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GEOMETRY
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Time: 2 ½ Hours
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OCTOBER - 1999
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Marks: 75
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| Q.1. |
Solve any Five sub-questions: |
10
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i) |
In
D ABC, a line parallel to side BC
cuts the sides AB and AC in points X and Y respectively such
that AX = 12, XB = 8, AY = 9. Find YC. |
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ii) |
Find
side of a square whose diagonal is16 cm. |
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iii) |
A
circle of radius 6 cm has two tangents AB and CD parallel
to each other. What is the distance between these tangents?
Why? |
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iv) |
Determine EA from the information given in the
following figure: |
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v) |
Draw
seg AB of length 9.7 cm. Take a point P on it such that A-P
B, AP = 3.5 cm. Construct line MN perpendicular to AB through
P. |
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vi) |
The
diameter of a circle is 14 cm. Find the area of its semi-circle. |
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vii) |
Find
the volume of a right-circular cylinder whose radius is 5
cm and whose height is 40 cm. (p = 3.14) |
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| Q.2.
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Solve any Five: |
15
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i) |
Using information given in figure below, answer
the following questions: |
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1) |
What
is the ratio BC : QR? |
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2) |
What
is the value of the ratio of heights AL and PS? |
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3) |
What
is the ratio of areas of D ABC and
D PQR? |
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ii) |
As
shown in the figure adjacent, P is the centre and A and B
are end-points of a diameter of a circle. C is a point of
the circle such that m D ABC = 35°.
Determine m Ð BAC, m Ð
PCB and m Ð PCA. |
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iii) |
In
the adjacent figure, the tangent drawn to a circle at point
T intersects the secant AB in point P.
Prove that D PTB ~ D PAT.
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iv) |
Draw
Ð ABC = 115°. Take point P on ray
BA such that BP = 5.4 cm. Draw seg PQ ^ line BC through point P. Measure
length PQ. |
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v) |
The
measure of arc of a circle, whose radius is 6 cm, is 60°.
Find area of minor sector.
(p = 3.14) |
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vi) |
The
radius (r) of a cone and its perpendicular height (h) are
3 cm and 4 cm respectively. Find the values of :
(a) Slant height (l) (b) Area of the
base (c) Curved surface
area. (p = 3.14) |
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vii) |
If
in D ABC, Ð B = 135°, AB = 15 and BC = 12, then
find the area of D ABC. |
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| Q.3.
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A) |
Solve any Two: |
6
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i) |
In
the figure adjacent, oBCDE is a parallelogram.
Show that : A (oBCDE) = 24 (D ABC). |
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ii) |
If
the angles of a triangle are 30°, 60° and 90° and the side
opposite 30 is half the hypotenuse, then prove that side opposite
60° is Ö3/2 times
the hypotenuse. |
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ii) |
Prove that Opposite angles of a cyclic quadrilateral
are supplementary. |
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B) |
Solve any One: |
4
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i) |
A
tangent at any point of a circle is perpendicular to the radius
at that point. Prove. |
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ii) |
In
the figure below, OP = 3.5 cm, QB = 1.4 cm and Ð AOB = 120°. Find area of shaded portion. |
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| Q.4. |
A) |
Solve any Two sub-questions: |
6
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i)
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Draw
D ABC with AB = 5 cm, BC = 6.4 cm
and AC = 7.8 cm. What is the radius of the circumcircle of
D ABC? Draw the circumcircle. |
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ii) |
Find
the volume of a sphere of diameter 4.2 cm. (p =  ) |
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iii) |
In
the adjacent figure, PM and PN are tangent segments. If PM
= 7 cm, find PN. Q is the center of the circle and if the
radius of the circle is also 7 cm, determine distance QP. |
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B) |
Solve any One: |
4
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i) |
If
a line divides any two sides of a triangle in the same ratio,
then the line is parallel to the third side. Prove. |
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ii) |
In
the following figure, a diameter AB and a chord CD intersects
each other at right angles in point P as shown. Prove : CP2
= AP X BP. |
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| Q.5. |
Solve any Three: |
15
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i) |
In
the adjacent figure, seg BD ^ side
AC, seg DE ^ side BC, then show that
DE x BD = DC x BE. If DE = 4, BD = 5, find BE and DC. |
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ii) |
P
is the centre of a circle. Three tangents AB, BC and AC of
this circle determine D ABC right
angled at B. If AB = 6, BC = 8, then determine the diameter
of the circle. |
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iii) |
In
the following figure, line AP is the tangent to circle at
A. Secant through P intersects chord AY in point X, such that
AP = PX = XY. If PQ = 1, QZ = 8, find AX. |
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iv) |
Two
congruent circles intersects each other in A and B. A transversal
through B intersects the two circles in C and D in such a
way that BC < BD. Show that AC = AD. |
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| Q.6.
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Solve any Three: |
15
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i) |
A
circle of radius 10 cm is passing through the vertices of
a regular hexagon. Find the area of shaded region. (p
= 3.14, Ö3 = 1.73) |
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ii) |
Circumference of base of a cone is 22 cm and its
height is equal to the diameter of its base. Find the volume
and total surface area of the cone. |
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iii) |
In
D ABC, Ð B = 90°. With AB as the diameter,
a semi-circle is drawn. This semi-circle cuts side AC in point
P. Prove that tangent to the semi-circle at point P bisects
side BC. |
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iv) |
In
D ABC, m Ð BAC = 60°, BC = 8 cm and AB2
+ AC2 = 82. Draw D ABC. D is the mid-point of BC. |
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In the adjoining figure, NQ is the bisector
of Ð MNP. If MN = 25, NP = 40, MQ = 12.5, then find the
length of seg PQ. 
In the adjoining figure, A is the point of contact.
If AB = 6 and KB = 4, determine BW. 
Observe the alongside figure and prove that
D ADC ~ D
BEC (Write only proof). 
In the alongside figure, P is the centre of
a circle, point A,B,C lie on the circle. P is between A
and B. BC = 10 and AC = 24. Determine the radius.
P is the centre of the circle. P is joined to
four points Q,R,S,T of the same circle as shown in the alongside
figure. Measures of some of the angles with vertex P are
indicated in the figure. Using them determine: 
As shown in the alongside figure, ray BE is
the bisector of Ð ABC. Line BC is tangent and BA is a secant
to the circle. m Ð ABC = 80°, then find out:
In the alongside figure, three semicircles of
diameters AB = 8 cm, BC = 4 cm and CD = 2 are drawn. Find
the following:
Given: 