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Together, we learn more! So here we go: Geometry and Mensuration Question Bank – CAT 2017
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1st edition: 17 September 2017 (Q.1 to Q.43)
Geometry and Mensuration Question Bank – CAT 2017
1. A ladder is inclined to a wall making an angle of 30° with it. A man is ascending the ladder at the rate of 2 m/s. How fast is he approaching the wall?
2. A circle of radius 6.5 cm is circumscribed around a right-angled triangle with the sides a, b and c cm where a, b and c are natural numbers. What is the perimeter of the triangle?
3. P (-sqrt3, 1) and Q (s, t) lie on the circumference of the circle with center O. What is the value of s?
4. The coordinates of the vertices of the triangle ΔABC are A(0, 1), B(4, 4) and C(5, 3). From which vertex the length of the median drawn to the opposite side is the minimum?
(1) A (2) B (3) C (4) All medians are of equal length
5. A point on a circle inscribed in a square is 1 and 2 units from the closest side of the square. What is the area of the square?
6. ABCD is a trapezium with AB parallel to CD. M is the midpoint of AD. Angle MCB = 150 degrees, BC = 4, MC = 12. Area of the trapezium ABCD will be?
7. A right circular cone of height h is cut by a plane parallel to its base and at a distance of h/3 from the base, then the volume of the resulting cone and the frustum are in the ratio?
8. ABC is an equilateral triangle with side length = 1 cm. P is any point on the circumcircle of this triangle. What is the value of AP^2 + BP^2 + CP^2 ?
9. The radii of the ends of a bucket 45 cm high, which is of the form of a frustum of a cone, are 28 cm and 7 cm. Determine its capacity.
1) 48510 cu.cm
2) 30400 cu.cm
3) 5610 cu.cm
4) 21020 cu.cm
10. The sides of a triangle are 5, 12 and 13 units respectively. A rectangle is constructed which is equal in area to the triangle and has a width of 10 units. Then the perimeter of the rectangle isTop of Form
11. A rhombus is drawn on the x-y plane by joining the points having coordinates A (0, 0), B (x1, y1), C (4, 6), D (x2, y2). What is the equation of the diagonal BD?
(1) 3x – 4y = 3
(2) 2y + 3x = 1
(3) 3y + 2x = 13
(4) 6x – 2y = 1
12. ABC is a triangle with AB = 14, BC = 10 and CA = 6. D and E are points on BC and CA respectively such that CD = 3 and CE = 2.5. A line passing through C and the point of intersection of AD and BE cut the side AB at F. Then AF = ?
13. Given points P1, P2, P3, … P7 on a straight line, in the order stated (not necessarily evenly spaced). Let P be an arbitrary point selected on the line and let s be the sum of undirected lengths PP1, PP2, PP3, … PP7. Then s is smallest if and only if the point P is
1. Midway between P1 and P7
2. Midway between P2 and P6
3. Midway between P3 and P5
4. At P4
5. At P1
14. Find the number of distinct integer-sided triangles with perimeter 1001.
15. The coordinates of the vertices of the triangle ΔABC are A(0, 1), B(4, 4) and C(5, 3). From which vertex the length of the median drawn to the opposite side is the minimum?
(1) A (2) B (3) C (4) All medians are of equal length
16. In a quadrilateral ABCD, E is a point on AB. If ∠ADE =∠DEC = ∠ECB= 30°, AD = 2 units and BC = 4 units, then what is the area of the ΔDEC?
17. Refer to the figure in the link. ABC, DEF and GHI are three equilateral triangles having the same area. BD = DG = GC. Find the ratio of EJKL to AKHIB.
18. The area of a right-angled triangle is 40 sq. cm and its perimeter is 40 cm. The length of its hypotenuse is
19. Find the equation of circle with center (2, 4) and equation of tangent as 3x + 4y = 12.
20. The area of the quadrilateral (in sq. units) formed by the points (1,8), (3,4), (5,8) and (–3,4) is
21. A farmer has a land in the shape of a triangle, the sides of which are 50 m, 120 m and 130 m. As it is a hilly area, the farmer can use only some portion in the middle of the field. To maximise his area of cultivation he draws a circle touching all the three sides. Now he plans to use the area covered inside the circle, only because it is more fertile and there exists a tube well in the centre. He draws perpendicular lines on the three sides from the tube well and divides the total cultivable area into three parts. He fixes the smallest portion for vegetables, the largest portion for wheat and the third portion for rice. Find the area (in sq.m.) in which vegetables are cultivated.
(d) Cannot be determined
22. A rectangular piece of paper is folded in such a way that one pair of diagonally opposite vertices coincide. If the dimensions of the rectangle is 40 cm * 30 cm, then find the length of the crease (fold) in cm.
2) 75 sqrt(3)
3) 37.5 cm
4) 60 sqrt(3)
23. A right angled triangle with sides 3 cm, 4 cm and 5 cm is drawn. Semicircles are drawn on all three sides with the side-length as the diameters. Find area of the shaded region (figure in the link)
24. A couple of questions on properties of intersecting chords and bisectors of chords
25. What is the maximum possible value of (21 Sin X + 72 Cos X)?
26. The area of a rectangle is thrice that of a square. If the length of the rectangle is 40 cm and its breadth is 3/2 times that of the side of the square, then the side of the square is :
(a) 15 cm
(b) 20 cm
(c) 30 cm
(d) 60 cm
27. The exterior angles of a quadrilateral are in the ratio 1:3:4:7. What is the sum of the largest and the smallest interior angles of the quadrilateral?
28. As a cone rolls on the surface of a table, it traces a circular arc centered on the vertex. While making a full circle, it rotates 17 times. Find the ratio of the height of the cone to its radius.
29. Isosceles trapezium is inscribed in a circle of radius 10 cm such that the non-parallel sides are 10 cm each. If one of the sides is the diameter of the circle that circumscribes the trapezium, what is the area of the trapezium?
30. Data Sufficiency – Geometry
31. Eight identical octagons are placed edge to edge to form a symmetrical star shaped figure. Find the area of the star shaped figure (figure in the link)
32. PQRS is a square, U and V are the midpoints of PS and SR. PV and UR meet at T. Find A(PQRS):A(PQRT).
33. D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC. What is the ratio of the circumradii of triangles DEF and ABC respectively?
34. The minimum number of straight lines required to obtain 16 non-overlapping parallelograms are
35. Consider a square ABCD of side 60 cm. lt contains arcs BD and AC drawn with centres at A and D respectively. A circle is drawn such that it is tangent to side AB, and the arcs BD and AC. What
is the radius of the circle?
A. 9 cm
B. 10 cm
C. 12 cm
D. 15 cm
E. None of the above
36. If you have a 4*6*8 cuboid from which the largest possible cube is cut out, what would be the minimum number of cubes into which the remaining figure could be divided assuming that there is no piece left over and the all the smaller cubes are of equal dimension?
37. A sphere is inscribed in a cube with an edge 10. What is the shortest distance from one of the vertices of the cube to the surface of the sphere?
38. In ABC, AB < AC < BC. CA is extended to a point P such that AP = AB. Point R is selected on CP such that CR = BC. If RP is equal to the diameter of the biggest circle that can be drawn inside ABC, and all angles are integers, which of the options is definitely true?
a. Angle B < 90 degrees
b. Angle C > 90 degrees
c. Angle A is not equal to 90 degrees
d. Angle C is 90 degrees
39. Find the difference between the lengths of the inradius and circumradius of a triangle with sides 11 units, 60 units and 61 units.Top of Form
40. Two tangents from an external point P touch a circle with center O at points A and B such that A is to the north of B. M(angle APB) is 70 degrees. Also, X and Y are points on the circumference of the circle such that A-Y-B-X-A in a clockwise manner and Y lies to the east of X. Find the measure of arc(AXB).
41. In a right-angled triangle, the sum of the squares of the three side lengths is 1800. The length of its hypotenuse is
42. A circle with center O and radius 8 cm is drawn, BD is a chord and A is a point on the minor arc BD. C is a point on BD such that AC is perpendicular to BD. AC = 4 cm and BC = 12 cm. Find CD. Figure in the link.
43. How many values of the integer k will make the triangle with sides 6, 8 and k obtuse?
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