Quadrilaterals – Live Class Discussion

1) Based on the figure below, what is the value of $x$, if $y$ = 10?

[CAT 2001]

(1) 0
(2) 11
(3) 12
(4) None of the above

2) Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90$\degree$ or concave if the internal angle is 270$\degree$. If the number of convex corners in such a polygon is 25, the number of concave corners must be
[CAT 2003 L]

(1) 20
(2) 0
(3) 21
(4) 22

3) In the figure below, ABCDEF is a regular hexagon and ∠AOF = 90$/degree$. FO is parallel to ED. What is the ratio of the area of the triangle AOF to that of the hexagon ABCDEF?

[CAT 2003 L]

(1) $\dfrac{1}{12}$

(2) $\dfrac{1}{6}$

(3) $\dfrac{1}{24}$

(4)$\dfrac{1}{18}$

4) Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?

[CAT 2003 R]

(1) $\dfrac{1}{3}$

(2) $\dfrac{1}{2}$

(3) $\dfrac{2}{3}$

(4) $\dfrac{5}{6}$

5) The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then the total area, in sq cm, of all six surfaces of the pillar is
[CAT 2017 Slot 2]

(1) 1300
(2) 1340
(3) 1480
(4) 1520

6) ABCD is a quadrilateral inscribed in a circle with centre O. If ∠ COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is
[CAT 2017 Slot 2]

Answer:

7) If three sides of a rectangular park have a total length 400 ft, then the area of the park is maximum when the length (in ft) of its longer side is
[CAT 2017 Slot 2]

Answer:

8) The area of a rectangle and the square of its perimeter are in the ratio 1 : 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio
[CAT 2018 Slot 2]

(1) 2 : 9
(2) 1 : 4
(3) 1 : 3
(4) 3 : 8

9) On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite sides. If the area of the sheet left unpainted is two-third of the painted area then the perimeter of the rectangle in inches is
[CAT 2020 Slot 1]

(1) $3 \sqrt{π} \left(5 + \dfrac{12}{π} \right)$

(2) $4 \sqrt{π} \left(3 + \dfrac{9}{π} \right)$

(3) $3 \sqrt{π} \left(\dfrac{5}{2} + \dfrac{6}{π} \right)$

(4) $5 \sqrt{π} \left(3 + \dfrac{9}{π} \right)$

10) If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm, then the length, in cm, of each side of the hexagon is
[CAT 2021 Slot 1]

(1) $6\sqrt{6}$
(2) $4\sqrt{6}$
(3) $2\sqrt{6}$
(4) $\sqrt{6}$

11) Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. If t is the mid point of CD, then the length of AT, in cm, is
[CAT 2021 Slot 1]

(1) $\sqrt{12}$
(2) $\sqrt{14}$
(3) $\sqrt{15}$
(4) $\sqrt{13}$

12) If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is
[CAT 2021 Slot 2]

(1) $\dfrac{\sqrt{13} + \sqrt{12}}{2}$

(2) $\sqrt{37} + \sqrt{13}$

(3) $\sqrt{13} + \sqrt{12}$

(4) $\dfrac{\sqrt{37} + \sqrt{13}}{2}$

13) The sides AB and CD of a trapezium ABCD are parallel, with AB being the smaller side. P is the midpoint of CD and ABPD is a parallelogram. If the difference between the areas of the parallelogram ABPD and the triangle BPC is 10 sq cm, then the area, in sq cm, of the trapezium ABCD is
[CAT 2021 Slot 2]

(1) 20
(2) 30
(3) 25
(4) 40

14) The cost of fencing a rectangular plot is ₹ 200 per ft along one side, and ₹ 100 per ft along the three other sides. If the area of the rectangular plot is 60000 sq. ft, then the lowest possible cost of fencing all four sides, in INR, is
[CAT 2021 Slot 3]

(1) 100000
(2) 160000
(3) 90000
(4) 120000

15) In a triangle ABC, ∠BCA = 50$\degree$. D and E are points on AB and AC, respectively, such that AD = DE. If F is a point on BC such that BD = DF, then [CAT 2021 Slot 3]

(1) 80
(2) 100
(3) 72
(4) 96

16) Let ABCD be a parallelogram.The Lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. If the angle ∠ADC is equal to 30$\degree$ then the area of the parallelogram, in sq cm ,is
[CAT 2021 Slot 3]

(1) $25 (\sqrt{3} + \sqrt{15})$

(2) $\dfrac{25(\sqrt{5} + \sqrt{15})}{2}$

(3) $\dfrac{25 (\sqrt{3} + \sqrt{15})}{2}$

(4) $25(\sqrt{5} + \sqrt{15})$

CAT 2025 Lesson : Quant: Mar '25 to Apr '25 - Algebra Live Solving Part 1 - 02 Mar 2025