# Question Bank - Geometry - Hemant Malhotra

• Q67) Find the minimum number of lines required to divide a plane in 79 distinct parts.

• One line divide plane in 2 parts
Now 2nd line in 4 parts
So new parts will be 4 - 2 = 2
Now 3 lines will divide in 6 parts so 6 - 3 = 3 increased parts
Same in 2N points = N parts

So 2 + 2 + 3 .. N
1 + 1 + 2 ... N
1 + (N) * (N + 1)/2

• R = abc / 4A
let side =x,2x/3 and 4x/9 so S=(a+b+c)/2=19x/18
S = 19x/18
A = (x/18)^2 * sqrt(7 * 209)
R = (8a^3/27) /(4 * (x/18)^2 * sqrt(7 * 209)]
R = 24x / sqrt (7 * 209)
now compare
R = 6sqrt(7/209) t
24x / sqrt (7*209) = 6sqrt(7/209)
24x = 42
x =7/4

• Q68) In the figure given below, ABCD is a rectangle and GB = DH = 2HC. What is the area (in sq cm)of the shaded region if AD = 1 cm and AB = 3 cm? • Q69) A triangle has vertex A at (0, 3), vertex B at (4, 0), and a vertex C at (x, 5) for some x where 0 < x < 4. If the area of the triangle is 8, what is the value of x?

• Q70) • Q71) • Q72) A cube of side 3 cm is cut into smaller cubes of side 1 cm. What is the ratio of Total surface area of the
larger cube and the sum of the total surfaces of all the smaller cubes?

• Q73) Which of the following points (x, y) in the co-ordinate plane does not lie on the line 31x + 13y - 75 = 0?
a) (2, 0)
b) (-24, 63)
c) (-11, 32)
d) (-37, 95)

• Q74) What is the number of distinct triangles possible with integer sides and having a perimeter = 11 ?

• Q74) What is the maximum distance of any point on the curve x^2 + y^2 - 6x + 14y - 23 = 0 from the point (10, 17)

• Q75) Find the area enclosed by the inequalities x + y ≥ 2 and x - y ≥ - 10 in the second quadrant

• Q76) Two altitudes of triangle are of 12 and 20 units respectively. Find the sum of all possible integral lengths of third altitude

• if a,b,c are three altitudes
then (a * c)/(a+c) < b < (ac/(a-c)
so (12 * 20)/(32) < b < 240/(20-12)
(240/32) < b < 240/8
30/4 < b < 30
so 7.5 < b < 30
0

• Q77) A circle is drawn in a Cartesian plane such that there is no point (x, y), where x and y are integers, that lie inside the circle. What is the maximum possible area of such circle

• Q78) The radius of the circle with centre O is under root 50cm. A and C are two points on the circle and B is a point inside the circle. the length of AB is 6cm and BC is 2cm. The angle ABC is a right angle. Find the square of the distance OB.

• Q79) A circle C1, of radius four units, touches another circle C2, of radius nine units, and the line ℓ1 is a direct common tangent to both the circles. Which of the following would be the radius of another circle C3, which touches both circles C1 and C2 and also the line ℓ1?
a) 1.32 units
b) 1.44 units
c) 1.58 units
d) 1.76 units

• Q80) What is the maximum number of chords of length R that can be drawn in a circle of radius R, such that no two chords intersect inside the circle?

123

109

47

207

30

42

53