Greatest Integer Function Understanding the value of $[x]$ for given ranges. For $-1 \le a Interpretation of the Table The phrasing and structure of the question require careful interpretation. The most plausible interpretation that leads to one of the options is summing the terms in each row that do not have the '+1' modifier. 3. Calculation Row 1: $ [b] + [c] = 0 + 1 = 1$. Row 2: $[a] + [c] = -1 + 1 = 0$. Row 3: $[a] + [b] = -1 + 0 = -1$. The sum of these results is $1 + 0 + (-1) = 0$. 406. Which of the following is not a convex set? - Question not attempted
Asked in Asst. Statistical Officer (Agri. Deptt.) 2024 · Topic: Mechanics (Newton's Laws of Motion and their applications)
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- Definition of Convex Set A set is convex if the line segment joining any two points in the set lies entirely within the set. 2. Convex Shapes Ellipses, circles, and half-planes are convex. 3. Non-Convex Shape Parabolas do not define convex regions when used in inequalities of the form $y \le f(x)$ or $y \ge f(x)$ where $f(x)$ is a strictly convex or concave function. 4. Testing Convexity Choose two points in the set and check if their midpoint (or any point on the segment connecting them) lies within the set. If even one such segment has a point outside the set, the set is not convex. 407. The Fourier transform of the function f(x) = - Question not attempted
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- Vieta's Formulas Relate the coefficients of a polynomial to the sums and products of its roots. 2. Roots of a Cubic Equation For $ax^3 + bx^2 + cx + d = 0$, the roots $\alpha, \beta, \gamma$ satisfy: $ \alpha + \beta + \gamma = -b/a$ $\alpha\beta + \alpha\gamma + \beta\gamma = c/a$ $\alpha\beta\gamma = -d/a$ 3. Condition of Equal Roots If two roots are equal, say $\alpha, \alpha, \beta$, then these relations can be used to form a system of equations. 4. Solving the System Express one root in terms of the other and substitute into the remaining equations to find the specific root values. 409. The particular integral of the differential equation (D² – 1)y = e²ˣ + 3 cos (4x + 3), where D = $\frac{d}{dx}$ is - - Question not attempted
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- The system of linear equations $x + y + z = 2$, $2x + y - z = 3$ and $3x + 2y + kz = 4$ has a unique solution, if - (1) $k \neq 0$ (2) $k = 0$ (3) $-2 < k < 2$ (4) $-1 < k < 1$
- Condition for Unique Solution A system of linear equations $Ax = b$ has a unique solution if and only if the determinant of the coefficient matrix $A$ is non-zero ($\det(A) \neq 0$). 2. Coefficient Matrix Identify the matrix $A$ formed by the coefficients of the variables in the system of equations. 3. Determinant Calculation Calculate the determinant of the coefficient matrix. 4. Solving for k Set the determinant to be non-zero and solve for the variable $k$. 412. A force of constant magnitude is applied on a particle which is always perpendicular to the motion of the particle. The particle take place in a plane. Which quantity/quantities will remain constant in this motion? - (5) Question not attempted
- A body of weight w₁ is suspended from the ceiling of a room through a chain of weight w₂. The ceiling pulls the chain by a force -
- If the tension in the cable supporting an elevator is equal to the weight of the elevator, the elevator may be (A) going up with increasing speed. (B) going down with increasing speed. (C) going up with uniform speed. (D) going down with uniform speed. Which of the above statements are true ? - Question not attempted