State which of the following statements are true and which are false. Justify your answer.
(i) 35 ∈ {x | x has exactly four positive factors}.
(ii) 128 ∈ {y | the sum of all the positive factors of y is 2y}
(iii) 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0}
(iv) 496 ∉ {y | the sum of all the positive factors of y is 2y}.
(i) True
According to the question,
35 ∈ {x | x has exactly four positive factors}
The possible positive factors of 35 = 1, 5, 7, 35
35 belongs to given set
Since, 35 has exactly four positive factors
⇒ The given statement 35 ∈ {x | x has exactly four positive factors} is true.
(ii) False
According to the question,
128 ∈ {y | the sum of all the positive factors of y is 2y}
The possible positive factors of 128 are 1, 2, 4, 8, 16, 32, 64, 128
The sum of them
= 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128
= 255
2y = 2 × 128 = 256
Since, the sum of all the positive factors of y is not equal to 2y
128 does not belong to given set
⇒ The given statement 128 ∈ {y | the sum of all the positive factors of y is 2y} is false.
(iii) True
According to the question,
3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0}
x4 – 5x3 + 2x2 – 112x + 6 = 0
On putting x = 3 in LHS:
(3)4 – 5(3)3 + 2(3)2 – 112(3) + 6
= 81 – 135 + 18 – 336 + 6
= –366
≠ 0
So, 3 does not belong to given set
⇒ The given statement 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0} is true.
(iv) False
According to the question,
496 ∉ {y | the sum of all the positive factors of y is 2y}
The possible positive factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
The sum of them
= 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496
= 996
2y = 2 × 496 = 992
Since, the sum of all the positive factors of y is equal to 2y
496 belongs to given set
⇒ The given statement 496 ∉ {y | the sum of all the positive factors of y is 2y} is false.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,