1040119 C1

1040119 - 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 : 11
·é¤Ü ÂëcÆUæð´ ·¤è ⴁØæ : 11
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 ·ð¤ 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 carry 1 mark each.
1.
If the HCF of 65 and 117 is expressible in the form 65m2117, then the value of m is :
(A)
2.
3.
5.
1
(D)
(B)
Composite number
(C)
An odd prime number
(D)
An odd composite number
3
1
1
If a, b are zeroes of x224x11, then 9: is :
9
:
3
(B)
5
(C)
25
(D)
23
One equation of a pair of dependent linear equations is 25x17y52, the second
equation can be :
(A)
10x114y1450
(B)
210x214y1450
(C)
210x114y1450
(D)
10x214y524
If sina5
1
and a is acute, then (3cosa24cos3a) is equal to :
2
0
If in DABC and DDEF
‘ B5 ‘ E
(B)
1
2
(C)
1
6
(D)
21
‘ B5 ‘ D
(D)
‘ A5 ‘ F
AB
BC
, then they will be similar if :
DE
FD
(B)
‘ A5 ‘ D
(C)
If xcos A51 and tan A5y, then x22y2 is equal to :
(A)
8.
(C)
Prime number
(A)
7.
2
(A)
(A)
6.
(B)
119221112 is :
(A)
4.
4
tan A
(B)
1
(C)
0
(D)
2tan A
2 cos 2A21
(C)
2 sin2A21
(D)
2 sin2A11
[cos4A2sin4A] is equal to :
(A)
2 cos2A11
1040119 - C1
(B)
2
9.
The value of the expression [(sec2u21)(12cosec2u)] is :
(A)
10.
21
(B)
1
(C)
0
(D)
1
2
(D)
None
If mode of a data is 45, mean is 27, then median is :
(A)
30
(B)
27
(C)
33
SECTION – B
Question numbers 11 to 18 carry 2 marks each.
11.
Find the LCM of 336 and 54 by prime factorisation method.
12.
Find the zeroes of the quadratic polynomial 4x227.
13.
Solve :
99x1101y5499
101x199y5501
14.
If D, E are points on the sides AB and AC of DABC such that AD56 cm, BD59 cm,
AE58 cm, EC512 cm. Prove that DE??BC.
15.
If the areas of two similar triangles are equal, prove that they are congruent.
16.
Find the value of the expression
cos30 sin60
1 cos60 sin30
OR
If ‘ A and ‘ B are acute angles such that cosA5cosB. Show that ‘ A5 ‘ B.
17.
Find the mode of the given data :
Class Interval
Frequency
18.
0 - 20
20 - 40
40 - 60
60 - 80
15
6
18
10
Find the median of following given data :
x
6
7
5
2
10
9
3
f
9
12
8
13
11
14
7
1040119 - C1
3
P.T.O.
SECTION – C
Question numbers 19 to 28 carry 3 marks each.
19.
Prove that 2 3 4 is an irrational number.
20.
Prove that the square of any positive integer is of the form 3m or 3m11 for some
integer m.
21.
Check whether the polynomial g (x)5x 3 23x11 is the factor of polynomial
p (x)5x 524x 31x 213x11
OR
Solve for x and y : mx2ny5m21n2 ; x2y52n
22.
Find values of a and b for which the system of linear equations has infinite number of
solutions :
(a1b)x22by55a12b11 ; 3x2y514
23.
In fig. 1, BL and CM are medians of DABC right angled at A.
4(BL21CM2)55BC2
1040119 - C1
4
Prove that
24.
ar (DFE)
Find ar (CFB)
In fig.2, DE??BC and AD : DB55 : 4.
OR
Two Isosceles triangles have equal vertical angles and their areas are in the ratio
16 : 25. Find ratio of their corresponding heights.
x
y
y2
y
x2
x
cosu1 sinu51 and sinu2 cosu51, prove that 2 2 2
a
b
b
a
a
b
25.
If
26.
C
§ AB ·
sec .
If A, B, C are interior angles of DABC, prove that cosec ¨
© 2 ¸¹
2
OR
If sin (2A1458)5cos (3082A), Find A.
27.
The daily expenditure of 100 families are given below. Calculate f1 and f2, if the mean
daily expenditure is Rs. 188.
Expenditure
140 - 160 160 - 180 180 - 200 200 - 220 220 - 240
No. of families
28.
5
25
5
f2
f1
Compute the median for the following data :
Class
Interval
Cumulative
Frequency
1040119 - C1
Less
than
20
Less
than
30
Less
than
40
Less
than
50
Less
than
60
Less
than
70
Less
than
80
Less
than
90
Less
than
100
0
4
16
30
46
66
82
92
100
5
P.T.O.
SECTION – D
Question numbers 29 to 34 carry 4 marks each.
29.
Prove that sin6u1cos6u5123sin2ucos2u.
30.
The sum of a 2 digit number and number obtained by reversing the order of digits is 99.
If the digits of the number differ by 3. Find the number.
OR
A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Find the speed
of sailor in still water and the speed of current.
31.
State and prove converse of Pythagoras theorem.
32.
Draw less than and more than ogive for the following distribution and hence obtain
the median.
Marks
No. of students
33.
30 - 40
14
40 - 50
6
50 - 60
10
60 - 70
20
70 - 80
30
Solve the following system of linear equations graphically :
2(x21)5y and x13y515.
Also find the coordinates of points where lines meet the y-axis.
34.
In DABC, ‘ B5908 AB53 cm and BC54cm. Find
(i)
sin C
(ii)
cos C
(iii)
sec A
OR
If tanA5
1
, DABC is right angled at B.
3
Find the value of sinA cosC1cosA sinC.
-o0o-
1040119 - C1
6
(iv)
cosec A
80 - 90
8
90 - 100
12
¹‡ÇU - ¥
Âýà٠ⴁØæ 1 âð 10 Ì·¤ ÂýˆØð·¤ ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ
1.
ØçÎ
(A)
2.
3.
5.
(C)
1
Öæ’Ø ⴁØæ
(C)
çßá× ¥Øæ’Ø ⴁØæ
(D)
çßá× Öæ’Ø ⴁØæ
ØçÎ a, b
(D)
3
(D)
23
ãñÐ
(B)
x224x11 ·ð¤
àæê‹Ø·¤ ãæð´ Ìæð
3
(B)
1
1
9:
9
:
5
·¤æ ×æÙ ãæð»æ Ñ
(C)
25
ØçÎ Îæð Øé‚× ÚñUç¹·¤ â×è·¤ÚU‡æ Áæð °·¤ ÎêâÚðU ÂÚU ¥æçŸæÌ ãñ´ ©Ù×ð´ âð °·¤ ØçÎ 25x17y52 ãæð Ìæð ÎêâÚUæ ãñ Ñ
(A)
10x114y1450
(B)
210x214y1450
(C)
210x114y1450
(D)
10x214y524
ØçÎ sina5
1
2
ÌÍæ a °·¤ ‹ØêÙ·¤æð‡æ ãæð ÌÕ (3cosa24cos3a) ãñ Ñ
0
DABC ÌÍæ DDEF ×ð´,
(B)
1
2
AB
BC
DE
FD
‘ B5 ‘ E
(B)
(C)
1
6
(D)
21
‘ B5 ‘ D
(D)
‘ A5 ‘ F
ÌÕ ßã ÎæðÙæð´ â×M¤Â ãæð´»ð ØçÎ Ñ
‘ A5 ‘ D
(C)
ØçÎ x cos A51 ÌÍæ tan A5y ÌÕ x22y2 ·ð¤ ÕÚUæÕÚU ãñ Ñ
(A)
8.
2
¥Öæ’Ø ⴁØæ
(A)
7.
(B)
M¤Â ×ð´ ÃØ€Ì ç·¤Øæ ÁæØ Ìæð m ·¤æ ×æÙ ãæð»æ Ñ
(A)
(A)
6.
4
119221112
(A)
4.
65 ÌÍæ 117 ·¤æ HCF 65 m2117 ·ð¤
tan A
[cos4A2sin4A] ÕÚUæÕÚU
(A)
2 cos 2A11
1040119 - C1
(B)
1
(C)
0
(D)
2tan A
2 cos 2A21
(C)
2 sin2A21
(D)
2 sin2A11
ãñ Ñ
(B)
7
P.T.O.
9.
[(sec2u21)(12cosec2u)] ·¤æ
(A)
10.
21
×æÙ ãñ Ñ
(B)
1
(C)
0
(D)
1
2
(D)
·¤æð§ü Ùãè´
ØçÎ ¥æ¡·¤ÇUæð´ ·¤æ ÕãéÜ·¤ 45 ÌÍæ ×æŠØ 27 ãæð Ìæð ×æçŠØ·¤æ ãæð»è Ñ
(A)
30
(B)
27
(C)
33
¹‡ÇU-Õ
Âýà٠ⴁØæ 11 âð 18 Ì·¤ ÂýˆØð·¤ ÂýàÙ 2 ¥´·¤æð´ ·¤æ ãñÐ
11.
¥Öæ’Ø »é‡æÙ¹´ÇU mæÚUæ
336 ÌÍæ 54 ·¤æ LCM (Ü.â.ß.)
™ææÌ ·¤ÚUæðÐ
12.
4x227 çmƒææÌ
13.
ãÜ ·¤ÚUæð Ñ
14.
ØçÎ
15.
ØçÎ Îæð â×M¤Â ç˜æÖéÁæð´ ·¤æ ÿæð˜æÈ¤Ü ÕÚUæÕÚU ãæð Ìæð çâh ·¤èçÁ° ç·¤ ßð âßæZ»â× ãñÐ
16.
çِ٠·¤æ ×æÙ ™ææÌ ·¤èçÁ° Ñ
ÕãéÂÎ ·ð¤ àæê‹Ø·¤ ™ææÌ ·¤ÚUæðÐ
99x1101y5499
101x199y5501
×ð´ D ÌÍæ E çՋÎé ÖéÁæ AB ÌÍæ AC ×ð´ §â Âý·¤æÚU ãæð´ ç·¤ AD56 âð.×è., BD59 âð.×è.,
AE58 âð.×è. ÌÍæ EC512 âð.×è. çâh ·¤èçÁ° Ñ DE??BC.
DABC
cos30 sin60
1 cos60 sin30
¥Íßæ
ØçÎ
17.
‘ A ÌÍæ ‘ B ‹ØêÙ·¤æð‡æ
çِ٠¥æ¡·¤ÇUæð´ ·¤æ ÕãéÜ·¤ ™ææÌ ·¤ÚUæðÐ
Æ¢ãŒr³Á<ÍÃ
¼ÍÁ›<¼ ÍÁ<³ Í
18.
§â Âý·¤æÚU ãñ ç·¤ cosA5cosB Ìæð Îàææü§°
0 - 20
20 - 40
40 - 60
60 - 80
15
6
18
10
çِ٠¥æ¡·¤ÇUæð´ ·¤è ×æçŠØ·¤æ ™ææÌ ·¤ÚUæðÐ
x
6
7
5
2
10
9
3
f
9
12
8
13
11
14
7
1040119 - C1
8
‘ A5 ‘ B.
¹‡ÇU - â
Âýà٠ⴁØæ 19 âð 28 Ì·¤ ÂýˆØð·¤ ÂýàÙ 3 ¥´·¤æð´ ·¤æ ãñÐ
19.
çâh ·¤èçÁ° ç·¤
20.
çâh ·¤èçÁ° ç·¤ ÏÙæˆ×·¤ Âê‡ææZ·¤ ·¤æ ß»ü 3m, 3m11 ·ð¤ M¤Â ×ð´ ãæð»æÐ Áãæ¡ m °·¤ Âê‡ææZ·¤ ãñÐ
21.
Áæ¡¿ ·¤èçÁ° ç·¤ g(x)5x323x11 ÕãéÂÎ p(x)5x524x31x213x11 ·¤æ »é‡æÙ¹´ÇU ãñÐ
2 3 4
°·¤ ¥ÂçÚU×ðØ â´Øæ ãñÐ
¥Íßæ
x
22.
ÌÍæ y ·ð¤ çÜØð ãÜ ·¤ÚUæð Ñ mx2ny5m21n2 ; x2y52n
a ÌÍæ b ·¤æ
×æÙ ™ææÌ ·¤ÚUæð çÁâ·ð¤ çÜØð ÚñUç¹·¤ â×è·¤ÚU‡æ ·ð¤ ¥Ù‹Ì ãÜ ãæð´Ð
(a1b)x22by55a12b11 ; 3x2y514
23.
¥æ·ë¤çÌ
(1)
×ð´ BL ÌÍæ CM, DABC ·¤è, çÁâ·¤æ
4(BL21CM2)55BC2
1040119 - C1
‘A
9
â×·¤æð‡æ ãñ, ç·¤ ×æçŠØ·¤æ ãñÐ çâh ·¤èçÁ° Ñ
P.T.O.
24.
¥æ·ë¤çÌ
(2) ×ð´ DE??BC ÌÍæ AD : DB55 : 4
æÍ×ͯ‹Ã (DFE)
æÍ×ͯ‹Ã (CFB)
Ìæð ™ææÌ ·¤èçÁ°
¥Íßæ
Îæð â×çmÕæãé ç˜æÖéÁæð´ ·ð¤ àæèáü ·¤æð‡æ â×æÙ ãñ, ÌÍæ ©Ù·ð¤ ÿæð˜æÈ¤Ü ·¤æ ¥ÙéÂæÌ 16 : 25 ãñÐ Ìæð ©Ù·¤è â´»Ì
ÖéÁæ¥æð´ ·¤æ ¥ÙéÂæÌ ™ææÌ ·¤ÚUæðÐ
x
y
cosu1 sinu51
a
b
ÌÍæ
y
x
sinu2 cosu51,
b
a
25.
ØçÎ
çâh ·¤èçÁ°
26.
ØçÎ A, B, C ç·¤âè DABC ·ð¤ ¥‹ÌÑ ·¤æð‡æ ãæð´ Ìæð çâh ·¤èçÁ°
y2
2
a2
b2
x2
C
§ AB ·
cosec ¨
sec .
¸
© 2 ¹
2
¥Íßæ
ØçÎ sin(2A1458)5cos(3082A) Ìæð A ·¤æ ×æÙ ™ææÌ ·¤èçÁ°Ð
27.
100 ÂçÚUßæÚUæð´
ãæðÐ
·¤æ ÎñçÙ·¤ ¹¿ü Ùè¿ð çÎØæ »Øæ ãñÐ
¦ã
©ÎÁ<ÆÍÁ<Í›× ž‹Ïɛh¿Í
28.
f1 ÌÍæ f2 ·¤æ
×æÙ ™ææÌ ·¤èçÁ° ØçΠ׊Ø×æÙ ¹¿ü 188 L¤ÂØæ
140 - 160 160 - 180 180 - 200 200 - 220 220 - 240
5
25
f1
5
f2
çِ٠¥æ¡·¤ÇUæð´ ·¤è ×æçŠØ·¤æ ™ææÌ ·¤ÚUæðÐ
Æ¢ãŒr³Á<ÍÃ
ɛ¦¿Ï¼ÍÁ›<¼ÍÁ<³Í
1040119 - C1
É×
ž‹¾
É×
ž‹¾
20
30
0
4
É×
ž‹¾
40
16
É×
ž‹¾
50
30
10
É×
ž‹¾
60
46
É×
ž‹¾
70
66
É×
ž‹¾
80
82
É×
ž‹¾
90
92
É×
ž‹¾
100
100
¹‡ÇU-Î
Âýà٠ⴁØæ 29 âð 34 Ì·¤ ÂýˆØð·¤ ÂýàÙ 4 ¥´·¤æð´ ·¤æ ãñÐ
29.
çâh ·¤èçÁ° sin6u1cos6u5123sin2ucos2u.
30.
2 ¥´·¤æð´
âð ÕÙè ⴁØæ ÌÍæ ©Ù·ð¤ ¥´·¤æð´ ·¤æð ÂÜÅUÙð ·ð¤ ÕæÎ ÕÙè ⴁØæ ·¤æ Øæð» 99 ãñÐ ØçÎ ©Ù·ð¤ ¥´·¤æð´ ·¤æ
¥‹ÌÚU 3 ãæð Ìæð ⴁØæ ™ææÌ ·¤ÚUæðÐ
¥Íßæ
°·¤ Ùæçß·¤ 8 ç·¤.×è. ÂæÙè ·ð¤ ¥Ùé·é¤Ü ÁæÙð ×ð´ 40 ç×ÙÅU Ü»æÌæ ãñ ÌÍæ ÂýçÌ·é¤Ü ¿ÜÙð ×ð´ 1 ƒæ´ÅUæ Ü»æÌæ ãñÐ
Ùæçß·¤ ·¤è »çÌ çSÍÚU ÂæÙè ×ð´ ÌÍæ ÁÜ Âýßæã ·¤è »çÌ Öè ™ææÌ ·¤èçÁ°Ð
31.
Âæ§Íæ»æðÚUâ Âý×ðØ çܹæð, ÌÍæ §â ·¤æð çâh ·¤èçÁ°Ð
32.
çِ٠ÕæÚ´UÕæÚUÌæ Õ´ÅUÙ ·ð¤ çÜØð Ò·¤× Âý·¤æÚU ·¤æÓ ÌÍæ Ò¥çÏ·¤ Âý·¤æÚU ·¤æÓ ÎæðÙæð´ ÌæðÚU‡æ ¹è´¿·¤ÚU ×æçŠØ·¤æ
çÙ·¤æçÜØðÐ
Œ›ž‹
ÎÆSÍδã¿Íכž‹Ïɛh¿Í
30 - 40
40 - 50
50 - 60
60 - 70
70 - 80
80 - 90
90 - 100
14
6
10
20
30
8
12
33.
çِ٠ÚñUç¹·¤ â×è·¤ÚU‡ææð´ ·¤æð »ýæȤ mæÚUæ ãÜ ·¤ÚUæð Ñ
2(x21)5y ÌÍæ x13y515.
ÌÍæ ©â çՋÎé ·ð¤ çÙÎðüàææ´·¤ ™ææÌ ·¤ÚUæð Áæð ÚðU¹æØð´ y-¥ÿæ ÂÚU ç×ÜÌè ãñÐ
34.
DABC ×𴠑 B5908, AB53 âð.×è.
(i)
sin C
(ii)
ÌÍæ BC54 âð.×è. ãñÐ ×æÙ ™ææÌ ·¤èçÁ°Ð
cos C
(iii)
sec A
(iv)
cosec A
¥Íßæ
UDABC â×·¤æð‡æ ç˜æÖéÁ ×ð´
‘ B â×·¤æð‡æ
ãñ ÌÍæ ØçÎ tanA5
1
3
Ìæð sinA cosC1cosA sinC ·¤æ ×æÙ ™ææÌ ·¤ÚUæðÐ
-o0o-
1040119 - C1
11
P.T.O.