10th 1040123_C1 Mathematics QP.p65

1040123 - C1
Class - X
MATHEMATICS
Time : 3 to 3½ hours
¥çÏ·¤Ì× â×Ø : 3 âð 3½ ƒæ‡ÅUð
Maximum Marks : 80
¥çÏ·¤Ì× ¥´·¤ : 80
Total No. of Pages : 13
·é¤Ü ÂëcÆUæð´ ·¤è ⴁØæ : 13
General Instructions :
1.
All questions are compulsory.
2.
The question paper consists of 34 questions divided into four sections A, B, C and D.
Section - A comprises of 10 questions of 1 mark each, Section - B comprises of 8 questions of
2 marks each, Section - C comprises of 10 questions of 3 marks each and Section - D comprises
of 6 questions of 4 marks each.
3.
Question numbers 1 to 10 in Section - A are multiple choice questions where you are to select
one correct option out of the given four.
4.
There is no overall choice. However, internal choice has been provided in 1 question of two
marks, 3 questions of three marks each and 2 questions of four marks each. You have to
attempt only one of the alternatives in all such questions.
5.
Use of calculator is not permitted.
6.
An additional 15 minutes time has been allotted to read this question paper only.
âæ×æ‹Ø çÙÎðüàæ Ñ
1.
âÖè ÂýàÙ ¥çÙßæØü ãñ´Ð
2.
§â ÂýàÙ˜æ ×ð´ 34 ÂýàÙ ãñ´, Áæð ¿æÚU ¹‡ÇUæð´ ×ð´ ¥, Õ, â ß Î ×ð´ çßÖæçÁÌ ãñÐ ¹‡ÇU - ¥ ×ð´ 10 ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤
ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ ¹‡ÇU - Õ ×ð´ 8 ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤ ÂýàÙ 2 ¥´·¤æð´ ·ð¤ ãñ´Ð ¹‡ÇU - â ×ð´ 10 ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤
ÂýàÙ 3 ¥´·¤æð´ ·¤æ ãñÐ ¹‡ÇU - Î ×ð´ 6 ÂýàÙ ãñ´ ¥æñÚU ÂýˆØð·¤ ÂýàÙ 4 ¥´·¤æð´ ·¤æ ãñÐ
3.
Âýà٠ⴁØæ 1 âð 10 Õãéçß·¤ËÂèØ ÂýàÙ ãñ´Ð çΰ »° ¿æÚU çß·¤ËÂæð´ ×ð´ âð °·¤ âãè çß·¤Ë ¿éÙð´Ð
4.
§â×ð´ ·¤æð§ü Öè âßæðüÂçÚU çß·¤Ë Ùãè´ ãñ, Üðç·¤Ù ¥æ´ÌçÚU·¤ çß·¤Ë 1 ÂýàÙ 2 ¥´·¤æð´ ×ð´, 3 ÂýàÙ 3 ¥´·¤æð´ ×ð´ ¥æñÚU 2 ÂýàÙ
4 ¥´·¤æð´ ×ð´ çΰ »° ãñ´Ð ¥æ çΰ »° çß·¤ËÂæð´ ×ð´ âð °·¤ çß·¤Ë ·¤æ ¿ØÙ ·¤Úð´UÐ
5.
·ñ¤Ü·é¤ÜðÅUÚU ·¤æ ÂýØæð» ßçÁüÌ ãñÐ
6.
§â ÂýàÙ-Â˜æ ·¤æð ÂɸÙð ·ð¤ çÜ° 15 ç×ÙÅU ·¤æ â×Ø çÎØæ »Øæ ãñÐ §â ¥ßçÏ ·ð¤ ÎæñÚUæÙ ÀUæ˜æ ·ð¤ßÜ ÂýàÙ-Â˜æ ·¤æð Âɸð´»ð
¥æñÚU ß𠩞æÚU-ÂéçSÌ·¤æ ÂÚU ·¤æð§ü ©žæÚU Ùãè´ çܹð´»ðÐ
1
P.T.O.
SECTION – A
Question numbers 1 to 10 are of 1 mark each.
1.
2.
Which of the following rational numbers have a terminating decimal expansion ?
(A)
125
441
(B)
77
210
(C)
15
1600
(D)
2
1208
(B)
The maximum value of
(A)
4.
0
908
(C)
458
(D)
608
(C)
1
(D)
3
2
1
is :
cosecM
(B)
21
Which of the following pairs of equations represent inconsistent system ?
(A)
3x22y58
(B)
2x13y51
5.
2 52 7 2
In DABC, if AB5 6 3 cm, AC512 cm and BC56 cm, then angle B is equal to :
(A)
3.
129
If tanA5
(A)
4
3
1040123 - C1
3x2y528
(C)
3x2y524
lx2y5m
(D)
x1my5l
5x2y510
10x22y520
3
and A1B5908 then the value of cot B is equal to :
4
(B)
1
2
(C)
2
3
4
(D)
1
6.
The lower limit of the modal class of the following data is :
C. I.
Frequency
(A)
7.
20 - 25
10
15
25 - 30
14
30 - 35
9
(C)
10
(D)
30
30
(B)
150
(C)
100
(D)
50
1
(B)
1
4
(D)
3
(C)
1
2
If a and b are the zeroes of the polynomial 4x213x17, then
(A)
10.
(B)
15 - 20
6
If x . tan 458 . cot 608 5 sin 308 . cosec 608, then the value of x is :
(A)
9.
10 - 15
15
The value of x in the factor tree is :
(A)
8.
25
5 - 10
5
7
3
(B)
7
3
(C)
3
7
1
1
is equal to :
9
:
(D)
3
7
A vertical stick 30 m long casts a shadow 15 m long on the ground. At the same time,
a tower casts a shadow 75 m long on the ground. The height of the tower is :
(A)
150 m
1040123 - C1
(B)
100 m
(C)
3
25 m
(D)
200 m
P.T.O.
SECTION – B
Question numbers 11 to 18 carry 2 marks each.
11.
Explain why 11313315317117 is a Composite Number.
12.
Check whether x213x11 is a factor of 3x415x327x212x12
13.
Solve for x and y
4
5 y 7
x
3
4 y 5
x
14.
The wickets taken by a bowler in 10 cricket matches are as follows :
2
6
4
5
0
2
1
3
2
3
Find the mode of the data.
15.
If A, B and C are interior angles of a triangle ABC, then show that
š·
š
§š
B
C
A
¸ cos
sin ¨
¨ 2 ¸
2
©
¹
16.
š
In DABC, right angled at B, AB55 cm and A C B 5308. Find BC and AC.
OR
If sinu 1 sin2u 5 1, then find the value of cos2u 1 cos4u.
17.
Prove that the area of the equilateral triangle described on the side of a square is half
the area of the equilateral triangle described on its diagonal.
1040123 - C1
4
18.
Find mode of the following data :
C. I.
Frequency
0 - 20
6
20 - 40
8
40 - 60
12
60 - 80
10
80 - 100
6
SECTION – C
Question numbers 19 to 28 carry 3 marks each.
19.
Use Euclid division lemma to show that cube of any positive integer is either of the
form 9m, 9m11, or 9m18.
20.
If the zeroes of the polynomial x323x21x11 are a2b, a and a1b, find a and b.
21.
P and Q are points on sides AB and AC respectively of DABC. If AP53 cm, PB56 cm,
AQ55 cm and QC510 cm, show that BC53PQ.
22.
In DABC, AD A BC such that AD25BD.CD. Prove that DABC is right angled at A.
OR
Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on
the same side of BC. If AC and BD intersect at P, prove that :
AP3PC5BP3DP
23.
If sinu1cosu5p and secu1cosecu5q, show that q(p221)52p
OR
If sinu1cosu5 2 sin(902u), then find the value of tanu.
24.
Evaluate :
sin39
2tan11 tan31 tan45 tan59 tan79 3(sin 2 21 sin 2 69 )
cos51
25.
Prove that 315 2 is irrational.
1040123 - C1
5
P.T.O.
26.
Find the median of the following data :
C.I.
Frequency
27.
1-4
6
4-7
30
7 - 10
40
10 - 13
16
13 - 16
4
16 - 19
4
Father’s age is 3 times the sum of ages of his two children. After 5 years his age will be
twice the sum of ages of two children. Find the age of father.
OR
Places A and B are 100 km apart on a highway. One car starts from A and another
from B at the same time. If the cars travel in the same direction at different speeds, they
meet in 5 hours. If they travel towards each other they meet in 1 hour. What are the
speeds of the two cars ?
28.
Draw the more than ogive for the following frequency distribution :
Class Interval 150 - 155 155 - 160 160 - 165 165 - 170 170 - 175 175 - 180
Frequency
6
10
22
34
16
12
SECTION – D
Question numbers 29 to 34 carry 4 marks each.
29.
Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the
squares of other two sides.
OR
Prove that the ratio of the areas of two similar triangles is equal to the square of the
ratio of their corresponding sides.
p 21
sinM .
30.
If secu1tanu5p, show that
31.
Solve graphically ; x2y51, 2x1y58. Shade the region bounded by these lines and
y-axis. Also find its area.
1040123 - C1
p 21
6
32.
The mean of the following data is 50. Find the missing frequencies f1 and f2.
C.I.
Frequency
0 - 20
20 - 40
40 - 60
17
f1
32
60 - 80 80 - 100
f2
19
Total
120
33.
If two zeroes of the polynomial p (x)5x426x3226x21138x235 are 2 3 , find the
other zeroes.
34.
If tanu 1 sinu 5 m and
tanu 2 sinu 5 n show that
m22n25 4 mn.
OR
If secu5x1
1
, then prove that tanu1secu52x or 1 .
4x
2x
-o0o-
1040123 - C1
7
P.T.O.
¹‡ÇU-¥
Âýà٠ⴁØæ 1 âð 10 Ì·¤ ÂýˆØð·¤ ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ
1.
2.
·¤æñÙ âè ÂçÚU×ðØ â´Øæ âæ´Ì Îàæ×Üß ×ð´ ãñ?
(A)
125
441
(B)
77
210
(C)
15
1600
(D)
2
DABC ×ð´,
(A)
3.
4.
1208
1
cosecM
(A)
ØçÎ AB5 6
3
2 527 2
âð.×è., AC512 âð.×è. ÌÍæ BC56 âð.×è., ÌÕ
(B)
‘ B ·¤æ
×æÙ ãæð»æ Ñ
908
(C)
458
(D)
608
21
(C)
1
(D)
3
2
lx2y5m
(D)
·¤æ ×ãžæ× ×æÙ ãæð»æ Ñ
0
(B)
çِÙçÜç¹Ì ÚñUç¹·¤ â×è·¤ÚU‡ææð´ Øé‚× ×ð´ ·¤æñÙ âæ ¥â´»Ì ãñ?
(A)
3x22y58
(B)
2x13y51
5.
129
ØçÎ tanA5
(A)
4
3
1040123 - C1
3
4
3x2y528
(C)
3x2y524
x1my5l
5x2y510
10x22y520
ÌÍæ A1B5908 ÌÕ cot B ·¤æ ×æÙ ãæð»æ Ñ
(B)
1
2
(C)
8
3
4
(D)
1
6.
çِ٠¥æ¡·¤Ç¸æð´ ×ð´ ×æçŠØ·¤æ ·¤ÿæ ·¤è çِ٠âè×æ ãæð»è Ñ
Æ¢ãŒr³Á<ÍÃ
¼ÍÁ›<¼ÍÁ<³Í
(A)
7.
20 - 25
25 - 30
30 - 35
5
15
6
10
14
9
(B)
15
30
(B)
150
1
(B)
1
4
(C)
10
(D)
30
(C)
100
(D)
50
(D)
3
(C)
ØçÎ a , b ÕãéÂÎ 4x213x17 ·ð¤ àæê‹Øæ´·¤ ãñ Ìæð
(A)
10.
15 - 20
ØçÎ x . tan 458 . cot 608 5 sin 308 . cosec 608 ÌÕ x ·¤æ ×æÙ ãæð»æ Ñ
(A)
9.
10 - 15
çِ٠»é‡æÙ¹´ÇU ßëÿæ ×ð´ x ·¤æ ×æÙ ãæð»æ Ñ
(A)
8.
25
5 - 10
7
3
(B)
1
1
9
:
7
3
1
2
ÕÚUæÕÚU ãæð»æ Ñ
(C)
3
7
(D)
3
7
°·¤ ©ŠßæüÏÚU 30 ×è. ܐÕð δÇU ·¤è ÀUæØæ ÏÚUæÌÜ ×ð´ 15 ×èÅUÚU ÂǸÌè ãñÐ ©âè â×Ø °·¤ ×èÙæÚU ·¤è ÀUæØæ 75
×èÅUÚU ܐÕè, ÏÚUæÌÜ ×ð´ ÂǸUÌè ãñÐ ×èÙæÚU ·¤è ª¤¡¿æ§ü ãæð»è Ñ
(A)
150
1040123 - C1
×è.
(B)
100
×è.
(C)
9
25
×è.
(D)
200
×è.
P.T.O.
¹‡ÇU-Õ
Âýà٠ⴁØæ 11 âð 18 Ì·¤ ÂýˆØð·¤ ÂýàÙ 2 ¥´·¤ ·¤æ ãñÐ
11.
SÂcÅU ·¤èçÁ° ç·¤ 11313315317117 °·¤ Öæ’Ø ⴁØæ ãñÐ
12.
Áæ¡¿ ·¤èçÁ° ç·¤ x213x11, 3x415x327x212x12 ·¤æ »é‡æÙ¹´ÇU ãñÐ
13.
x ÌÍæ y ·ð¤
×æÙ ·ð¤ çÜØð ãÜ ·¤Úð´UÐ
4
5 y 7
x
3
4 y 5
x
14.
°·¤ »ð´ÎÕæÁ mæÚUæ 10 ç·ý¤·ð¤ÅU ×ñ¿ ×ð´ çÜØð »Øð çßç·¤ÅU §â Âý·¤æÚU ãñ Ñ
2
6
4
5
0
2
1
3
2
3
¥æ¡·¤Ç¸æð´ ·¤æ ÕãéÜ·¤ ™ææÌ ·¤ÚUæðÐ
15.
ØçÎ A, B, C ç·¤âè ç˜æÖéÁ ABC ·ð¤ ¥æ‹ÌçÚU·¤ ·¤æð‡æ ãæð´ Ìæð Îàææü§Øð Ñ
A
§ BC ·
sin ¨
¸ cos
2
© 2 ¹
16.
â×·¤æð‡æ DABC çÁâ·¤æ
çÙ·¤æçÜ°Ð
‘ B â×·¤æð‡æ
ãñ, ØçÎ AB55 âð.×è. ÌÍæ
‘ ACB5308 Ìæð
ÖéÁæ
BC ÌÍæ AC
Øæ
ØçÎ sinu 1 sin2u 5 1, ÌÕ cos2u 1 cos4u ·¤æ ×æÙ ™ææÌ ·¤ÚUæðÐ
17.
çâh ·¤èçÁ° ç·¤ °·¤ ß»ü ·¤è ç·¤âè ÖéÁæ ÂÚU ÕÙæØð »° â×Õæãé ç˜æÖéÁ ·¤æ ÿæð˜æÈ¤Ü ©âè ß»ü ·ð¤ °·¤ çß·¤‡æü
ÂÚU ÕÙæØð »Øð â×Õæãé ç˜æÖéÁ ·ð¤ ÿæð˜æÈ¤Ü ·¤æ ¥æÏæ ãæðÌæ ãñÐ
18.
çِÙçÜç¹Ì ¥æ¡·¤Ç¸æð´ ·¤æ ÕãéÜ·¤ ™ææÌ ·¤èçÁ°Ð
Æ¢ãŒr³Á<ÍÃ
¼ÍÁ›<¼ÍÁ<³Í
1040123 - C1
0 - 20
20 - 40
40 - 60
60 - 80
80 - 100
6
8
12
10
6
10
¹‡ÇU-â
Âýà٠ⴁØæ 19 âð 28 Ì·¤ ÂýˆØð·¤ ÂýàÙ 3 ¥´·¤ ·¤æ ãñÐ
19.
Øêç€ÜÇU Öæ» Âý×ðçØ·¤æ mæÚUæ Îàææü§Øð ç·¤ ç·¤âè ÏÙæˆ×·¤ Âê‡ææZ·¤ ·¤æ ƒæÙ 9m, 9m11, or 9m18 ·ð¤ M¤Â ×ð´
ãæðÌæ ãñÐ
20.
ØçÎ ÕãéÂÎ x323x21x11 ·ð¤ àæê‹Øæ´·¤ a2b, a ÌÍæ
21.
ç·¤âè ç˜æÖéÁ ABC ·ð¤ ÖéÁæ AB ÌÍæ AC ×ð´ P ¥æñÚU Q Îæð çՋÎé ãñÐ ØçÎ AP53 âð.×è., PB56 âð.×è.
AQ55 âð.×è. ÌÍæ QC510 âð.×è. ãæð Ìæð Îàææü§Øð BC53PQ.
22.
ç·¤âè DABC ×ð´,
â×·¤æð‡æ ãñÐ
AD A BC
a1b ãæð´
Ìæð a ÌÍæ b ·ð¤ ×æÙ ™ææÌ ·¤ÚUæðÐ
§â Âý·¤æÚU ãñ ç·¤ AD25BD.CD Ìæð çâh ·¤èçÁ° ç·¤ DABC ·¤æ ·¤æð‡æ
Øæ
Îæð â×·¤æð‡æ DABC ÌÍæ DDBC ·¤‡æü BC ·ð¤ °·¤ ãè ÌÚUȤ ¹è´¿ð »ØðÐ ØçÎ
P çՋÎé ÂÚU Âýç̑ÀðUÎ ·¤Úð´U Ìæð çâh ·¤ÚUæð Ñ
AC ÌÍæ BD
A
°·¤ ÎêâÚðU ·¤æð
AP3PC5BP3DP
23.
ØçÎ sinu1cosu5p ÌÍæ secu1cosecu5q, Îàææü§Øð q(p221)52p.
Øæ
ØçÎ sinu1cosu5
2 sin(902u) ÌÕ tanu ·¤æ
×æÙ ™ææÌ ·¤ÚUæðÐ
24.
sin39
2tan11 tan31 tan45 tan59 tan79 3(sin 2 21 sin 2 69 )
cos51
25.
çâh ·¤èçÁ° ç·¤ 315
26.
çِ٠¥æ¡·¤Ç¸æð´ ·¤è ×æçŠØ·¤æ ™ææÌ ·¤ÚUæðÐ
Æ¢ãŒr³Á<ÍÃ
¼ÍÁ›<¼ÍÁ<³Í
1040123 - C1
2
·¤æ ×æÙ ™ææÌ ·¤Úð´UÐ
°·¤ ¥ÂçÚU×ðØ â´Øæ ãñÐ
1-4
4-7
7 - 10
10 - 13
13 - 16
16 - 19
6
30
40
16
4
4
11
P.T.O.
27.
çÂÌæ ·¤è ©×ý ¥ÂÙð ÎæðÙæð´ Փææð´ ·¤è ©×ý ·ð¤ Øæð» ·¤è ÌèÙ »éÙè ãñÐ Âæ¡¿ âæÜ ÕæÎ çÂÌæ ·¤è ©×ý ¥ÂÙð Փææð´ ·¤è
©×ý ·¤è Øæð» âð Îæð »éÙè ãæð ÁæØð»èÐ çÂÌæ ·¤è ©×ý ™ææÌ ·¤ÚUæðÐ
Øæ
A ÌÍæ B âǸ·¤ ãæ§üßð ×ð´ 100 ç·¤.×è. ÎêÚUè ÂÚU ãñÐ Îæð ·¤æÚð´U A ÌÍæ B âð °·¤ ãè â×Ø ×ð´ ÌÍæ °·¤ ãè çÎàææ ×ð´
¥Ü» ¥Ü» »çÌ âð ¿ÜÌè ãñ Ìæð ßã ÎæðÙæð´ 5 ƒæ´ÅðU ÂÚU ç×ÜÌè ãñÐ ØçÎ ßã °·¤ ÎêâÚðU ·¤è ÌÚUȤ ¿ÜÌè ãñ´ Ìæð
1 ƒæ´ÅðU ÕæÎ ç×ÜÌè ãñÐ ÎæðÙæð´ ·¤æÚUæð´ ·¤è »çÌ ™ææÌ ·¤èçÁ°Ð
28.
Ùè¿ð çܹð ÕæÚ´UÕæÚUÌæ Õ´ÅUÙ ·¤æ ÌæðÚU‡æ Òâð ¥çÏ·¤ Âý·¤æÚUÓ ¹è´ç¿°Ð
Æ¢ãŒr³Á<ÍÃ
¼ÍÁ›<¼ÍÁ<³Í
150 - 155 155 - 160 160 - 165 165 - 170 170 - 175 175 - 180
6
10
22
34
16
12
¹‡ÇU - Î
Âýà٠ⴁØæ 29 âð 34 Ì·¤ ÂýˆØð·¤ ÂýàÙ 4 ¥´·¤ ·¤æ ãñÐ
29.
çâh ·¤èçÁ° ç·¤ â×·¤æð‡æ ç˜æÖéÁ ×ð´ ·¤‡æü ·¤æ ß»ü àæðá ¥‹Ø ÎæðÙæð´ ÖéÁæ¥æð´ ·ð¤ ß»æðZ ·ð¤ Øæð» ·ð¤ ÕÚUæÕÚU ãæðÌæ ãñÐ
Øæ
çâh ·¤èçÁ° Îæð â×M¤Â ç˜æÖéÁæð´ ·ð¤ ÿæð˜æȤÜæð´ ·¤æ ¥ÙéÂæÌ §Ù·¤è â´»Ì ÖéÁæ¥æ𴠷𤠥ÙéÂæÌ ·ð¤ ß»ü ·ð¤ ÕÚUæÕÚU
ãæðÌæ ãñÐ
30.
ØçÎ secu1tanu5p, Îàææü§Øð
31.
»ýæȤèØ çßçÏ mæÚUæ ãÜ ·¤ÚUæð Ñ
p 21
p 21
sinM .
x2y51, 2x1y58
©â ÿæð˜æ ·¤æð ÀUæØæ´ç·¤Ì ·¤èçÁ° Áæð ©ÂØéü€Ì ÚðU¹æØð´ ÌÍæ y-¥ÿæ ·ð¤ Õè¿ ×ð´ ãæðÐ §â ÿæð˜æ ·¤æ ÿæð˜æÈ¤Ü Öè ™ææÌ
·¤Úð´UÐ
32.
çِÙçÜç¹Ì ¥æ¡·¤Ç¸æð´ ·¤æ ׊Ø×æÙ 50 ãñ, ¥™ææÌ ÕæÚ´UÕæÚUÌæ f1 ÌÍæ f2 ™ææÌ ·¤ÚUæðÐ
Æ¢ãŒr³Á<ÍÃ
¼ÍÁ›<¼ÍÁ<³Í
1040123 - C1
0 - 20
20 - 40
40 - 60
17
f1
32
60 - 80 80 - 100
f2
12
19
Total
120
33.
ØçÎ ÕãéÂÎ p(x)5x426x3226x21138x235 ·ð¤ àæê‹Øæ´·¤
34.
ØçÎ tanu 1 sinu 5 m ÌÍæ
tanu 2 sinu 5 n ãæð
2 3
ãæð Ìæð ¥‹Ø àæê‹Øæ´·¤ ™ææÌ ·¤ÚUæðÐ
Ìæð Îàææü§Øð
m22n25 4 mn.
Øæ
ØçÎ secu5x1
1
4x
ÌÕ çâh ·¤ÚUæð tanu1secu52x Øæ
1 .
2x
-o0o-
1040123 - C1
13
P.T.O.