JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
If the system of equations \(x+y+z=6 \,; \,2 x+5 y+\alpha z=\beta \,; \, x+2 y+3 z=14\) has infinitely many solutions, then \(\alpha+\beta\) is equal to.
- A \(8\)
- B \(36\)
- C \(44\)
- D \(48\)
Answer & Solution
Correct Answer
(C) \(44\)
Step-by-step Solution
Detailed explanation
\(x+y+z=6\) \(2 x+5 y+\alpha z=\beta \quad(1)\) \(x+2 y+3 z=14\) \(x+y=6-z\) \(x+2 y=14-3 z\) On solving \(x=z-2 \Rightarrow y=8-2 z\) in \((2)\) \(2(z-2)+5(8-2 z)+\alpha z=\beta\) \((\alpha-8) z=\beta-36\) For having infinite solutions \(\alpha-8=0 \quad \& \quad \beta-36=0\)…
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
- If \(\alpha, \beta\) are the roots of the equation \(x^{2}-\left(5+3 \sqrt{\log _{3} 5}-5 \sqrt{\log _{5} 3}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0\) then the equation, whose roots are \(\alpha+\frac{1}{\beta} \text { and } \beta+\frac{1}{\alpha} \text {, }\)JEE Mains 2022 Hard
- Let \(A, B\) and \(C\) be three events, which are pair-wise independence and \(\bar E\) denotes the complement of an event \(E\) . If \(P(A \cap B \cap C) = 0\) and \(P(C) > 0,\) then \(P[(\bar A \cap \bar B)|\,C]\) is equal toJEE Mains 2018 Hard
- If \(\mathrm{x}=2 \sin \theta-\sin 2 \theta\) and \(\mathrm{y}=2 \cos \theta-\cos 2 \theta\) ; \(\theta \in[0,2 \pi],\) then \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) at \(\theta=\pi\) is :JEE Mains 2020 Hard
- A tangent to the hyperbola \(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{2} = 1\) meets \(x-\) axis at \(P\) and \(y-\) axis at \(Q\). Lines \(PR\) and \(QR\) are drawn such that \(OPRQ\) is a rectangle (where \(O\) is the origin). Then \(R\) lies onJEE Mains 2013 Hard
- If \([\mathrm{x}]\) be the greatest integer less than or equal to \(\mathrm{x}\), then \(\sum_{\mathrm{n}=8}^{100}\left[\frac{(-1)^{n} \mathrm{n}}{2}\right]\) is equal to:JEE Mains 2021 Easy
- Let \(y=y(x)\) be the solution curve of the differential equation
\(x\left(x^2+e^x\right) d y+\left(e^x(x-2) y-x^3\right) d x=0, x \gt 0\) passing through the point \((1,0)\). Then \(y(2)\) is equal to :JEE Mains 2025 Medium
More PYQs from JEE Mains
- The area (in sq. units) of the region bounded by the parabola, \(y = x^2 + 2\) and the lines, \(y = x + 1, x = 0\) and \(x = 3\), isJEE Mains 2019 Hard
- The value of the integral \(\int \frac{\sin \theta \cdot \sin 2 \theta\left(\sin ^{6} \theta+\sin ^{4} \theta+\sin ^{2} \theta\right) \sqrt{2 \sin ^{4} \theta+3 \sin ^{2} \theta+6}}{1-\cos 2 \theta} d \theta\) is (where \(c\) is a constant of integration)JEE Mains 2021 Hard
- Bag \(I\) contains \(3\) red,\(4\) black and \(3\) white balls and Bag \(II\) contains \(2\) red,\(5\) black and \(2\) white balls. One ball is transferred from Bag \(I\) to Bag \(II\) and then a ball is draw from Bag \(II\). The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red,is.JEE Mains 2022 Medium
- If the value of the integral \(\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,d x=\alpha e^{-1}+\beta\) where \(\alpha, \beta \in R, 5 \alpha+6 \beta=0\), and \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(x\); then the value of \((\alpha+\beta)^{2}\) is equal to :JEE Mains 2021 Hard
- Let a line \(L: 2 x+y=k, k\,>\,0\) be a tangent to the hyperbola \(x^{2}-y^{2}=3 .\) If \(L\) is also a tangent to the parabola \(y^{2}=\alpha x\), then \(\alpha\) is equal to :JEE Mains 2021 Hard
- If \(\sin x+\sin ^2 x=1, x \in\left(0, \frac{\pi}{2}\right)\), then \(\left(\cos ^{12} x+\tan ^{12} x\right)+3\left(\cos ^{10} x+\tan ^{10} x+\cos ^8 x+\tan ^8 x\right)+\left(\cos ^6 x+\tan ^6 x\right)\) is equal to :JEE Mains 2025 Medium