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GEOMETRY
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Time: 2 ½ Hours
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MARCH – 2000
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Marks: 75
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| Q.1. |
Solve any Five from the following: |
10 |
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i) |
The
radius of a right circular cylinder is 5 cm and height is
7 cm. Find the curved surface area of cylinder. |
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ii) |
If
the lengths of sides of D PQR are
p = 8, q = 15, r = 17, state whether given triangle is right-angled
triangle or not. If yes, which is the right angle? |
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iii) |
 In the following figure,  ABCD
is a trapezium, side
AB || side DC. Diagonals AC and BD cut each other at
O.
If AB = 6, DC = 20, OD = 15, find OB. |
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iv) |
A
circle has radius = 20 cm. Find length of the greatest chord
in the circle. |
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v) |
 In the following figure, m Ð RQS = 40° . Find m Ð QPR. |
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vi) |
Construct equilateral D
ABC of length of sides equal to 4.5 cm. Draw a perpendicular
AM on side BC at point M. (Do not write construction.) |
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vii) |
Find
volume of sphere whose radius is 14 cm. |
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| Q.2. |
Solve any Five: |
15 |
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i) |
The
radius of a circle is 3.5 cm and the area of sector is 3.85
cm2. Find the length of corresponding arc and measure
of that arc. |
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ii) |
 In the following figure, seg AM is the medium
of D ABC. l (BC) =
16 cm and AB2 + AC2 = 200 cm. Using
Apollonius’ theorem, find the length of seg AM. |
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iii) |
The
radii of 2 circles are 6 cm and
8 cm. Find the radius of a circle whose area is equal to sum
of areas of the two circles. |
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iv) |
In
the following figure, seg BE ^ side
AC, seg CF ^ side AB, A-E-C and A-F-B.
Prove: D ABE
~ D ACF. |
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v) |
Construct D PRS with PR = 5.4 cm, RS = 4.5 cm,
PS = 6.7 cm. Construct circumcircle of D PRS. (Do not write
construction) |
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vi) |
 The length of side of a square is 28 cm. Find
the area of portion between the square and its incircle. |
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vii) |
Find
the volume and curved surface area of a cone whose base radius
is 12 cm and perpendicular height is 16 cm. (p
= 3.14) |
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| Q. 3. |
A) |
Solve any Two: 6 |
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6 |
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i) |
 Three congruent circles with centres A, B and
C and radius 5 cm touch each other in points D, E, F as shown
in the following figure. Find the perimeter of D
ABC |
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ii) |
The
circumference of a right circular cylinder is 88 cm and its
height is 15 cm. Find the volume of the cylinder. |
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ii) |
Construct a circle with point O as centre and radius
4.5 cm. Take a chord AB of length 6 cm. Construct the tangents
to the circle at A and B and let them intersect in point P.
(Do not write construction.) |
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B) |
Solve any One: 4 |
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4 |
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i) |
 In the following figure, D ABC is an equilateral triangle of
side 10 cm. Seg BC is the diameter of a semicircle. Find the
area of the shaded region.
(p = 3.14 Ö3
= 1.73) |
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ii) |
Prove: "Angles inscribed in the same arc are congruent."
(Figure is essential). |
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| Q.4. |
A) |
Solve any Two sub-questions:
6 |
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6 |
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i) |
Areas of two similar triangles are 144 cm2
and 81 cm2. If one side of the first triangle is
6 cm, find the corresponding side of the second triangle.
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ii) |
Prove: If the angle of a triangle are 300,
600 and 900, then the side opposite
to 300 angle is half of the hypotenuse. (Figure
is essential.) |
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iii) |
In
a circle of radius 24 cm, two parallel chords are drawn on
the same side of the center and whose lengths are 14 cm and
48 cm respectively. Find the distance between them.
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B) |
Solve any One: 4 |
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4 |
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i) |
Prove: Line joining center of a circle and mid-point
of a chord is perpendicular to the chord. (Figure is essential.) |
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ii) |
 In the following figure, ![]() ABCD is a parallelogram. E is any point
on side AD. Prove:
1. D AEF
~ D DEC;
2. D AEF ~ D BCF.
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| Q. 5. |
Solve any Three sub-questions; |
15 |
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i) |
 In the following figure, seg BD ^ side AC, seg DE ^ side BC.
Show that: DE x BD = DC x BE.
If DE = 4, BD = 5, find BE and DC. |
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ii) |
In
ABC, m Ð B = 90°, AB = 6, BC = 8.
What is the length of the median drawn from point B on side
AC. (Use Apollonius theorem.) |
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iii) |
 In the following figure, P is the center of
a circle. E is a point on a diameter. Two chords AB and CD
intersect the diameter in point E such that Ð AEP @ Ð
DEP.
Prove : AB = CD. |
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iv) |
In
the following figure, .gif) ABCD is a cyclic trapezium.
seg AB || seg DC. m Ð
ABC = 80°.
Find the values of the following: |
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a) m (arc ADC); |
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b) m (arc ABC); |
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c) m Ð BCD; |
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d) m (arc DCB); |
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e) Prove : m (arc BXC) = m (arc AYD). |
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| Q. 6. |
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Solve any Three: |
15 |
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i) |
 In the following figure, m
Ð ABC = 90°. Sides of triangle touch circle with centre
O. If the perpendicular sides of triangle are 4x and 2x –
5 and the hypotenuse is 25 units, find the area of the shaded
portion. |
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ii) |
If
the radius of circular is increased by 30% keeping its height
constant, find the percentage increase in its volume.
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iii) |
Draw
Ð ABC = 60°. Draw bisector of Ð
ABC and take a point Q on it such that d (B, Q) = 8 cm. Draw
a circle which touches ray BA and ray BC from centre Q. Measure
radius of circle and length of tangent segment drawn from
B. (Do not write construction.) |
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iv) |
 Given: Chord BD @ Chord CEm Ð CBA
= 50° Find m Ð BCA and m Ð BDE and prove that:seg BC || seg
DE. |
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