KCET · Maths · Binomial Theorem
The value at \(x = 2\) for \(\dfrac{x^3 + 3x^2 + 3x + 1}{x^4 + 4x^3 + 6x^2 +4x + 1}\)
- A \(3\)
- B \(\dfrac{25}{61}\)
- C \(\dfrac{1}{3}\)
- D \(\dfrac{19}{73}\)
Answer & Solution
Correct Answer
(C) \(\dfrac{1}{3}\)
Step-by-step Solution
Detailed explanation
The given expression is \(\dfrac{x^3 + 3x^2 + 3x + 1}{x^4 + 4x^3 + 6x^2 + 4x + 1}\)
Using the binomial expansion, the numerator can be written as \((x + 1)^3\) and the denominator can be written as \((x + 1)^4\).
The expression simplifies to \(\dfrac{(x + 1)^3}{(x + 1)^4} = \dfrac{1}{x + 1}\)
Substituting \(x = 2\), we get \(\dfrac{1}{2 + 1} = \dfrac{1}{3}\)
Answer: \(\dfrac{1}{3}\)
Using the binomial expansion, the numerator can be written as \((x + 1)^3\) and the denominator can be written as \((x + 1)^4\).
The expression simplifies to \(\dfrac{(x + 1)^3}{(x + 1)^4} = \dfrac{1}{x + 1}\)
Substituting \(x = 2\), we get \(\dfrac{1}{2 + 1} = \dfrac{1}{3}\)
Answer: \(\dfrac{1}{3}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \((g \circ f)(x)=\sin x\) and \(f \circ g(x)=(\sin \sqrt{x})^2\) Then,KCET 2024 Easy
- \(\int \operatorname{cosec}(x-a) \operatorname{cosec} x d x\) is equal toKCET 2009 Hard
- If the parabola \(y=\alpha x^{2}-6 x+\beta\) passes through the point \((0,2)\) and has its tangent at \(x=\frac{3}{2}\) parallel to \(X\)-axis, thenKCET 2021 Medium
- If \(f(x)\) is a function such that \(f^{\prime \prime}(x)+f(x)=0\) and \(g(x)=[f(x)]^{2}+\left[f^{\prime}(x)\right]^{2}\) and \(g(3)=8\), then \(g(8)\) is equal toKCET 2013 Hard
- If \(a>b>0, \sec ^{-1} \frac{a+b}{a-b}=2 \sin ^{-1} x\), then \(x\) isKCET 2010 Hard
- The remainder when, \(10^{10} \cdot\left(10^{10}+1\right)\left(10^{10}+2\right)\) is divided by 6 isKCET 2013 Medium
More PYQs from KCET
- The negation of the statement
"For every real number \(x ; x^2+5\) is positive" isKCET 2024 Easy - \(\lim _{n \rightarrow \infty} \frac{3 \cdot 2^{n+1}-4 \cdot 5^{n+1}}{5 \cdot 2^{n}+7 \cdot 5^{n}}\) is equal toKCET 2009 Easy
- The standard reduction potential at \(298 \mathrm{~K}\) for the following half cell reaction
\(Z n^{2+}(a q)+2 e^{-} \rightarrow Z n(s) ; \quad E^{\circ}=-0.762 V\)
\(C r^{3+}(a q)+3 e^{-} \rightarrow C r(s) ; \quad E^{\circ}=0.740 V\)
\(2 H^{+}(a q)+2 e^{-} \rightarrow H_{2}(g) ; \quad E^{\circ}=0.0 V\)
\(F_{2}(g)+2 e^{-} \rightarrow 2 F^{-}(a q) ; \quad E^{\circ}=2.87 V\)
Which of the following is strongest reducing agent ?KCET 2017 Easy - If \(f(x)\) and \(g(x)\) are two functions with \(g(x)=x-\frac{1}{x}\) and \(f \circ g(x)=x^3-\frac{1}{x^3}\), then \(f^{\prime}(x)\) is equals toKCET 2023 Medium
- Of the total incident solar radiation, the percentage of photosynthetically active radiation (PAR) captured by the plants isKCET 2019 Hard
- The sum of the series, \(\frac{1}{2.3} \cdot 2+\frac{2}{3.4} \cdot 2^{2}+\frac{3}{4.5} \cdot 2^{3}+\ldots\) upto \(n\) terms isKCET 2013 Hard