Geometry and Mensuration: Level 3 Test 6

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Geometry and Mensuration: Level 3 Test 6

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Question 1

The length of the common chord of two circles of radii 15 cm and 20 cm whose centers are 25 cm apart, is (in cm):

A	24
B	25
C	15
D	20

Question 1 Explanation:

$ \begin{array}{l}Let\,\,the\,\,total\,\,length\,\,be\,\,2x\,\,units\\\frac{1}{2}x\times 25=\frac{1}{2}\times 15\times 20\\x=12units\\thus\,\,2x=24\,units\\Thus\,\,the\,\,correct\,\,option\,\,is\,\,a\end{array}$

Question 2

In a triangle ABC, the internal bisector of the angle A meets BC at D. If AB =4, AC= 3 and ∠A= 60^o, then, the length of AD is:

A	$ \displaystyle 2\sqrt{3}$
B	$ \displaystyle \frac{12\sqrt{3}}{7}$
C	$ \displaystyle \frac{15\sqrt{3}}{8}$
D	$ \displaystyle \frac{6\sqrt{3}}{7}$

Question 2 Explanation:

$ \begin{array}{l}let\,\,the\,\,length\,\,of\,\,AD\,\,be\,\,x\,\,units\\\frac{1}{2}\times 4x\times Sin30=\frac{4}{7}\times 4\times 3\times Sin60\\x=\frac{12}{7}\sqrt{3}\\Correct\,\,option\,\,is\,\,b\end{array}$

Question 3

In the figure (not drawn to scale) given below, P is a point on AB such that AP: PB= 4: 3. PQ is parallel to AC and QD is parallel to CP. In ∠ARC, ∠ARC= 90^o, and in ΔPSQ, ∠PSQ= 90^o. The length of QS is 6 cms. What is ratio AP: PD?

A	10: 3
B	2: 1
C	7: 3
D	8: 3

Question 3 Explanation:

$ \displaystyle \begin{array}{l}AP:\text{ }PB\text{ }=\text{ }4:3\\Thus\text{ }AP=\text{ }4x\text{ }and\text{ }PB=3x\\Again,\text{ }PD:DB\text{ }=\text{ }CQ:AB\text{ }=\text{ }4:3\\PD=\frac{4}{7}\times BP=\frac{4}{7}\times \frac{3}{4}AP=\frac{3}{7}AP\end{array}$

Question 4

In the figure (not drawn to scale) given below, if AD = CD = BC, and BCE = 96^o, how much is DBC?

A	32^o
B	84^o
C	64^o
D	Cannot be determined

Question 4 Explanation:

Let angle CAD= DCA = x
and thus CDB = 2x and CBD = 2x.
Thus, 180-2x+x+96= 180
=> 2x= 64
Correct option is (c)

Question 5

In the figure given below (not drawn to scale), A, B and C are three points on a circle with centre O, The chord BA is extended to a points T such that CT becomes a tangent at the circle at point C. If ∠ATC= 30^o and ∠ACT= 50^o, then the angle BOA is: