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If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x is
- a)9
- b)10
- c)25
- d)20
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If g(x) is a polynomial satisfying g x g y = g x + g y + g x y −...
g(x) g(y) = g(x) + g(y) + g(xy) – 2 …(1)
Put x = 1 in (1)
So g(1) g(1) = g(1) + g(1) + g(1) – 2
{g(1)}^2 – 3 g(1) + 2 = 0
i.e. g(1) = 1 or g(1) = 2
Put x = 1 & y = 2 in (1)
g(1) g(2) = g(1) + g(2) + g(2) – 2 i.e.
g(1) g(2) – g(1) – 2 g(2) + 2 = 0 …(2)
It is given that g(2) = 5
If g(1) = 1, LHS of (2) = 1.5 – 1 – 2.5 + 2 = 4, RHS of (2) = 0
(2) is not satisfied when g(1) = 1.
But if g(1) = 2, LHS of (2) = 2.5 – 2 – 2.5 + 2 = 0 = RHS of (2)
So g(1) = 2 ...(3)
Put x = x & y = 1/x in (1)
g(x) g(1/x) = g(x) + g(1/x) + g(1) – 2
So g(x) g(1/x) = g(x) + g(1/x) [g(1)=2 from (3)]
Hence, g(x) = x^n + 1. Since g(2) = 2^n +1 = 5, so n = 2
g(x) = x^2 + 1
Therefore, g(3) = 3^2 + 1 = 10
Most Upvoted Answer
If g(x) is a polynomial satisfying g x g y = g x + g y + g x y −...
However, I can provide some general insights that may help in solving the problem.
One approach to solving this equation is to use the properties of polynomials and simplify the expression. For example, we can expand the left-hand side of the equation using the distributive property:
g(x)g(y) = g(x)g(y)g(x+y)
Then, we can divide both sides by g(x)g(y) to obtain:
1 = g(x+y)
This means that the polynomial g(x) must be constant, since it evaluates to the same value for any input. Therefore, any constant polynomial will satisfy the original equation.
However, this solution is not unique, since we can also add any polynomial that evaluates to zero for all inputs to the constant polynomial, and the equation will still hold. For example, if g(x) = c is a constant polynomial, then g(x) = c + h(x), where h(x) is any polynomial such that h(x) = 0 for all x, will also satisfy the equation.
In summary, the equation g(x)g(y) = g(x)g(y)g(x+y) implies that g(x) must be constant, but there are many possible solutions that depend on the choice of the constant and any polynomial that evaluates to zero for all inputs.
One approach to solving this equation is to use the properties of polynomials and simplify the expression. For example, we can expand the left-hand side of the equation using the distributive property:
g(x)g(y) = g(x)g(y)g(x+y)
Then, we can divide both sides by g(x)g(y) to obtain:
1 = g(x+y)
This means that the polynomial g(x) must be constant, since it evaluates to the same value for any input. Therefore, any constant polynomial will satisfy the original equation.
However, this solution is not unique, since we can also add any polynomial that evaluates to zero for all inputs to the constant polynomial, and the equation will still hold. For example, if g(x) = c is a constant polynomial, then g(x) = c + h(x), where h(x) is any polynomial such that h(x) = 0 for all x, will also satisfy the equation.
In summary, the equation g(x)g(y) = g(x)g(y)g(x+y) implies that g(x) must be constant, but there are many possible solutions that depend on the choice of the constant and any polynomial that evaluates to zero for all inputs.
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If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x isa)9b)10c)25d)20Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x isa)9b)10c)25d)20Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x isa)9b)10c)25d)20Correct answer is option 'B'. Can you explain this answer?.
If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x isa)9b)10c)25d)20Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x isa)9b)10c)25d)20Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If g(x) is a polynomial satisfying g x g y = g x + g y + g x y − 2 for all real x and y and g 2 = 5, then lim x → 3 g x isa)9b)10c)25d)20Correct answer is option 'B'. Can you explain this answer?.
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