Domain Calculation for Function $f(x)$
The function is given by $f(x) = \frac{1}{\sqrt{10+3x-x^2}} + \frac{1}{\sqrt{x+|x|}}$. To find the domain $(a, b)$, we need to consider the conditions under which each term is defined.
First Term Domain: $\frac{1}{\sqrt{10+3x-x^2}}$
- For the square root in the denominator to be defined and non-zero, the expression inside must be strictly positive: $10+3x-x^2 > 0$.
- Rewrite as: $x^2 - 3x - 10 < 0$.
- Factor the quadratic: $(x-5)(x+2) < 0$.
- This inequality holds true when $x$ is between the roots -2 and 5.
- So, the domain for the first term is $-2 < x < 5$.
Second Term Domain: $\frac{1}{\sqrt{x+|x|}}$
- The expression inside the square root must be strictly positive: $x+|x| > 0$.
- Consider cases for $|x|$:
- If $x > 0$, then $|x| = x$. The inequality becomes $x+x > 0 \implies 2x > 0 \implies x > 0$.
- If $x \leq 0$, then $|x| = -x$. The inequality becomes $x+(-x) > 0 \implies 0 > 0$, which is false.
- Therefore, the domain for the second term is $x > 0$.
Combined Domain $(a, b)$
The domain of the function $f(x)$ is the intersection of the domains of both terms.
- Condition 1: $-2 < x < 5$
- Condition 2: $x > 0$
- The intersection is $0 < x < 5$.
Thus, the domain is $(a, b) = (0, 5)$. This implies $a=0$ and $b=5$.
Calculation of $(1+a)^2+b^2$
Substitute the values of $a$ and $b$ into the expression:
- $(1+a)^2+b^2 = (1+0)^2 + 5^2$
- $= 1^2 + 25$
- $= 1 + 25$
- $= 26$
The value of the expression is 26.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
