A polygon is convex if, for every pair of points inside the polygon, the line segment joining them lies completely inside or on the polygon. Which one of the following is NOT a convex polygon?
A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like \(\underline{\hspace{2cm}}\)
Details of prices of two items P and Q are presented in the above table. The ratio of cost of item P to cost of item Q is 3:4. Discount is calculated as the difference between the marked price and the selling price. The profit percentage is calculated as the ratio of the difference between selling price and cost, to the cost. The formula for Profit Percentage is: \[ \text{Profit \%} = \frac{\text{Selling Price} - \text{Cost}}{\text{Cost}} \times 100 \] The discount on item Q, as a percentage of its marked price, is:
Let \( f: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \to \mathbb{R} \) be given by \( f(x) = \frac{\pi}{2} + x - \tan^{-1}x \). Consider the following statements: P: \( |f(x) - f(y)| < |x - y| \text{ for all } x, y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \).Q: \( f \) has a fixed point. Then:
Consider the following statements: P: \( d_1(x,y) = \left| \log \left( \frac{x}{y} \right) \right| \) is a metric on \( (0, 1) \). Q: \( d_2(x, y) = \begin{cases} |x| + |y|, & \text{if } x \neq y \\ 0, & \text{if } x = y \end{cases} \) is a metric on \( (0, 1) \). Then:
Let \( \Gamma \) denote the boundary of the square region \( R \) with vertices \( (0,0), (2,0), (2,2), (0,2) \), oriented in the counter-clockwise direction. Then \( \int_{\Gamma} (1 - y^2) dx + x \, dy = \underline{\hspace{1cm}}. \)
The number of 5-Sylow subgroups in the symmetric group \( S_5 \) of degree 5 is \( \underline{\hspace{1cm}}\).
Let \( I \) be the ideal generated by \( x^2 + x + 1 \) in the polynomial ring \( R = \mathbb{Z}_3[x] \), where \( \mathbb{Z}_3 \) denotes the ring of integers modulo 3. Then the number of units in the quotient ring \( R/I \) is \(\underline{\hspace{1cm}} \).
If the polynomial \[ p(x) = \alpha + \beta (x+2) + \gamma (x+2)(x+1) + \delta (x+2)(x+1)x \] interpolates the data \[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 2 \\ -1 & -1 \\ 0 & 8 \\ 1 & 5 \\ 2 & -34 \\ \hline \end{array} \] then \( \alpha + \beta + \gamma + \delta = \underline{\hspace{1cm}}\) .
Consider the Linear Programming Problem \( P \): \[ \text{Maximize } 2x_1 + 3x_2 \] subject to \[ 2x_1 + x_2 \leq 6, \] \[ -x_1 + x_2 \leq 1, \] \[ x_1 + x_2 \leq 3, \] \[ x_1 \geq 0 \text{ and } x_2 \geq 0. \] Then the optimal value of the dual of \( P \) is equal to \(\underline{\hspace{1cm}}\).
Consider the Linear Programming Problem \( P \): \[ \text{Minimize } 2x_1 - 5x_2 \] subject to \[ 2x_1 + 3x_2 + s_1 = 12, \] \[ -x_1 + x_2 + s_2 = 1, \] \[ -x_1 + 2x_2 + s_3 = 3, \] \[ x_1 \geq 0, x_2 \geq 0, s_1 \geq 0, s_2 \geq 0, \text{ and } s_3 \geq 0. \] If \[ \left[ \begin{array}{c} x_1 \\ s_1 \\ s_2 \\ s_3 \end{array} \right] \] is a basic feasible solution of \( P \), then \( x_1 + s_1 + s_2 + s_3 = \underline{\hspace{1cm}}. \)
Let \( H \) be a complex Hilbert space. Let \( u, v \in H \) be such that \( \langle u, v \rangle = 2 \). Then \[ \frac{1}{2\pi} \int_0^{2\pi} \| u + e^{it} v \|^2 e^{it} dt = \underline{\hspace{1cm}}. \]
