The domain of the function f(x) = \(\frac{1}{\sqrt{[x]^2-3[x]-10}}\) is (where [x] denotes the greatest integer less than or equal to x)
- (-∞,-2) ∪ (6, ∞)
- (-∞,-2) ∪ (5, ∞)
- (-∞,-3) ∪ (6, ∞)
- (-∞,-3) ∪ (5, ∞)
The Correct Option is A
Solution and Explanation
Given Function:
\[ F(x) = \frac{1}{\lfloor x \rfloor^2 - 3\lfloor x \rfloor - 10}. \]
To ensure \( F(x) \) is defined, the denominator must be positive:
\[ \lfloor x \rfloor^2 - 3\lfloor x \rfloor - 10 > 0. \]
Step 1: Solve the Inequality
Factorize the quadratic expression:
\[ \lfloor x \rfloor^2 - 3\lfloor x \rfloor - 10 = (\lfloor x \rfloor + 2)(\lfloor x \rfloor - 5). \]
The inequality becomes:
\[ (\lfloor x \rfloor + 2)(\lfloor x \rfloor - 5) > 0. \]
Step 2: Analyze the Intervals
The roots of the quadratic are \( \lfloor x \rfloor = -2 \) and \( \lfloor x \rfloor = 5 \). Using a sign chart:
| Interval | Sign of \( (\lfloor x \rfloor + 2)(\lfloor x \rfloor - 5) \) |
|---|---|
| \( (-\infty, -2) \) | + |
| \( (-2, 5) \) | - |
| \( (5, \infty) \) | + |
The inequality is satisfied in the intervals:
\[ \lfloor x \rfloor < -2 \quad \text{or} \quad \lfloor x \rfloor > 5. \]
Step 3: Refine the Solution
Since \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), the solution must be refined to:
\[ \lfloor x \rfloor \leq -3 \quad \text{or} \quad \lfloor x \rfloor \geq 6. \]
Step 4: Write the Solution for \( x \)
The corresponding intervals for \( x \) are:
\[ x \in (-\infty, -2) \cup [6, \infty). \]
Final Answer:
\( x \in (-\infty, -2) \cup [6, \infty) \).
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Functions Questions
The domain of \(y= cos^{-1}|\frac{2-|x|}{4}| log(3 - x)^{-1}\) is [α, β) - {y} then the value of α+β-y =?
- If the domain of the function \(f(x)\) = \(\frac{\sqrt{x^2-25}}{(4-x^2)}+ \log(x^2+2x-15)\) is \((-∞, α) ∪ (β, ∞)\), then \(α^2+β^2\) is equal to \(b\)
- If the domain of the function \[ f(x) = \cos^{-1} \left( \frac{2 - |x|}{4} \right) + \left( \log_e (3 - x) \right)^{-1} \] is \([-\alpha, \beta) - \{ \gamma \}\), then \( \alpha + \beta + \gamma \) is equal to:
- If the function \( f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases} \) is differentiable on \( \mathbb{R} \), then \( 48 (a + b) \) is equal to \(\_\_\_\_\_\_\_\_\_\).
- Consider the function \( f : (0, 2) \to \mathbb{R} \) defined by \[ f(x) = \frac{x}{2} + \frac{2}{x} \] and the function \( g(x) \) defined by \[ g(x) = \begin{cases} \min\{f(t)\}, & 0 < t \leq x \text{ and } 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases} \] Then:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.