Question Bank  Geometry  Hemant Malhotra

Q6) In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

Q7) Given 10 distinct points, A_1 through A_5 on line R, and B_1 through B_5 on line S, draw the 25 possible line segments A_i B_j. Excluding the 10 given points themselves, what is the minimum number of distinct points at which these 25 segments can intersect?

Q8) PQ and RS are perpendicular chords of a circle intersecting at T. RT= 2.5, TS=11, PT= 3. Find the radius of the circle (approximately).
a) 4.5
b) 12.1
c) 7.4
d) 15.6

Q9) For a trapezium, S1 denotes the sum of the squares of the sides and S2 denotes the sum of the squares of the diagonals. S1 – S2 = 576. If the longer parallel side is 50 cm, the shorter parallel side is _______.
a) 26 cm
b) 22 cm
c) 18 cm
d) 30 cm

Q10) 2x + 3y + 5 = 0 and 6x – 4y + 9 = 0 are two lines. If a point is at a distance of 1 unit from each of the two given lines, then how many such points exist?
a) 1
b) 3
c) 4
d) 2

Q11) A triangle, whose sides are 20 cm, 48 cm and 52 cm, is to be cut into two pieces of equal area by a single straight cut, which passes through one of the vertices of the triangle. What is the approximate maximum value of the sum of the perimeters of the two pieces

Q12) A goat is tied at the corner of a rectangular shed of dimensions 14 m x 7 m, with a rope 21 m long. If the goat can graze on the ground around and outside the shed as far as it is permitted by the rope, find the area (in sq.m.) on which it can graze.
a) 1232
b) 1560.5
c) 1260
d) 1543.5

Q13) The 2 adjacent sides of a quadrilateral are 6cm and 8cm. What is the maximum area of the quadrilateral (in sq cm) if it is inscribed in a circle of radius 5cm ?

Q14) A hexagon with sides of length 2, 7, 2, 11, 7, 11 is inscribed in a circle. Find the radius of the circle.

Q15) Four cubes of volume 1, 8, 27, and 125 are glued together at their faces. What is the smallest possible surface area of the resulting solid figure?

Q16) If ΔPAB is formed by three tangents to a circle having centre O and ∠APB = 40º, find ∠AOB.

Q17) A green colour cube is cut into smaller and identical cubes with 6 cuts. The cube is cut with a knife which applies red colour to the surfaces of the cubes while cutting. The smaller cubes are further cut with 6 cuts into smaller and identical cubes, with a knife which does not apply any colour to the surface. The questions are based on the cubes obtained after these cuts. How many smaller cubes have three faces coloured red?
a) 48
b) 52
c) 56
d) 64

@hemant_malhotra right angled triangle is of form 81517
area of triangle = (1/2)xhx17 = 1/2x8x15
h = 7

Q18) A, B, C, ..... till K are 11 points, spaced on the circumference of a circle in that order. What is the sum of all the corner angles of the ‘star’ formed by joining the points AEIBFJCGKDHA  in that order?

Q19) In a cyclic quadrilateral ABCD, if AB = 2, BC = 3, CD = 4 and DA = 5, what is the ratio of the lengths of the diagonals?
(1) 7 : 11
(2) 11 : 13
(3) 10 : 11
(4) 13 : 15

@hemant_malhotra
simple ..centroid will be 1,3

@hemant_malhotra decrease by 1 percent

Q20) What is the least possible number of cuts required to cut a cube into 34 identical pieces ?

let number of cuts along length , width and height is a,b,c and total number of cuts=a+b+c
number of cubes =(a+1)(b+1)(c+1)
If product is cnstant sum will be minimum if they are very close to each other
here (a+1)(b+1)(c+1)=1 * 2 * 17
so a+1+b+1+c+1=1+2+17 so a+b+c=17

Q21) A right circular cylindrical vessel having height 20 cm and inner diameter 10 cm is filled with water upto 15 cm of it's height as measured from the base. Find the angle by which the cylinder must be tilted so that the water in the cylinder is just about to fall out?