Question:
If $f(x) = \begin{cases} x+1, & x \le 1 \\ 3 - ax^2, & x>1 \end{cases}$ is continuous at $x = 1$, then $a$ is
If $f(x) = \begin{cases} x+1, & x \le 1 \\ 3 - ax^2, & x>1 \end{cases}$ is continuous at $x = 1$, then $a$ is
Show Hint
Continuity condition: LHL $=$ RHL $= f(a)$. Always check all three for piecewise functions.
Updated On: May 2, 2026
- 1
- 2
- $-3$
- $-2$
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
For continuity at $x = 1$, the left-hand limit, right-hand limit, and function value must all be equal.
Step 2: Detailed Explanation:
$f(1) = 1 + 1 = 2$.
RHL $= \lim_{x\to1^+}(3-ax^2) = 3-a$.
Continuity: $3 - a = 2 \Rightarrow a = 1$.
Step 3: Final Answer:
$a = 1$.
For continuity at $x = 1$, the left-hand limit, right-hand limit, and function value must all be equal.
Step 2: Detailed Explanation:
$f(1) = 1 + 1 = 2$.
RHL $= \lim_{x\to1^+}(3-ax^2) = 3-a$.
Continuity: $3 - a = 2 \Rightarrow a = 1$.
Step 3: Final Answer:
$a = 1$.
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Calculus Questions
- The minimum value of $x^{2} + \dfrac{1}{1 + x^{2}}$ is at}
- Derivative of $x^{x}$ is
- The value of $a$ for which $f(x) = a\sin x + \dfrac{1}{3}\sin 3x$ has an extremum at $x = \dfrac{\pi}{3}$ is
- If $|f(x)|$ is continuous at $x = a$, then $f(x)$
- The area in square units enclosed by the curve $x^{2}y = 36$, the $x$-axis and the lines $x = 6$ and $x = 9$ is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions