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.
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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:
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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’?
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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?
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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:
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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?
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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?
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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.