Let me go through the questions and provide solutions for each one.

7. If one of the roots of the equation $x^2 - 3x + k = 0$ is 1, then the value of $k$ is:

The quadratic equation is given as: $x^2 - 3x + k = 0$ One of the roots is 1. By Vieta's formulas, for a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$, and the product of the roots is $\frac{c}{a}$.

• The sum of the roots is $1 + r = 3$ (since $a = 1$ and $b = -3$).
  • $r = 2$, so the roots are 1 and 2.
• The product of the roots is $1 \times 2 = k$.

Thus, $k = 2$.

Answer: (A) 2

8. If one of the roots of the cubic equation $3x^3 - 5x^2 - 11x - 3 = 0$ is $\frac{1}{3}$, then the other two roots are:

We are given that $\frac{1}{3}$ is a root of the cubic equation: $3x^3 - 5x^2 - 11x - 3 = 0$ We can use synthetic division or factorization to find the other roots.

Using synthetic division:

• Divide $3x^3 - 5x^2 - 11x - 3$ by $x - \frac{1}{3}$.
  • After performing synthetic division, the quotient will give a quadratic equation, whose roots can be found using the quadratic formula.

The two roots are $1$ and $-3$.

Answer: (C) 1 & -3

9. The shaded area is represented by which of the following options?

Looking at the graph, the shaded region is bound by the following lines:

• $x + y = 6$ (above the line),
• $2x - y = 0$ (below the line),
• $x > 0$.

The correct system of inequalities representing this region is: $x + y > 6, \quad 2x - y > 0, \quad x > 0$

Answer: (A) $x + y > 6; 2x - y > 0; x > 0$

10. A dietician recommends a mixture of two kinds of foods. How can the recommendation be expressed?

We are given the nutrients for two types of food:

The mixture should contain:

• At least 45 units of carbs,
• At least 25 units of protein,
• At least 15 units of fat,
• At least 15 units of fibre.

If $x$ units of Food-1 and $y$ units of Food-2 are mixed, the system of inequalities becomes: