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Question 1

Which of the following are the first three days of 2010? Indicate all correct options.

  1. Sunday
  2. Tuesday
  3. Friday
  4. Thursday
  5. Saturday

Correct Answer: A, C and E

Explanation:

2010 = 1600 + 400 + 9
= 0 odd days + 1 odd day + 2 leap years + 7 ordinary years
= 1 odd day + 2*2 odd days + 7 odd days
= 1 odd day + 4 odd days + 0 odd days
= 5 odd days
The year started on Friday.
Options A, C and E are correct.

Question 2

Which of the following statements is true? Indicate all true statements.

  1. The sides of a regular polygon are equal
  2. A polygon has 5 sides
  3. The interior angles of a regular polygon are equal
  4. A cyclic quadrilateral is always regular
  5. The number of diagonals in a polygon is equal to the number of vertices

Correct Answer: A and C

Explanation:

The sides of a regular polygon are equal.
Option A is true.

A polygon has 4 or more sides.
Option B is false.

The interior angles of a regular polygon are equal.
Option C is true.

A cyclic rectangle is not regular.
Option D is false.

Two vertices are joined to form a diagonal. Hence, the number of diagonals in a polygon is less than the number of vertices.
Option E is false.

Question 3

Two horses are tethered to two adjacent corners of a rectangular field of dimensions 7m by 10m. The length of the rope is equal to the breadth of the field. Which of the following statements are true? Indicate all true statements.

  1. The two horses cannot graze in any common area
  2. An area of (70-49*π/4) sq m is common to both the horses
  3. Each horse can graze in area 49π/4 sq m
  4. An area of (70-49*π/2) sq m is common to both the horses
  5. The two horses can graze in equal area

Correct Answer: C, E

Explanation:

The length of the triangles are 10m and the breadth is 7m
The horses are tied to adjacent corners and the length of the rectangle is less than twice the breadth. Hence they can graze in common area.
Option A is false.

Area of rectangle = 7*10=70 sq m
Area in which a horse can graze = 1/4*π*7^2
= 49π/4
Option C is true and option E is trivially true.

70-49π/4=area of rectangle-area in which one horse can graze.
Option B is false.
70-49π/2=area of rectangle- 2*area in which one horse can graze.
Option D is false.

Question 4

The volumes of two semi-spheres are in ratio 8:27. Which of the following statements is true? Indicate all such statements.

  1. The radii of the two spheres are in ratio 2:3
  2. The radii of the two spheres are in ratio 4:9
  3. The radii of the two spheres are in ratio 64:729
  4. If the sum of the radii is 10cm, then the smaller radius is 2cm
  5. If the sum of the radii is 10cm, then the longer radius is 6cm

Correct Answer: A and E

Explanation:

Let the two radii be r1 and r2
The two volumes will be 2/3*π*r13 and 2/3*π*r23
(2/3*π*r3)/(2/3*π*r23) = 8:27
r13/r23=8:27
r1/r2=2:3
Option A is true and B and C are false.

Let the radii be 2x and 3x
2x+3x=10
5x=10
x=2
The two radii are 4cm and 6cm
Option D is false and E is true.
[π=22/7]

Question 5

If 8x=4/(2x), then which of the following is true? Indicate all such options.

  1. [x/2]x=1/2
  2. 9(2x)=3
  3. 5(x-1/2)=0
  4. [1/4](-x)=2
  5. [(16)x](-4) = 1/4

Correct Answer: A and D

Explanation:

8x=4/(2x)
(23)x=22/(2x)
2(3x)*2x=22
2(3x+x)=22
2(4x)=22
4x=2
x=1/2

[x/2]x= [1/2/2](1/2)
= (1/4)(1/2) = 1/2
Option A is true.

9(2x)= 9(2*1/2)
=91=9
Option B is false.

5(x-1/2)=5(1/2-1/2)
= 50=1
Option C is false.

[1/4](-x)=(1/4)(-1/2)
= 4(1/2) = 2
Option D is true.

[(16)x](-4) = (16)(x*-4)
=16(-4x)
=16(-4*1/2)
= 16-2
= 1/162 = 1/256
Option E is false.

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