Further Mathematics | Study Mode

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Question 1
A
B
C
-4 
D
-5 
Explanation.
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\(\frac{1}{5^{-y}} = 25(5^{4-2y})\)

\(\implies 5^{y} = (5^{2})(5^{4-2y})\)

\(5^{y} = 5^{2+4-2y}\)

Comparing bases, we have

\(y = 6 - 2y\)

\(3y = 6 \implies y = 2\)


Correct Option:
B
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Question 2
A
\(\sin^{2} \theta\) 
B
\(\sec^{2} \theta\) 
C
\(\tan^{2} \theta\) 
D
\(\cos^{2} \theta\) 
Explanation.
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\((1 + \sin\theta)(1 - \sin\theta) = 1 - \sin \theta + \sin \theta - \sin^{2} \theta\)

\(= 1 - \sin^{2} \theta\)

Recall, \(\cos^{2} \theta + \sin^{2} \theta = 1\)

\(\therefore 1 - \sin^{2} \theta = \cos^{2} \theta\).


Correct Option:
D
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Question 3
A
B
C
\(\frac{1}{3}\) 
D
\(\frac{1}{4}\) 
Explanation.
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When you have two lines, \(y_{1}, y_{2}\), perpendicular to each other, the product of their slopes = -1.

\(3x + 4y + 6 = 0 \implies 4y = -6 - 3x\)

\(\therefore y = \frac{-6}{4} - \frac{3}{4}x\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{-3}{4}\)

Also, \(4x - by + 3 = 0 \implies by = 4x + 3\)

\(y = \frac{4}{b}x + \frac{3}{b}\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{4}{b}\)

\(\frac{-3}{4} \times \frac{4}{b} = -1 \implies \frac{4}{b} = \frac{4}{3}\)

\(b = 3\)


Correct Option:
B
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Question 4
A
B
C
D
Explanation.
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\((x * y) = \frac{x+y}{2}\)

\((3 * b) = \frac{3+b}{2}\)

\(x \circ y = \frac{x^{2}}{y}\)

\((\frac{3+b}{2}) \circ 48 = \frac{(\frac{3+b}{2})^{2}}{48} = \frac{1}{3}\)

\(\frac{(3+b)^{2}}{48 \times 4} = \frac{1}{3}\)

\((3 + b)^{2} = \frac{48 \times 4}{3} = 64\)

\(b^{2} + 6b + 9 = 64 \implies b^{2} + 6b +9- 64 = 0\)

\(b^{2} + 6b - 55 = 0 \implies b^{2} - 5b + 11b - 55 = 0\)

\(b(b - 5) + 11(b - 5) = 0 \implies (b - 5) = \text{0 or (} b + 11) = 0\)

Since b > 0, b - 5 = 0

b = 5.


Correct Option:
C
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Question 5
A
-67 
B
-61 
C
61 
D
67 
Explanation.
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\(f(x) = 3x^{3} + 8x^{2} + 6x + k\)

\(f(2) = 3(2^{3}) + 8(2^{2}) + 6(2) + k = 1\)

\(\implies 24 + 32 + 12 + k = 1\)

\(68 + k = 1 \therefore k = 1 - 68 = -67\)


Correct Option:
A
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Question 6
A
\(\frac{5}{4}\) 
B
\(\frac{3}{5}\) 
C
\(1\) 
D
\(\frac{2}{3}\) 
Explanation.
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\(8^{x} (\frac{1}{4})^{y} = 1\)

\((2^{3})^{x} (2^{-2})^{y} = 2^{0}\)

\(2^{3x - (-2y)} = 2^{0}\)

\(\implies 3x + 2y = 0 .... (1)\)

\(\log_{2}(x - 2y) = 1\)

\( x - 2y = 2^{1} = 2 ..... (2)\)

Solving equations 1 and 2,

\(x = \frac{1}{2}, y = \frac{-3}{4}\)

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


Correct Option:
A
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Question 7
A
\(7 + \sqrt{2}\) 
B
\(7 + 7\sqrt{2}\) 
C
\(1 - 7\sqrt{2}\) 
D
\(1 + \sqrt{2}\) 
Explanation.
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\(\frac{1 + \sqrt{8}}{3 - \sqrt{2}}\)

Rationalizing by multiplying through with \(3 + \sqrt{2}\),

\((\frac{1 + \sqrt{8}}{3 - \sqrt{2}})(\frac{3 + \sqrt{2}}{3 + \sqrt{2}}) = \frac{3 + \sqrt{2} + 3\sqrt{8} + 4}{9 - 2}\)

= \(\frac{3 + \sqrt{2} + 3\sqrt{4 \times 2} + 4}{7} \)

= \(\frac{7 + 7\sqrt{2}}{7} = 1 + \sqrt{2}\)


Correct Option:
D
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Question 8
A
64.245 
B
61.255 
C
60.255 
D
60.245 
Explanation.
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\((1.98)^{6} = (1 + 0.98)^{6} = 1 + 6(0.98) + 15(0.98)^{2} + 20(0.98)^{3} + 15(0.98)^{4} + 6(0.98)^{5} + (0.98)^{6}\)

\(\approxeq 1 + 5.88 + 14.406 + 18.823 + 13.836 + 5.424 + 0.886 \)

= \(60.255\)


Correct Option:
C
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Question 9
A
\(2x - 3\) 
B
\(3x + 1\) 
C
\(x - 2\) 
D
\(3x + 2\) 
Explanation.
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To get the third factor, take the product of the other 2 factors and then divide the main equation by their product.


Correct Option:
A
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Question 10
A
-3 and 5 
B
5 and -5 
C
3 and -3 
D
-5 and 3 
Explanation.
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Given an exponential sequence, say \(a, b, c,...\), as consecutive terms, then \(\sqrt{a \times c} = b\).

\(\therefore 2, (k+1), 8 \implies \sqrt{2 \times 8} = k + 1\)

\(k + 1 = \pm{4} \implies k = \text{-5 or 3}\)


Correct Option:
D
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