SET-1
Series ONS
·¤æðÇU Ù´.
Code No.
ÚUæðÜ Ù´.
65/1/C
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 11 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 11 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
65/1/C
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.
65/1/C
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 x e N ¥æñÚU
If x e N and
2.
x 13 2 2
58
2 3x 2x
ãñ, Ìæð x ·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð
x 13 2 2
5 8 , then find the value of x.
2 3 x 2x
ÂýæÚUçÖ·¤ SÌ´Ö â´ç·ý¤Øæ C2 ® C212C1 çِ٠¥æÃØêã â×è·¤ÚU‡æ ÂÚU ܻ槰 Ñ
2 1  3 1   1 0

5


 2 0   2 0  21 1 
Use elementary column operation C2 ® C212C1 in the following matrix
equation :
2 1  3 1   1 0

5


 2 0   2 0  21 1 
3.
·¤æðçÅU 232 ·ð¤ âÖè âÖ´ß ¥æÃØêãæð´ ·¤è ⴁØæ, çÁÙ·¤æ ÂýˆØð·¤ ¥ßØß
çÜç¹°Ð
1, 2
¥Íßæ
3
ãñ,
Write the number of all possible matrices of order 232 with each entry 1, 2
or 3.
65/1/C
3
P.T.O.
4.
©â çÕ´Îé ·¤æ çSÍçÌ âçÎàæ çÜç¹° Áæð çÕ´Îé¥æð´, çÁÙ·ð¤ çSÍçÌ âçÎàæ
ãñ, ·¤æð ç×ÜæÙð ßæÜð ÚðU¹æ¹´ÇU ·¤æð 2 : 1 ·ð¤ ¥ÙéÂæÌ ×ð´ Õæ¡ÅUÌæ ãñÐ
→
→
3 a 22 b
→
→
ÌÍæ 2 a 13 b
Write the position vector of the point which divides the join of points with
→
→
→
→
position vectors 3 a 22 b and 2 a 13 b in the ratio 2 : 1.
5.
©Ù ×æ˜æ·¤ âçÎàææð´ ·¤è ⴁØæ çÜç¹° Áæð âçÎàææð´
→
∧
∧
∧
a 52 i 1 j 12 k
ÌÍæ
→
∧
∧
b 5j 1k
ÎæðÙæð´ ÂÚU
Ü´Õ ãñ´Ð
Write the number of vectors of unit length perpendicular to both the vectors
→
∧
∧
∧
→
∧
∧
a 52 i 1 j 12 k and b 5 j 1 k .
6.
©â â×ÌÜ ·¤æ âçÎàæ â×è·¤ÚU‡æ ™ææÌ ·¤èçÁ°, Áæð ç·¤ x, y ¥æñÚU z-¥ÿæ ÂÚU ·ý¤×àæÑ 3, 24 ¥æñÚU
2 ¥´Ìѹ´ÇU ·¤æÅUÌæ ãñÐ
Find the vector equation of the plane with intercepts 3, 24 and 2 on x, y and
z-axis respectively.
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.
©â çÕ´Îé ·ð¤ çÙÎðüàææ´·¤ ·¤èçÁ° Áãæ¡ ÂÚU çÕ´Îé¥æð´ A(3, 4, 1) ¥æñÚU B(5, 1, 6) âð ãæð·¤ÚU ÁæÙð ßæÜè
ÚðU¹æ XZ â×ÌÜ ·¤æð Âýç̑ÀðUÎ ·¤ÚUÌè ãñÐ ßã ·¤æð‡æ Öè ™ææÌ ·¤èçÁ° Áæð Øã ÚðU¹æ XZ â×ÌÜ
·ð¤ âæÍ ÕÙæÌè ãñÐ
Find the coordinates of the point where the line through the points A(3, 4, 1)
and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes
with the XZ plane.
65/1/C
4
8.
°·¤ â×æ´ÌÚU ¿ÌéÖéüÁ ·¤è Îæð ¥æâóæ ÖéÁæ°¡
∧
∧
∧
2 i 2 4 j 25 k
¥æñÚU
∧
∧
∧
2 i 1 2 j 13 k
ãñ´Ð §â·ð¤ ÎæðÙæð´
çß·¤‡ææðZ ·ð¤ â×æ´ÌÚU Îæð ×æ˜æ·¤ âçÎàæ ™ææÌ ·¤èçÁ°Ð çß·¤‡ææðZ ·ð¤ âçÎàææð´ ·¤æ ÂýØæð» ·¤ÚU·ð¤ â×æ´ÌÚU
¿ÌéÖéüÁ ·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ°Ð
∧
∧
∧
∧
∧
∧
The two adjacent sides of a parallelogram are 2 i 24 j 25 k and 2 i 12 j 13 k .
Find the two unit vectors parallel to its diagonals. Using the diagonal vectors,
find the area of the parallelogram.
9.
°·¤ Âæâæ Èð´¤·¤Ùð ·ð¤ ¹ðÜ ×ð´ °·¤ ÃØç€Ì ` 5 ÁèÌÌæ ãñ ØçÎ ©âð 4 âð ÕǸè ⴁØæ ÂýæŒÌ ãæðÌè
ãñ ¥‹ØÍæ ßã ` 1 ãæÚU ÁæÌæ ãñÐ ßã ÃØç€Ì 3 ÕæÚU Âæâæ Èð´¤·¤Ùð ·¤æ çهæüØ ÜðÌæ ãñ Üðç·¤Ù ¿æÚU
âð ÕǸè ⴁØæ ÂýæŒÌ ·¤ÚUÙð ÂÚU ¹ðÜ ÀUæðǸ ÎðÌæ ãñÐ ×ÙécØ mæÚUæ ÁèÌè/ãæÚUè ÁæÙð ßæÜè ÚUæçàæ ·¤è
ÂýˆØæàææ ™ææÌ ·¤èçÁ°Ð
¥Íßæ
°·¤ ÍñÜð ×ð´ 4 »ð´Îð´ ãñ´Ð ØæÎë‘ÀUØæ Îæð »ð´Îð´ çÕÙæ ÂýçÌSÍæÂÙæ ·ð¤ çÙ·¤æÜè »§ü ¥æñÚU ÎæðÙæð´ âÈð¤Î Âæ§ü
»§üÐ §â·¤è €Øæ ÂýæçØ·¤Ìæ ãñ ç·¤ ÍñÜð ×ð´ âÖè »ð´Îð´ âÈð¤Î ãñ´?
In a game, a man wins ` 5 for getting a number greater than 4 and loses
` 1 otherwise, when a fair die is thrown. The man decided to throw a die thrice
but to quit as and when he gets a number greater than 4. Find the expected
value of the amount he wins/loses.
OR
A bag contains 4 balls. Two balls are drawn at random (without replacement)
and are found to be white. What is the probability that all balls in the bag
are white ?
65/1/C
5
P.T.O.
10. xsinx1(sinx)cosx
·¤æ
x
·ð¤ âæÂðÿæ ¥ß·¤ÜÙ ·¤èçÁ°Ð
¥Íßæ
ØçÎ y52 cos(logx)13 sin(logx) ãñ, Ìæð çâh ·¤èçÁ° ç·¤
d2 y
dy
2
x
1x
1 y 50
2
dx
dx
ãñÐ
Differentiate xsinx1(sinx)cosx with respect to x.
OR
d2 y
dy
If y52 cos(logx)13 sin(logx), prove that x2 2 1 x
1 y 50 .
dx
dx
11.
ØçÎ x5a sin 2t(11cos 2t) ÌÍæ y5b cos 2t(12cos 2t) ãñ, Ìæð t 5 p ÂÚU
4
If x5a sin 2t(11cos 2t) and y5b cos 2t(12cos 2t), find
12.
ØçÎ ß·ý¤ y25ax31b ·ð¤ çÕ´Îé
b ·ð¤ ×æÙ ™ææÌ ·¤èçÁ°Ð
(2, 3)
ÂÚU SÂàæü ÚðU¹æ ·¤æ â×è·¤ÚU‡æ
dy
dx
™ææÌ ·¤èçÁ°Ð
dy
p
at t 5 .
dx
4
y54x25
ãñ, Ìæð a ÌÍæ
The equation of tangent at (2, 3) on the curve y25ax31b is y54x25. Find
the values of a and b.
13.
2
™ææÌ ·¤èçÁ° Ñ ∫ 4 x 2 dx
x 1 x 22
x2
dx
Find : ∫ 4
x 1 x2 2 2
65/1/C
6
14.
p
2
2
x
×æÙ ™ææÌ ·¤èçÁ° Ñ ∫ sinsin
x 1 cosx
dx
0
¥Íßæ
3
2
×æÙ ™ææÌ ·¤èçÁ° : ∫ x cos px dx
0
Evaluate :
p
2
sin2 x
∫ sinx 1 cosx dx
0
OR
Evaluate :
3
2
∫ x cos px dx
0
15.
™ææÌ ·¤èçÁ° Ñ ∫ (3x 1 1)
4 2 3x 2 2x 2 d x
Find : ∫ (3x 1 1) 4 2 3x 2 2x2 dx
16.
¥ß·¤Ü â×è·¤ÚU‡æ ·¤æð ãÜ ·¤èçÁ° Ñ
y1x
dy
dy
5x2y
dx
dx
Solve the differential equation :
y1x
65/1/C
dy
dy
5x2y
dx
dx
7
P.T.O.
17.
çmÌèØ ¿ÌéÍæZàæ ×ð´ °ðâð ßëžææ𴠷𤠷é¤Ü ·¤æ ¥ß·¤Ü â×è·¤ÚU‡æ ™ææÌ ·¤èçÁ° Áæð çÙÎðüàææ´·¤ ¥ÿææð´
·¤æð SÂàæü ·¤ÚUÌð ãñ´Ð
Form the differential equation of the family of circles in the second quadrant
and touching the coordinate axes.
18. x
·ð¤ çÜ° ãÜ ·¤èçÁ° Ñ sin21x1sin21(12x)5cos21x
¥Íßæ
ØçÎ
y
x
cos21 1 cos21 5a
a
b
ãñ, Ìæð çâh ·¤èçÁ° ç·¤
xy
y2
x2
2
2
cos
a1
5sin2 a
2
2
ab
a
b
ãñÐ
Solve the equation for x : sin21x1sin21(12x)5cos21x
OR
y
xy
y2
x
x2
cosa1 2 5sin2 a
If cos21 1 cos21 5a , prove that 2 22
a
b
ab
a
b
19.
°·¤ ÅþUSÅU Ùð Îæð Âý·¤æÚU ·ð¤ Õæ‡ÇU (¥ÙéÕ´Ï Â˜æ) ×ð´ ÏÙ çÙßðçàæÌ ç·¤ØæÐ ÂãÜð Õæ‡ÇU ×ð´ 10%
ŽØæÁ ¥æñÚU ÎêâÚðU Õæ‡ÇU ×ð´ 12% ŽØæÁ ç×ÜÌæ ãñÐ ÅþUSÅU ·¤æð ` 2,800 ·é¤Ü ŽØæÁ ÂýæŒÌ ãé¥æÐ
ÂÚU‹Ìé ØçÎ ÅþUSÅU Ùð ¥ÎÜæ-ÕÎÜè ·¤ÚU·ð¤ Õæ‡ÇUæð´ ×ð´ ÏÙ Ü»æØæ ãæðÌæ, Ìæð ©âð ` 100 ·¤× ŽØæÁ
ÂýæŒÌ ãæðÌæÐ ¥æÃØêã çßçÏ âð ™ææÌ ·¤èçÁ° ç·¤ ÅþUSÅU Ùð ç·¤ÌÙæ ÏÙ çÙßðçàæÌ ç·¤ØæÐ ÂýæŒÌ
ŽØæÁ ·¤æð ãðË°ðÁ §´çÇUØæ ×ð´ ÎæÙ ç·¤Øæ Áæ°»æÐ §â ÂýàÙ ×ð´ ·¤æñÙ-âæ ×êËØ ÎàææüØæ »Øæ ãñ?
A trust invested some money in two type of bonds. The first bond pays
10% interest and second bond pays 12% interest. The trust received ` 2,800
as interest. However, if trust had interchanged money in bonds, they would
have got ` 100 less as interest. Using matrix method, find the amount invested
by the trust. Interest received on this amount will be given to Helpage India
as donation. Which value is reflected in this question ?
65/1/C
8
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.
Îæð Âý·¤æÚU ·ð¤ ¹æÎ ‘A’ ¥æñÚU ‘B’ ãñ´Ð ‘A’ ×ð´ 12% Ùæ§ÅþUæðÁÙ ¥æñÚU 5% ȤæSȤæðçÚU·¤ °çâÇU ãñ ÁÕç·¤
‘B’ ×ð´ 4% Ùæ§ÅþUæðÁÙ ¥æñÚU 5% ȤæSȤæðçÚU·¤ °çâÇU ãñÐ ç×^è ·ð¤ çSÍçÌ ÂÚUèÿæ‡æ ·ð¤ ÕæÎ ç·¤âæÙ
·¤æð ™ææÌ ãé¥æ ç·¤ ©âð ȤâÜ ·ð¤ çÜ° ·¤× âð ·¤× 12 ç·¤.»ýæ. Ùæ§ÅþUæðÁÙ ¥æñÚU 12 ç·¤.»ýæ.
ȤæSȤæðçÚU·¤ °çâÇU ·¤è ¥æßàØ·¤Ìæ ãñÐ ØçÎ ‘A’ ·¤æ ×êËØ ` 10 ÂýçÌ ç·¤.»ýæ. ¥æñÚU ‘B’ ·¤æ ×êËØ
` 8 ÂýçÌ ç·¤.»ýæ. ãñ Ìæð ¥æÜð¹ mæÚUæ ÂçÚU·¤çÜÌ ·¤èçÁ° ç·¤ ©âð ÂýˆØð·¤ Âý·¤æÚU ·¤è ç·¤ÌÙè ¹æÎ
ÂýØæð» ·¤ÚUÙè ¿æçã° ç·¤ ·¤× âð ·¤× ·¤è×Ì ×ð´ Âæðá·¤ ̈ßæð´ ·¤è ¥æßàØ·¤Ìæ ÂêÚUè ãæð Áæ°Ð
There are two types of fertilisers ‘A’ and ‘B’. ‘A’ consists of 12% nitrogen and
5% phosphoric acid whereas ‘B’ consists of 4% nitrogen and 5% phosphoric acid.
After testing the soil conditions, farmer finds that he needs at least 12 kg of
nitrogen and 12 kg of phosphoric acid for his crops. If ‘A’ costs ` 10 per kg
and ‘B’ cost ` 8 per kg, then graphically determine how much of each type of
fertiliser should be used so that nutrient requirements are met at a minimum
cost.
¥‘ÀðU â´ÌÚUæð´ ×ð´ 5 ¹ÚUæÕ â´ÌÚð ¥ÂýˆØæçàæÌ ·¤æÚU‡æ âð ç×Ü »° ãñ´Ð ¿æÚU â´ÌÚðU ©žæÚUæðžæÚU
ÂýçÌSÍæÂÙæ ·ð¤ âçãÌ çÙ·¤æÜð »°, Ìæð ¹ÚUæÕ â´ÌÚUæð´ ·¤æð çÙ·¤æÜÙð ·¤è ⴁØæ ·¤æ ÂýæØç·¤Ìæ Õ´ÅUÙ
™ææÌ ·¤èçÁ°Ð Õ´ÅUÙ ·¤æ ×æŠØ ÌÍæ ÂýâÚU‡æ Öè ™ææÌ ·¤èçÁ°Ð
21. 20
Five bad oranges are accidently mixed with 20 good ones. If four oranges are
drawn one by one successively with replacement, then find the probability
distribution of number of bad oranges drawn. Hence find the mean and
variance of the distribution.
65/1/C
9
P.T.O.
22.
çÕ´Îé P, çÁâ·¤æ çSÍçÌ âçÎàæ
∧
∧
∧
2 i 13 j 1 4 k
ãñ âð â×ÌÜ
(
→
∧
∧
∧
)
r . 2 i 1 j 1 3 k 2 26 5 0
¹è´¿ð »° ÜÕ ·ð¤ ÂæÎ ·¤æ çSÍçÌ âçÎàæ ÌÍæ ܐÕßÌ÷ ÎêÚUè ™ææÌ ·¤èçÁ°Ð ÌÜ ×ð´
ÂýçÌçÕÕ Öè ™ææÌ ·¤èçÁ°Ð
P
ÂÚU
·¤æ
Find the position vector of the foot of perpendicular and the perpendicular
∧
∧
∧
distance from the point P with position vector 2 i 13 j 14 k to the plane
→
(
∧
∧
∧
)
r . 2 i 1 j 1 3 k 2 26 5 0 . Also find image of P in the plane.
23.
Îàææü § ° ç·¤ °·¤ çm¥æÏæÚU è â´ ç ·ý ¤ Øæ Áæð A5R2{21} ÂÚU âÖè a, b e A ·ð ¤ çÜ°
a*b5a1b1ab mæÚUæ ÂçÚUÖæçáÌ ãñ ·ý¤× çßçÙ×ðØ ÌÍæ âæã¿üØ ãñÐ A ×ð´ * ·¤æ ̈â×·¤
¥ßØß ™ææÌ ·¤èçÁ° ÌÍæ çâh ·¤èçÁ° ç·¤ A ·¤æ ÂýˆØð·¤ ¥ßØß ÃØ鈷ý¤×‡æèØ ãñÐ
Show that the binary operation * on A5R2{21} defined as a*b5a1b1ab
for all a, b e A is commutative and associative on A. Also find the identity
element of * in A and prove that every element of A is invertible.
24.
çâh ·¤èçÁ° ç·¤ â×çmÕæãé ç˜æÖéÁ, çÁâ×ð´ r ç˜æ’Øæ ·¤æ °·¤ ¥´ÌßëÌ ¹è´¿æ »Øæ ãñ, ·¤æ ‹ØêÙÌ×
ÂçÚU×æÂ
6 3r
ãñÐ
¥Íßæ
ØçÎ °·¤ â×·¤æð‡æ ç˜æÖéÁ ×ð´ ·¤‡æü ÌÍæ °·¤ ÖéÁæ ·¤æ Øæð» çÎØæ »Øæ ãæð, Ìæð Îàææü§° ç·¤ ç˜æÖéÁ
·¤æ ÿæð˜æÈ¤Ü ¥çÏ·¤Ì× ãæð»æ ÁÕç·¤ ©Ù·ð¤ Õè¿ ·¤æ ·¤æð‡æ
p
3
ãæð»æÐ
Prove that the least perimeter of an isosceles triangle in which a circle of radius
r can be inscribed is 6 3 r .
OR
If the sum of lengths of hypotenuse and a side of a right angled triangle is
given, show that area of triangle is maximum, when the angle between them
is
65/1/C
p
.
3
10
25.
çâh ·¤èçÁ° ç·¤ ß·ý¤ y254x ¥æñÚU x254y, ©â ß»ü ·ð¤ ÿæð˜æÈ¤Ü ·¤æð ÌèÙ ÕÚUæÕÚU Öæ»æð´ ×ð´
Õæ¡ÅUÌð ãñ´ Áæð ç·¤ ÚðU¹æ¥æð´ x50, x54, y54 ¥æñÚU y50 mæÚUæ ÂçÚUÕh ãñÐ
Prove that the curves y254x and x254y divide the area of square bounded
by x50, x54, y54 and y50 into three equal parts.
26.
âæÚUç‡æ·¤æ𴠷𤠻é‡æÏ×æðZ ·¤æ ÂýØæð» ·¤ÚU çâh ·¤èçÁ° ç·¤
1
11 cosA
1
11 cosB
1
11 cosC
DABC
50
°·¤ â×çmÕæãé ç˜æÖéÁ ãñ ØçÎ
ãñÐ
cos2 A 1 cosA cos2B 1 cosB cos2C 1 cosC
¥Íßæ
°·¤ Îé·¤æÙÎæÚU ·ð¤ Âæâ ÌèÙ çßçÖóæ Âý·¤æÚU ·ð¤ ÂðÙ ‘A’, ‘B’ ¥æñÚU ‘C’ ãñ´Ð ×èÙê Ùð ÂýˆØð·¤ Âý·¤æÚU
·¤æ °·¤-°·¤ ÂðÙ ·é¤Ü ` 21 ×ð´ ¹ÚUèÎæÐ ÁèßÙ Ùð ‘A’ Âý·¤æÚU ·ð¤ 4 ÂðÙ, ‘B’ Âý·¤æÚU ·ð¤ 3 ÂðÙ ¥æñÚU
‘C’ Âý·¤æÚU ·ð¤ 2 ÂðÙ ` 60 ×ð´ ¹ÚUèÎð ÁÕç·¤ çàæ¹æ Ùð ‘A’ Âý·¤æÚU ·ð¤ 6 ÂðÙ, ‘B’ Âý·¤æÚU ·ð¤ 2 ÂðÙ
¥æñÚU ‘C’ Âý·¤æÚU ·ð¤ 3 ÂðÙ ` 70 ×ð´ ¹ÚUèÎðÐ ¥æÃØêã çßçÏ âð ÂýˆØð·¤ Âý·¤æÚU ·ð¤ ÂðÙ ·¤æ ×êËØ ™ææÌ
·¤èçÁ°Ð
Using properties of determinants, show that DABC is isosceles if :
1
11 cosA
1
11 cosB
1
11 cosC
50
cos2 A 1 cosA cos2B 1 cosB cos2C 1 cosC
OR
A shopkeeper has 3 varieties of pens ‘A’, ‘B’ and ‘C’. Meenu purchased 1 pen
of each variety for a total of ` 21. Jeevan purchased 4 pens of ‘A’ variety,
3 pens of ‘B’ variety and 2 pens of ‘C’ variety for ` 60. While Shikha purchased
6 pens of ‘A’ variety, 2 pens of ‘B’ variety and 3 pens of ‘C’ variety for ` 70.
Using matrix method, find cost of each variety of pen.
65/1/C
11
P.T.O.