041/IX/SA2/06/B1
Class - IX
MATHEMATICS
Time : 3 to 3½ hours
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, 4 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 calculators is not permitted.
6.
An additional 15 minutes time has been allotted to read this question paper only.
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1.
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4.
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1
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SECTION - A
1.
Question numbers 1 to 10 carry 1 mark each.
The number of cubes of 3 cm side which can be cut out of a cuboid of dimensions
l518 cm, b515 cm and h52 cm is.
(A) 180
(B) 60
(C) 20
(D) Can’t be cut
2.
The linear equation 2y2350, represented as ax1by1c50, has.
(A) a unique solution
(B) infinitely many solutions
(C) two solutions
(D) no solution
3.
If P (E) denotes the probability of an event E, then :
(A) P(E) < 0
(B) P(E) > 1
(C) 0[P(E)[1
(D) 21[P(E)[1
4.
The total surface area of a cone whose radius is 2 r and slant height
(A)
5.
2pr (l1r)
(B)
2pr (
l
1r)
2
(C)
2pr (l14r)
l
is :
2
(D)
pr (l14r)
The graph of the equation ax1by1c50 may be of the form :
(A)
(B)
(C)
(D)
6.
Given three collinear points, then the number of circles which can be drawn through
three points is :
(A) Zero
(B) One
(C) Two
(D) Infinite
7.
In the given figure, ABCD is a parallelogram. If ∠ B 5 1008 , then (∠ A 1 ∠ C) is
equal to :
(A)
3608
041/IX/SA2/06/B1
(B)
2008
(C)
2
1808
(D)
1608
8.
Any point on the line x1y50 is of the form :
(A)
9.
(B)
(a, a)
(C)
(0, a)
(D)
(a, 0)
In a class, there are x girls and y boys, a student is selected at random, then the probability
of selecting a boy is :
(A)
10.
(2a, a)
x/y
(B)
x/(x1y)
(C)
y/(x1y)
(D)
y/x
A right circular cylinder just encloses a sphere of radius a cm, then the curved surface
area of the cylinder is :
(A)
4 p a2 cm
(B)
2 p a2 cm2
(C)
3 p a2 cm
(D)
4 p a2 cm2
SECTION - B
Question number 11 to 18 carry 2 marks each.
11.
A die is thrown 500 times, the frequency of outcomes 1, 2, 3, 4, 5 and 6 are noted in the
following frequency distribution table :
Outcome
Frequency
1
90
2
70
3
75
4
95
5
88
6
82
Find the probability of occurrence of a prime number.
12.
13.
Give one example of a situation in which
(i)
the mean is an appropriate measure of central tendency
(ii)
the mean is not an appropriate measure of central tendency but the median is an
appropriate measure of central tendency
Prove that equal chords of a circle subtend equal angles at the centre.
OR
Suppose you are given a circle. Give a construction to find its centre.
14.
Find a value of k so that x521 and y521 is a solution of the linear equation
9kx112ky563
15.
Give the equation of one line passing through (2, 14). How many more such lines are
there and why ?
041/IX/SA2/06/B1
3
16.
P and Q are any two points lying on the sides DC and AD respectively of a parallelogram
ABCD. Show that ar (APB)5ar (BQC).
17.
In the given figure, PQRS is a parallelogram and line segments PA and RB bisect the
angles P and R respectively. Show that PA??RB.
18.
Savitri had to make a model of a cylindrical kaleidoscope for her science project. She
wanted to use chart paper to make the curved surface of the kaleidoscope. What
would be the area of chart paper required by her, if she wanted to make a kaleidoscope
of length 25 cm with a 3.5 cm radius ?
SECTION - C
Question number 19 to 28 carry 3 marks each.
19.
If the diagonals of a parallelogram are equal, then show that it is a rectangle.
20.
The following frequency distribution table gives the weights of 38 students of a class
Weight in kg Number of students
30-35
10
35-40
5
40-45
15
45-50
5
50-55
1
55-60
2
Total
38
Find the probability that the weight of a student is
(i)
more than or equal to 45 kg
(ii)
less than 30 kg
(iii)
more than or equal to 30 kg but less than 60 kg
OR
041/IX/SA2/06/B1
4
The following table shows the performance of two sections of students in a mathematics
test of 100 marks :
Marks Number of students
0-20
7
20-30
10
30-40
10
40-50
20
50-60
20
60-70
15
70 above
8
Total
90
Find
(i)
(ii)
(iii)
the probability that a student obtained
less than 20% in the mathematics test.
marks 60 or above
marks more than or equal to 40 but less than 60
21.
In the figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that
OB5OD. If AB5CD then show that
ar (DDOC) 5 ar (DAOB)
22.
In the figure, AP ?? BQ ?? CR. Prove that ar (AQC) 5 ar (PBR)
OR
In the figure ar (DRC) 5 ar (DPC) and ar (BDP) 5 ar (ARC). Show that both the
quadrilaterals ABCD and DCPR are trapeziums.
041/IX/SA2/06/B1
5
23.
Find the mean for the weekly pocket money (in rupees) using the following data.
Pocket money (in Rs.)
Frequency
55
8
50
3
49
10
81
7
48
3
57
7
65
2
24.
Express the linear equation 253x in the form ax1by1c50 and indicate the values of
a, b and c. Also give the geometrical representation of above equation in two variables.
25.
ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C . Show that
(i)
ABCD is a square.
(ii)
Diagonal BD bisects ∠ B as well as ∠ D
26.
In the given figure, AB is a diameter of the circle ; CD is a chord equal to the radius of
the circle. AC and BD when extended intersect at a point E. Prove that ∠ AEB 5 608 .
27.
A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12
cm. Find the volume of the solid so obtained.
28.
The diameter of a metallic ball is 21 cm. What is the mass of the ball, if the density of
the metal is 5 gm per cm3 ?
OR
The pillars of a temple are cylindrically shaped. If each pillar has a circular base of
radius 20 cm and height 10 m, how much concrete mixture would be required to build
14 such pillars ?
SECTION - D
Question number 29 to 34 carry 4 marks each.
29.
The taxi fare in a city is as follows : For the first kilometer, the fare is Rs. 8 and for the
subsequent distance it is Rs. 5 per km. Taking the distance covered as x km and total
fare as Rs. y, write a linear equation for this information, and draw its graph.
OR
If the work done by a body on application of a constant force is directly proportional to
the distance travelled by the body, express this in the form of an equation in two variables
and draw the graph of the same by taking the constant force as 5 units. Also read from
the graph the work done when the distance travelled by the body is 2 units.
041/IX/SA2/06/B1
6
30.
If two intersecting chords of a circle make equal angles with the diameter passing
through their point of intersection, prove that the chords are equal.
31.
A hemispherical dome of a building needs to be painted. If the circumference of the
base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is
Rs. 5 per 100 cm2.
32.
Prove that the parallelograms on the same base and between the same parallels are
equal in area.
33.
Construct a triangle with perimeter 10 cm and base angles 608 and 458.
OR
Construct a triangle ABC in which BC57.5 cm, ∠ B 5 458 and AC2AB52.5 cm.
34.
The following table presents the number of literate females in a town :
Age group
Number of females
10-15
300
15-20
980
20-25
800
25-30
580
Draw a frequency polygon for the above data.
-o0o-
041/IX/SA2/06/B1
7
30-35
290
35-40
50
¹‡ÇU - ¥
1.
Âýà٠ⴁØæ 1 âð 10 Ì·¤ ÂýˆØð·¤ ÂýàÙ 1 ¥´·¤ ·¤æ ãñÐ
3
âð×è ÖéÁæ ßæÜð ƒæÙæð´ ·¤è ⴁØæ çÁÙ·¤æð °·¤ ƒæÙæÖ çÁâ·¤è çß×æ°´
âð×è ãñ´, ×ð´ âð ·¤æÅUæ Áæ â·¤Ìæ ãñÐ
(A)
(C)
2.
3.
ÚñUç¹·¤ â×è·¤ÚU‡æ
6.
7.
çÁâð
ax1by1c50
60
·¤æÅUæ ÁæÙæ â´Öß Ùãè´ ãñ
·ð¤ M¤Â ×ð´ ÃØ€Ì ç·¤Øæ »Øæ ãñ, ·ð¤ çÜ° Ñ
¥çmÌèØ ãÜ ãñ´
(B)
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(C)
Îæð ãÜ ãñ´
(D)
·¤æð§ü ãÜ Ùãè´ ãñ
ØçÎ ç·¤âè ƒæÅUÙæ
E
P (E) ãñ, ÌÕ
(B) P(E) > 1
·¤è ÂýæçØ·¤Ìæ
P(E) < 0
°·¤ àæ´·é¤, çÁâ·¤è ç˜æ’Øæ 2 r ß çÌØü·¤ ª¡¤¿æ§ü
(A)
5.
2y2350,
(B)
(D)
âð×è, b515 âð×è ¥æñÚU h52
(A)
(A)
4.
180
20
l518
2pr (l1r)
(B)
2pr (
Ñ
(C)
0[P(E)[1 (D)
21[P(E)[1
l
ãñ, ·¤æ âÂê‡æü ÂëDèØ ÿæð˜æÈ¤Ü ãñ Ñ
2
l
1r)
2
(C)
2pr (l14r) (D)
pr (l14r)
â×è·¤ÚU‡æ ax1by1c50 ·¤æ ¥æÜð¹ çِ٠Âý·¤æÚU ·¤æ ãæð»æ Ñ
(A)
(B)
(C)
(D)
ÌèÙ â´ÚðU¹èØ çՋÎé çΰ »° ãñ´, §Ù ÌèÙ çՋÎé¥æð´ âð »éÁÚUÙð ßæÜð ßëžææð´ ·¤è ⴁØæ ãñ Ñ
(A) àæê‹Ø
(B) °·¤
(C) Îæð
(D)
¥Ù´Ì
çΰ »° 翘æ ×ð´, ABCD °·¤ â×æ´ÌÚU ¿ÌéÖüéÁ ãñÐ ØçÎ ∠ B 5 1008 ãæð Ìæð (∠ A 1 ∠ C) ·¤æ ×æÙ ãñ Ñ
(A)
3608
041/IX/SA2/06/B1
(B)
2008
(C)
8
1808
(D)
1608
8.
ÚðU¹æ
x1y50
(A)
9.
(2a, a)
(B)
(a, a)
(C)
(0, a)
(D)
(a, 0)
°·¤ ·¤ÿææ ×ð´ ÜǸ緤Øæð´ ·¤è ⴁØæ x ¥æñÚU ÜǸ·¤æð´ ·¤è ⴁØæ y ãñÐ §Ù×ð´ âð °·¤ çßlæÍèü ·¤æ ØÎë‘ÀUØæ ¿ØÙ
ç·¤Øæ »ØæÐ ¿éÙð »° çßlæÍèü ·ð¤ ÜǸ·¤æ ãæðÙð ·¤è ÂýæçØ·¤Ìæ ãñ Ñ
(A)
10.
ÂÚU çSÍÌ ·¤æð§ü çՋÎé çِ٠M¤Â ·¤æ ãæð»æ Ñ
y/x
(B)
x/(x1y)
(C)
y/(x1y)
(D)
x/y
°·¤ ÜÕ ßëžæèØ ÕðÜÙ, ç˜æ’Øæ a âð×è ßæÜð °·¤ »æðÜð ·¤æð Âê‡æüÌØæ ƒæðÚðU ãé° ãñ, ÌÕ ÕðÜÙ ·¤æ ß·ý¤ ÂëDèØ ÿæð˜æȤÜ
ãñ Ñ
4 p a2
(A)
âð×è
2 p a2
(B)
âð×è2
(C)
3 p a2
âð×è
(D)
4 p a2
âð×è2
¹‡ÇU - Õ
Âýà٠ⴁØæ 11 âð 18 Ì·¤ ÂýˆØð·¤ ÂýàÙ 2 ¥´·¤ ·¤æ ãñÐ
11.
°·¤ Âæâð ·¤æð 500 ÕæÚU ©ÀUæÜæ ÁæÌæ ãñ ¥æñÚU çِ٠ÕæÚ´UÕæÚUÌæ Õ´ÅUÙ âæÚU‡æè ×ð´ ÂçÚU‡ææ×æð´ 1, 2, 3, 4, 5 ¥æñÚU 6 ·¤è
ÕæÚ´UÕæÚUÌæ¥æð´ ·¤æð çÜ¹æ »Øæ Ñ
¬Á⁄UáÊÊ◊
’Ê⁄Uê’Ê⁄UÃÊ
1
2
3
4
5
6
90
70
75
95
88
82
°·¤ ¥Öæ’Ø ⴁØæ Âýæ# ãæðÙð ·¤è ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ°Ð
12.
13.
çِ٠çSÍçÌ ÂÚU ¥æÏæçÚUÌ °·¤ ©ÎæãÚU‡æ ÎèçÁ° Ñ
(i)
×æŠØ ãè ·ð¤‹ÎýèØ Âýßëçžæ ·¤æ ©ÂØé€Ì ×æ ãñÐ
(ii)
×æŠØ ·ð¤‹ÎýèØ Âýßëçžæ ·¤æ ©ÂØé€Ì ×æ Ùãè´ ãñ, ÁÕç·¤ ×æŠØ·¤ °·¤ ©ÂØé€Ì ×æ ãñÐ
çâh ·¤èçÁ° ç·¤ °·¤ ßëžæ ·¤è ÕÚUæÕÚU Áèßæ°´ ßëžæ ·ð¤ ·ð¤‹Îý ÂÚU ÕÚUæÕÚU ·¤æð‡æ ¥´ÌçÚUÌ ·¤ÚUÌè ãñ´Ð
¥Íßæ
×æÙ ÜèçÁ° ¥æ·¤æð °·¤ ßëžæ çÎØæ ãñÐ §â·ð¤ ·ð¤‹Îý ·¤æð ™ææÌ ·¤ÚUÙð ·ð¤ çÜ° °·¤ ÚU¿Ùæ ÎèçÁ°Ð
14.
k ·¤æ
×æÙ ™ææÌ ·¤èçÁ° çÁâ·ð¤ çÜ° x521 ¥æñÚU y521 çِÙ
ÚñUç¹·¤ â×è·¤ÚU‡æ ·¤æ °·¤ ãÜ ãñ Ñ
9kx112ky563
15.
çÕ´Îé
(2, 14)
âð »éÁÚUÌè ãé§ü °·¤ ÚðU¹æ ·¤æ â×è·¤ÚU‡æ çÜç¹°Ð
¥æñÚU €Øæð´?
041/IX/SA2/06/B1
9
§â Âý·¤æÚU ·¤è ¥æñÚU ç·¤ÌÙè ÚðU¹æ°´ ãæð â·¤Ìè ãñ´,
16.
P ¥æñÚU Q ·ý¤×àæÑ
Îàææü§° ç·¤ ÿæð.
â×æ´ÌÚU ¿ÌéÖüéÁ
ABCD
·¤è ÖéÁæ¥æð´
DC ¥æñÚU AD ÂÚU
çSÍÌ çՋÎé ãñ´ (¥æ·ë¤çÌ Îðç¹°)Ð
(APB)5ÿæð. (BQC)
17.
Îè »§ü ¥æ·ë¤çÌ ×ð´, PQRS °·¤ â×æ´ÌÚU ¿ÌéÖüéÁ ãñ ¥æñÚU ÚðU¹æ ¹‡ÇU PA ¥æñÚU RB ·ý¤×àæÑ ·¤æð‡ææð´ P ¥æñÚU R ·¤æð
â×çmÖæçÁÌ ·¤ÚUÌð ãñ´Ð Îàææü§° ç·¤ Ñ PA??RB
18.
âæçߘæè ·¤æð ¥ÂÙð çߙææÙ ·ð¤ ÂýæðÁð€ÅU ·ð¤ çÜ° °·¤ ÕðÜÙæ·¤æÚU ·ð¤çÜÇUæðS·¤æð ·¤æ ×æòÇUÜ ÕÙæÙæ ÍæÐ ßã §â
·ð¤çÜÇUæðS·¤æð ·¤è ß·ý¤ ÂëD ÕÙæÙð ·ð¤ çÜ° ¿æÅüU ·¤æ»Á ·¤æ ÂýØæð» ·¤ÚUÙæ ¿æãÌè ÍèÐ ØçÎ ßã 25 âð×è ܐÕæ§ü
¥æñÚU 3.5 âð×è ç˜æ’Øæ ·¤æ ·ð¤çÜÇUæðS·¤æð ÕÙæÙæ ¿æãÌè ãñ, Ìæð ©âð ¿æÅüU ·¤æ»Á ·ð¤ ç·¤ÌÙð ÿæð˜æÈ¤Ü ·¤è ¥æßàØ·¤Ìæ
ãæð»è?
¹‡ÇU - â
Âýà٠ⴁØæ 19 âð 28 Ì·¤ ÂýˆØð·¤ ÂýàÙ 3 ¥´·¤ ·¤æ ãñÐ
19.
ØçÎ °·¤ â×æ´ÌÚU ¿ÌéÖüéÁ ·ð¤ çß·¤‡æü ÕÚUæÕÚU ãæð´, Ìæð Îàææü§° ç·¤ ßã °·¤ ¥æØÌ ãñÐ
20.
çِ٠ÕæÚ´UÕæÚUÌæ Õ´ÅUÙ âæÚU‡æè °·¤ ·¤ÿææ ·ð¤ 38 çßlæçÍüØæð´ ·¤æ ÖæÚU ÎàææüÌè ãñ Ñ
÷Ê⁄U (Á∑§.ª˝Ê. ◊¥)
ÁfllÊÁÕ¸ÿÊ¥ ∑§Ë ‚¥ÅÿÊ
30-35
35-40
40-45
45-50
50-55
55-60
10
5
15
5
1
2
∑È ‹
38
°·¤ çßlæÍèü ·¤æ ÖæÚU çِ٠ãæðÙð ·¤è ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° Ñ
(i)
45 ç·¤Üæð»ýæ× âð ¥çÏ·¤ Øæ ©â·ð¤ ÕÚUæÕÚU
(ii) 30 ç·¤Üæð»ýæ× âð ·¤×
(iii) 30 ç·¤Üæð»ýæ× âð ¥çÏ·¤ Øæ ©â·ð¤ ÕÚUæÕÚU Üðç·¤Ù 60 ç·¤Üæð»ýæ× âð ·¤×
¥Íßæ
041/IX/SA2/06/B1
10
çِ٠âæÚU‡æè
100 ¥´·¤
ßæÜð »ç‡æÌ ·ð¤ °·¤ ÅðUSÅU ×ð´ Îæð çßÖæ»æð´ ·ð¤ çßlæçÍüØæð´ ·ð¤ ÂýÎàæüÙ ·¤æð ÎàææüÌè ãñ Ñ
•¥∑§
ÁfllÊÁÕ¸ÿÊ¥ ∑§Ë ‚¥ÅÿÊ
0-20
20-30
30-40
40-50
50-60
60-70
7
10
10
20
20
15
70 •ÊÒ⁄U
•Áœ∑§
8
∑ȧ‹
90
ÂýæçØ·¤Ìæ ™ææÌ ·¤èçÁ° ç·¤ °·¤ çßlæÍèü »ç‡æÌ ÅðUSÅU ×ð´ Ñ
21.
(i)
20% âð
(ii)
60 Øæ
©ââð ¥çÏ·¤ ¥´·¤ Âýæ# ·¤ÚUÌæ ãñÐ
(iii)
40 Øæ
©ââð ¥çÏ·¤ Üðç·¤Ù
·¤× Âýæ# ·¤ÚUÌæ ãñÐ
60
âð ·¤× ¥´·¤ Âýæ# ·¤ÚUÌæ ãñÐ
ABCD ·ð¤ çß·¤‡æü AC ¥æñÚU BD ÂÚUSÂÚU
OB5OD ãñÐ ØçÎ AB5CD ãñ, Ìæð Îàææü§° ç·¤ Ñ
ar (DDOC) 5 ar (DAOB)
Îè »§ü ¥æ·ë¤çÌ ×ð´, ¿ÌéÖüéÁ
çÕ´Îé
O
ÂÚU §â Âý·¤æÚU Âýç̑ÀðUÎ ·¤ÚUÌð ãñ´
ç·¤
22.
Îè »§ü ¥æ·ë¤çÌ ×ð´, AP ?? BQ ?? CR
ãñÐ
çâh ·¤èçÁ° ç·¤
ar (AQC) 5 ar (PBR)
¥Íßæ
Îè »§ü ¥æ·ë¤çÌ ×ð´, ar (DRC) 5 ar (DPC) ¥æñÚU ar (BDP) 5 ar (ARC)
ABCD ¥æñÚU DCPR â×Ü´Õ ãñ´Ð
041/IX/SA2/06/B1
11
ãñÐ
Îàææü§° ç·¤ ÎæðÙæð´ ¿ÌéÖüéÁ
23.
çِ٠¥æ´·¤Ç¸æð´ ·ð¤ çÜ° ×æŠØ âæ#æçã·¤ ÁðÕ ¹¿ü (L¤ÂØæð´ ×ð´) ™ææÌ ·¤èçÁ° Ñ
¡’ πø¸ (UL§¬ÿÊ¥ ◊¥)
’Ê⁄Uê’Ê⁄UÃÊ
55
50
49
81
48
57
65
8
3
10
7
3
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