Mathematics (2015) | Study Mode

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Question 61
A

3(2 + \(\sqrt{5}\))

B

2(3 + \(\sqrt{5}\))

C

5(2 + \(\sqrt{3}\))

D

\(3\sqrt{3}\) + 1

Explanation

5 (2 - \(\sqrt{3}\))

Using conjugate surds

[5 (2 - \(\sqrt{3}\))]\(\times\)(2 + \(\sqrt{3}\)) (2 + \(\sqrt{3}\))]

[5(2+\(\sqrt{3}\))((2- \(\sqrt{3}\)))2]

[5(2 +\(\sqrt{3}\)) (2)2- (\(\sqrt{3}\))2]

[5(2 +\(\sqrt{3}\)) (4- 3)]

= 5(2 + \(\sqrt{3}\))


Correct option: C
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Question 62
A

(a + b)(a - b)

B

(a - 2 + b)

C

(a + 1)(a - 2 + b)

D

(a + b) 2

Explanation

The trinomial = \( a^2 - 4a + 4 \\

a^2 - b^2- 4a + 4 = (a^2 - 4a + 4) - b^2 \\

(a^2 - 2a - 2a + 4) - b^2 \\

[a(a - 2)- 2(a - 2)] - b^2 \\

(a - 2)2 - b^2 \\

(a - 2 + b)(a - 2 - b )\)


Correct option: B
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Question 63
A
\(\frac{2\sqrt{13}}{13}\)
B
\(\frac{3\sqrt{13}}{13}\)
C
\(\frac{4\sqrt{13}}{13}\)
D
\(\frac{6\sqrt{13}}{13}\)
Explanation
tan x = \(\frac{2}{3}\)(given), is illustrated in a right-angled \(\Delta\)

thus m2 = 22 + 32

= 4 + 9 = 13

m = \(\sqrt{13}\)

Hence, 2sin x = 2 x \(\frac{2}{m}\)

2 x\(\frac{2}{\sqrt{13}}\)

= \(\frac{4}{\sqrt{13}}\)

= \(\frac{4}{\sqrt{13}} = \frac{\sqrt{13}}{\sqrt{13}}\)

= \(\frac{4\sqrt{13}}{13}\)

Correct option: C
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Question 64
A

10.5cm

B

5cm

C

10cm

D

12cm

Explanation

The area of an ellipse (e) =ab/4

Let a rep. the length of its major axis = 14cm

Let b rep. the length of its minor axis = ?

= 3.142

Area of an ellipse = 132cm2

132 = (14 \(\times\) b) 4

132 = (3.142 \(\times\) 14b) 4

3.142 \(\times\) 14b =132 \(\times\) 4

b = (1324 \(\times\) 4) (3.142 \(\times\) 14)
= 528/43.988

= 12cm

The length of its minor axis (b) = 12cm


Correct option: D
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Question 65
A

1/2

B

1/3

C

2/5

D

3/4

Explanation

Prob.(A outcome) = \(\frac{1}{6}\), Prob.(B outcome) =

P(A B) = 1/12, Prob.(A \(\cup\) B)

Prob.(A \(\cup\) B) = Prob.(A) + Prob.(B) \(\cup\) Prob.(A B)

1/6 + 1/4 + 1/12

= (2 + 3 ? 1) 12

= 4/12

= 1/3

Prob.(A \(\cup\) B) = 1/3


Correct option: B
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Question 66
A

x = 1, y = 1/2

B

x = 2, y = 1/2

C

x =0, y = 1

D

x = 2, y = 1

Explanation

\(27^x 81^{(x + 2y)} = 9 \)

\(27^x = 9 \times 81^{(x+2y)} \)

\({(3^{3} )}^x =32 \times 3^{4(x + 2y)} \)

\( =3^{(2 + 4x + 8y)}\)

\(3^{3x} = 3^{ (2 + 4x + 8y)}\)

3x = 2 + 4x + 8y

3x - 4x - 8y = 2 ----- (1)

x + 4y = 0 ------- (2)

- 4y = 2

y = (- 2) 4 = -

y = -

Substitute the value of y into equation (2)

i.e x + 4y = 0

x + 4( - 1/2) = 0

x - 2 = 0

x = 2

- x = 2,y = -

Method II

\( 27^x 31^{(x + 2y) }= 9\)

3^{3x} \(\times\) \(3^{( - 4x - 8y)} = 32\)

\(3^{(3x - 8y)} = 32 \)

- x - 8y=2 ---- (1)

x + 4y = 0 ----- (2)

- 4 = 2

y= 2/4 =

y =

Substitute the value of y into equation 2

x + 4y=0

x + 4 (- 1) 2) = 0

x - 2 = 0

x = 2

x = 2, y =


Correct option: B
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Question 67
A

x = 1 twice

B

x = 0 or 1

C

x = 3 or 1

D

x = 2 twice

Explanation

-(2x + 2 ) - -x = 1

-(2x + 2) = 1 + -x

Square both sides

-(2x + 2)2 = (1 + -x)2

((2x + 2)2) = 1 + -x + -x+ x

2x + 2 = 1 + 2-x +x

Collect the like term together

2x - x + 2 - 1 = 2 -x
x + 1 = 2-x

Square both sides

(x + 1)2 = (2-x)2

x2 + 2x + 1 = 4x

x2 + 2x - 4x + 1 = 0

(x2- x)(x + 1)= 0

x(x - 1) -1(x - 1)

(x - 1)(x -1)

Either (x - 1) = 0 or (x - 1) = 0

x = 1 or 1

x = 1 twice


Correct option: A

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