 GEOMETRY - CAT 2013

Q1:

Triangle ABC is right-angled at A. D is a point on AB such that CD = 1. AE is the altitude from A to BC. If BD = BE = 1, what is the length of AD? (1) 2^1/3 - 1 (2) (5^1/2 - 1)/2 (3) (5^1/2 + 1)/4 (4) 2^1/2 - 1 (5) none of these

Q2: The two adjacent sides of a quadrilateral are 6 cm and 8 cm long. What is the maximum possible area of the quadrilateral (in sq cm) if it is inscribed in a circle of radius 5 cm?

Q3: In a circle AB and CD are two chords intersecting at a point ,P . given
AP=4
PB=6
PC=3
find PD ?

Q4: PQRS is a cyclic quad. PQR is an equilateral triangle. find angle RSP.

Q5:
A 3 cm tall man finds that angle of elevation of the top of a 15 cm high pillar and the angle of depression of its base are complementary angle.distance of the man from the pillar is ?

Q6: A circle of maximum possible size is cut from a square sheet of board. Subsequently, a square of maximum possible size is cut from the resultant circle. What will be area of the final square?

Q7: two line segments PQ and RS intersects at X in sucha way XP=XR. if PSX=RQX, then one must have-
1. PR=QS
2. PS=RQ
3. XSQ=XRP
4. ar(triangle PXR)=ar(triangle QXS)

Q8: In triangle ABC ,the internal bisector of angle a meets BC at D.If AB=4 AC=3 and LA(angle A)=60 degree, then length of AD is what??

Q9 :The lenght of the common chord of two circles of radii 15 and 20cm respectively,whose centeres are 25cm apart is
1)24cm 2)25cm 3) 15cm 4)20cm

Q10: 4 horses are yied at four corners of a square plot of side 14m so that the adjacent horses can just reach one another. There is a small circular pond of area 20m^2 at the centre.find the ungrazed area??
1)22 2)42 3)84 4)168

Q11: In a unit square ABCD, points E and F are marked on the sides AB and BC respectively such that AE = BF. G is the point of intersection of the line segments DE and AF. What is the minimum possible length (in units) of the line segment BG?

(1)1/SQRT(2)

(2)1/2

(3)SQRT - 1/7

(4)1/3

(5)1/6

Q12: Two parallel cords AB and CD are drawn on opposite sides of diameter of a circle.AB is 16cm long and is at a distance of 5cm from CD.If CD is 14cm long, then find the radius of the circle.
A.Sqrt(60)
B.Sqrt(55)
C.Sqrt(85)
D. NOT

Q13: Three Circles each of diameter 4 cm , are kept touching each other . The smallest circle circumscribing these circles is drawn. What it the length of the side of the square.
a. 7.14
b. 7.58
c. 7.86
d. 7.92

Q14: In a square ABCD, point M lies on side AB and point N lies on side CD such that MB = 2 cm and ND = 1 cm. Line RS is drawn parallel to side BC and PQ is drawn parallel to side AB. Point P is on side AD, point Q is on side BC, point R is on side AB and point S is on side CD of the square and point O is the intersection point of lines PQ, RS and MN. If mONS = 40 and mMSC = 60 then what is mOMS?

Q15: A triangle is divided into four partsby two straight lines from two corners. the areas of three parts are 8 , 5 , 10..find the area of fourth part .

Q16: In a triangular base pyramid with base ABC and vertex S, all plane angles with vertex S are 90. The areas of lateral faces SAB,SAC and SBC are 3,4 and 6 respectively. Find the volume of SABC.
a. 4
b. 5
c. 6
d. 12

Q17: In an isosceles triangle ABC AB = AC = 7 cm . AD produced on BC such that D is closer to B , AD= 5 cm . The sum of lengths of BD and CD , if BD and CD (in cms) are distinct integers could beA) 11 cm B) 28 cm C) 14 cm D) 9 cm

Q18: Triangle ABC is right angled at A . DEFG is a square of side 4 cm such that DE lies on the hypotenuse of this triangle . Find the sum of BD and EC if it is known that their lengths in cm are distinct integers greater than 1 .

A)10 B) 8 C) 6 D) 12

Q19: What is the area (in cm2) of an isosceles triangle ABC with sides AB = 5 cm, AC = 5 cm and BC = 8 cm?
1)12

2)12.5

3)24

4)20

Q20: In ABC, D is a point on side BC such that BC = 4 BD and A(ABC) is 100 sq. units. What is the area of ABD in sq. units?
1)20

2)25

3)100

4)12.5