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Question 1

Two chords of lengths 32 cm and 112 cm lie in a circle of radius 65cm. Which of the following is true? Indicate all true statements

  1. If the chords be parallel, then the maximum possible distance between them is 96 cm
  2. The two chords cannot be parallel to each other
  3. If the chords be parallel, then they are at least 30 cm apart
  4. If the chords be parallel, then the maximum possible distance between them is 144 cm
  5. If the chords be parallel, then there are only two possible distances between them

Correct Answer: A, B and E

Explanation:

The two chords can be parallel to each other.
Option B is false.
They can either be on the same side or on opposite sides of the centre. Since their lengths are fixed, they can only be at fixed distances from the centre and hence there are only two possible distances between them.
Option E is true.

When the two chords are parallel, half the chord, a line from the centre of the circle perpendicular to the chord and the radius form a right triangle.
The line from the centre and perpendicular to the chord bisects it.
Let the chord of length 32cm be x cm from the centre of the circle and let the chord of length 112 cm be at a distance y
cm
Applying Pythagoras theorem to the triangle so formed, we get
65 = Sqrt[(32/2)2+x2]
652=162+x2
x2= 4225-256 = 3969
x=63

65 = √[(112/2)2+y2]
652=562+y2
x2= 4225-3136 = 1089
y=33

The maximum distance is possible when the two chords are on opposite sides of the centre and the least distance is when the two chords are on the same side.
The maximum distance = 63+33=96cm
The minimum distance = 63-33=30cm
Options A and B are true and C is false.

Question 2

A solid sphere of radius 10cm is melted and molded into 8 spheres of equal radius. Which of the following statements is true? Indicate all such options.

  1. The surface area of the big sphere is the sum of the surface areas of the small spheres
  2. The surface area of each small sphere is 80π
  3. The volume of the big sphere is the sum of the volume of the small spheres
  4. The radius of the big sphere is 8 times the radius of a small sphere
  5. The surface area of each small sphere is 100π

Correct Answer: C and E

Explanation:

Let the radius of each small sphere be r cm.
Surface area of bigger sphere = 4pi*r^2
=4*π*10^2 = 4*π*100 = 400π

Volume of bigger sphere = 4/3*π*r3
= 4/3*π*103
= 4000π/3 cc

Volume of each small sphere = (4000π/3)/8
= 4/3*π(1000/8)
= 4/3*π(10/2)3

Hence, radius of each small sphere is 10/2=5cm
Option D is false.

Surface area of one small sphere = 4*π*52
= 100π
Sum of surface areas of the 8 spheres = 8*100π=800π

Option A and B are false.
Option C and E are true.

Question 3

Which of the following are factors of (x-y+z)2+(y-z+x)2+2(x-y+z)(y-z+x)? Indicate all the correct options.

  1. 2x
  2. y
  3. x
  4. x2
  5. 4

Correct Answer: A, C, D and E

Explanation:

(x-y+z)2+(y-z+x)2+2(x-y+z)(y-z+x) = (x-y+z)2+2(x-y+z)(y-z+x)+(y-z+x)2
= [(x-y+z)+(y-z+x)]2
= (2x)2=4x2
2x, x, x2 and 4 are factors
Options A, C, D and E are true.

Question 4

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

  1. There are 9 one-digit natural numbers with distinct digits
  2. There are 81 two-digit natural numbers with distinct digits
  3. There are 5040 three-digit natural numbers with distinct digits
  4. There are 4536 four-digit natural numbers with distinct digits
  5. There are 27216 five-digit natural numbers with distinct digits

Correct Answer: A, B, D and E

Explanation:

The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
There are 9 natural numbers with one-digit.
Option A is correct.

Two-digit natural numbers:
Numbers of two-digits with distinct digits = P(10,2)
Numbers with 0 in the tens place = numbers with 0 in tens place and one of the remaining digits in the units place =
P(9,1)
Required two-digit numbers = P(10,2) - P(9,1)
= 10!/8!-9 = 10*9-9 = 81

Three-digit natural numbers:
Number of three-digits with distinct digits = P(10,3)
Numbers with 0 in the hundreds place = P(9,2)
Required two-digit numbers = P(10,3) - P(9,2)
= 10!/7! - 9!/7! = 10*9*8-9*8
= 720-72=648

Four-digit natural numbers:
Numbers of four-digits with distinct digits = P(10,4)
Numbers with 0 in the thousands place = P(9,3)
Required two-digit numbers = P(10,4) - P(9,3)
= 5040-504=4536

Five-digit natural numbers:
Numbers of five-digits with distinct digits = P(10,5)
Numbers with 0 in the ten-thousands place = P(9,4)
Required two-digit numbers = P(10,5) - P(9,4)
= 10!/5! - 9!/5!
= 30240-3024
= 27216

Options A, B, D and E are true.

Question 5

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

  1. C(10,4) = C(10,6)
  2. C(10,10) = C(10,9)
  3. Five different letters can be put in six different envelops is C(6,5) ways
  4. A cricket team of 11 can be chosen from 15 players in C(15,11) ways
  5. A cricket team of 11 can be chosen from 15 players in P(15,11) ways

Correct Answer: A and D

Explanation:

C(10,4) = 10!/(4!6!) and C(10,6) = 10!/(6!4!)
Option A is true.

C(10,10) = 10!/(10!0!) =1 and C(10,9) = 10!/(9!1!) = 10
Option B is false.

Five different letters can be put in six different envelops in P(6,5) ways.
Option C is false.

A cricket team of 11 can be chosen in C(15,11) ways.
Option D is true and E is false.

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