Further Mathematics | Study Mode

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Question 1
A
\({x : x \in R, x \neq 3}\) 
B
\({x : x \in R, x \neq 1}\) 
C
\({x : x \in R, x \neq 0}\) 
D
\({x : x \in R, x\neq -3}\) 
Explanation.
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\(f(x) = \frac{x}{3 - x} \)

f(x) has a defined value except at x = 3 where the function is undefined.


Correct Option:
A
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Question 2
A
\(\frac{6 + \sqrt{2}}{4}\) 
B
\(\frac{3 + \sqrt{6}}{4}\) 
C
\(\frac{\sqrt{2} - \sqrt{6}}{4}\) 
D
\(\frac{3 - \sqrt{6}}{4}\) 
Explanation.
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\(\cos (x + y) = \cos x \cos y - \sin x \sin y \)

\(\cos (60 + 45) = \cos 60 \cos 45 - \sin 60 \sin 45\)

= \(\frac{1}{2} \times \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}\)

= \(\frac{\sqrt{2} - \sqrt{6}}{4}\)


Correct Option:
C
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Question 3
A
\(1\frac{1}{2}\) 
B
\(1\frac{1}{4}\) 
C
\(2\frac{1}{4}\) 
D
\(2\frac{1}{2}\) 
Explanation.
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\(\frac{5}{\sqrt{2}} = \frac{5 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{5\sqrt{2}}{2}\)

\(\frac{\sqrt{8}}{8} = \frac{2\sqrt{2}}{8} = \frac{\sqrt{2}}{4}\)

\(\frac{5}{\sqrt{2}} - \frac{\sqrt{8}}{8} = (\frac{5}{2} - \frac{1}{4})\sqrt{2}\)

= \(\frac{9}{4}\sqrt{2} \)

= \(2\frac{1}{4}\sqrt{2}\)


Correct Option:
C
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Question 4
A
\(-1\) 
B
\(\frac{-1}{3}\) 
C
\(\frac{-3}{7}\) 
D
\(\frac{-5}{19}\) 
Explanation.
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\(16^{3x} = \frac{1}{4}(32^{x - 1})\)

\((2^{4})^{3x} = (2^{-2})((2^{5})^{x - 1})\)

\(2^{12x} = 2^{-2 + 5x - 5}\)

\(12x = -7 + 5x\)

\(7x = -7 \implies x = -1\)


Correct Option:
A
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Question 5
A
-2 
B
\(\frac{-1}{2}\) 
C
\(\frac{1}{2}\) 
D
Explanation.
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\(\frac{\log_{5} 8}{\log_{5} \sqrt{8}} = \frac{\log_{5} 8}{\log_{5} 8^{\frac{1}{2}}}\)

= \(\frac{\log_{5} 8}{\frac{1}{2}\log_{5} 8}\)

= \(\frac{1}{\frac{1}{2}} \)

= 2


Correct Option:
D
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Question 6
A
\(\frac{560}{243}\) 
B
\(\frac{841}{243}\) 
C
\(\frac{1120}{243}\) 
D
\(\frac{4481}{243}\) 
Explanation.
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\(^{10}C_{7 - 1} (2^{10 - 6}) (\frac{-1}{3})^{6}\)

\(\frac{10!}{(10 - 6)! 6!} \times 16 \times \frac{1}{243} \)

= \(210 \times 16 \times \frac{1}{729} \)

= \(\frac{1120}{243}\)


Correct Option:
C
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Question 7
A
\(x^{2} - 6x - 9 = 0\) 
B
\(x^{2} - 6x + 6 = 0\) 
C
\(x^{2} + 6x - 9 = 0\) 
D
\(x^{2} + 6x + 6 = 0\) 
Explanation.
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\((x - \alpha)(x - \beta) = 0\)

\((x - (3 - \sqrt{3}))(x - (3 + \sqrt{3})) = 0\)

\((x^{2} - (3 - \sqrt{3})x - (3 + \sqrt{3})x + (9 + 3\sqrt{3} - 3\sqrt{3} - 3) = 0\)

\(x^{2} - 3x - x\sqrt{3} - 3x + x\sqrt{3} + 6 = 0\)

\(x^{2} - 6x + 6 = 0\)


Correct Option:
B
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Question 8
A
-12 
B
-6 
C
D
Explanation.
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Using remainder theorem, since x - 3 is a factor, then

given \(2x^{2} - 2x + p\), f(3) = 0

\(2(3^{2}) - 2(3) + p = 0 \implies 18 - 6 = -p\)

\(p = -12\)


Correct Option:
A
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Question 9
A
110, 250 
B
110, 290 
C
200, 250 
D
250, 290 
Explanation.
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The valueof the sine of an angle is negative in the third and fourth quadrant. Hence options A and B are not the options.

\(\sin 250 = -\sin (250 - 180) = - \sin 70\)

\(\sin 290 = - \sin (360 - 290) = - \sin 70\)


Correct Option:
D
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Question 10
A
6 or 3 
B
-18 or -9 
C
-7 or 5 
D
-5 or 7 
Explanation.
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An fractionis undefined when the denominator has value = 0.

\(\frac{x^{2} - 9x + 18}{x^{2} + 2x - 35}\) is undefined when \(x^{2} + 2x - 35 = 0\)

\(x^2 + 7x - 5x - 35 = 0 \implies (x + 7)(x - 5) = 0\)

\(x = \text{-7 or 5}\)


Correct Option:
C
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