Q.4. Consider the system of linear equations in x, y, z :,,,Where is a differentiable function. If

Q: What is the correct answer for JEE Main Mathematics Q4?

The correct answer is B. Sol. For infinitely many solutions, the determinant of the coefficient matrix must be zero for all $t \in R$.$\begin{vmatrix} 1 & 2 & t \\ 6 & 1 & 5t \\ 3 & t^2 & f(t) \end{vmatrix} = 0$Expanding the determinant: $1(f(t) - 5t^3) - 2(6f(t) - 15t) + t(6t^2 - 3) = 0$$f(t) - 5t^3 - 12f(t) + 30t + 6t^3 - 3t = 0$$-11f(t) + t^3 + 27t = 0$$11f(t) = t^3 + 27t$$f(t) = \frac{t^3 + 27t}{11}$Now, we need to ch

Q: Q.4. Consider the system of linear equations in x, y, z :,,,Where is a differentiable function. If this system has infinitely many solutions for all , then f

The correct answer is B. Sol. For infinitely many solutions, the determinant of the coefficient matrix must be zero for all $t \in R$.$\begin{vmatrix} 1 & 2 & t \\ 6 & 1 & 5t \\ 3 & t^2 & f(t) \end{vmatrix} = 0$Expanding the determinant: $1(f(t) - 5t^3) - 2(6f(t) - 15t) + t(6t^2 - 3) = 0$$f(t) - 5t^3 - 12f(t) + 30t + 6t^3 - 3t = 0$$-11f(t) + t^3 + 27t = 0$$11f(t) = t^3 + 27t$$f(t) = \frac{t^3 + 27t}{11}$Now, we need to ch

Q: What are the options for this JEE Main Mathematics question?

A. is a constant function · B. is strictly increasing on R · C. is strictly decreasing on R · D. has two critical points

Question

Q.4. Consider the system of linear equations in x, y, z :,,,Where  is a differentiable function. If this system has infinitely many solutions for all , then f

FirstInTest · Accepted Answer

Sol. For infinitely many solutions, the determinant of the coefficient matrix must be zero for all $t \in R$.$\begin{vmatrix} 1 & 2 & t \ 6 & 1 & 5t \ 3 & t^2 & f(t) \end{vmatrix} = 0$Expanding the determinant: $1(f(t) - 5t^3) - 2(6f(t) - 15t) + t(6t^2 - 3) = 0$$f(t) - 5t^3 - 12f(t) + 30t + 6t^3 - 3t = 0$$-11f(t) + t^3 + 27t = 0$$11f(t) = t^3 + 27t$$f(t) = \frac{t^3 + 27t}{11}$Now, we need to check the nature of $f(t)$.$f'(t) = \frac{1}{11}(3t^2 + 27)$Since $3t^2 \ge 0$, $3t^2 + 27 > 0$ for all $t \in R$.Therefore, $f'(t) > 0$ for all $t \in R$.This means $f(t)$ is strictly increasing on R.

Answer

is a constant function

Answer

is strictly increasing on R

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is strictly decreasing on R

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has two critical points

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