If \(\displaystyle\lim _{n \rightarrow \infty}\left(\sqrt{n^2-n-1}+n \alpha+\beta\right)=0 \) then \(8(\alpha+\beta)\) is equal to :
- 4
- $-8$
-4
- 8
The Correct Option is C
Solution and Explanation
Top Questions on linear inequalities
- For the system of linear equations
\(x+y+z=6\)
\(\alpha x+\beta y+7 z=3\)
\(x+2 y+3 z=14\).
which of the following is NOT true ?- JEE Main - 2023
- Mathematics
- linear inequalities
If [x+6]+[x+3] ≤ 7 and let call its solution as set A and set B is the solution of inequality 35x-8 < 3-3x.
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- The inequality \(\frac{2x-1}{3}\geq\frac{3x-2}{4}-\frac{(2-x)}{5}\) holds for x belonging to
- KEAM - 2023
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Let α,β, and γ be real numbers. Consider the following system of linear equations:
x + 2y + z = 7
x + αz = 11
2x - 3y + βz = γ
Match each entry in List I to the correct entries in List IIList I List II (P) If β=\(\frac{1}{2}\)(7α - 3) and \(\gamma\)=28, then the system has (1) a unique solution (Q) If β=\(\frac{1}{2}\)(7α - 3) and \(\gamma\)\(\neq\)28, then the system has (2) no solution (R) If β\(\neq\)\(\frac{1}{2}\)(7α - 3) where \(\alpha\)=1 and \(\gamma\)\(\neq\)28, then the system has (3) infinitely many solutions (S) If β\(\neq\)\(\frac{1}{2}\)(7α - 3) where \(\alpha\)=1 and \(\gamma\)=28, then the system has (4) x = 11, y = - 2 and z = 0 as a solution (5) x = -15 , y = 4 and z = 0 as a solution - JEE Advanced - 2023
- Mathematics
- linear inequalities
- The shaded region in the figure given is the solution of which of the inequations?
- KCET - 2023
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- linear inequalities
Questions Asked in JEE Main exam
- A convex lens of focal length 40 cm forms an image of an extended source of light on a photo-electric cell. A current \( I \) is produced. The lens is replaced by another convex lens having the same diameter but focal length 20 cm. The photoelectric current now is:
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- Dual nature of matter
- For a sparingly soluble salt \( \text{AB}_2 \), the equilibrium concentrations of \( \text{A}^{2+} \) ions and \( \text{B}^- \) ions are \( 1.2 \times 10^{-4} \, \text{M} \) and \( 0.24 \times 10^{-3} \, \text{M} \), respectively. The solubility product of \( \text{AB}_2 \) is:
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- Method of Preparation of Diazoniun Salts
The portion of the line \( 4x + 5y = 20 \) in the first quadrant is trisected by the lines \( L_1 \) and \( L_2 \) passing through the origin. The tangent of an angle between the lines \( L_1 \) and \( L_2 \) is:
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- Tangents and Normals
- Let \[ \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \quad \vec{b} = -\hat{i} + \hat{k}, \quad \vec{c} = \beta \hat{j} - \hat{k}, \] where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha, \beta) \) for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_1, \beta_1) \) and \( (\alpha_2, \beta_2) \). Then \( \alpha_1^2 + \beta_1^2 - \alpha_2 \beta_2 \) is equal to:
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- The molecular formula of second homologue in the homologous series of monocarboxylic acid is
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Concepts Used:
Linear Inequalities
The expressions where any two values are compared by the inequality symbols such as, ‘<’, ‘>’, ‘≤’ or ‘≥’ are called linear inequalities. These values could be numerical or algebraic or a combination of both expressions. A system of linear inequalities in two variables involves at least two linear inequalities in the identical variables. After solving linear inequality we get an ordered pair. So generally, in a system, the solution to all inequalities and the graph of the linear inequality is the graph representing all solutions of the system.
How to Solve Inequalities in Maths?
Follow the below steps to solve all types of inequalities:
- At the vert first, write the inequality as an equation.
- Solve the provided equation for one or more values.
- Now, display all the values obtained in the number line.
- Use open circles to show the excluded values on the number line.
- Find the interim.
- At the moment, take any random value from the interval and substitute it in the inequality equation to check whether the values reassure the inequality equation.
- Intervals that reassure the inequality equation are the solutions of the given inequality equation.