1. Equation of a line if 4x – 5y + 6 = 0. What is the slope of the line perpendicular to the given line?
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Correct answer is option - 1
Explanation: The given line can be rearranged as 5y = 4x + 6 which gives y = (4/5) x + 6/5. This is in the form of y = mx + b where ‘m’ is the slope and ‘b’ is the y-intercept.
Hence the slope of the given line, m_{1} = 4/5.
If two lines are perpendicular to each other, then the product of the slopes m_{1} * m_{2} = -1.
This gives the slope of the second line, m_{2} = -5/4.
2. For what value of ‘m’ will the ratio (4 + m)/ (10 + m) = 3/5?
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Correct answer is option - 3
Explanation: Cross -multiply the denominators to the other side of the equation.
5 (4 + m) = 3 (10 + m) which gives 20 + 5m = 30 + 3m.
Combining the like terms gives: 5m – 3m = 30 – 20.
2m = 10 ==> m = 5.
3. By how much does the expression 5x^{3} + y^{2} lesser than 5y^{2} + x^{3}if x = 2 and y = 3?
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Correct answer is option - 4
Explanation: When x = 2 and y = 3, 5x^{3} + y^{2} = 5(2)^{3} + 3^{2} = 40 + 9 = 49
And, 5y^{2} + x^{3} = 5(3)^{2} + 2^{3} = 45 + 8 = 53.
Therefore the difference is 53 – 49 = 4.
4. Which of the following is a factor of the polynomial, 3x^{2} – 10x + 8?
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Correct answer is option - 3
Explanation: The given quadratic expression can be expanded as: 3x^{2} – 6x – 4x + 8.
This gives 3x (x – 2) – 4 (x – 2).
This implies (x – 2) (3x – 4) are the factors of the given polynomial.
Since x – 2 is not in the options, 3x - 4 is the answer!
5. What is the value of √-(-5)^{2}?
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Correct answer is option - 2
Explanation:
√-(-5)^{2} = √-(25) = √25 * √-1 = √25 * i = ± 5i.
Since there is no -5i in the options, the answer is 5i.
6. A circle has the center at the origin (0, 0) and passes through the point (6,8). Which of the following gives the equation of the circle?
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Correct answer is option - 5
Explanation: Equation of a circle with center at (h, k) is (x – h)^{2} + (y – k)^{2} = r^{2}.
(x – 0)^{2} + (y – 0)^{2} = r^{2} ==> x^{2} + y^{2} = r^{2}.
Substituting the point (6, 8) we get, 36 + 64 = r^{2}. This gives r = 10.
Therefore the equation of the circle is x^{2} + y^{2} = 100.
7. ABCD is an isosceles trapezoid and the angle CAB = 115°. The angle ‘x’ as shown in the diagram is:
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Correct answer is option - 4
Explanation: An isosceles trapezoid has side AB parallel to side CD and side AC = side BD.
Since it is an isosceles triangle, angle CAB = angle DBA = 115° and angle ACD = angle BDC = x
This gives 115° + 115° + x + x = 360° which implies 230° + 2x = 360°.
2x = 360° - 230° = 130° which gives, x = 130°/2 = 65°.