Wednesday’s GATE CS set

Let $r$ be a root of the equation $x^2-4x+1=0$.

The value of $r^3+\frac{1}{r^3}$ is

A synchronous circuit has three positive-edge-triggered flip-flops $A$, $B$ and $C$ driven by a common clock, wired as follows ($\overline{X}$ is the complement output of flip-flop $X$):

The JK flip-flop holds for $J=K=0$, resets for $J=0,K=1$, sets for $J=1,K=0$ and toggles for $J=K=1$. Initially $(A,B,C)=(0,0,0)$, and the circuit is then clocked indefinitely.

Which ONE of the following values of $(A,B,C)$ can NEVER occur?

Consider the following C program. Assume a 2-D array is stored contiguously in row-major order, so pointer arithmetic that runs past the end of one row continues into the next row.

#include <stdio.h>
int main(void) {
    int a[3][4] = { {2, 5, 8, 1},
                    {7, 3, 6, 4},
                    {9, 0, 11, 10} };
    printf("%d %d", *(*(a + **a) + a[0][3]), *(a[0] + 6));
    return 0;
}

What does the program print?

An unordered array holds $64$ distinct integers. What is the minimum number of comparisons between array elements that suffices, in the worst case, to find both the largest and the second-largest element?

Let $w^R$ denote the reverse of a string $w$. Which ONE of the following languages is context-free but NOT deterministic context-free?