JEE Mains · Maths · STD 12 - 5. continuity and differentiation
If the function \(f(x)\, = \left\{ {\begin{array}{*{20}{c}}{ - x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x < 1\,\,\,\,}\\{a + {{\cos }^{ - 1}}(x + b),\,\,\,\,\,\,\,\,\,1 \le x \le 2} \end{array}} \right.\) is differentiable at \(x = 1 ,\) then \(\frac {a}{b}\) is equal to
- A \(\frac {\pi + 2}{2}\)
- B \(\frac {\pi - 2}{2}\)
- C \(\frac {-\pi - 2}{2}\)
- D \(-1-cos^{-1}\,(2)\)
Answer & Solution
Correct Answer
(A) \(\frac {\pi + 2}{2}\)
Step-by-step Solution
Detailed explanation
\(f\left( x \right) = \left\{ \begin{array}{l} - x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x < 1\\ a + {\cos ^{ - 1}}\left( {x + b} \right)\,\,\,1 \le x \le 2 \end{array} \right.\) \(f(x)\) is continuous…
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
- The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and \(\mathrm{x}= \pm \frac{4}{\sqrt{3}}\), respectively. Let the line \(y-\sqrt{3} \mathrm{x}+\sqrt{3}=0\) touch this hyperbola at \(\left(\mathrm{x}_0, \mathrm{y}_0\right)\). If \(\mathrm{m}\) is the product of the focal distances of the point \(\left(\mathrm{x}_0, \mathrm{y}_0\right)\), then \(4 \mathrm{e}^2+\mathrm{m}\) is equal to ...........JEE Mains 2024 Hard
- Let \(S=\left\{x \in R: 0 < x < 1\right.\) and \(\left.2 \tan ^{-1}\left(\frac{1-x}{1+x}\right)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}\). If \(n ( S )\) denotes the number of elements in \(S\) then:JEE Mains 2023 Hard
- The equation of the line through the point \((0,1,2)\) and perpendicular to the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}\) isJEE Mains 2021 Medium
- Consider a triangular plot \(ABC\) with sides \(AB = 7\, m\), \(BC = 5\, m\) and \(CA = 6\, m\). A vertical lamp-post at the mid point \(D\) of \(AC\) subtends an angle \(30^o\) at \(B\). The height (in \(m\)) of the lamp-post isJEE Mains 2019 Hard
- Let \(\mathrm{P}\) be a point on the hyperbola \(\mathrm{H}: \frac{\mathrm{x}^2}{9}-\frac{\mathrm{y}^2}{4}=1\), in the first quadrant such that the area of triangle formed by \(\mathrm{P}\) and the two foci of \(\mathrm{H}\) is \(2 \sqrt{13}\). Then, the square of the distance of \(\mathrm{P}\) from the origin isJEE Mains 2024 Hard
- If \(m\) is the minimum value of \(k\) for which the function \(f\left( x \right) = x\sqrt {kx - {x^2}} \) is increasing in the interval \([0,3]\) and \(M\) is the maximum value of \(f\) in \([0, 3]\) when \(k = m\), then the ordered pair \((m, M)\) is equal toJEE Mains 2019 Hard
More PYQs from JEE Mains
- In an examination, there are \(5\) multiple choice questions with \(3\) choices, out of which exactly one is correct There are \(3\) marks for each correct answer, \(-2\) marks for each wrong answer and \(0\) mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets \(5\) marks is. . . . . ... . .JEE Mains 2022 Hard
- Let \(\mathrm{A}(-2,-1), \mathrm{B}(1,0), \mathrm{C}(\alpha, \beta)\) and \(\mathrm{D}(\gamma, \delta)\) be the vertices of a parallelogram \(A B C D\). If the point \(C\) lies on \(2 x-y=5\) and the point \(D\) lies on \(3 x-2 y=6\), then the value of \(|\alpha+\beta+\gamma+\delta|\) is equal to ...........JEE Mains 2024 Hard
- The function \(f : N \to N\) defined by \(f\left( x \right) = x - 5\left[ {\frac{x}{5}} \right]\) , where \(N\) is set of natural numbers and \([x]\) denotes the greatest integer less than or equal to \(x\), isJEE Mains 2017 Hard
- If the matrix \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{array}\right]\) satisfies the equation \(A ^{20}+\alpha A ^{19}+\beta A =\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{array}\right]\) for some real numbers \(\alpha\) and \(\beta\), then \(\beta-\alpha\) is equal to ........ .JEE Mains 2021 Hard
- Let \(A (0,1), B (1,1)\) and \(C (1,0)\) be the mid - points of the sides of a triangle with incentre at the point D. If the focus of the parabola \(y^2=4 a x\) passing through \(D\) is \((\alpha+\beta \sqrt{2}, 0)\), where \(\alpha\) and \(\beta\) are rational numbers, then \(\frac{\alpha}{\beta^2}\) is equal toJEE Mains 2023 Hard
- Let \(\mathrm{g}: \mathrm{N} \rightarrow \mathrm{N}\) be defined as \(g(3 n+1)=3 n+2\) \(g(3 n+2)=3 n+3\) \(g(3 n+3)=3 n+1, \text { for all } n \geq 0\) Then which of the following statements is true?JEE Mains 2021 Hard