This is a 3 hour sample paper for cbse class 12 board exam. This covers all chapters of ncert 12th math book. For more such papers visit clay6.com/papers/.
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
This is a 3 hour sample paper for cbse class 12 board exam. This covers all chapters of ncert 12th math book. For more such papers visit clay6.com/papers/.
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
1. SAMPLE QUESTION PAPER
MATHEMATICS
CLASS –XII (2013-14)
BLUE PRINT
Unit
I.
VSA (1)
LA (6)
Total
1 (1)
Relations and Functions
SA (4)
4 (1)
–
5 (2)
10 (4)
II.
Matrices
2 (2)
Determinants
III.
–
–
VB 6 (1)
5 (2)
8 (3)
13 (5)
–
5 (2)
–
8 (2)
–
18 (4)
*4 (1)
6 (1)
2 (2)
*4 (1)
6 (1)
18 (5)
–
–
6 (1)
–
–
8 (2)
–
8 (2)
2 (2)
*4 (1)
–
6 (3)
1 (1)
4 (1)
*6 (1)
11 (3)
du
4 (1)
–
Continuity and Differentiability
.e
Integrals
w
w
Application of Integrals
w
Differential equations
Vectors
*4 (1)
1 (1)
Application of Derivatives
IV.
1 (1)
rit
e.
co
m
Inverse Trigonometric Functions
3–dim geometry
44 (11)
17 (6)
V.
Linear Programming
–
–
VB 6 (1)
–
6 (1)
VI.
Probability
–
VB 4 (1)
*6 (1)
–
10 (2)
2. SECTION–A
Question number 1 to 10 carry 1 mark each.
1.
Write the smallest equivalence relation R on Set A = {1, 2, 3}.
2.
If | ⃗|
3.
If ⃗ and ⃗⃗ are two unit vectors inclined to x–axis at angles 300 and 1200
, then find the value of | ⃗ ̂̇|
respectively, then write the value of | ⃗
–
√ –
–
(– )
5.
Evaluate
6.
If A = (
7.
For what value of k, the matrix (
8.
If |
9.
If ∫ (
10.
Evaluate: ∫
̂|
⃗⃗|
Find the sine of the angle between the line
– 5 = 0.
|⃗
–
–
–
and the plane 2x – 2y + z
rit
e.
co
m
4.
| ⃗ ̂̇|
.e
) is skew symmetric?
, where , are acute angles, then write the value of +.
w
w
|
du
), then what is A. (adj. A)?
)
write the value of k.
– )
w
(
SECTION–B
Questions numbers 11 to 22 carry 4 marks each.
11.
Let S be the set of all rational numbers except 1 and * be defined on S by
a * b = a + b – ab,
a, b S. Prove that:
a)
* is a binary on S.
b)
* is commutative as well as associative. Also find the identity element of *.
3. 12.
If a + b + c o and |
|
, then using properties of determinants, prove
that a = b = c.
13.
Evaluate: ∫(
)√ –
–
–
dx
OR
Evaluate: ∫ (
)
Find a unit vector perpendicular to the plane of triangle ABC where the vertices
rit
e.
co
m
14.
)(
are A (3, –1, 2) B (1,–1, –3) and C (4, –3, 1).
OR
Find the value of , if the points with position vectors ̂̇– ̂̇– ̂ ̂̇
15.
̂ are coplanar.
̂
(̂̇
Show that the lines ⃗
̂– ̂ )
( ̂̇– ̂) and ⃗
du
̂ and ̂̇
( ̂̇ – ̂ )
( ̂̇
̂̇– ̂ – ̂̇
̂̇
̂ ) intersect.
–
–
–
w
w
Prove that
–
OR
w
16.
.e
Also find their point of intersection.
Find the greatest and least values of (
17.
Show that the differential equation
–
–
)
(
√
–
)
is homogeneous, and
solve it.
18.
Find the particular solution of the differential equation cos x dy = sin x (cos x – 2y)
dx, given that y = 0 when
19.
.
Out of a group of 8 highly qualified doctors in a hospital, 6 are very kind and
cooperative with their patients and so are very popular, while the other two
4. remain reserved. For a health camp, three doctors are selected at random. Find
the probability distribution of the number of very popular doctors.
What values are expected from the doctors?
20.
Show that the function g(x) = |x–2|, xR, is continuous but not differentiable at
x = 2.
(
) with respect to x.
21.
Differentiate
22.
Show that the curves xy = a2 and x2 + y2 = 2a2 touch each other.
Separate the interval [
rit
e.
co
m
OR
] into sub–intervals in which f (x) =
increasing or decreasing.
is
SECTION–D
Find the equation of the plane through the points A (1, 1, 0), B (1, 2, 1) and C (–2, 2,
.e
23.
du
Question numbers 23 to 29 carry 6 marks each.
w
w
–1) and hence find the distance between the plane and the line
–
–
–
.
OR
w
A plane meets the x, y and z axes at A, B and C respectively, such that the centroid
of the triangle ABC is (1, –2, 3). Find the Vector and Cartesian equation of the
plane.
24.
A company manufactures two types of sweaters, type A and type B. It costs Rs. 360
to make one unit of type A and Rs. 120 to make a unit of type B. The company can
make atmost 300 sweaters and can spend atmost Rs. 72000 a day. The number of
sweaters of type A cannot exceed the number of type B by more than 100. The
company makes a profit of Rs. 200 on each unit of type A but considering the
5. difficulties of a common man the company charges a nominal profit of Rs. 20 on a
unit of type B. Using LPP, solve the problem for maximum profit.
(
–
)
25.
Evaluate: ∫
26.
Using integration find the area of the region {(
27.
A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a
)
}
mixture where the proportions are 4:4:2 respectively. The germination rates of
rit
e.
co
m
three types of seeds are 45%, 60% and 35%. Calculate the probability.
a)
of a randomly chosen seed to germinate.
b)
that it is of the type A2, given that a randomly chosen seed does not
germinate.
OR
du
Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls.
One ball is transferred from Bag I to Bag II and then two balls are drawn at
.e
random (without replacement) from Bag II. The balls so drawn are found to be
28.
w
w
both red in colour. Find the probability that the transferred ball is red.
Two schools A and B want to award their selected teachers on the values of
w
honesty, hard work and regularity. The school A wants to award Rs. x each, Rs. y
each and Rs. z each for the three respective values to 3, 2 and 1 teachers with a
total award money of Rs. 1.28 lakhs. School B wants to spend Rs. 1.54 lakhs to
award its 4, 1 and 3 teachers on the respective values (by giving the same award
money for the three values as before). If the total amount of award for one prize on
each value is Rs. 57000, using matrices, find the award money for each value.
29.
A given rectangular area is to be fenced off in a field whose length lies along a
straight river. If no fencing is needed along the river, show that the least length of
fencing will be required when length of the field is twice its breadth.
6. MARKING SCHEME
SECTION–A
1.
R = {(
2.
2a2
1
3.
√
1
4.
)(
)(
)}
1
1
√
–
6.
–
(
7.
k=–
1
rit
e.
co
m
5.
)
–
8.
10.
–cot x ex + C
du
k = –2
1
1
1
1
.e
9.
1
a)
let
–
w
11.
w
w
SECTION–B
( – )(
Since
–
–
– )
–
1
b)
is commutative
also, (
)
(
=
–
–
)
–
–
1
7. (
)
(
–
–
=
)
(
)
)
(
=
–
–
is associative
1
Let e be the identity,
Then a * e = a a + e – ae = a e (1–a) = 0
1
Since 1 – a 0 e = 0.
|
|
(
)|
(
–
|
)|
du
–
rit
e.
co
m
12.
= (a + b + c) (–b2–c2 + 2bc –a2 + ac + ab – bc) = 0
–
–
1
– |
–
1
w
w
.e
= – (a + b + c) (a2 + b2 + c2 – ab – bc – ca) = 0
= – (a + b + c) [(a–b)2 + (b–c)2 + (c–a)2] = 0
1
w
Since a + b + c = 0 a–b = 0, b–c = 0, c–a = 0
or a = b = c.
13.
∫(
∫(
–
– ) (√
∫( – )√ –
1
) (√ –
–
–
) dx
)
where sin x = t
1
½
8. ∫ √( – )
= ∫( – )√ –
(
=
) ⁄
–
= [
[
–
√ –
]
–
[√
½
|( – )
⁄
(
–
√ –
√
–
|
]
1
– )
– )|
|(
]
1
∫(
)(
)
–
)
– ∫
|
(
–
)
1
1
w
w
A vector perpendicular to the plane of ABC
= ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗
w
14.
|
2
.e
–
∫
du
(
)(
rit
e.
co
m
OR
̂
̂
̂
= |–
½
– |
–
̂– ̂
̂
1+1
√
1
–
or
̂̇
|⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗|
̂– ̂
√
Unit vector to plane of ABC =
√
(
OR
̂
̂ – ̂)
½
9. Let the points be A (3, –2, –1), B (2, 3, –4), C (–1, 1, 2) and D (4, 5, )
⃗⃗⃗⃗⃗⃗
̂– ̂
– ̂̇
⃗⃗⃗⃗⃗⃗
̂
̂
̂
)̂
1½
A, B, C, D are coplanar if [⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗] = 0
½
̂
̂
[⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗]
(
–
|–
–
|
½
rit
e.
co
m
⃗⃗⃗⃗⃗⃗
1(15+9) – 7 (–3 – 12) + (+1) (–3+20) = 0
=–
(̂
̂– ̂ )
du
Let, any point on the line ⃗
( ̂– ̂ )
( ̂
̂ ) be Q (
.e
and any point on line ⃗
( ̂– ̂) be P (
–
– )
½
)
½
½
w
w
If the lines intersect, P and Q must coincide for some and .
–
½
1+3 = 4+2 ..... (i)
1– = 0 ..... (ii)
w
15.
1
–1 = –1 + 3 ..... (iii)
Solving (ii) and (iii) we get = 1 and = 0
1
Putting in (i) we get 4 = 4, hence lines intersect.
½
P or Q (4, 0, –1) is the point of intersection.
1
10. 16.
–
LHS =
–
=
–
=
–
–
–
½
–
–
=
–
–
1
–
–
½
–
–
1½
–
= cot –1 3=RHS
½
(
–
)
–
(
)
–
=(
–
–
=( ) –
–
–
–
–
and greatest value = 2 [(
17.
x dy – y dx = √
=
x
,y
+
√
–
–
)
½
½
)
]
]
1
[ ]
w
least value =
–
(
– )
w
w
= [(
–
( –
) –
.e
= [(
–
) –
du
=
rit
e.
co
m
OR
1
)
]
1
dx
+√
=
=
+√
( ) =f( )
½
( )
=
[
(
)
√
( ) ]
1
11. differential equation is homogeneous.
let
= v or y = vx
v+x
=v+x
or ∫
= v+√
log|v + √
y+√
=∫
√
1
v+√
| = log cx
= cx
1
= cx2
1
Given differential equation can be written as
cot x
+ 2y = cos x or
+ 2 tan xy = sin x
∫
Integrating factor =
=
=∫
1
½
x dx
w
w
When x = , y = 0
= sec2 x.
.e
y. sec2x = sec x + c 1
½
du
the solution is y. sec2x = ∫
y = cosx + c cos2x.
rit
e.
co
m
18.
½
0= + C
C = -2
½
19.
½
w
Hence the solution is y = cosx–2cos2 x.
let x be the random variable representing the number of very popular doctors.
x:
1
P(x)
2
6C1 . 2C2
6C2. 2C1
8C3
=
3
6C3
8C3
=
½
8C3
=
1½
12. It is expected that a doctor must be
Qualified
Very kind and cooperative with the patients
20.
2
g(x) = |x-2| = {
½
LHL = lim (2-x) = 0
x2-
lim (x-2) = 0 and g (2) = 0
x2+
1
rit
e.
co
m
RHL =
g(x) is continuous at x = 2……………………………..
RHD = lim
h0
LHD
(
(
)
)
( )
= lim
h0
(
)
)
= -1
1
=1
½
g(x) is not differentiable at x = 2
½
.e
Let y = log (xsinx + cot2 x)
(xsinx + cot2x)……………………..
1
w
w
=
Let u =
and v = cot2x.
log u = sinx. log x ,
=
or
(
= lim
h0
w
21.
RHD
( )
du
LHD = lim
h0
½
= 2 cot x (–cosec2x)
½
+ logx. cosx
= xsinx [
=
] ………………………………..
[
(
)
1
]
1½
13. Solving xy = a2 and x2+y2 = 2a2 to get x =
a
for x = a, y = a and x = -a, y = -a
i.e the two curves intersect at P (a, a) and Q (-a, -a)
xy = a2
+y=0
x
x2+y2 = 2a2
2x+2y
=
–
1
= -1 at P and Q
1
=
1
=0
= -1 at P and Q
rit
e.
co
m
Two curves touch each other at P
as well as at Q.
1
OR
f(x) = sin4x + cos4x
du
f ‘(x) = 4 sin3x cosx -4 cos3x sinx
.e
= -4 sinx cosx (cos2x – sin2x)
f'(x) = 0
w
w
= –2 sin2x cos2x = - sin 4x
sin4x = 0
x = 0,
,
4x = 0,
1
2 , 3 , ………………
1
w
22.
) (
Sub Intervals are (
f’(x) < 0 in (0,
And f’(x) > 0 in (
)
)
f(x) is decreasing in (0,
,
)
f(x) in increasing in (
)
1
,
)
1
14. SECTION – D
to the plane is parallel to ⃗⃗⃗⃗⃗⃗
A vector
⃗⃗ = |
̂
̂
̂
| = – ̂̇ –
̂
̂
⃗⃗⃗⃗⃗⃗
1
̂̇– ̂
̂̇
1½
Equation of plane is ⃗. (2 ̂ + 3 ̂ – 3 ̂ ) = 5
( ⃗ ⃗⃗
1
⃗ ⃗⃗)
rit
e.
co
m
Since, (2 ̂ + 3 ̂ – 3 ̂ ). (3 ̂ – ̂ + ̂ ) = 0, so the given line is parallel to the plane.
½
Distance between the point (on the line) (6, 3, -2) and the plane ⃗.
(2 ̂ + 3 ̂ – 3 ̂ ) – 5 = 0 is
d=
|
|
√
=
√
=√
1½
du
Let the coordinates of points A, B and C be (a, 0, 0), (0, b, 0) and (0, 0, c)
= 1,
a = 3,
+ = 1 and
= -2
b = -6
Equation of plane is
or
+
w
w
Equation of plane is
½
.e
respectively.
w
23.
and
and
+
1
+ =1
6x – 3y + 2z – 18 = 0
=3
1½
c=9
1
1
which in vector form is
⃗ (6 ̂ – 3 ̂ + 2 ̂ ) = 18
1
15. 24.
let the company manufactures sweaters of type A = x, and that of type B = y. daily
LPP is Maximise P = 200x + 20y
s.t. x+y
300
360 x + 120y
x
72000
100
0y
1½
0
Correct Graph
1½
rit
e.
co
m
x–y
1
Getting vertices of the feasible region as
A (100, 0), B (175, 75), C (150, 150) and D (0, 300)
Maximum profit at B
= 200 (175) + 20 (75)
du
So Maximum Profit
let I = ∫ (
=
=
[(
) . x dx
w
25.
w
w
= Rs. 36500
1½
.e
= 35000 + 1500
)
]
1
∫
∫
½
x = tan
=
-∫
d
½
-∫
=
dx = sec2
∫
1
17. let E1 : randomly selected seed is A1 type P(E1) =
E2 : randomly selected seed is A2 type P(E2) =
1
E3 : randomly selected seed is A3 type P (E3) =
(i)
let A : selected seed germinates
P (A/E1) =
P (A/E3) =
1
rit
e.
co
m
P (A/E2) =
½
P(A/E3) =
½
P(A) = P(E1) P(A/E1) + P(E2) P(A/E2) + P (E3). P(A/E3)
=
=
du
let A : selected seed does not germinate
P(A/E1) =
.e
P(E2/A) =
, P(A/E2) =
w
w
(ii)
or 0.49
(
) ( |
)
(
(
) ( | )
) ( | )
(
) ( |
)
=
½
1
½
1
w
27.
OR
Let
E1 : transferred ball is red.
E2 : transferred ball is black.
½
A : Getting both red from 2nd bag (after transfer)
P(E1) =
P(E2) =
1
18. 5
P(A/E1)
=
P(A/E2)
=
P(E1/A)
=
C2
=
C2
4
C2
=
C2
1
or
10
1
) ( | )
) ( | ) ( ) ( |
(
(
=
½
)
=
1+1
The three equations are 3x+2y+z = 1.28
rit
e.
co
m
28.
or
10
4x + y + 3z = 1.54
x + y + z = 0.57
(
) ( )= (
i.e AX = B
du
)
)
1
)(
)=(
.e
w
w
w
( )= (
x = 25000, y = 21000, z = 11000
29.
½
1
|A| = -5 and X = A-1B
A-1 = (
1½
)
½
1½
let length be x m and breadth be y m.
length of fence L = x+2y
Let given area = a
L=x+
xy = a or y =
1
19. =1
=o
=
1
x2 = 2a
x=√
1
>o
½
for minimum length L = √
x=√
and breadth y =
√
+
=
√
√
= √
= x
1
½
w
w
w
.e
du
rit
e.
co
m
x = 2y
1