JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \( \alpha, \beta \in \mathbb{R} \) be such that the function
\( f(x)=\begin{cases}2\alpha(x^{2}-2)+2\beta x&,x<1\\ (\alpha+3)x+(\alpha-\beta)&,x\ge1\end{cases} \)
be differentiable at all \( x \in \mathbb{R} \). Then \( 34(\alpha+\beta) \) is equal to
- A 84
- B 48
- C 36
- D 24
Answer & Solution
Correct Answer
(B) 48
Step-by-step Solution
Detailed explanation
\(f(x)=\left\{\begin{array}{ccc}2 \alpha x^2+2 \beta x-4 \alpha & ; & x<1 \\ (\alpha+3) x+\alpha-\beta & ; & x \geq 1\end{array}\right.\) \(f\left(1^{+}\right)=2 \alpha-\beta+3, f\left(1^{-}\right)=-2 \alpha+2 \beta\)…
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 the circle \(C_1: x^2+y^2-2(x+y)+1=0\) and \(C_2\) be a circle having centre at \((-1,0)\) and radius \(2\) . If the line of the common chord of \(\mathrm{C}_1\) and \(\mathrm{C}_2\) intersects the \(\mathrm{y}\)-axis at the point \(\mathrm{P}\), then the square of the distance of \(\mathrm{P}\) from the centre of \(\mathrm{C}_1\) is :JEE Mains 2024 Hard
- The mean and variance of seven observations are \(8\) and \(16\), respectively. If \(5\) of the observations are \(2, 4, 10, 12, 14,\) then the product of the remaining two observations isJEE Mains 2019 Hard
- Let R be a relation defined on the set \( \{1,2,3,4\}\times\{1,2,3,4\} \) by \( R=\{((a,b),(c,d)):2a+3b=3c+4d\} \).
Then the number of elements in R isJEE Mains 2026 Medium - Let \(PQ\) be a focal chord of the parabola \(y^{2}=4 x\) such that it subtends an angle of \(\frac{\pi}{2}\) at the point \((3, 0)\). Let the line segment \(PQ\) be also a focal chord of the ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}\). If \(e\) is the eccentricity of the ellipse \(E\), then the value of \(\frac{1}{e^{2}}\) is equal toJEE Mains 2022 Hard
- Consider the following system of equations : \(x+2 y-3 z=a\) ; \(2 x+6 y-11 z=b\) ; \(x-2 y+7 z=c\) where \(a , b\) and \(c\) are real constants. Then the system of equations :JEE Mains 2021 Medium
- The equation of the plane passing through the point \((1,2,-3)\) and perpendicular to the planes \(3 x+y-2 z=5\) and \(2 x-5 y-z=7,\) isJEE Mains 2021 Medium
More PYQs from JEE Mains
- Let \(f(x) = log_e\,(sin\,x),\) \((0\,<\,x\,< \pi )\) and \(g(x) = sin^{-1}\,(e^{-x}),\) \((x\, \ge \,0)\). If \(\alpha \) is a positive real number such that \(a\) \( = (fog)’(\alpha )\) and \(b = (fog)(\alpha ),\) thenJEE Mains 2019 Hard
- Let the tangents at two points \(A\) and \(B\) on the circle \(x ^{2}+ y ^{2}-4 x +3=0\) meet at origin \(O (0,0)\). Then the area of the triangle of \(OAB\) is.JEE Mains 2022 Hard
- The value of \(\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3\) isJEE Mains 2023 Hard
- Suppose \(y=y(x)\) be the solution curve to the differential equation \(\frac{d y}{d x}-y=2-e^{-x}\) such that \(\lim _{x \rightarrow \infty} y(x)\) is finite. If \(a\) and \(b\) are respectively the \(x-\) and \(y\)-intercepts of the tangent to the curve at \(x=0\), then the value of \(a-4 b\) is equal to\(....\)JEE Mains 2022 Hard
- Let \(\mathrm{P}(\mathrm{h}, \mathrm{k})\) be a point on the curve \(\mathrm{y}=\mathrm{x}^{2}+7 \mathrm{x}+2\) nearest to the line, \(y=3 x-3 .\) Then the equation of the normal to the curve at \(\mathrm{P}\) isJEE Mains 2020 Medium
- The lines \(\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}\) are distinct. For \(\mathrm{n}=1,2,3, \ldots, 10\) all the lines \(\mathrm{L}_{2 \mathrm{n}-1}\) are parallel to each other and all the lines \(L_{2 n}\) pass through a given point \(P\). The maximum number of points of intersection of pairs of lines from the set \(\left\{\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}\right\}\) is equal to :JEE Mains 2024 Medium