JEE Mains · Maths · STD 12 - 12. linear programming
If the set of all solutions of \(|x^2 + x - 9| = |x| + |x^2 - 9|\) is \([\alpha, \beta] \cup [\gamma, \infty)\), then \((\alpha^2 + \beta^2 + \gamma^2)\) is equal to:
- A \(9\)
- B \(18\)
- C \(36\)
- D \(72\)
Answer & Solution
Correct Answer
(B) \(18\)
Step-by-step Solution
Detailed explanation
The given equation is \(|x^2 + x - 9| = |x| + |x^2 - 9|\). Let \(A = x\) and \(B = x^2 - 9\). Then \(A + B = x^2 + x - 9\). The equation is of the form \(|A + B| = |A| + |B|\), which holds true if and only if \(A \cdot B \ge 0\). Substituting the values of \(A\) and \(B\):…
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
- Let \(A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right),\) where \([t]\) denotes the greatest integer less than or equal to \(\mathrm{t}\). If \(\operatorname{det}(\mathrm{A})=192\), then the set of values of \(\mathrm{x}\) is the intervalJEE Mains 2021 Hard
- The number of positive integers \(k\) such that the constant term in the binomial expansion of \(\left(2 x^{3}+\frac{3}{x^{k}}\right)^{12}, x \neq 0\) is \(2^{8} \cdot \ell\), where \(\ell\) is an odd integer, is......JEE Mains 2022 Medium
- If the sum of the series \(20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .\) upto \(n ^{ th }\) term is \(488\) and the \(n^{\text {th }}\) term is negative, thenJEE Mains 2020 Hard
- If \(A\) is a \(3×3\) non-singular matrix such that \(AA’=A’A \) and \( B=A^{-1}A’\) then \(BB’ \) equalsJEE Mains 2014 Medium
- Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\) if the diagonal \(\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}\) then the area of the parallelogram is equal toJEE Mains 2024 Medium
- Let a curve \(y = y ( x )\) pass through the point \((3,3)\) and the area of the region under this curve, above the \(x\)-axis and between the abscissae \(3\) and \(x(>3)\) be \(\left(\frac{y}{x}\right)^{3}\). If this curve also passes through the point \((\alpha, 6 \sqrt{10})\) in the first quadrant, then \(\alpha\) is equal to \(........\)JEE Mains 2022 Hard
More PYQs from JEE Mains
- If a tangent to the ellipse \(x^{2}+4 y^{2}=4\) meets the tangents at the extremities of its major axis at \(\mathrm{B}\) and \(\mathrm{C}\), then the circle with \(\mathrm{BC}\) as diameter passes through the point:JEE Mains 2021 Hard
- If the perpendicular bisector of the line segment joining the points \(P (1,4)\) and \(Q ( k , 3)\) has \(y\)- intercept equal to \(-4,\) then a value of \(k\) isJEE Mains 2020 Medium
- For any real number \(x\), let \([ x ]\) denote the largest integer less than equal to \(x\). Let \(f\) be a real valued function defined on the interval \([-10,10]\) by \(f(x)=\left\{\begin{array}{cl}x-[x], & \text { if }(x) \text { is odd } \\ 1+[x]-x & \text { if }(x) \text { is even }\end{array}\right.\) Then the value of \(\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x d x\) is.JEE Mains 2022 Hard
- Let \(\overrightarrow{ a }=\hat{ i }+2 \hat{ j }-\hat{ k }, \overrightarrow{ b }=\hat{ i }-\hat{ j }\) and \(\overrightarrow{ c }=\hat{ i }-\hat{ j }-\hat{ k }\) be three given vectors. If \(\overrightarrow{ r }\) is a vector such that \(\overrightarrow{ r } \times \overrightarrow{ a }=\overrightarrow{ c } \times \overrightarrow{ a }\) and \(\overrightarrow{ r } \cdot \overrightarrow{ b }=0,\) then \(\overrightarrow{ r } \cdot \overrightarrow{ a } \quad\) is equal to ...........JEE Mains 2021 Medium
- If \(f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6\), then \(f(1)\) is equal to :JEE Mains 2025 Medium
- Let slope of the tangent line to a curve at any point \(P ( x , y )\) be given by \(\frac{ xy ^{2}+ y }{ x } .\) If the curve intersects the line \(x+2 y=4\) at \(x=-2,\) then the value of \(y ,\) for which the point \((3, y )\) lies on the curve, is ..... .JEE Mains 2021 Hard