MHT CET · Maths · Continuity and Differentiability
If
\(\begin{aligned} f(x) &=6 \beta-3 \propto x, \text { if }-4 \leq x < -2 \ \end{aligned}\) \(=4 x+1, \text { if }-2 \leq x \leq 2 \)
is continuous on \([-4,2]\), then \(\propto+\beta=\)
- A \(\frac{-7}{6}\)
- B \(\frac{4}{7}\)
- C \(\frac{-4}{7}\)
- D \(\frac{7}{6}\)
Answer & Solution
Correct Answer
(A) \(\frac{-7}{6}\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow-2^{-}} f(x)=\lim _{x \rightarrow-2^{-}}(6 \beta-3 \alpha x)\)
\(=6 \beta+6 \alpha\) ...(1)
\(\lim _{x \rightarrow-2^{+}} f(x)=\lim _{x \rightarrow-2^{+}}(4 x+1)=-7\) ...(2)
Given \(f(x)\) is continuous on \([-4,2]\)
\(\therefore 6 \beta+6 \alpha=-7\)
\(\therefore \alpha+\beta=\frac{-7}{6}\)
\(=6 \beta+6 \alpha\) ...(1)
\(\lim _{x \rightarrow-2^{+}} f(x)=\lim _{x \rightarrow-2^{+}}(4 x+1)=-7\) ...(2)
Given \(f(x)\) is continuous on \([-4,2]\)
\(\therefore 6 \beta+6 \alpha=-7\)
\(\therefore \alpha+\beta=\frac{-7}{6}\)
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
- Three fair coins numbered 1 and 0 are tossed simultaneously. Then variance \(\operatorname{Var}(\mathrm{X})\) of the probability distribution of random variable \(\mathrm{X}\), where \(X\) is the sum of numbers on the uppermost faces, isMHT CET 2023 Medium
- If \(A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]\), then \(A(\operatorname{adj} A)=\)MHT CET 2021 Medium
- If \(\int \frac{x+1}{\sqrt{2 x-1}} \mathrm{~d} x=\mathrm{f}(x) \sqrt{2 x-1}+\mathrm{c}\), (where c is a constant of integration), then \(\mathrm{f}(x)\) is equal toMHT CET 2024 Easy
- If \(x=3 \sin \theta, y=3 \cos \theta \cos \phi, z=3 \cos \theta \sin \emptyset\), then \(x^{2}+y^{2}+z^{2}=\)MHT CET 2020 Easy
- The value of \(\sin \left(2 \sin ^{-1} 0.8\right)\) is equal toMHT CET 2022 Medium
- \(\begin{aligned} & \text { If } \bar{a}=4 \hat{i}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=\hat{i}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \\ & \text { then } \bar{a} \times(\bar{a} \times(\bar{a} \times(\bar{a} \times \overline{\mathrm{b}})))=\end{aligned}\)MHT CET 2025 Hard
More PYQs from MHT CET
- An alternating e.m.f. is given by \(e=e_0 \sin \omega t\). In how much time the e.m.f. will have half its maximum value, if e starts from zero ?
\(\left(\mathrm{T}=\text { time period, } \sin 30^{\circ}=\frac{1}{2}\right)\)MHT CET 2025 Easy - Calculate osmotic pressure of solution of 0.025 mole glucose in \(100 \mathrm{~mL}\) water at \(300 \mathrm{~K}\). \(\left[\mathrm{R}=0.082 \mathrm{~atm} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]\)MHT CET 2023 Easy
- How many pairs of sympathetic ganglia are present in ANS?MHT CET 2015 Hard
- The Cartesian equation of the lien passing through the points \(\mathrm{A}\) \((2,2,1)\) and \(B(1,3,0)\) isMHT CET 2021 Easy
- Let \(y=y(x)\) be the solution of the differential equation \(x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=x \log x,(x\gt1)\)
If \(2(y(2))=\log 4-1\) then the value of \(y(\mathrm{e})\) isMHT CET 2024 Hard - The percentage (by weight) of sodium hydroxide in a \(1.25\) molal \(\mathrm{NaOH}\) solution isMHT CET 2009 Medium