Let the point (p, p+1) lie inside the region \(E = \{(x,y):3-x≤y≤\sqrt{9-x^2}, 0 ≤ x ≤ 3\} \)If the set of all values of p is the interval (a,b). then b2 + b - a2 is equal to_____
Correct Answer: 3
Solution and Explanation
The region \(E\) is defined by the inequalities: \[ 3 - x \leq y \leq \sqrt{9 - x^2}, \quad 0 \leq x \leq 3. \] The point \(A = (p, p+1)\) lies inside the region. For \(A\) to lie above the line \(L: y = 3 - x\): \[ p + p + 1 - 3 > 0 \implies 2p > 2 \implies p > 1. \] For \(A\) to lie below the semicircle \(S: y = \sqrt{9 - x^2}\): \[ p^2 + (p+1)^2 - 9 < 0. \] Solving: \[ p^2 + p^2 + 2p + 1 - 9 < 0 \implies 2p^2 + 2p - 8 < 0 \implies p^2 + p - 4 < 0. \] The roots of \(p^2 + p - 4 = 0\) are: \[ p = \frac{-1 \pm \sqrt{17}}{2}. \] So, the valid interval is: \[ 0 < p < \frac{\sqrt{17} - 1}{2}. \] Combining with \(p > 1\): \[ 1 < p < \frac{\sqrt{17} - 1}{2}. \] Calculating: \[ b^2 + b - a^2 = 3. \] ✅ Therefore, the final answer is: \(3\).
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 types of differential equations Questions
- Let the system of equations \(x+2y+3z = 5\), \(2x+3y+z = 9\), \(4x+3y+λz = μ\) have an infinite number of solutions. Then \(λ + 2μ\) is equal to
- Let \(y=y(t)\) be a solution of the differential equation\(\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}\) where, \(\alpha > 0, \beta>0\) and \(\gamma > 0\). Then \(\displaystyle\lim _{t \rightarrow \infty} y(t)\)
The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$.
Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is
The statement
\((p⇒q)∨(p⇒r) \)
is NOT equivalent toLet α, β(α > β) be the roots of the quadratic equation x2 – x – 4 = 0.
If \(P_n=α^n–β^n, n∈N\) then \(\frac{P_{15}P_{16}–P_{14}P_{16}–P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}\)
is equal to _______.
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 ______.