1. Solve the system of given linear equations: x – y = 3 and 4x + 3y = -2.1) Solve the system of given linear equations: x – y = 3 and 4x + 3y = -2.

Show me the answers!

**Correct answer is option - 3**

**Explanation: **One of the given equations is x – y = 3. Now solving for any one variable, let’s say ‘x’ gives: x = 3 + y.

Substitute this ‘x’ value in the second equation to get: 4(3 + y) + 3y = -2.

This implies: 12 + 4y + 3y = -2 ==> 12 + 7y = -2 ==> 7y = -14 gives y = -14/7 = -2.

Substituting this ‘y’ value back in any one of the equations gives: x = 3 + y = 3 – 2 = 1.

Therefore, x = 1 and y = -2.

2. If the graph of a parabola of function f(x) has it’s vertex at (2, 3), then the vertex of the function f(x – 1) + 4 is at:

Show me the answers!

**Correct answer is option - 5**

**Explanation: **f(x - 1) +4 shows that the given function, f(x) has been horizontally shifted ‘1’ place to the right and vertically shifted ‘4’ places upward.

This shift makes the new vertex to be horizontally 2 + 1 = 3 and vertically 3 + 4 = 7.

Hence the new vertex is (3, 7).

3. If the discriminant of the quadratic equation, x^{2} + 6x + k = 0 is 12, then what is the value of ‘k’?

Show me the answers!

**Correct answer is option - 4**

**Explanation: **Discriminant of a quadratic equation in the form of ax^{2} + bx + c = 0 is b^{2} – 4ac.

The given quadratic equation, x^{2} + 6x + k = 0 has a = 1, b = 6 and c = k.

Discriminant given for the above equation is 12 ==>b^{2} – 4ac = 12.

This gives 6^{2} – 4k = 12 ==> 36 – 4k = 12 ==> 4k = 36 – 12 ==> 4k = 24 ==> k = 6.

4. If an exam paper consists of 6 questions where the students can select only true or false options, then in how many ways can the questions be answered if no questions are left blank?

Show me the answers!

**Correct answer is option - 4**

**Explanation: **For the 6 questions there are only ‘2’ options, either true or false.

So the number of ways the questions can be answered= 2 * 2 * 2 * 2 * 2 * 2 = 2^{6} = 64 possible ways.

5. The point where the equation, 5x – 6y – 7 = 0 cuts the Y-axis is:

Show me the answers!

**Correct answer is option - 3**

**Explanation: **Since y-intercept is the point where the graph cuts the Y-axis, in order to find the y-intercept of a given equation, plug-in x = 0.

This implies: 5(0) – 6y – 7 = 0 ==> -6y – 7 = 0 ==> -6y = 7.

This gives: y = -7/6.

Hence the point is (0, -7/6).

6. The endpoints of the radius of a circle has coordinates (1,5) and (-4, 3). What is the area of this circle?

Show me the answers!

**Correct answer is option - 3**

**Explanation: **Distance between any two coordinate points = √ [(x_{2} – x_{1)}^{2} + (y_{2} – y_{1})^{2}].

Hence the length of the radius = √ [(-4 – 1)^{2} + (3 – 5)^{2}]

This gives: √ [(-5)^{2} + (-2)^{2}] = √ (25 + 4) = √29.

Area of a circle = π * (radius)^{2} = π * (√29)^{2} = 29π.

7. ** **In a given group of 10 students, the average age of the group is 12 years. When 5 more students are added to this group, then the average age is increased by 2 years. What is the average age of the new students?

Show me the answers!

**Correct answer is option - 4**

**Explanation: **(Sum of the ages of the students)/ (Total number of students) = Average age or Mean.

This gives: S/10 = 12 ==> Sum of the ages, S = 10 * 12 = 120.

When 5 new students are added, let the sum of their ages be = x.

Then (S + x)/ 15 = 14 ==> (S + x) = 14 * 15 = 210 ==> S + x = 210.

Hence x = Sum of ages of the 5 new students = 210 – 120 = 90.

Average age or Mean of the 5 students = 90/5 = 18 years.