A value of c for which the minimum value of f(x) = x2 – 4cx + 8c is greater than the maximum value of g(x) = – x2 + 3cx – 2c, is
\(\dfrac{1}{2}\)
Minimum of \(f\): \(f(x)=(x-2c)^{2}-4c^{2}+8c \Rightarrow f_{\min}=8c-4c^{2}\).
Maximum of \(g\): \(g(x)=-\left(x-\tfrac{3c}{2}\right)^{2}+\tfrac{9c^{2}}{4}-2c \Rightarrow g_{\max}=\tfrac{9c^{2}}{4}-2c\).
\(8c-4c^{2}>\tfrac{9c^{2}}{4}-2c \Rightarrow 10c>\tfrac{25c^{2}}{4} \Rightarrow c(25c-40)<0 \Rightarrow 0<c<1.6\).
Among the given values only \(c=\tfrac{1}{2}\) lies in this range.
Hence, \(\dfrac{1}{2}\).
Similar Questions
The number of district pairs of integers (x, y) satisfying the inequalities
x > y ≥ 3 and x + y < 14 is
A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is
The number of non-negative integer values of \(k\) for which the quadratic equation x2 – 5x + \(k\) = 0 has only integer roots, is
In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is
Stocks A, B and C are priced at rupees 120, 90 and 150 per share, respectively. A trader holds a portfolio consisting of 10 shares of stock A, and 20 shares of stocks B and C put together. If the total value of her portfolio is rupees 3300, then the number of shares of stock B that she holds, is
If a – 6b + 6c = 4 and 6a + 3b – 3c = 50, where a, b and c are real numbers, the value of 2a + 3b – 3c is
In the set of consecutive odd numbers {1, 3, 5, ….., 57}, there is a number of \(k\) such that the sum of all the elements less than \(k\) is equal to the sum of all the elements greater than \(k\). Then, \(k\) equals.
The number of distinct integers \(n\) for which \(\log_{\left(\frac{1}{4}\right)}\left(n^{2}-7n+11\right)>0\), is
Let 3 ≤ x ≤ 6 and [x2] = [x]2, where [x] is the greatest integer not exceeding x. If set S represents all feasible values of x, then a possible subset of S is
For any natural number \(k\), let \(a_{k}=3^{k}\). The smallest natural number \(m\) for which \(\left\{(a_{1})^{1}\times(a_{2})^{2}\times\ldots\times(a_{20})^{20}\right\}<\left\{a_{21}\times a_{22}\times\ldots\times a_{(20+m)}\right\}\), is
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
