JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
If the system of linear equations:
\(x+y+z=6\),
\(x+2y+5z=10\),
\(2x+3y+\lambda z=\mu\)
has infinitely many solutions, then the value of \(\lambda+\mu\) equals:
- A \(12\)
- B \(16\)
- C \(22\)
- D \(28\)
Answer & Solution
Correct Answer
(C) \(22\)
Step-by-step Solution
Detailed explanation
For the system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix \(\Delta\) must be zero, and \(\Delta_x = \Delta_y = \Delta_z = 0\). \(\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 2 & 3 & \lambda \end{vmatrix} = 0\) Expanding…
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 \(f(\mathrm{x})=\left|2 \mathrm{x}^2+5\right| \mathrm{x}|-3|, \mathrm{x} \in \mathrm{R}\). If \(\mathrm{m}\) and \(\mathrm{n}\) denote the number of points where \(f\) is not continuous and not differentiable respectively, then \(m+n\) is equal to :JEE Mains 2024 Medium
- If \(A=\left(\begin{array}{ll}{2} & {2} \\ {9} & {4}\end{array}\right)\) and \(I=\left(\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right),\) then \(10 A^{-1}\) is equal toJEE Mains 2020 Hard
- If the length of the perpendicular from the point \((\beta , 0, \beta )\, (\beta \neq 0)\) to the line \(\frac{x}{1} = \frac{{y - 1}}{0} = \frac{{z + 1}}{{ - 1}}\) is \(\sqrt {\frac{3}{2}} \), then \(\beta \) is equal toJEE Mains 2019 Medium
- If \(tan\, A\) and \(tan\, B\) are the roots of the quadratic equation, \(3x^2 - 10x - 25 = 0\) then the value of \(3\, sin^2\, (A +B)- 10\, sin\,(A +B). cos\,(A+ B)- 25\, cos^2\, (A+B)\) isJEE Mains 2018 Hard
- The tangent and the normal lines at the point \((\sqrt 3,1)\) to the circle \(x^2 + y^2 = 4\) and the \(x -\) axis form a triangle. The area of this triangle (in square units) isJEE Mains 2019 Hard
- The remainder when \((2023)^{2023}\) is divided by \(35\) is \(..........\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- Line \(L_1\) of slope 2 and line \(L_2\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \(\mathrm{P}_1, \mathrm{P}_2, \ldots . \mathrm{P}_{12}\) are 12 points on line \(L_1\) and \(Q_1, Q_2, \ldots . . Q_9\) are 9 points on line \(L_2\). Then the total number of triangles, that can be formed having vertices at three of the 22 points \(\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots \mathrm{P}_{12}\), \(\mathrm{Q}_1, \mathrm{Q}_2, \ldots . \mathrm{Q}_9\), is:JEE Mains 2025 Easy
- If \(\int \limits_{\frac{1}{3}}^3\left|\log _e x\right| d x=\frac{m}{n} \log _e\left(\frac{n^2}{e}\right)\), where \(m\) and \(n\) are coprime natural numbers, then \(m ^2+ n ^2-5\) is equal to \(............\).JEE Mains 2023 Hard
- Let \(A=\{1,2,3, \ldots, 10\}\) and \(B=\left\{\frac{m}{n}: m, n \in A, m \lt n\right.\) and \(\left.\operatorname{gcd}(m, n)=1\right\}\). Then \(n(B)\) is equal to :JEE Mains 2025 Medium
- Two equal sides of an isosceles triangle are along \(-x+2 y=4\) and \(x+y=4\). If m is the slope of its third side, then the sum, of all possible distinct values of \(m\), is :JEE Mains 2025 Medium
- If \(y=y(x)\) is the solution of the differential equation \(x \frac{d y}{d x}+2 y=x e^{x}, y(1)=0\) then the local maximum value of the function \(z(x)=x^{2} y(x)-e^{x}\), \(x \in R\) isJEE Mains 2022 Hard
- Let \(\mathrm{f}:[0,3] \rightarrow \mathrm{R}\) be defined by \(f(x)=\min \{x-[x], 1+[x]-x\}\) where \([\mathrm{x}]\) is the greatest integer less than or equal to \(\mathrm{x}\). Let \(\mathrm{P}\) denote the set containing all \(x \in[0,3]\) where \(f\) is discontinuous, and \(Q\) denote the set containing all \(x \in(0,3)\) where \(f\) is not differentiable. Then the sum of number of elements in \(\mathrm{P}\) and \(\mathrm{Q}\) is equal to \(......\)JEE Mains 2021 Hard