JSR 030 Set 2 Mathematics.p65

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JSR 030 Set 2 Mathematics.p65
SET-2
Series JSR
·¤æðÇU Ù´.
Code No.
ÚUæðÜ Ù´.
30/2
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.
·
·
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·
·
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·ë¤ÂØæ Áæ¡¿ ·¤ÚU Üð´ ç·¤ §â ÂýàÙ-˜æ ×ð´ ×éçÎýÌ ÂëcÆU 15 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´ 31 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 15 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 31 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.
â´·¤çÜÌ ÂÚUèÿææ - II
SUMMATIVE ASSESSMENT - II
»ç‡æÌ
MATHEMATICS
çÙÏæüçÚUÌ â×Ø Ñ
3 ƒæ‡ÅðU
Time allowed : 3 hours
30/2
¥çÏ·¤Ì× ¥´·¤ Ñ 90
Maximum Marks : 90
1
P.T.O.
âæ×æ‹Ø çÙÎðüàæ Ñ
(i)
âÖè ÂýàÙ ¥çÙßæØü ãñ´Ð
(ii)
§â ÂýàÙ-˜æ ×ð´
(iii)
¹‡ÇU ¥ ×ð´ °·¤-°·¤ ¥´·¤ ßæÜð 4 ÂýàÙ ãñ´Ð ¹‡ÇU Õ ×ð´ 6 ÂýàÙ ãñ´ çÁâ×ð´ âð ÂýˆØð·¤ 2 ¥´·¤ ·¤æ
ãñÐ ¹‡ÇU â ×ð´ 10 ÂýàÙ ÌèÙ-ÌèÙ ¥´·¤æð´ ·ð¤ ãñ´Ð ¹‡ÇU Î ×ð´ 11 ÂýàÙ ãñ´ çÁÙ×ð´ âð ÂýˆØð·¤ 4 ¥´·¤
31
ÂýàÙ ãñ´ Áæð ¿æÚU ¹‡ÇUæð´ - ¥, Õ, â ¥æñÚU Î ×ð´ çßÖæçÁÌ ãñ´Ð
·¤æ ãñÐ
(iv)
·ñ¤Ü·é¤ÜðÅUÚU ·¤æ ÂýØæð» ßçÁüÌ ãñÐ
General Instructions :
(i)
All questions are compulsory.
(ii)
The question paper consists of 31 questions divided into four sections – A, B, C
and D.
(iii) Section A contains 4 questions of 1 mark each. Section B contains 6 questions of
2 marks each, Section C contains 10 questions of 3 marks each and Section D
contains 11 questions of 4 marks each.
(iv) Use of calculators is not permitted.
30/2
2
¹‡ÇU - ¥
SECTION - A
Âýà٠ⴁØæ 1 âð 4 Ì·¤ ÂýˆØð·¤ ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ
Question numbers 1 to 4 carry 1 mark each.
1.
°·¤ ÎèßæÚU ·ð¤ âæÍ Ü»è âèɸè ÿæñçÌÁ ·ð¤ âæÍ 608 ·¤æ ·¤æð‡æ ÕÙæÌè ãñÐ ØçÎ âèÉ¸è ·¤æ ÂæÎ
ÎèßæÚU âð 2.5 ×è. ·¤è ÎêÚUè ÂÚU ãñ, Ìæð âèÉ¸è ·¤è ܐÕæ§ü ™ææÌ ·¤èçÁ°Ð
A ladder, leaning against a wall, makes an angle of 608 with the horizontal.
If the foot of the ladder is 2.5 m away from the wall, find the length of the
ladder.
2.
k
·ð¤ ç·¤â ×æÙ ·ð¤ çÜ°
k19, 2k21
ÌÍæ
2k17
°·¤ â×æ´ÌÚU ŸæðÉ¸è ·ð¤ ·ý¤×æ»Ì ÂÎ ãñ´?
For what value of k will k19, 2k21 and 2k17 are the consecutive terms of
an A.P. ?
3.
¥æ·ë¤çÌ 1 ×ð´ O ·ð¤‹Îý ßæÜð ßëžæ ·ð¤ çÕ´Îé C ÂÚU PQ °·¤ SÂàæü ÚðU¹æ ãñÐ ØçÎ
ãñ ÌÍæ ÐCAB5308 ãñ, Ìæð ÐPCA ™ææÌ ·¤èçÁ°Ð
¥æ·ë¤çÌ
AB
°·¤ ÃØæâ
1
In fig.1, PQ is a tangent at a point C to a circle with centre O. If AB is a
diameter and ÐCAB5308, find ÐPCA.
Figure 1
30/2
3
P.T.O.
4.
žææð´ ·¤è ¥‘ÀUè Âý·¤æÚU Èð´¤ÅUè »§ü Ìæàæ ·¤è »aè ×ð´ âð ØæÎë‘ÀUØæ °·¤ žææ çÙ·¤æÜæ »ØæÐ
ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° ç·¤ çÙ·¤æÜæ »Øæ žææ Ù Ìæð ÜæÜ Ú´U» ·¤æ ãñ ¥æñÚU Ù ãè °·¤ Õð$»× ãñÐ
52
A card is drawn at random from a well shuffled pack of 52 playing cards. Find
the probability of getting neither a red card nor a queen.
¹‡ÇU - Õ
SECTION - B
Âýà٠ⴁØæ 5 âð 10 Ì·¤ ÂýˆØð·¤ ÂýàÙ ·ð¤ 2 ¥´·¤ ãñÐ
Question numbers 5 to 10 carry 2 marks each.
5.
¥æ·ë¤çÌ 2 ×ð´ °·¤ ¿ÌéÖüéÁ ABCD, O ·ð´¤Îý ßæÜð ßëžæ ·ð¤ ÂçÚU»Ì §â Âý·¤æÚU ÕÙæ§ü »§ü ãñ ç·¤
ÖéÁæ°¡ AB, BC, CD ÌÍæ DA ßëžæ ·¤æð ·ý¤×àæÑ çÕ´Îé¥æð´ P, Q, R ÌÍæ S ÂÚU SÂàæü ·¤ÚUÌè ãñ´Ð
çâh ·¤èçÁ° ç·¤ AB1CD5BC1DA Ð
¥æ·ë¤çÌ
2
In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with
centre O, in such a way that the sides AB, BC, CD and DA touch the circle at
the points P, Q, R and S respectively. Prove that. AB1CD5BC1DA.
Figure 2
30/2
4
6.
°·¤ â×æ´ÌÚU ŸæðÉ¸è ·¤æ ¿æñÍæ ÂÎ àæê‹Ø ãñÐ çâh ·¤èçÁ° ç·¤ §â·¤æ
ÂÎ ·¤æ ÌèÙ »éÙæ ãñÐ
25
ßæ´ ÂÎ, ©â·ð¤
11
ßð´
The 4th term of an A.P. is zero. Prove that the 25th term of the A.P. is three
times its 11th term.
7.
¥æ·ë¤çÌ 3 ×ð´ °·¤ Õæs çÕ´Îé P âð, O ·ð¤‹Îý ÌÍæ r ç˜æ’Øæ ßæÜð ßëžæ ÂÚU Îæð SÂàæü ÚðU¹æ°¡ PT ÌÍæ
PS ¹è´¿è »§ü ãñ´Ð ØçÎ OP52r ãñ, Ìæð Îàææü§° ç·¤ ÐOTS5ÐOST5308Ð
¥æ·ë¤çÌ
3
In Fig. 3, from an external point P, two tangents PT and PS are drawn to a
circle with centre O and radius r. If OP52r, show that ÐOTS5ÐOST5308.
Figure 3
8.
çâh ·¤èçÁ° ç·¤ çÕ´Îé (3, 0), (6, 4) ÌÍæ (21, 3) °·¤ â×çmÕæãé â×·¤æð‡æ ç˜æÖéÁ ·ð¤ àæèáü ãñ´Ð
Prove that the points (3, 0), (6, 4) and (21, 3) are the vertices of a right angled
isosceles triangle.
9.
×æÙæ P ÌÍæ Q, A(2, 22) ÌÍæ B(27, 4) ·¤æð ç×ÜæÙð ßæÜð ÚðU¹æ¹´ÇU ·¤æð §â Âý·¤æÚU â×ç˜æÖæçÁÌ
·¤ÚUÌð ãñ´ ç·¤ P, çÕ´Îé A ·ð¤ Âæâ ãñÐ P ÌÍæ Q ·ð¤ çÙÎðüàææ´·¤ ™ææÌ ·¤èçÁ°Ð
Let P and Q be the points of trisection of the line segment joining the points
A(2, 22) and B(27, 4) such that P is nearer to A. Find the coordinates of
P and Q.
30/2
5
P.T.O.
10. x
·ð¤ çÜ° ãÜ ·¤èçÁ° :
Solve for x :
2 x19 1 x 513
2 x19 1 x 513
¹‡ÇU - â
SECTION - C
Âýà٠ⴁØæ 11 âð 20 Ì·¤ ÂýˆØð·¤ ÂýàÙ 3 ¥´·¤ ·¤æ ãñÐ
Question numbers 11 to 20 carry 3 marks each.
11.
ØçÎ çՋÎé P(x, y) çÕ´Îé¥æð´
·¤èçÁ° ç·¤ bx5ay.
A(a1b, b2a)
ÌÍæ
B(a2b, a1b)
âð â×ÎêÚUSÍ ãñ, Ìæð çâh
If the point P(x, y) is equidistant from the points A(a1b, b2a) and
B(a2b, a1b). Prove that bx5ay.
12.
°·¤ à洀Ãææ·¤æÚU ÕÌüÙ, çÁâ·ð¤ ¥æÏæÚU ·¤è ç˜æ’Øæ 5 âð×è ÌÍæ ª¡¤¿æ§ü 24 âð×è ãñ, ÂæÙè âð ÂêÚUæ
ÖÚUæ ãñÐ ©â ÂæÙè ·¤æð °·¤ ÕðÜÙæ·¤æÚU ÕÌüÙ, çÁâ·¤è ç˜æ’Øæ 10 âð×è ãñ, ×ð´ ÇUæÜ çÎØæ ÁæÌæ
ãñÐ ÕðÜÙæ·¤æÚU ÕÌüÙ ×ð´ ç·¤ÌÙè ª¡¤¿æ§ü Ì·¤ ÂæÙè ÖÚU ÁæØð»æ? ( p5
22
7
ÜèçÁ°
)
A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This
water is emptied into a cylindrical vessel of base radius 10 cm. Find the height
to which the water will rise in the cylindrical vessel. (Use p5
30/2
6
22
)
7
13.
¥æ·ë¤çÌ 4 ×ð´ O ·ð¤‹Îý ßæÜð ßëžæ ·¤æ ÃØæâ AB513 âð×è ãñ ÌÍæ AC512 âð×è ãñÐ
ç×ÜæØæ »Øæ ãñÐ ÀUæØæ´ç·¤Ì ÿæð˜æ ·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ°Ð ( p53.14 ÜèçÁ° )
¥æ·ë¤çÌ
BC
·¤æð
4
In fig.4, O is the centre of a circle such that diameter AB513 cm and
AC512 cm. BC is joined. Find the area of the shaded region. (Take p53.14)
Figure 4
14. 12 âð×è
ÃØæâ ßæÜæ °·¤ »æðÜæ, °·¤ Ü´Õ ßëžæèØ ÕðÜÙæ·¤æÚU ÕÌüÙ ×ð´ ÇUæÜ çÎØæ ÁæÌæ ãñ, çÁâ×ð´
·é¤ÀU ÂæÙè ÖÚUæ ãñÐ ØçÎ »æðÜæ Âê‡æüÌØæ ÂæÙè ×ð´ ÇêÕ ÁæÌæ ãñ, Ìæð ÕðÜÙæ·¤æÚU ÕÌüÙ ×ð´ ÂæÙè ·¤æ
SÌÚU
3
5
9
âð×è ª¡¤¿æ ©ÆU ÁæÌæ ãñÐ ÕðÜÙæ·¤æÚU ÕÌüÙ ·¤æ ÃØæ⠙ææÌ ·¤èçÁ°Ð
A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel,
partly filled with water. If the sphere is completely submerged in water, the
water level in the cylindrical vessel rises by 3
cylindrical vessel.
30/2
7
5
cm. Find the diameter of the
9
P.T.O.
15.
¥æ·ë¤çÌ 5 ×ð´ °·¤ Åñ´UÅU ÕðÜÙ ·ð¤ ª¤ÂÚU Ü»ð ©âè ÃØæâ ßæÜð àæ´·é¤ ·ð¤ ¥æ·¤æÚU ·¤æ ãñÐ ÕðÜÙæ·¤æÚU
Öæ» ·¤è ª¡¤¿æ§ü ÌÍæ ÃØæâ ·ý¤×àæÑ 2.1 ×è. ÌÍæ 3 ×è. ãñ´ ÌÍæ à洀Ãææ·¤æÚU Öæ» ·¤è çÌÚUÀUè ª¡¤¿æ§ü
2.8 ×è. ãñÐ Åñ´UÅU ·¤æð ÕÙæÙð ×ð´ Ü»ð ·ñ¤Ùßæâ ·¤æ ×êËØ ™ææÌ ·¤èçÁ°, ØçÎ ·ñ¤Ùßæâ ·¤æ Öæß
` 500
ÂýçÌ ß»ü ×è ãñÐ
(p5
22
7
ÜèçÁ°
)
¥æ·ë¤çÌ
5
In fig. 5, a tent is in the shape of a cylinder surmounted by a conical top of
same diameter. If the height and diameter of cylindrical part are 2.1 m and
3 m respectively and the slant height of conical part is 2.8 m, find the cost of
canvas needed to make the tent if the canvas is available at the rate of
` 500/sq.metre. (Use p5
22
)
7
Figure 5
30/2
8
16.
¥æ·ë¤çÌ
6
×ð´, Îæð â·ð¤‹ÎýèØ ßëžææð´, çÁâ·¤è ç˜æ’Øæ°¡
ÀUæØæ´ç·¤Ì ÿæð˜æ ·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ° ÁÕç·¤
¥æ·ë¤çÌ
7
âð×è ÌÍæ
ÐAOC5408
14
âð×è ãñ´, ·ð¤ Õè¿ çƒæÚðU
ãñÐ (p5
22
7
ÜèçÁ°)
6
In fig. 6, find the area of the shaded region, enclosed between two concentric
circles of radii 7 cm and 14 cm where ÐAOC5408. (Use p5
22
)
7
Figure 6
17.
°·¤ ÃØç€ˆæ °·¤ ÁÜØæÙ ·ð¤ ÇñU·¤, Áæð ÂæÙè ·ð¤ SÌÚU âð 10 ×è. ª¡¤¿æ ãñ, âð °·¤ ÂãæÇ¸è ·ð¤
çàæ¹ÚU ·¤æ ©óæØÙ ·¤æð‡æ 608 ÌÍæ ÂãæÇ¸è ·ð¤ ÌÜ ·¤æ ¥ßÙ×Ù ·¤æð‡æ 308 ÂæÌæ ãñÐ ÂãæǸè âð
ÁÜØæÙ ·¤è ÎêÚUè ÌÍæ ÂãæÇ¸è ·¤è ª¡¤¿æ§ü ™ææÌ ·¤èçÁ°Ð
A man standing on the deck of a ship, which is 10 m above water level, observes
the angle of elevation of the top of a hill as 608 and the angle of depression of
the base of hill as 308. Find the distance of the hill from the ship and the
height of the hill.
30/2
9
P.T.O.
18.
ÌèÙ ¥´·¤æð´ ·¤è °·¤ ÏÙæˆ×·¤ ⴁØæ ·ð¤ ¥´·¤ â×æ´ÌÚU Ÿæðɸè ×ð´ ãñ ÌÍæ ©Ù·¤æ Øæð» 15 ãñÐ ¥´·¤æð´
·¤æ SÍæÙ ÂÜÅUÙð´ ÂÚU ÕÙÙð ßæÜè ⴁØæ ×êÜ â´Øæ âð 594 ·¤× ãñРⴁØæ ™ææÌ ·¤èçÁ°Ð
The digits of a positive number of three digits are in A.P. and their sum is 15.
The number obtained by reversing the digits is 594 less than the original
number. Find the number.
19.
ØçÎ çmƒææÌè â×è·¤ÚU‡æ
ç·¤ 2a5b1c ãñÐ
(a2b)x21(b2c)x1(c2a)50
·ð¤ ×êÜ â×æÙ ãñ´, Ìæð çâh ·¤èçÁ°
If the roots of the quadratic equation (a2b)x21(b2c)x1(c2a)50 are equal,
prove that 2a5b1c.
20. 52 žææð´
·¤è Ìæàæ ·¤è »aè ×ð´ âð ÜæÜ Ú´U» ·ð¤ »éÜæ×, Õð»× ÌÍæ ÕæÎàææã çÙ·¤æÜ çÜ° »°Ð àæðá
»aè ×ð´ âð ØæÎë‘ÀUØæ °·¤ žææ çÙ·¤æÜæ »ØæÐ ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° ç·¤ çÙ·¤æÜæ »Øæ žææ
(i) °·¤ ·¤æÜæ ÕæÎàææã ãñ (ii) °·¤ ÜæÜ Ú´U» ·¤æ žææ ãñ (iii) °·¤ ·¤æÜæ žææ ãñ
From a pack of 52 playing cards, Jacks, Queens and Kings of red colour are
removed. From the remaining, a card is drawn at random. Find the probability
that drawn card is :
(i) a black King (ii) a card of red colour (iii) a card of black colour
¹‡ÇU - Î
SECTION - D
Âýà٠ⴁØæ 21 âð 31 Ì·¤ ÂýˆØð·¤ ÂýàÙ 4 ¥´·¤ ·¤æ ãñÐ
Question numbers 21 to 31 carry 4 marks each.
21.
ç·¤âè ÚUæ’Ø ×ð´ ÖæÚUè Õæɸ ·ð¤ ·¤æÚU‡æ ãÁæÚUæð´ Üæð» ÕðƒæÚU ãæð »°Ð 50 çßlæÜØæð´ Ùð ç×Ü·¤ÚU ÚUæ’Ø
âÚU·¤æÚU ·¤æð 1500 Åñ´ÅU Ü»æÙð ·ð¤ çÜ° SÍæÙ ÌÍæ ·ñ¤Ùßâ ÎðÙð ·¤æ ÂýSÌæß ç·¤Øæ çÁâ×ð´ ÂýˆØð·¤
çßlæÜØ ÕÚUæÕÚU ·¤æ ¥´àæÎæÙ Îð»æÐ ÂýˆØð·¤ Åñ´ÅU ·¤æ çÙ¿Üæ Öæ» ÕðÜÙæ·¤æÚU ãñ, çÁâ·ð¤ ¥æÏæÚU
·¤è ç˜æ’Øæ 2.8 ×è. ÌÍæ ª¡¤¿æ§ü 3.5 ×è. ãñÐ ÂýˆØð·¤ Åñ´UÅU ·¤æ ª¤ÂÚUè Öæ» àæ´·é¤ ·ð¤ ¥æ·¤æÚU ·¤æ
ãñ çÁâ·ð¤ ¥æÏæÚU ·¤è ç˜æ’Øæ 2.8 ×è. ÌÍæ ª¡¤¿æ§ü 2.1 ×è. ãñÐ ØçÎ ÅñU´ÅU ÕÙæÙð ßæÜð ·ñ¤Ùßæâ
·¤æ ×êËØ ` 120 ÂýçÌ ß»ü ×è. ãñ, Ìæð ÂýˆØð·¤ çßlæÜØ mæÚUæ ·é¤Ü ÃØØ ×ð´ ¥´àæÎæÙ ™ææÌ ·¤èçÁ°Ð
§â ÂýàÙ mæÚUæ ·¤æñÙ âæ ×êËØ ÁçÙÌ ãæðÌæ ãñ?
30/2
( p5
10
22
7
ÜèçÁ° )
Due to heavy floods in a state, thousands were rendered homeless. 50 schools
collectively offered to the state government to provide place and the canvas
for 1500 tents to be fixed by the government and decided to share the whole
expenditure equally. The lower part of each tent is cylindrical of base radius
2.8 m and height 3.5 m, with conical upper part of same base radius but of
height 2.1 m. If the canvas used to make the tents costs ` 120 per sq.m, find
the amount shared by each school to set up the tents. What value is generated
by the above problem ? (Use p5
22.
22
)
7
¥æ·ë¤çÌ 7 ×ð´ Îæð â×æÙ ç˜æ’Øæ ·ð¤ ßëžæ, çÁٷ𤠷ð¤‹Îý O ÌÍæ O' ãñ´ ÂÚUSÂÚU çÕ´Îé X ÂÚU SÂàæü ·¤ÚUÌð
ãñ´Ð OO' ÕɸæÙð ÂÚU O' ·ð¤‹Îý ßæÜð ßëžæ ·¤æð çÕ´Îé A ÂÚU ·¤æÅUÌæ ãñÐ çÕ´Îé A âð O ·ð¤‹Îý ßæÜð
ßëžæ ÂÚU
AC
°·¤ SÂàæü ÚðU¹æ ãñ ÌÍæ
O'D ^ AC
¥æ·ë¤çÌ
ãñÐ
DO'
CO
·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð
7
In Fig. 7, two equal circles, with centres O and O', touch each other at X.OO'
produced meets the circle with centre O' at A. AC is tangent to the circle with
centre O, at the point C. O'D is perpendicular to AC. Find the value of
DO'
.
CO
figure 7
30/2
11
P.T.O.
23.
ⴁØæ¥æð´ 1, 2, 3 ÌÍæ 4 ×ð´ âð ·¤æð§ü ⴁØæ x ØæÎë‘ÀUØæ ¿éÙè »§ü ÌÍæ ⴁØæ¥æð´ 1, 4, 9 ÌÍæ 16
×ð´ âð ·¤æð§ü ⴁØæ y ØæÎë‘ÀUØæ ¿éÙè »§ü ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° x ÌÍæ y ·¤æ »é‡æÙÈ¤Ü 16 âð ·¤×
ãñÐ
A number x is selected at random from the numbers 1, 2, 3 and 4. Another
number y is selected at random from the numbers 1, 4, 9 and 16. Find the
probability that product of x and y is less than 16.
24.
¥æ·ë¤çÌ 8 ×ð´ °·¤ ç˜æÖéÁ ABC ·ð¤ àæèáü A(4, 6), B(1, 5) ÌÍæ C(7, 2) ãñÐ °·¤ ÚðU¹æ¹´ÇU
DE ÖéÁæ¥æð´ AB ÌÍæ AC ·¤æð ·ý¤×àæÑ çÕ´Îé¥æð´ D ÌÍæ E ÂÚU §â Âý·¤æÚU ·¤æÅUÌæ ¹è´¿æ »Øæ ãñ
ç·¤
AD
AE
1
5
5
AB
AC
3
ãñÐ
DADE
·¤æ ÿæð˜æÈ¤Ü ™ææÌ ·¤èçÁ° ÌÍæ ©â·¤è
DABC
·ð¤ ÿæð˜æȤÜ
âð ÌéÜÙæ ·¤èçÁ°Ð
¥æ·ë¤çÌ
8
In fig. 8, the vertices of DABC are A(4, 6), B(1, 5) and C(7, 2). A line-segment
DE is drawn to intersect the sides AB and AC at D and E respectively such
that
AD
AE
1
5
5 . Calculate the area of DADE and compare it with area of
AB
AC
3
DABC.
Figure 8
30/2
12
25.
¥æ·ë¤çÌ 9 ×ð´, O ·ð´¤Îý ßæÜð ßëžæ ·¤æ °·¤ ç˜æ’ع´ÇU OAP ÎàææüØæ »Øæ ãñ çÁâ·¤æ ·ð¤‹Îý ÂÚU
¥Ì´çÚUÌ ·¤æð‡æ u ãñÐ AB ßëžæ ·¤è ç˜æ’Øæ OA ÂÚU Ü´Õ ãñ Áæð OP ·ð¤ ÕɸæÙð ÂÚU çÕ´Îé B ÂÚU
·¤æÅUÌæ ãñÐ çâh ·¤èçÁ° ç·¤ ÚðU¹æ´ç·¤Ì Öæ» ·¤æ ÂçÚU×æÂ
¥æ·ë¤çÌ

r  tanu 1 secu 1

pu
21
180

ãñÐ
9
In Fig. 9, is shown a sector OAP of a circle with centre O, containing ∠ u. AB
is perpendicular to the radius OA and meets OP produced at B. Prove that
pu

21
the perimeter of shaded region is r  tanu 1 secu 1
180


Figure 9
26.
°·¤ âèÏè ÚðU¹æ ×ð´ çSÍÌ ƒæÚUæð´ ÂÚU 1 âð 49 Ì·¤ ·¤è ⴁØæ°¡ (·ý¤×æÙéâæÚU) ¥´ç·¤Ì ãñ´Ð Îàææü§°
ç·¤ §Ù ¥´ç·¤Ì ⴁØæ¥æð´ ×ð´ °·¤ °ðâè ⴁØæ X ¥ßàØ ãñ ç·¤ X âð ÂãÜð ¥æÙð ßæÜð ƒæÚUæð´ ÂÚU
·¤è ¥´ç·¤Ì ⴁØæ¥æð´ ·¤æ Øæð», X ·ð¤ ÕæÎ ¥æÙðßæÜè ¥´ç·¤Ì ⴁØæ¥æð´ ·ð¤ Øæð» ·ð¤ ÕÚUæÕÚU ãñÐ
The houses in a row are numbered consecutively from 1 to 49. Show that there
exists a value of X such that sum of numbers of houses proceeding the house
numbered X is equal to sum of the numbers of houses following X.
30/2
13
P.T.O.
27.
°·¤ ×æðÅÚU ÕæðÅU, çÁâ·¤è çSÍÚU ÁÜ ×ð´ ¿æÜ 24 ç·¤×è/ƒæ´ÅUæ ãñ, ÏæÚUæ ·ð¤ ÂýçÌ·ê¤Ü 32 ç·¤×è ÁæÙð
×ð´, ßãè ÎêÚUè ÏæÚUæ ·ð¤ ¥Ùé·ê¤Ü ÁæÙð ·¤è ¥Âðÿææ 1 ƒæ´ÅUæ ¥çÏ·¤ â×Ø ÜðÌè ãñÐ ÏæÚUæ ·¤è ¿æÜ
™ææÌ ·¤èçÁ°Ð
A motor boat whose speed is 24 km/h in still water takes 1 hour more to go
32 km upstream than to return downstream to the same spot. Find the speed
of the stream.
28.
°·¤ â×çmÕæãé ç˜æÖéÁ ABC ·¤è ÚU¿Ùæ ·¤èçÁ° çÁâ×ð´ BC55.5 âð×è, ÌÍæ çÁâ·ð¤ àæèáü Ü´Õ
AL ·¤è Ü´Õæ§ü 3 âð×è ãñÐ çȤÚU °·¤ ¥‹Ø ç˜æÖéÁ ·¤è ÚU¿Ùæ ·¤èçÁ° çÁâ·¤è ÖéÁæ°¡ DABC
·¤è â´»Ì ÖéÁæ¥æð´ ·¤æ
3
4
Öæ» ãæð´Ð
Draw an isosceles DABC in which BC55.5 cm and altitude AL53 cm. Then
construct another triangle whose sides are
3
of the corresponding sides of
4
DABC.
29.
çâh ·¤èçÁ° ç·¤ ßëžæ ·ð¤ ç·¤âè çÕ´Îé ÂÚU ¹è´¿è »§ü SÂàæü ÚðU¹æ SÂàæü çÕ´Îé âð ãæð·¤ÚU ÁæÙð ßæÜè
ç˜æ’Øæ ÂÚU Ü´Õ ãæðÌè ãñÐ
Prove that tangent drawn at any point of a circle is perpendicular to the radius
through the point of contact.
30.
°·¤ Âý·¤æàæ SÌ´Ö ·ð¤ çàæ¹ÚU âð, Áæð â×éÎý ·ð¤ ÁÜ SÌÚU âð 100 ×è. ª¡¤¿æ ãñ, °·¤ ÁÜØæÙ, Áæð
âèÏæ Âý·¤æàæ SÌ´Ö ·¤è ¥æðÚU ¥æ ÚUãæ ãñ, ·¤æ ¥ßÙ×Ù ·¤æð‡æ 308 âð ÕÎÜ·¤ÚU 608 ãæð ÁæÌæ ãñÐ
Âýðÿæ‡æ ·ð¤ ¥´ÌÚUæÜ ×ð´ ÁÜØæÙ mæÚUæ ÌØ ·¤è »§ü ÎêÚUè ™ææÌ ·¤èçÁ°Ð
( 3 51.73
ÜèçÁ° )
As observed from the top of a light house, 100 m high above sea level, the
angles of depression of a ship, sailing directly towards it, changes from 308 to
608. Find the distance travelled by the ship during the period of observation.
( Use
30/2
3 51.73 )
14
31.
°·¤ °ðâð ¥æØÌæ·¤æÚU Âæ·ü¤ ·¤æð ÕÙæÙæ ãñ çÁâ·¤è ¿æñǸæ§ü §â·¤è Ü´Õæ§ü âð 3 ×è. ·¤× ãñÐ §â·¤æ
ÿæð˜æÈ¤Ü ÂãÜð âð çÙç×üÌ â×çmÕæãé ç˜æÖéÁæ·¤æÚU Âæ·ü¤ çÁâ·¤æ ¥æÏæÚU ¥æØÌæ·¤æÚU Âæ·ü¤ ·¤è
¿æñǸæ§ü ·ð¤ ÕÚUæÕÚU ãñ ÌÍæ ª¡¤¿æ§ü 12 ×è. ãñ, âð 4 ß»ü ×èÅUÚU ¥çÏ·¤ ãñÐ ¥æØÌæ·¤æÚU Âæ·ü¤ ·¤è
Ü´Õæ§ü ¥æñÚU ¿æñǸæ§ü ™ææÌ ·¤èçÁ°Ð
A rectangular park is to be designed whose breadth is 3 m less than its length.
Its area is to be 4 square metres more than the area of a park that has already
been made in the shape of an isosceles triangle with its base as the breadth of
the rectangular park and of altitude 12 m. Find the length and breadth of
the rectangular park.
30/2
15
P.T.O.

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