(i) Find the general solution of differential equation
(1+x2 )
dy
= (1+y2 )
dx
(ii) Find the general solution of differential equation
(iii) Find the general solution of differential equation
dy
dx
x+1
= 2−y
𝑑𝑦
𝑑𝑥
1−cos 𝑥
= 1+cos 𝑥
Chapter 10: Vectors
Q No.
1
Question(s)
The position vector of the point which divides the join of points with position vectors
a⃗ + ⃗b and 2a⃗ − ⃗b in the ratio 1: 2 is
⃗
⃗ +2b
3a
2
3
4
5
6
7
9
10
11
12
13
14
15
16
⃗
⃗ −b
5a
√34
2
1
1
1
1
1
1
√48
2
(C) √18
(D) None of these.
The projection of vector a⃗ = 2î − ĵ + k̂ along ⃗b = î + 2ĵ + 2k̂ is
2
1
(A) 3
(B) 3
(C) 2
(D) √6
If a⃗ and ⃗b are unit vectors, then what is the angle between a⃗ and ⃗b for √3 a⃗ − ⃗b to be a
unit vector?
(A) 30°
(B) 45°
(C) 60°
(D) 90°
The unit vector perpendicular to the vectors î − ĵ and î + ĵ forming a right handed
system is
î−ĵ
î+ĵ
(A) k̂
(B) −k̂
C) √2
(D) √2
(B)
1
⃗
⃗ +b
4a
(A)
(B) a⃗
(C) 3
(D) 3
3
The vector with initial point P (2, –3, and 5) and terminal point Q (3, –4, and 7) is
(A) î − ĵ + 2k̂ (B) 5î − 7ĵ + 12k̂ (C) −î + ĵ − 2k̂ (D) None of these.
The angle between the vectors î − ĵ and ĵ − k̂ is
π
2π
π
5π
(A) 3
(B) 3
(C) − 3
(D) 3
The value of λ for which the two vectors 2î − ĵ + 2k̂ and 3î + λĵ + k̂ are perpendicular
is
(A) 2
(B) 4
(C) 6
(D) 8
̂
The area of the parallelogram whose adjacent sides is î + k and 2î + ĵ + k̂ is
(A) √2
(B) √3
(C) 3
(D) 4
⃗
⃗
⃗
If |a⃗| = 8 , |b| = 3 and |a⃗ × b| = 8 , then value of a⃗. b is
(A) 6 √3
(B) 8√3
(C) 12√3
(D) None of these.
The two vectors ĵ + k̂ and 3î − ĵ + 4k̂ represents the two sides AB and AC,
respectively of a ∆ABC. The length of the median through A is
(A)
8
Marks
If |a⃗| = 3 and −1 ≤ k ≤ 2, then |ka⃗| lies in the interval
(A) [0,6]
(B) [−3,6]
C) [3,6]
(D) [1,2]
2
2
2
For any vector a⃗ , the value of (a⃗ × î) + (a⃗ × ĵ) + (a⃗ × k̂) is equal to
(A) a⃗
(B)3a⃗
C) 4a⃗
(D) 2a⃗
The number of vectors of unit length perpendicular to the vectors a⃗ = 2î + ĵ + 2k̂ and
a⃗ = ĵ + k̂ is
(A) One
(B) Two
C) Three
(D) Infinity
Find the values of x, y and z so that the vectors 𝑎 = x î + 2ĵ + zk̂ and 𝑏⃗ = 2 î + y ĵ + k̂
are equal.
(A) x = 2, y = -2, z = 1
(B)x = -2, y = -2, z = -1
(C) x = 2, y = 2
(D) x = 2, y = 2, z = 1
If θ be the angle between two vectors a⃗ and ⃗b, then a⃗. ⃗b ≥ 0 only when
π
π
(A) 0 < θ < 2 (B) 0 ≤ θ ≤ 2
C) 0 < θ < π
(D) 0 ≤ θ ≤ π
Let a⃗ and ⃗b be two unit vectors and θ be the angle between them. Then a⃗ + ⃗b is a unit
vector if
π
π
π
2π
(A) θ = 4
(B) θ = 3
C) θ = 2
(D) θ = 3
1
1
1
1
1
1
1
1
17
18
The value of î. (ĵ × k̂) + ĵ. (î × k̂) + k̂. (î × ĵ) is
(A) 0
(B) –1
(C) 1
(D) 3
If θ be the angle between two vectors a⃗ and ⃗b, then |a⃗. ⃗b| = |a⃗ × ⃗b| when θ is equal to
π
π
(A) θ = 0
(B) θ = 4
C) θ = 2
(D) θ = π
1
1
ASSERTION-REASON BASED QUESTIONS
In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose
the correct answer out of the following choices.
(A) Both (A) and (R) are true and (R) is the correct explanation of (A).
(B) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(C) (A) is true but (R) is false.
(D) (A) is false but (R) is true.
Q No.
Question(s)
Mark
19
Assertion (A): The points A(−2î + 3ĵ + 5k̂), B(î + 2ĵ + 3k̂) and C(7î − k̂) are collinear.
1
20.
Reason (R): The points A(−2î + 3ĵ + 5k̂), B(î + 2ĵ + 3k̂) and C(7î − k̂) hold the relation
⃗⃗⃗⃗⃗ | = |AB
⃗⃗⃗⃗⃗ | + |BC
⃗⃗⃗⃗⃗ |
|AC
⃗ , we always have |a⃗. ⃗b| ≤ |a⃗||b
⃗|
Assertion (A): For any two vectors a⃗ andb
1
⃗ , we always have |a⃗ + b
⃗ | ≤ |a⃗| + |b
⃗|
Reason (R): For any two vectors a⃗ andb
2-Marks Questions
Q No.
Question(s)
Q21
If â and b̂ are unit vectors, then prove that |â + b̂ | = 2 cos 2 , where θ is the angle between
them.
Write the direction cosines of the vectors −2î + ĵ − 5 k̂ .
2
If a⃗ and ⃗b are two vectors of magnitude 3 and 3 respectively such that a⃗ × ⃗b is a unit vector,
write the angle between a⃗ and ⃗b.
If a⃗ , ⃗b and c⃗ are three mutually perpendicular unit vectors, then prove that |a⃗ + ⃗b + c| =
Q22
Q23
Q24
Marks
θ
2
2
2
2
Q26
√3.
⃗ perpendicular to each other? Where a⃗ = 2î +
For what value of λ are the vectors a⃗ and b
3ĵ + 4 k̂ and ⃗b = 3î + 2ĵ − λk̂.
If 𝑎 = 𝑖̂ + 𝑗̂ + 2𝑘̂ and 𝑏⃗ = 2 𝑖̂ + 𝑗̂ − 2𝑘̂ , find the unit vector in the direction 6𝑏⃗.
Q27
Q28
Find the value of 𝜆 when the projevtion of 𝑎 = 𝜆𝑖̂ + 𝑗̂ + 4𝑘̂ on 𝑏⃗ = 2𝑖̂ + 6𝑗̂ + 3𝑘̂ is 4 units.
2
2
2
For any two vectors 𝑎 and 𝑏⃗ prove that : |𝑎 + 𝑏⃗| + |𝑎 − 𝑏⃗| = 2 (|𝑎|2 + |𝑏⃗| ).
2
2
Q29
2
Q30
⃗ represent two adjacent sides of a parallelogram, then write vectors representing
If a⃗ and b
its diagonals.
If the points A (m, - 1), B(2, 1) and C(4,5) are collinear, find the value of m.
Q31
For what value of' a the vectors 2î − 3ĵ + 4 k̂ and aî + 6ĵ − 8 k̂ are collinear?
2
Q32
⃗.
Find 𝜆 and 𝜇 if (2𝑖̂ + 6𝑗̂ + 27𝑘̂) × (𝑖̂ + 𝜆𝑗̂ + 𝜇𝑘̂) = 𝑂
2
Q25
2
2
2
3-Marks Questions
Q No.
Question(s)
Marks
Q33
For what value of 𝜆 are the vectors 𝑎 = 2î + 𝜆ĵ + k̂ and 𝑏⃗ = î − 2ĵ + 3k̂ perpendicular to
each other.
Three vectors 𝑎, 𝑏⃗ & 𝑐 satisfy the condition 𝑎 + 𝑏⃗ + 𝑐 = ⃗0. Evaluate the quantity 𝜇 = 𝑎. 𝑏⃗ +
𝑏⃗ . 𝑐 + 𝑐. 𝑎 𝑖𝑓 |𝑎| = 3, |𝑏⃗| = 4 & |𝑐| = 2
2
For any vector a⃗ , prove that |𝑎 × 𝑖̂|2 + |𝑎 × 𝑗̂|2 + |𝑎 × 𝑘̂| = 2|𝑎|2 .
If 𝑎 = 2𝑖̂ + 2𝑗̂ + 3𝑘̂ , 𝑏⃗ = −𝑖̂ + 2𝑗̂ + 𝑘̂ and 𝑐 = 3𝑖̂ + 𝑗̂ are such that 𝑎 + 𝜆𝑏⃗ is
perpendicular to 𝑐 , then find the value of 𝜆.
Find the area of the triangle with vertices 𝐴(1,1,3), 𝐵(2,3,5) and 𝐶(1,5,5).
3
Q38
Given that 𝑏⃗ = 2𝑖̂ + 4𝑗̂ − 5𝑘̂ and 𝑐 = 𝜆 𝑖̂ + 2𝑗̂ + 3𝑘̂ , such that the scalar product of 𝑎 =
𝑖̂ + 𝑗̂ + 𝑘̂ and unit vector along sum of the given two vectors 𝑏⃗ and 𝑐 is unity.
5-Marks Questions
3
Q No.
Question(s)
Marks
Q34
Q35
Q36
Q37
3
3
3
3
Q39
If with reference to the right handed system of mutually perpendicular unit vectors 𝑖̂, 𝑗̂ and
𝑘̂, 𝛼 = 3î − ĵ, 𝛽 = 2𝑖̂ + 𝑗̂ − 3𝑘̂, then express 𝛽 in the form 𝛽 = 𝛽1 + 𝛽2, where 𝛽1 is
parallel to 𝛼 and 𝛽2 is perpendicular to 𝛼 .
5
Q40
⃗ = 3î − 2ĵ + 7k̂ and c = 2î − ĵ + 4k̂ .Find a vector 𝑑 which is
Let a⃗ = î + 4ĵ + 2k̂ , b
perpendicular to both 𝑎 and 𝑏⃗, and 𝑐. 𝑑 = 15.
5
Chapter 11: Three Dimensional Geometry
Q.
No.
1
Question
Marks
A bullet shot from the gun travels a straight line path which makes angles 90°, 60° and 30° with
the positive direction of x-axis, y-axis and z-axis respectively. Its direction cosines are
(a) 1,
√3 1
,
2 2
(b)
√3 1 1
, ,
2 √2 2
1 √3
2
(c) 0, 2 ,
1
(d) none of these
2
The direction cosines of an electricity straight wire with direction ratios 2,-3, 4 are
2
−3
4
4
−6
8
(a) 2, -3, 4
(b) 4, -6, 8
(c)
,
,
(c)
,
,
1
3
Three stars in sky are positioned at A(2, -4, 6), B(4, 6, -8) and C(6, 16, -22) with respect to a
common reference point O(0, 0, 0). A student is confused whether those three stars are in same
line or not. He asks his teacher to help him to solve this problem. Help him to answer this
question.
(a) Three stars are collinear
(b) Three stars are not in a same line
(c) AB is perpendicular to BC
(d) none of these
Find the direction ratios of a ray of light passing through the points (1, 2, 3) and (-1, -3, 5).
(a) -2, 5, 2
(b) -2, -5, 2
(c) -2, -5, 8
(d) 2, -5, 8
What are direction ratios of the line 𝑟 = (3𝑖̂ + 4𝑗̂ - 5𝑘̂) + m (7𝑗̂ + 3𝑘̂)?
(a) 3, 4, -5
(b) -3, -4, 5
(c) 3, 11, -2
(d) 0, 7, 3
What are the direction cosines of the line having direction ratios 0, -3, 4?
(a) 0, -3, 4
(b) 0, -8, 10
(c) 0, 3/5, 4/5
(d) 0, -3/5, 4/5
Find the Cartesian equation of a line parallel to y-axis and passing through the point (1, -2, 7)
𝑥−1 𝑦+2
𝑧−7
𝑥−1 𝑦+2 𝑧−7
(a) 1 = −2 = 7
(b) 1 = 0 = 1
1
√29 √29 √29
4
5
6
7
(c)
8
9
10
11
12
13
14
15
𝑥+1
1
=
𝑦−2
−2
=
𝑧+7
7
(d)
𝑥−1
0
=
𝑦+2
1
=
√29 √29 √29
1
1
1
1
𝑧−7
0
𝑥−6
𝑦−4
𝑧−1
Write down the vector form of the following equation of line 2 = 1 = −3
(a) 𝑟 = (6𝑖̂ + 4𝑗̂ +1𝑘̂) + 𝛼 (2𝑖̂ + 𝑗̂ - 3𝑘̂)
(b) 𝑟 = (2𝑖̂ + 𝑗̂ - 3𝑘̂) + 𝛼 (6𝑖̂ + 4𝑗̂ +1𝑘̂)
(c) 𝑟 = (-2𝑖̂ - 𝑗̂ + 3𝑘̂) + 𝛼 (6𝑖̂ + 4𝑗̂ +1𝑘̂)
(d) 𝑟 = (-6𝑖̂ - 4𝑗̂ -1𝑘̂) + 𝛼 (2𝑖̂ + 𝑗̂ - 3𝑘̂)
Two lines with direction ratios a, b, c and p, q, r respectively are said to be ………… if ap + bq +
cr = 0.
(a) Parallel
(b) Perpendicular
(c) Coincident
(d) Skew
For what value of p, given two lines are parallel?
𝑥−1 𝑦+2
𝑧−7
𝑥−8 𝑦−2 𝑧+2
= −2 = 7 and
= 𝑝 = 14
1
2
(a) p = -2
(b) p = 4
(c) p = -4 (d) can’t be determined
If a line has direction ratios 2, – 1, – 2, determine its direction cosines:
(a). ⅓, ⅔, -⅓
(b). ⅔, -⅓, -⅔
(c). -⅔, ⅓, ⅔
(d). None of the above
The direction ratios of the line 6x – 2 = 3y + 1 = 2z – 2 are:
(a) 6, 3, 2
(b) 1, 1, 2
(c) 1, 2, 3
(d) 1, 3, 2
What are the direction cosines of x axis?
(a) (1,0,0)
(b) (0,1,1)
(c) (0,0,1)
(d) (0,1,1)
What is the angle between the lines 2x = 3y = – z and 6x = – y = – 4z ?
𝜋
𝜋
𝜋
(a) 0 (b) 4
(c) 3
(d) 2
Assertion (A): The vector form of the line through the point (5, 2, – 4) which is parallel to the
vector 2𝑖̂ + 𝑗̂ – 6𝑘̂ is
𝑟 = 2𝑖̂ + 𝑗̂ – 6𝑘̂ + s (5𝑖̂ +2 𝑗̂ – 4𝑘̂)
1
1
1
1
1
1
1
1
16
17
18
19
20
Reason (R) : Vector equation of a line passing through the given point with position vector 𝑎 and
parallel to the given vector 𝑏⃗ is 𝑟 = 𝑎 + 𝑠 𝑏⃗
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Assertion (A): Skew lines are non-intersecting and non-parallel lines.
Reason (R) : They exist in 3D space only.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
1
Assertion (A) : The angle between the diagonals of a cube is 3
Reason (R) : The D.R.s of the diagonals of a cube are proportional to a, a, a and –a, a, a
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Assertion (A) : The image of (0, 2, 0) in x-axis is (0, –2, 0)
Reason (R) : x-axis is perpendicular to y-axis and with reference to (0, 2, 0), (0, 0, 0) is foot of the
perpendicular on x-axis.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
𝜋
Assertion (A) : If a line makes an angle of 4 with each of y and z axis then it makes a right angle
with x-axis
Reason (R) : The sum of the angles made by a line with the coordinate axes is 𝜋
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
𝜋
Assertion (A) : The acute angle between the line 𝑟 = (𝑖̂ + 𝑗̂ +2𝑘̂) + 𝛽 (𝑖̂ - 𝑗̂) and x-axis is 4
Reason (R) : If 𝜃 is the acute angle between 𝑟 = ⃗⃗⃗⃗
𝑎1 + 𝛽 𝑏⃗1 and 𝑟 = ⃗⃗⃗⃗
𝑎2 + 𝛽 𝑏⃗2 , then 𝑐𝑜𝑠 𝜃 =
1
1
1
1
1
⃗ .𝑏
⃗
𝑏
||𝑏⃗ 1||𝑏⃗2 ||
1
21
22
23
24
25
2
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Check whether the given two lines are coincident, skew, parallel or perpendicular?
𝑟 = (6𝑖̂ + 4𝑗̂ +1𝑘̂) + 𝛽 (2𝑖̂ + 𝑗̂ - 3𝑘̂)
𝑟 = (-2𝑖̂ - 𝑗̂ + 3𝑘̂) + 𝛼 (6𝑖̂ + 3𝑗̂ - 9𝑘̂)
Find the angle between the pair of lines: 𝑟 = (6𝑖̂ + 4𝑗̂ - 8𝑘̂) + 𝛾 (2𝑖̂ + 4𝑗̂ + 4𝑘̂) and 𝑟 = (10𝑖̂ - 4𝑗̂) +
̂ + 4𝑗̂ +12𝑘̂)
𝛿 (6𝑖
Find the direction cosines of the sides of the triangle with vertices A (1, 3, 5), B (2, 5, 7) and C (1, -4, 3)
Show that the line passing through two points (2, 3, 5) and (5, 6, 8) is parallel to the line through
the points (1, 6, -5) and (4, 9, -2).
Find the angle between the pair of lines:
2
2
2
2
2
26
27
28
29
30
31
32
𝑟 = (8𝑖̂ + 4𝑗̂ +9𝑘̂) + 𝛽 (2𝑖̂ + 𝑗̂ - 3𝑘̂)
𝑟 = (7𝑖̂ - 3𝑗̂ + 1𝑘̂) + 𝛼 (6𝑖̂ + 3𝑗̂ - 9𝑘̂)
Find the value of ‘m’ so that the given lines are perpendicular.
−𝑥+1 𝑦+2 𝑧−7
𝑥−8
𝑦−2 𝑧+2
=
=
and
=
= 6
1
2
4
2
𝑚
2
Find the value of  , so that the following lines are perpendicular to each other
x  5 2  y 1 z
x 2 y 1 1 z




5  2
5
1 and 1
4
3
Find the shortest distance between the following lines:
𝑟 = (2𝑖̂ + 4𝑗̂ - 8𝑘̂) + 𝛽 (2𝑖̂ + 3𝑗̂ + 6𝑘̂)
𝑟 = (𝑖̂ - 2𝑗̂ - 4𝑘̂) + 𝛼 (4𝑖̂ + 6𝑗̂ +12𝑘̂)
Find the equation of a line parallel to
=(
3
+ 2 + 3 ) + (2 + 3 + 4 ) and passing through
2 + 4 + 5 . Also find the S.D. between these lines.
𝑥+1 𝑦+1 𝑧+1
Find the shortest distance between the lines 7 = −6 = 1 and
Find the shortest distance between the lines
2
=(
𝑥−3
1
=
𝑦−5
−2
=
𝑧−7
and
=(2
Find the equation of the line passing through (1, –1, 1) and perpendicular to the lines joining the
points(4, 3, 2), (1, –1, 0) and (1, 2, –1), (2, 2, 1).
4
at a distance 5 units from the point P(1, 3, 3)
34
4
Show that the lines
intersection.
and
intersect. Find their point of
35
38
3
4
Find the point on the line
37
3
1
33
36
3
4
Find the image of the point (1, 6, 3) in the line
.
Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8,4) to the line
joining the points B(0, -1,3) and C(2, -3,-1).
The equation of motion of a missile are x = 2t, y = 3t, z = t, where the time ‘t’ is given in seconds
and distance is measured in kilometers. Based on it, answer the following question;
(i)
What is the path of the missile?
(a) Straight line
(b) Parabola
(c) Circle
(d) Ellipse
(ii)
Which of the following points lie on the path of missile?
(a) (1, 2, 3)
(b) (2, 3, 1) (c) (4, 1, -2)
(d) (1, -2, 3)
(iii) At what distance will the missile be in 10 seconds from the starting point (0, 0, 0)?
(a) 10√14 km (b) 20√14 km (c) 10√7 km (𝑑)20√14 km
(iv)
The position of missile at a certain instant of time is (2,-8, 15) then what will be height
of the missile from the ground if ground is considered as xy-plane?
(a) 2 km
(b) 8 km
(c) 15 km
(d) 7 km
In a class, teacher asks students what they know about space or three dimensional system. He asks
students some basic questions. Help students to answer the following;
(i)
What is the equation of x-axis in space?
(a) x = 0, y = 0
(b) y = 0, z = 0
(c) x = 0
(d) none of these
(ii)
What are direction ratios of y-axis?
(a) 0,0,1
(b) 1,0,0
(c) 0,1,0
4
4
4
39
40
\
(d) 1,0,1
(iii) Direction cosines of a line are < m, m, m >, then
(a) m > 0
(b) m < 0
(c) m < 1
1
−1
(d) m = or
√3
√3
(iv)
Which of the following statement is correct?
(a) Direction ratios of a line are equal to its direction cosines.
(b) Direction ratios of two perpendicular lines are proportional.
(c) Direction ratios of two parallel lines are proportional.
(d) All of these are correct.
Find the coordinates of the image of the point (2, 3, 4) with respect to the line 𝑟 = (2𝑗̂ + 4𝑘̂) + 𝛾
(2𝑖̂ + 4𝑗̂ + 1𝑘̂); where 𝛾 is a scalar. Also, find the distance of the image from the origin.
An aeroplane is flying along the line 𝑟 = 𝛼 (2𝑖̂ + 3𝑗̂ + 4𝑘̂); where 𝛼 is a scalar and another
aeroplane is flying along the line 𝑟 = (𝑖̂ + 𝑗̂)+ 𝛾 (3𝑗̂ + 2𝑘̂); where 𝛾 is a scalar. At what points on
the line should they reach, so that the distance between them is shortest. Find the shortest possible
distance between them.
4
4
Chapter 12: Linear Programming Problems
Q.
No.
1
2
3
4
5
6
QUESTION
Feasible region is the set of points which satisfy
(a) The objective functions
(b) Some the given constraints
(c) All of the given constraints
(d) None of these
The solution set of the inequality 4x + 5y > 6 is
(a) an open half-plane not containing the origin.
(b) an open half-plane containing the origin.
(c) the whole XY-plane not containing the line inequality 4x + 5y = 6.
(d) a closed half plane containing the origin.
Maximize Z = 10 x1 + 25 x2, subject to 0 ≤ x1 ≤ 3, 0 ≤ x2 ≤ 3, x1 + x2 ≤ 5
(a) 80 at (3, 2)
(b) 75 at (0, 3)
(c) 30 at (3, 0)
(d) 95 at (2, 3)
Based on the given shaded region as the feasible region in the graph, at which point(s) is
the objective function Z = 3x + 9y maximum?
a) Point B
b) Point C
c) Point D
d) every point on the line segment CD
The feasible, region for an LPP is shown
shaded in the figure. Let Z = 3x – 4y be the
objective function. A minimum of Z occurs at
(a) (0, 0)
(b) (0, 8)
(c) (5, 0)
(d) (4, 10)
Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F
= 4x+ 6y be the objective function. The Minimum value of F occurs at
MARK
1
1
1
1
1
1
7
8
9
10
11
12
(a) only (0, 2)
(b) only (3, 0)
(c) the mid-point of the line segment joining the points (0, 2) and (3, 0) only
(d) any point on the line segment joining the points (0, 2) and (3, 0).
The maximum value of Z = x+ 4y subject to the constraints 3x+ 6y ≤ 6, 4x+ 8y ≥ 16, x ≥ 0, y
≥ 0 is
(a) 4
(b) 8
(c) unbounded feasible region
(d) Does not exist feasible region
The feasible region of the inequality x+ y ≤ 1 and x– y ≤ 1 lies in ......... quadrants.
(a) Only I and II
(b) Only I and III
(c) Only II and III
(d) All the four
The position of the points O (0, 0) and P (2, –1) is ........, in the region of the inequality 2y–
3x < 5.
(a) O is inside the region and P is outside the region
(b) O and P both are inside the region
(c) O and P both are outside the region
(d) O is outside the region and P is inside the region
The point at which the maximum value of Z = 3x+ 2y subject to the constraints x+ 2y ≤ 2, x
≥ 0, y ≥ 0 is
(a) (0, 0)
(b) (1.5, – 1.5)
(c) (2, 0)
(d) (0, 2)
The vertices of the feasible region determined by some linear constraints are (0, 2), (1, 1),
(3, 3), (1, 5). Let Z = px+ qy where p, q > 0. The condition on p and q so that the maximum
of Z occurs at both the points (3, 3) and (1, 5) is
(a) p= q
(b) p= 2q
(c) q= 2p
(d) p= 3q
The feasible solution for a LPP is shown in Figure Let z = 3x – 4y be the objective function.
Minimum of Z occurs at
1
1
1
1
1
1
(a) (0, 0)
(b) (0, 8)
(c) (5, 0)
(d) (4, 10)
13
The solution set of the following system of inequations: x + 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1,
is
(a) bounded region
(b) unbounded region
(c) only one point
1
14
15
16
17
18
19
(d) empty set
Which of the following statement is correct?
(a) Every L.P.P. admits an optimal solution
(b) A L.P.P. admits a unique optimal solution
(c) If a L.P.P. admits two optimal solutions, it has an infinite number of optimal solutions
(d) The set of all feasible solutions of a L.P.P. is not a convex set.
Assertion (A): Feasible region is the set of points which satisfy all of
the given constraints.
Reason (R): The optimal value of the objective function is attained at the
points on X-axis only.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false.
D. A is false but R is true.
Assertion (A): It is necessary to find objective function value at every
point in the feasible region to find optimum value of the objective
function.
Reason (R): For the constrains2x+3y 6, 5x+3y 15, x 0 and y 0 corner
points of the feasible region are (0,2), (0,0) and (3,0).
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false.
D. A is false but R is true.
Assertion (A): It is necessary to find objective function value at every
point in the feasible region to find optimum value of the objective
function.
Reason(R): For the constrains2x+3y ≤ 6, 5x+3y ≤ 15, x ≥ 0 and y ≥ 0 corner
points of the feasible region are (0,2), (0,0) and (3,0).
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false.
D. A is false but R is true.
Assertion (A): For the constraints of linear optimizing function Z = x1+
X2 given by x1+ x2 ≤ 1, 3x1 + x2 ≥ 1, x ≥ 0 and y ≥ 0 there is no feasible
region.
Reason (R): Z = 7x + y, subject to 5x + y ≤ 5, x + y ≥ 3, x ≥ 0, y ≥ 0.
1 5
The corner points of the feasible region are (2, 2) (0,3) and (0,5).
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false.
D. A is false but R is true.
Assertion (A): The maximum value of Z = 11x+7y
Subject to the constraints are
2x+y ≤ 6,
x ≤ 2,
x, y ≥ 0.
Occurs at the point (0,6).
Reason (R): If the feasible region of the given LPP is bounded, then the
maximum and minimum values of the objective function occur at corner points.
1
1
1
1
1
20
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false.
D. A is false but R is true.
Assertion (A): If an LPP attains its maximum value at two corner points of
the feasible region then it attains maximum value at infinitely many points.
Reason (R): if the value of the objective function of a LPP is same at two
corners then it is same at every point on the line joining two corner points.
A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false.
D. A is false but R is true.
1
Chapter 13: Probability
S.No.
QUESTION
Mark
1
1
The value of k, for which the following distribution is a probability distribution
X
30
10
–10
P(X)
1/5
3/10
k
(a) 1/3
(b) 1/2
(c) 1/10
(d) 1/5
2
Ramesh is playing with a dice, and he supposed that event A is getting a number greater than 6 1
and event B is getting an odd prime number. Further he finds that 𝑃(𝐴) = 0 and 𝑃(𝐵) =
1
3
, then 𝑃(𝐵/𝐴) 𝑖𝑠
(a) 0
3
33
56
6
1
10
3
16
37
221
1
(c)
(d)
14
3
28
2
9
(𝑏) 5
1
1
(𝑐) 20
(𝑑) 3
(𝑏)
5
11
14
(𝑐) 16
16
(𝑑) 16
5
1
(𝑏) 13
2
(𝑐) 13
(𝑑) 13
A bag contains 10 good and 6 bad mangoes. One of the mangoes is selected. The probability 1
that it is either good or bad
64
(a)
9
9
(b) 64
1
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of kings 1
obtained. Then the expected value of E is
(a)
8
(d) not defined
1
Archaeological Survey of India has found coins at one of the sites of Indus
Valley civilization. While studying these coins for historical evidence faces of
the one of coin is labelled as head and tail. These coins are flipped in the air
and result is noted. If events A and B are defined as A= two heads come, B=
last should be head. Then, A and B are
(a) Independent (b) not independent (c) mutually exclusive (d)none of these
A box contains 6 pens and 10 pencils. Half of the pens and half of the pencils are of blue colour. 1
If one of the items is chosen at random, the probability that it is of blue colour or is a pen is
(a)
7
(c) 1
3
In a boy’s college, 30% students play Cricket, 25% play Football and
10% students play both Cricket and Football. One student is selected
at random. The probability that he likes Cricket if he also like Football
is
(a)
5
1
A rocket has 8 engines out of which 3 are not working. If the two
engines are selected without replacement and tested, the probability
that both are not working.
(a)
4
(b)
64
(𝑏)
49
64
(𝑐)
40
(𝑑)
64
24
64
If A and B are two independent events such that P( A)  0.4 , P( B)  p and P( A  B)  0.6 ,
then the value of ‘p’ is ?
(a)
1
2
(b)
1
3
(c)
2
3
(d)
1
5
10.
11
12
13
14
15
16
17
18
Q19
Q20
Three balls are drawn from a bag contains 2 red and 5 black balls. If the random variable X
represent the number of red balls drawn, then X can take values
(a) 0, 1, 2
(b) 0, 1, 2, 3
(c) 0
(d) 1, 2
If A and B be two given events such that P(A) = 0.6, P(B) = 0.2 and
P(A/B) = 0.5 Then P(A’/B’)=
(a) 1/10
(b) 3/10
(c) 3/8
(d) 6/7
If A and B are two independent events such that P(A) = 1/7, P(B) = 1/6 then 𝑃(𝐴’ ∩ 𝐵’) =
(a) 5/7
(b) 6/7
(c) 5/6
(d) 1/6
If A and B are two independent events such that P(B/A)= 2/5 , then P(B’) is
(a) 3/5
(b) 2/5
(c) 1/5
(d) 4/5
In a throw of a fair dice event E = {1,3, 6} and event F = {4, 6} then P(E/F) is
(a) 1/6
(b) 1/3
(c) 1/2
(d) 2/3
Three persons A, B and C, fire a target in turn. Their probabilities of hitting the target are 0.2,
0.3 and 0.5 respectively, the probability that target is hit, is
(a) 0.993
(b) 0.94
(c) 0.72
(d) 0.90
Let A and B be two given independent events such that P(A) = P, and P(B)=Q and P(exactly
one of A and B) = 2/3, then value of 3P+3Q–6PQ
(a) 2
(b) –2
(c) 4
(d) –4
Two numbers are selected at random at random (without replacement) from positive integers 2,
3, 4, 5, 6, 7. Let X denotes the larger of two numbers obtained. Then value of X may be
(a) 3,4,5
(b) 4, 5
(c) 5, 6, 7
(d) All (a),(b),(c)
Bag A contains 3 red and 5 black balls and bag B contains 2 Red and 4 black balls. A ball is
drawn from one of the bag. The probability that the ball drawn is red :
(a) 17/24
(b) 17/48
(c) 3/8
(d) 1/3
ASSERTION-REASON BASED QUESTIONS:
In the following questions, a statement of assertion (A) is followed by a statement of Reason
(R). Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
𝐵
Assertion (A): 𝐴 and 𝐵 two events, then𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃 (𝐴).
1
Reason (R) : Two events are said to be exhaustive if probability of the one of the events
Assertion (A): The probability of getting either a king or an ace from a pack of 52 playing cards 1
is
2
.
13
Reason (R): For any two events 𝐴 & B, 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵).
21
22
23
2
If E and F are two independent events, then prove that E and F are also independent events.
A box contains 12 black and 24 white balls. Two balls are drawn from the box one after the 2
other without replacement. What is the probability that both drawn balls are black?
Two cards are drawn successively, without replacement from a pack of 52 well shuffled cards. 2
What is the probability that first card is king, and the second one is an ace?
24
25
Bag one contains 3 red and 4 black balls second another Bag II contains 5 red and 6 black
2
balls. One ball is drawn at random from one of the bags and it is found to be red. Find the
probability that it was drawn from second Bag.
A group consists of an equal number of girls and boys. Out of this group 20% of boys and 30% 2
of the girls are unemployed. If a person is selected at random from this group, then find the
probability of the selected person being employed.
26
Abraham speaks truth in 80% cases and Bhavesh speaks truth in 90% cases. In What percentage 2
of cases are they likely to agree with each other in stating the same fact?
27
3
A urn contains 4 white and 6 red balls. Four balls are drawn at random from the urn. Find the
probability distribution of number of white balls.
A bag contains (2𝑛 + 1) coins. It is known that (𝑛 − 1) of these coins have a head on both 3
sides, whereas the rest of the coins are fair. A coin is picked up at random from the bag and is
31
tossed. If the probability that the toss results in a head is , determine the value of 𝑛.
42
1
2
The probability that Abraham hits the target is and the probability that Bhavesh hits it, is . If 3
28
29
30
31
32
33
3
5
both try to hit the target independently, find the probability that target is hit.
An electric shop has two types of LED bulbs of equal quantity. The probability of an LED bulb 3
lasting more than 6 months given that it is of type 1 is 0.7 and is given that it is of type 2 is 0.4.
Then find the probability that on LED bulb chosen uniformly at random lasts more than 6
months.
The reliability of a COVID PCR test is
4
specified as follows:
Of people having COVID, 90% of the
test
detects the disease but 10% goes
undetected. Of people free of COVID,
99% of the test is judged COVID
negative but 1% are diagnosed as
showing COVID positive. From a large
population of which only 0.1% have
COVID, one person is selected at
random, given the COVID PCR test, and the pathologists reports him/her as COVID positive.
(a) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is
actually having COVID’?
(b) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is
actually not having COVID’?
(c) What is the probability that the person is actually not having COVID’?
There are two antiaircraft guns, named A and B. The 4
probabilities that the shell fired from them hitting an
airplane are 0.3 and 0.2 respectively. Both of them fired
one shell at an airplane at the same time.
(a) What is the probability that the shells fired from,
exactly one of them hit the plane?
(b) If it is known that the shell fired from exactly one
of them hit the plane, then what is the probability that
it was fired from B?
Read the following passage and answer the questions given below:
An MNC (Multi-National Company) made it compulsory for its employees to get themselves
insured against injury, accident or illness etc. for financial back-up. 2000 people insured
themselves from ICICI Prudential, 4000 people insured themselves from MAX LIFE
Insurance and 6000 people insured themselves from STAR Health insurance. The probability
of an accident, injury or illness involving a person insured from ICICI Prudential, MAX Life
Insurance and STAR Health Insurance are 0.01, 0.03 and 0.15 respectively. One of the insured
persons meets with an accident, injury or illness.
(i)
What is the probability that an insured person has taken insurance from STAR
Health insurance?
(ii)
What is the probability that an insured person has taken insurance from MAX Life
Insurance?
(iii) Find the probability that an insured person meets with an accident.
34
35
Ramesh is going to play a game of chess against one of four opponents in an inter school sports 5
competition. Each opponent is equally likely to be paired against him. The table below shows
the chances of Ramesh losing, where paired against each opponent.
Opponent
Chance of losing
Opponent 1
12%
Opponent 2
60%
Opponent 3
𝑥%
Opponent4
84%
1
If the probability that Ramesh loses the game that day is 2, find the probability for Ramesh to
be losing when paired against opponent 3.
In a factory, machine A produces 30% of total output, machine B produces 25% and the 5
machine C produces the remaining output. The defective items produced by machines A, B and
C are 1%, 1.2%, 2% respectively. An item is picked at random from a day’s output and found
to be defective. Find the probability that it was produced by machine B?
KENDRIYA VIDYALAYA SANGATHAN, GUWAHATI REGION
Sample Paper for Practice SESSION 2023-24
CLASS :XII
SUBJECT: MATHEMATICS (041)
Time Allowed: 3 Hours
Maximum Marks: 80
General Instructions:
1. This question paper contains five sections - A, B, C, D and E. Each section is compulsory. However,
there are internal choices in some questions.
2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.
3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each)
with sub parts.
Q.No.
1.
SECTION – A
(Multiple Choice Questions) Each question carries 1 mark
10 19 2
What is the value of the minor of element 9 in 0 13 1 ?
9
2.
3.
4.
5.
6.
7.
Mark
s
1
24 2
(a) 9
(b) 7
(c) 7
(d) 0
If A is a square matrix of order 3 and 2A  k A , the the value of k is ?
(a) 8
(b) 8
(c) 4
(d) 4
The value of  for which the projection of a   i  j  4k on b  2i  6 j  3k is 4
units is ?
(a) 4
(b) 5
(c) 3
(d) 6
The number of points of discontinuity of the function f ( x)  x  1  x  2  sin x ,
x   0, 4 is ?
(a) 1
(b) 2
(c) 3
(d) 0
1
The derivative of sec(tan x) w.r.t x is ?
1
x
x
(a)
(b)
(c)
(d) x 1  x 2
2
2
2
1 x
1 x
1 x
If ‘m’ and ‘n’ respectively are the degree and order of the differential equation
3
d  dy  
    0 , then m  n  ?
dx  dx  
(a) 1
(b) 2
(c) 3
(d) 4
For an L.P.P the objective function is Z  4 x  3 y , and
the feasible region determined by a set of constraints (
linear inequalities ) is shown in the graph
Which one of the following is true ?
(a) Maximum value of Z is at R.
(b) Maximum value of Z is at Q.
(c) Value of Z at R is less than the value at P.
(d) Value of Z at Q is less than the value at R.
1
1
1
1
1
8.
The area of a parallelogram whose adjacent sides are i  2 j  3k and 2 i  j  4k is ?
1
(a) 5 3 sq. units (b) 10 3 sq. units (c) 5 6 sq. units (d) 10 6 sq. units
9.
10.
1
1 1  x 2 dx is

2

(a)
(b)
(c)
3
3
6
T
Let A be a square matrix the A. A is
(a) Singular matrix
(c) skew-symmetric matrix
11.
The value of
(a)
12.
13.
14.
15.
16.
17.
1
3
The value of
If

 x  1 x  log x 
 x  log x 
x 1
3
x 1

+C
4 1
(b)

12
1
(b) symmetric matrix
(d) non-singular matrix
1
2
x
3
(d)
dx is
 x  log x 
2
2
 C (c)
1
C
x  log x
(d)
1
 x  log x 
2
C
, then the value of x is
x 3 x  2 1 3
(a) 4
(b) 3
(c) 0
(d) 2
If A is a square matrix of order 3  3 such that adj A  25 , A  ?
1
(a) 125
(b) 9
(c) 5
(d)
5
If A and B are two independent events such that P( A)  0.4 , P( B)  p and
P( A  B)  0.6 , then the value of ‘p’ is ?
1
1
2
1
(a)
(b)
(c)
(d)
2
3
3
5
The corner points of the feasible region determined by the system of linear constraints
are (0,3), (1,1) and (3,0). Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where 𝑝, 𝑞 > 0. Conditions on 𝑝 and 𝑞 so
that the minimum of 𝑧 occurs at (3,0) and (1,1).
(a) 𝑝 = 3𝑞
(b) 2𝑝 = 𝑞
(c) 𝑝 = 3𝑞
(d) 𝑝 = 𝑞
2
3
4
2
x
x
x
d y
If y  1  x     ...... then
is equal to
2! 3! 4!
dx 2
(a)  x
(b)  y
(c) x
(d) y
If ‘  ’ is the angle between two vectors a and b , then a .b  a  b when ‘  ’ is equal
1
1
1
1
1
1
to


(c)
(d) 
4
2
The direction ratios of the line 4 x  12  2 x  4  3z  3 are
18.
(a) 4,6,3
(b) 6,3,4
(c) 4,6,3
(d) 3,6,4
ASSERTION-REASON BASED QUESTIONS:
In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
19.
Assertion (A): The domain of the function sec1 (2 x  1) is  , 1  0,   .
(a) 0
(b)
1
1
20.
21.
22.
23.
24.
25.
26.
27.
28.
 2  5
Reason (R): sec1  

3 6

x 5 y  2 z
x y z
Assertion (A): The lines

 and   are perpendicular.
7
5
1
1 2 3
x  x1 y  y1 z  z1
x  x2 y  y2 z  z2




Reason (R): Two lines
and
are
a1
b1
c1
a2
b2
c2
perpendicular if a1a2  b1b2  c1c2  0 .
SECTION –B
(This section comprises of very short answer type questions (VSA) of 2 marks each)

  3 
Find the value of sin   sin 1 
 
2
2



OR
Show that the function f : R  R given by f ( x)  x3  x , is an injective function.
2
A particle moves along the curve 6 y  x3  2 . Find the points on the curve at which ycoordinate is changing 8 times as fast as the x-coordinate.
Evaluate : ∫ 𝑠𝑖𝑛3 𝑥 𝑑𝑥
OR
𝑑𝑥
Evaluate : ∫ 𝑒 𝑥 +𝑒 −𝑥
2
d2y
 49 y .
dx 2
Find the maximum profit that a company can make , if the profit function is given by
𝑃(𝑥) = 72 + 42𝑥 − 𝑥 2 , where x is the number of units and P is profit in rupees.
SECTION –C
(This section comprises of very short answer type questions (SA) of 3 marks each)
2
If y  500 e7 x  600 e7 x , then show that
1
Evaluate : 
dx
5  4 x  x2
Let a pair of dice is thrown and X denote the sum of the numbers that appear on the two
dice. Find the probability distribution and mean of X.
OR
Let A and B be two students seeking admission in a college. The probability that A is
selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the
probability that B is selected.

ecos x
Evaluate:  cos x  cos x dx
e e
0
OR

Evaluate:
2
2
3
3
3
2


29.
1
sin x dx
2
Solve the differential equation:  tan 1 y  x  dy  1  y 2  dx .
OR
Show that the differential equation is homogeneous and solve it:
x
x


y
2 y e dx   y  2 xe y  dy  0




3
30.
Solve the following Linear Programming Problem graphically:
Maximize Z  7 x  10 y subject to 4 x  6 y  240 ; 6 x  3 y  240 ; x, y  0 .
3
31.
 1  sin x 
Find:  e x 
dx
 1  cos x 
3
SECTION-D
(This section comprises of long answer type questions (LA) of 5 marks each)
32.
Using integration, find the area of the triangle ABC with vertices A  1, 0  , B 1,3 and C
5
 3, 2  .
33.
34.
Show that the relation R on the set A  x  Z :0  x  12 , given by R   a, b  : a  b
is a multipleof 4 is an equivalence relation. Also write equivalence class 1 .
OR
Let R be a relation on N  N defined by  a, b  R  c, d   ad  b  c   bc  a  d  .
Show that R is an equivalence relation.
 4 4 4   1 1 1 
Determine the product  7 1 3  1 2 2  and use it to solve the system of

 

 5 3 1  2 1
3 
5
5
equations x  y  z  4 ; x  2 y  2 z  9 and 2 x  y  3z  1.
35.
Find the co-ordinates of the foot of the perpendicular drawn from the point A 1,8, 4  to
5
the line joining the points B  0, 1,3 and C  2, 3, 1 .
OR


Show that the lines r  (2i  3 j )   4i  6 j  12k and r  (2i  3 j  2k )


  2i  3 j  6k are parallel. Hence find the shortest distance between them.
SECTION-E
This section contains three Case-study / Passage based questions. First two questions have three
sub-parts (i), (ii) and (iii) of marks 1, 1 and 2 respectively. Third question has two sub-parts of 2 marks
each.
36.
4
A shopkeeper sells three types of flowers seeds A1 , A2 and A3 .
These are sold as mixture, where their proportions are 4:4:2 respectively.
Also their germination rates are 45%, 60% and 35% respectively.
Let A1 : seed A1 is chosen, A2 : seed A2 is chosen and A3 : seed A3 is chosen.
Also let E: seed germinates.
(i) Find P( A1 ), P( A2 ) and P( A3 ) .
(ii) Write P(E | A1 ) +P(E | A2 ) + P(E | A3 ).
37.
38.
(iii) Calculate the probability of a randomly chosen seed to germinate. Express the
answer in %.
OR
(iii) Calculate the probability that seed is of the type A2 given that a randomly chosen
seed does not germinate.
CASE STUDY II: Read the following passage and answer the questions given below.
A sports stadium is elliptical in shape. The district sports administration wants to design
a rectangular football field with the maximum possible area. The football field is given
x2 y 2
by the graph of

1.
25 9
(i) If the length and the breadth of the rectangular field be ‘2x’ and ‘2y’ respectively,
then find the area function in terms of ‘x’.
(ii) Find the critical point of the function obtained in (i).
(iii) Use first derivative test to find the length ‘2x’ and width ‘2y’ of the soccer field, that
will maximize its area.
OR
(iii) Use second derivative test to find the length ‘2x’ and width ‘2y’ of the soccer field,
that will maximize its area.
CASE STUDY III : Read the following passage and answer the questions given below.
An aeroplane is flying along the line 𝑟 = 𝛼 (2𝑖̂ + 3𝑗̂ + 4𝑘̂); where 𝛼 is a scalar and another
aeroplane is flying along the line 𝑟 = (𝑖̂ + 𝑗̂)+ 𝛾 (3𝑗̂ + 2𝑘̂); where 𝛾 is a scalar.
(i)
At what points on the line should they reach, so that distance between them is
shortest
(ii)
Find the shortest possible distance between them.
4
4