SET-3
Series ONS
·¤æðÇU Ù´.
Code No.
ÚUæðÜ Ù´.
65/3/S
ÂÚUèÿææÍèü ·¤æðÇU ·¤æ𠩞æÚU-ÂéçSÌ·¤æ ·ð¤ ×é¹-ÂëcÆU
ÂÚU ¥ßàØ çܹð´Ð
Roll No.
Candidates must write the Code on the
title page of the answer-book.
·
·
·
·
·
·
·
·
·
·
·ë¤ÂØæ Áæ¡¿ ·¤ÚU Üð´ ç·¤ §â ÂýàÙ-˜æ ×ð´ ×éçÎýÌ ÂëcÆU 12 ãñ´Ð
ÂýàÙ-˜æ ×ð´ ÎæçãÙð ãæÍ ·¤è ¥æðÚU çΰ »° ·¤æðÇU ِÕÚU ·¤æð ÀUæ˜æ ©žæÚU-ÂéçSÌ·¤æ ·ð¤ ×é¹-ÂëcÆU ÂÚU çܹð´Ð
·ë¤ÂØæ Áæ¡¿ ·¤ÚU Üð´ ç·¤ §â ÂýàÙ-˜æ ×ð´ 26 ÂýàÙ ãñ´Ð
·ë¤ÂØæ ÂýàÙ ·¤æ ©žæÚU çܹÙæ àæéM¤ ·¤ÚUÙð âð ÂãÜð, ÂýàÙ ·¤æ ·ý¤×æ´·¤ ¥ßàØ çܹð´Ð
§â ÂýàÙ-Â˜æ ·¤æð ÂɸÙð ·ð¤ çÜ° 15 ç×ÙÅU ·¤æ â×Ø çÎØæ »Øæ ãñÐ ÂýàÙ-Â˜æ ·¤æ çßÌÚU‡æ Âêßæüq ×ð´
10.15 ÕÁð ç·¤Øæ Áæ°»æÐ 10.15 ÕÁð âð 10.30 ÕÁð Ì·¤ ÀUæ˜æ ·ð¤ßÜ ÂýàÙ-Â˜æ ·¤æð Âɸð´»ð ¥æñÚU §â
¥ßçÏ ·ð¤ ÎæñÚUæÙ ß𠩞æÚ-ÂéçSÌ·¤æ ÂÚU ·¤æð§ü ©žæÚU Ùãè´ çܹð´»ðÐ
Please check that this question paper contains 12 printed pages.
Code number given on the right hand side of the question paper should be written
on the title page of the answer-book by the candidate.
Please check that this question paper contains 26 questions.
Please write down the Serial Number of the question before attempting
it.
15 minute time has been allotted to read this question paper. The question paper
will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will
read the question paper only and will not write any answer on the answer-book
during this period.
»ç‡æÌ
MATHEMATICS
çÙÏæüçÚUÌ â×Ø Ñ
3
ƒæ‡ÅðU
¥çÏ·¤Ì× ¥´·¤ Ñ 100
Time allowed : 3 hours
65/3/S
Maximum Marks : 100
1
P.T.O.
âæ×æ‹Ø çÙÎðüàæ Ñ
(i)
âÖè ÂýàÙ ¥çÙßæØü ãñ´Ð
(ii)
·ë¤ÂØæ Áæ¡¿ ·¤ÚU Üð´ ç·¤ §â ÂýàÙ-˜æ ×ð´
(iii)
¹‡Ç U¥ ·ð¤ ÂýàÙ 1 - 6 Ì·¤ ¥çÌ Üƒæé-©žæÚU ßæÜð ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤ ÂýàÙ ·ð¤ çÜ° 1 ¥´·¤ çÙÏæüçÚUÌ
ãñÐ
(iv)
¹‡ÇU Õ ·ð¤ ÂýàÙ
ÂýàÙ ãñ´Ð
26
7 - 19
Ì·¤ Îèƒæü-©žæÚU
I
Âý·¤æÚU ·ð¤ ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤ ÂýàÙ ·ð¤ çÜ°
4
¥´·¤
20 - 26
Ì·¤ Îèƒæü-©žæÚU
II
Âý·¤æÚU ·ð¤ ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤ ÂýàÙ ·ð¤ çÜ°
6
¥´·¤
çÙÏæüçÚUÌ ãñ´Ð
(v)
¹‡ÇU â ·ð¤ ÂýàÙ
çÙÏæüçÚUÌ ãñ´Ð
(vi)
©žæÚU çܹÙæ ÂýæÚUÖ ·¤ÚUÙð âð ÂãÜð ·ë¤ÂØæ ÂýàÙ ·¤æ ·ý¤×æ´·¤ ¥ßàØ çÜç¹°Ð
General Instructions :
(i)
All questions are compulsory.
(ii)
Please check that this question paper contains 26 questions.
(iii) Questions 1 - 6 in Section A are very short-answer type questions carrying 1 mark
each.
(iv) Questions 7 - 19 in Section B are long-answer I type questions carrying 4 marks
each.
(v)
Questions 20 - 26 in Section C are long-answer II type questions carrying 6 marks
each.
(vi) Please write down the serial number of the question before attempting it.
65/3/S
2
¹‡ÇU - ¥
SECTION - A
Âýà٠ⴁØæ 1 âð 6 Ì·¤ ÂýˆØð·¤ ÂýàÙ ·¤æ 1 ¥´·¤ ãñÐ
Question numbers 1 to 6 carry 1 mark each.
1.
ØçÎ
A
°·¤ °ðâæ ß»ü ¥æÃØêã ãñ ç·¤
?A ? 5 5
ãñ, Ìæð
?AA T ?
·¤æ ×æÙ çÜç¹°Ð
If A is a square matrix such that ?A?5 5 , write the value of ?AA T ? .
2.
∧
→
∧
∧
â×ÌÜæð´ r . (2 i 23 j 1 6 k )24 5 0 ÌÍæ
·¤èçÁ°Ð
∧
→
∧
∧
r . (6 i 2 9 j 118 k )130 5 0
→
∧
∧
·ð¤ Õè¿ ÎêÚUè ™ææÌ
∧
Find the distance between the planes r . (2 i 23 j 1 6 k )2 4 5 0 and
∧
→
∧
∧
r . (6 i 2 9 j 118 k )130 5 0 .
3.
→
→
ØçÎ a ÌÍæ b Îæð ×æ˜æ·¤ âçÎàæ ã´ñ Ìæð
°·¤ ×æ˜æ·¤ âçÎàæ ãæð?
→
→
a
ÌÍæ
→
b
·ð¤ Õè¿ ·¤æð‡æ €Øæ ãæð»æ, ØçÎ
→
→
→
→
a2 2 b
→
If a and b are unit vectors, then what is the angle between a and b for
→
→
a 2 2 b to be a unit vector ?
65/3/S
3
P.T.O.
4.
ØçÎ
1 2 
A 5

 3 21 
ÌÍæ
 1 24 

 3 22 
ãñ´, Ìæð
B5 
?AB?
·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð
2
 1 24 
 , find ?AB?.
 and B 5 
 3 21 
 3 22 
1
If A 5
5.
ØçÎ
0
3 

 2 25 
A5 
ÌÍæ
 0 4a 
KA 5 

28 5b 
ãñ´, Ìæð
k
ÌÍæ a ·ð¤ ×æÙ ™ææÌ ·¤èçÁ°Ð
0
3 
 0 4a 
 and KA 5 
 find the values of k and a.
 2 25 
28 5b 
If A 5 
6.
ØçÎ âçÎàæ
→
a
¥æñÚU
→
b
§â Âý·¤æÚU ã´ñ ç·¤
4
1 →
a5 , b5
3
2
→
¥æñÚU
1
→
a3b 5 3
→
ã´ñ, Ìæð
→ →
a.b
™ææÌ ·¤èçÁ°Ð
1
→
1 →
→ →
4
→
→
If vectors a and b are such that a 5 , b 5
and a 3 b 5 3 , then find
2
3
→ →
a.b .
¹‡ÇU - Õ
SECTION - B
Âýà٠ⴁØæ 7 âð 19 Ì·¤ ÂýˆØð·¤ ÂýàÙ ·ð¤ 4 ¥´·¤ ãñ´Ð
Question numbers 7 to 19 carry 4 marks each.
7.
ØçΠȤÜÙ
p

k sin 2 ( x 11), x [ 0
x50
f ( x )5 
 tan x 2 sin x , x > 0

x3
ÂÚU â´ÌÌ ãñ, Ìæð
k
·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð
p

k sin 2 ( x 11), x [ 0
Find k, if f ( x )5 
is continuous at x50.
 tan x 2 sin x , x > 0

x3
65/3/S
4
8.
( sin 2x )x1 sin21 3x ·¤æ x ·ð¤ âæÂðÿæ ¥ß·¤ÜÙ ·¤èçÁ°Ð
¥Íßæ
 1 1 x2 2 1 2 x2
tan21 
 1 1 x 2 1 1 2 x2





·¤æ
cos21x2
·ð¤ âæÂðÿæ ¥ß·¤ÜÙ ·¤èçÁ°Ð
Differentiate ( sin 2x )x1 sin21 3x with respect to x.
OR
 1 1 x2 2 1 2 x2
Differentiate tan21 
 1 1 x2 1 1 2 x2

9.
→ →
çÎØæ ãñ ç·¤ ÌèÙ âçÎàæ
∧
→
§â Âý·¤æÚU °·¤ ç˜æÖéÁ ÕÙæÌð ãñ´ ç·¤
™ææÌ ·¤èçÁ° ç·¤ ç˜æÖé Á ·¤æ ÿæð ˜ æȤÜ
p, q, r, s
→
a , b ÌÍæ c

 with respect to cos21x2.


∧
∧
b 5 s i 13 j 1 4 k
ÌÍæ
→
∧
∧
∧
c 53 i 1 j 22 k
5 6
ãñ Áãæ¡
→
→
→
a5b1 c
∧
→
Ð °ðâð
∧
∧
a 5 p i 1 q j 1 r k,
ãñ´Ð
→ → →
→
→
→
Given that vectors a , b , c form a triangle such that a 5 b 1 c . Find p, q, r, s
→
∧
∧
∧ →
∧
∧
∧
such that area of triangle is 5 6 where a 5 p i 1 q j 1 r k, b 5s i 13 j 14 k and
→
∧
∧
∧
c 53 i 1 j 22 k .
10.
Îæð ÍñÜð A ¥æñÚU B ãñ´Ð ÍñÜð A ×ð´ 3 âÈð¤Î ¥æñÚU 4 ÜæÜ »ð´Îð ãñ´ ÌÍæ B ÍñÜð ×ð´ 4 âÈð¤Î ¥æñÚU
3 ÜæÜ »ð´Îð ãñ´Ð ÌèÙ »ð´Îð ØæÎë‘ÀUØæ ç·¤âè °·¤ ÍñÜð ×ð´ âð (ÂýçÌSÍæÂÙæ çÕÙæ) çÙ·¤æÜè »Øè
¥æñÚU ÂæØè´ »Øè´ ç·¤ ßã Îæð âÈð¤Î ÌÍæ °·¤ ÜæÜ ãñ´ ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° ç·¤ ßã ÍñÜð B ×ð´
âð çÙ·¤æÜè »§ü ÍèÐ
There are two bags A and B. Bag A contains 3 white and 4 red balls whereas
bag B contains 4 white and 3 red balls. Three balls are drawn at random
(without replacement) from one of the bags and are found to be two white and
one red. Find the probability that these were drawn from bag B.
65/3/S
5
P.T.O.
11.
§üàææÙ ¥ÂÙð »æ¡ß ×ð´ °·¤ ¥æØÌæ·¤æÚU Ö깇ÇU, çßlæÜØ ·ð¤ çÜ° ÎæÙ ÎðÙæ ¿æãÌæ ãñÐ ÁÕ ©ââð
Ö깇ÇU ·¤è çß×æ°¡ ÂêÀUè »§Z Ìæð ©âÙð ÕÌæØæ ç·¤ ØçÎ §â·¤è Ü´Õæ§ü 50 ×è. ·¤× ÌÍæ ¿æñǸæ§ü
50 ×è. Õɸæ Îè Áæ°, Ìæð §â·¤æ ÿæð˜æÈ¤Ü â×æÙ ÚUãð»æ ÂÚU‹Ìé ØçÎ §â·¤è Ü´Õæ§ü 10 ×è. ·¤× ·¤ÚU
Îè Áæ° ÌÍæ ¿æñǸæ§ü 20 ×è. ·¤× ·¤ÚU Îè Áæ° Ìæð §â·¤æ ÿæð˜æÈ¤Ü 5300 ×è2 ·¤× ãæð Áæ°»æÐ
¥æÃØêãæð´ ·ð¤ ÂýØæð» âð §â ŒÜæÅU ·¤è çß×æ°¡ ™ææÌ ·¤èçÁ°Ð ·¤æÚU‡æ Öè ÎèçÁ° ç·¤ ß㠌ÜæÅU €Øæð´
ÎæÙ ÎðÙæ ¿æãÌæ ãñ?
Ishan wants to donate a rectangular plot of land for a school in his village.
When he was asked to give dimensions of the plot, he told that if its length is
decreased by 50 m and breadth is increased by 50 m, then its area will remain
same, but if length is decreased by 10 m and breadth is decreased by 20 m,
then its area will decrease by 5300 m2. Using matrices, find the dimensions
of the plot. Also give reason why he wants to donate the plot for a school.
12.
¥ß·¤Ü â×è·¤ÚU‡æ ·¤æð ãÜ ·¤èçÁ° Ñ
x
x


2y e y dx 1  y 22x e y  dy 50


x


x

y
y
Solve the differential equation : 2y e dx 1  y 22x e  dy 50

13.

¥ß·¤Ü â×è·¤ÚU‡æ ·¤æð ãÜ ·¤èçÁ° Ñ ( x 1 1)
dy
2 y 5 e3x ( x 1 1)3
dx
Solve the differential equation : ( x 1 1)
dy
2 y 5 e3x ( x 1 1)3
dx
65/3/S
6
14.
™ææÌ ·¤èçÁ° Ñ ∫[log (log x )1
Find : ∫ [log (log x )1
15.
çâh ·¤èçÁ° ç·¤
1
( log x )2
1
( log x )2
] dx
] dx
3
 17  p
2sin21   2 tan21   5
5
 31  4
¥Íßæ
â×è·¤ÚU‡æ ·¤æð
x
·ð¤ çÜ° ãÜ ·¤èçÁ° Ñ
3
cos (tan21x )5sin  cot21 
4

3
 17  p
Prove that 2sin21   2 tan21   5
5
 31  4
OR
 21 3 
21
Solve the equation for x : cos (tan x )5sin  cot
4 

16.
12sin x
™ææÌ ·¤èçÁ° Ñ ∫ sin x (11sin x) dx
Find :
65/3/S
12sin x
∫ sin x (11sin x) dx
7
P.T.O.
p
17.
sin2 x
dx
sin x 1 cos x
0
2
×æÙ ™ææÌ ·¤èçÁ° Ñ ∫
¥Íßæ
1
×æÙ ™ææÌ ·¤èçÁ° Ñ ∫ cot21 (1 2 x 1 x2 ) dx
0
p
sin2 x
∫ sin x 1 cos x dx
0
Evaluate :
2
OR
1
(
)
Evaluate : ∫ cot21 1 2 x 1 x2 dx
0
18.
ß·ý¤ ay25x3 ·ð¤ ©â çÕ´Îé çÁâ·¤æ
·¤èçÁ°Ð
x
çÙÎðüàææ´·¤
am2
ãñ, ÂÚU ¥çÖÜ´Õ ·¤æ â×è·¤ÚU‡æ ™ææÌ
Find equation of normal to the curve ay25x3 at the point whose x coordinate
is am2.
19.
çÕ´Îé¥æð́ A (3, 2, 1), B (4, 2, 22) ÌÍæ C (6, 5, 21) âð ãæð·¤ÚU ÁæÙð ßæÜð â×ÌÜ ·¤æ â×è·¤ÚU‡æ
™ææÌ ·¤èçÁ°Ð ¥ÌÑ l ·¤æ ×æÙ ™ææÌ ·¤èçÁ° çÁâ·ð¤ çÜ° A (3, 2, 1), B (4, 2, 22),
C (6, 5, 21) ÌÍæ D (l, 5, 5) â×ÌÜèØ ãæð´Ð
¥Íßæ
©â çÕ´Îé ·ð¤ çÙÎðüàææ´·¤ ™ææÌ ·¤èçÁ° Áãæ¡ ÚðU¹æ
â×ÌÜ ·¤æð ç×ÜÌè ãñ Áæð âçÎàæ
→
∧
→
∧
∧
n 5 i 1 j 13k
ÎêÚUè ÂÚU ãñÐ
65/3/S
∧
∧
∧
∧
∧
∧
©â
4
11
·¤è
r 5(2i 22 j 23 k )1l(3 i 14 j 13 k )
8
ÂÚU ÜÕ´ßÌ ãñ ÌÍæ ×êÜ çÕ´Îé âð
Find the equation of plane passing through the points A (3, 2, 1), B (4, 2, 22)
and C (6, 5, 21) and hence find the value of l for which A (3, 2, 1),
B (4, 2, 22), C (6, 5, 21) and D (l, 5, 5) are coplanar.
OR
Find
→
the
∧
∧
co-ordinates
∧
∧
∧
of
the
point
where
the
line
∧
r 5(2i 22 j 23 k )1l(3 i 14 j 13 k ) meets the plane which is perpendicular to
→
∧
∧
∧
the vector n 5 i 1 j 13k and at a distance of
4
from origin.
11
¹‡ÇU - â
SECTION - C
Âýà٠ⴁØæ 20 âð 26 Ì·¤ ÂýˆØð·¤ ÂýàÙ ·ð¤ 6 ¥´·¤ ãñ´Ð
Question numbers 20 to 26 carry 6 marks each.
20.
ÂýÍ× ÀUÑ ÏÙ Âê‡ææZ·¤æð´ ×ð´ âð Ìè٠ⴁØæ°´ ØæÎë‘ÀUØæ (çÕÙæ ÂýçÌSÍæÂÙæ) ¿éÙè »§ZÐ ×æÙ Üð´ X
ÌèÙæð´ ⴁØæ¥æð´ ×ð´ âð âÕâð ÀUæðÅUè ⴁØæ ·¤æð ÃØ€Ì ·¤ÚUÌæ ãñ, Ìæð X ·¤æ ÂýæçØ·¤Ìæ Õ´ÅUÙ ™ææÌ
·¤èçÁ°Ð Õ´ÅUÙ ·¤æ ×æŠØ ÌÍæ ÂýâÚU‡æ Öè ™ææÌ ·¤èçÁ°Ð
Three numbers are selected at random (without replacement) from first six
positive integers. If X denotes the smallest of the three numbers obtained, find
the probability distribution of X. Also find the mean and variance of the
distribution.
21.
°·¤ â´ÌéçÜÌ ¥æãæÚU ×ð´ ·¤× âð ·¤× 80 ×æ˜æ·¤ çßÅUæç×Ù A ¥æñÚU 100 ×æ˜æ·¤ ¹çÙÁ ãæðÙæ
¿æçã°Ð Îæð ¹æl ÂÎæÍü F1 ¥æñÚU F2 ©ÂÜŽÏ ãñ´ çÁÙ·¤è Üæ»Ì ·ý¤×àæÑ ` 5 ÂýçÌ ×æ˜æ·¤ ¥æñÚU
` 6 ÂýçÌ ×æ˜æ·¤ ãñÐ Öæð’Ø ÂÎæÍü F1 ·¤è °·¤ §·¤æ§ü ×ð´ çßÅUæç×Ù A ·ð¤ 4 ×æ˜æ·¤ ¥æñÚU ¹çÙÁ
ÂÎæÍü ·ð¤ 3 ×æ˜æ·¤ âç×çÜÌ ãñ´, ÁÕç·¤ Öæð’Ø ÂÎæÍü F2 ·¤è °·¤ §·¤æ§ü ×ð´ çßÅUæç×Ù A ·ð¤
3 ×æ˜æ·¤ ¥æñÚU ¹çÙÁ ÂÎæÍü ·ð¤ 6 ×æ˜æ·¤ âç×çÜÌ ãñ´Ð §âð ÚñUç¹·¤ Âýæð»ýæ×٠ⴁØæ ·ð¤ M¤Â
×ð´ çÙM¤çÂÌ ·¤èçÁ°Ð â´ÌéçÜÌ ¥æãæÚU ·¤è ‹ØêÙÌ× Üæ»Ì ™ææÌ ·¤èçÁ° Áæð ç·¤ §Ù ÎæðÙæð´ ¹æl
ÂÎæÍæðZ ·¤æ çןæ‡æ ãæð ¥æñÚU ‹ØêÙÌ× Âæðá‡æ ·¤è ¥æßàØ·¤Ìæ ·¤æð ÂêÚUè ·¤ÚUÌæ ãæðÐ
65/3/S
9
P.T.O.
A diet is to contain at least 80 units of Vitamin A and 100 units of minerals.
Two foods F1 and F2 are available costing ` 5 per unit and ` 6 per unit
respectively. One unit of food F1 contains 4 units of vitamin A and 3 units of
minerals whereas one unit of food F2 contains 3 units of vitamin A and 6 units
of minerals. Formulate this as a linear programming problem. Find the
minimum cost of diet that consists of mixture of these two foods and also meets
minimum nutritional requirement.
22.
âæÚUç‡æ·¤æ𴠷𤠻é‡æÏ×æðZ ·¤æ ÂýØæð» ·¤ÚU çâh ·¤èçÁ° ç·¤
:
(b 1 c)2
a2
bc
(c 1 a)2
b2
ca 5(a 2 b) (b 2 c) (c 2 a) (a 1 b 1 c) (a2 1 b2 1 c2 )
(a 1 b)2
c2
ab
¥Íßæ
ÂýæÚ´UçÖ·¤ ´ç€Ì â´ç·ý¤Øæ¥æð´ ·¤æ ÂýØæð» ·¤ÚU·ð¤ çِ٠¥æÃØêã ·¤æ ÃØ鈷ý¤× ™ææÌ ·¤èçÁ° Ñ
 2 21 3 


A 5  25 3 1 
 23 2 3 


Using properties of determinants, prove that :
65/3/S
(b 1 c)2
a2
bc
(c 1 a)2
b2
ca 5(a 2 b) (b 2 c) (c 2 a) (a 1 b 1 c) (a2 1 b2 1 c2 )
(a 1 b)2
c2
ab
10
OR
Using elementary row operations, find the inverse of the following matrix :
 2 21 3 


A 5  25 3 1 
 23 2 3 


23.
Îæð â×æ´ÌÚU ÚðU¹æ¥æð´
x 21 y 11 z
5
5
21
2
3
ÌÍæ
x y 22 z 11
5
5
4
22
6
·¤æð ¥´ÌçßücÅU ·¤ÚUÙð ßæÜð
â×ÌÜ ·¤æ â×è·¤ÚU ‡ æ ™ææÌ ·¤èçÁ°Ð ¥ÌÑ Îàææü § ° ç·¤ €Øæ Âý æ ŒÌ â×ÌÜ, Úð U ¹ æ
x22 y 21 z 22
5
5
3
1
5
·¤æð ¥´ÌçßücÅU ·¤ÚUÌæ ãñ ¥Íßæ Ùãè´?
Find the equation of the plane containing two parallel lines
x 21 y 11 z
x y 22 z 11
5
5 and 5
5
. Also, find if the plane thus obtained
21
4
22
6
2
3
contains the line
24.
x22 y 21 z 22
5
5
or not.
3
1
5
â×æ·¤ÜÙæð´ ·¤æ ÂýØæð» ·¤ÚU·ð¤ ÿæð˜æ
·¤èçÁ°Ð
{(x, y) : y2[6ax
ÌÍæ
x21y2[16a2}
·¤æ ÿæð˜æÈ¤Ü ™ææÌ
Using integration find the area of the region {(x, y) : y2[6ax and x21y2[16a2}
25.
×æÙæ ç·¤
°·¤ ȤÜÙ f (x)54x2112x115 mæÚUæ ÂçÚUÖæçáÌ ãñÐ Îàææü§° ç·¤
f : N ® S ÃØ鈷ý¤×‡æèØ ãñ (ÁÕç·¤ S, f ·¤æ ÂçÚUâÚU ãñ)Ð f ·¤æ ÂýçÌÜæð× ™ææÌ ·¤èçÁ°Ð ¥ÌÑ
f21(31) ÌÍæ f21(87) ™ææÌ ·¤èçÁ°Ð
f : N ® N
Let f : N ® N be a function defined as f (x)54x2112x115. Show that
f : N ® S is invertible (where S is range of f). Find the inverse of f and hence
find f21(31) and f21(87).
65/3/S
11
P.T.O.
26.
¥´ÌÚUæÜ ™ææÌ ·¤èçÁ° Áãæ¡ ÂÚU ȤÜÙ f(x)5x428x3122x2224x121 çÙÚU´ÌÚU ßÏü×æÙ ¥Íßæ
çÙÚ´UÌÚU Oæâ×æÙ ãñÐ
¥Íßæ
ȤÜÙ f (x)5sec x1log cos2 x, 0 < x < 2p ·ð¤ ¥çÏ·¤Ì× ÌÍæ ‹ØêÙÌ× ×æÙ ™ææÌ ·¤èçÁ°Ð
Determine the intervals in which the function f (x)5x428x3122x2224x121
is strictly increasing or strictly decreasing.
OR
Find the maximum and minimum values of f (x)5sec x1log cos 2 x,
0 < x < 2p.
65/3/S
12