JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
The least positive value of \(a\) for which the equation \(2 \mathrm{x}^{2}+(\mathrm{a}-10) \mathrm{x}+\frac{33}{2}=2 \mathrm{a}\) has real roots is
- A \(4\)
- B \(5\)
- C \(8\)
- D \(9\)
Answer & Solution
Correct Answer
(C) \(8\)
Step-by-step Solution
Detailed explanation
\(D \geq 0 \Rightarrow(a-10)^{2}-4 \times 2 \times\left(\frac{33}{2}-2 a\right) \geq 0\) \(\Rightarrow a^{2}-4 a-32 \geq 0\) \(\Rightarrow a \in(-\infty, 4] \cup[8, \infty)\)
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 \(A\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right] .\) If \(A^{-1}=\alpha I+\beta A, \alpha, \beta \in R, I\) is a \(2 \times 2\) identity matrix, then \(4(\alpha-\beta)\) is equal to:JEE Mains 2021 Medium
- If the constant term in the binomial expansion of \(\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9\) is \(-84\) and the Coefficient of \(x^{-3 \ell}\) is \(2^\alpha \beta\), where \(\beta < 0\) is an odd number, Then \(|\alpha \ell-\beta|\) is equal toJEE Mains 2023 Hard
- Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a twice differentiable function such that \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbf{R}\). If \(f^{\prime}(0)=4 \mathrm{a}\) and \(f\) satisfies \(f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0\), \(\mathrm{a}\gt0\), then the area of the region \(\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(\mathrm{a} x), 0 \leq x \leq 2\}\) is:JEE Mains 2025 Hard
- The value of \(36(4 \cos ^2 9^{\circ}-1)(4 \cos ^2 27^{\circ}-1) (4\cos ^2 81^{\circ}-1) (4 \cos ^2 243^{\circ}-1)\) isJEE Mains 2023 Hard
- Let three real numbers \(a, b, c\) be in arithmetic progression and \(\mathrm{a}+1, \mathrm{~b}, \mathrm{c}+3\) be in geometric progression. If \(\mathrm{a}>10\) and the arithmetic mean of \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) is \(8\) , then the cube of the geometric mean of \(a, b\) and \(c\) isJEE Mains 2024 Medium
- Let \(A=\left\{n \in N \mid n^{2} \leq n+10,000\right\}, B=\{3 k+1 \mid k \in N\}\) and \(C=\{2 k \mid k \in N\}\), then the sum of all the elements of the set \(A \cap(B-C)\) is equal to \(.....\)JEE Mains 2021 Medium
More PYQs from JEE Mains
- The absolute difference between the squares of the radii of the two circles passing through the point \((-9,4)\) and touching the lines \(x+y=3\) and \(x-y=3\), is equal to ______.JEE Mains 2025 Medium
- Let the vectors \((2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}\) \((1+\mathrm{b}) \hat{i}+2 \mathrm{b} \hat{j}-\mathrm{b} \hat{k}\) and \((2+\mathrm{b}) \hat{i}+2 \mathrm{b} \hat{j}+(1-\mathrm{b}) \hat{k}\) \(\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R}\) be co-planar. Then which of the following is true?JEE Mains 2021 Hard
- The number of \(3-\)digit numbers, formed using the digits \(2,3,4,5\) and \(7\) , when the repetition of digits is not allowed, and which are not divisible by \(3\) , is equal to ..........JEE Mains 2024 Medium
- Let \(f : R \rightarrow R\) and \(g : R \rightarrow R\) be two functions defined by \(f(x)=\log _{e}\left(x^{2}+1\right)-e^{-x}+1\) and \(g(x)=\frac{1-2 e^{2 x}}{e^{x}}\). Then, for which of the following range of \(\alpha\), the inequality \(f\left(g\left(\frac{(\alpha-1)^{2}}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)\) holds?JEE Mains 2022 Hard
- Let \(A=\{(x, y): 2 x+3 y=23, x, y \in N\}\) and \(B=\{x:(x, y) \in A\}\). Then the number of one-one functions from \(\mathrm{A}\) to \(\mathrm{B}\) is equal to ................JEE Mains 2024 Medium
- Let \(A=\{-4,-3,-2,0,1,3,4\}\) and \(R =\{( a , b ) \in A\) \(\times A : b =| a |\) or \(\left.b ^2= a +1\right\}\) be a relation on \(A\). Then the minimum number of elements, that must be added to the relation \(R\) so that it becomes reflexive and symmetric, is \(........\).JEE Mains 2023 Medium