Further Mathematics | Study Mode

Join Discuss Share Question Share to WhatsApp
Question 2
A
\(-\tan \theta\) 
B
\(-\cos \theta\) 
C
\(\tan \theta\) 
D
\(\cos \theta\) 
Explanation.
Share Answer

\(\frac{\cos 2\theta - 1}{\sin 2\theta}\)

\(\cos (x + y) = \cos x \cos y - \sin x \sin y \implies \cos 2\theta = \cos^{2} \theta - \sin^{2} \theta\)

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

\(\sin 2\theta = 2\sin \theta \cos \theta\)

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

= \(\frac{-2 \sin^{2} \theta}{2\sin \theta \cos \theta} = \frac{- \sin \theta}{\cos \theta}\)

= \(-\tan \theta\)


Correct Option:
A
Join Discuss Share Question Share to WhatsApp
Question 3
A
\(-1 \leq x \leq 3\) 
B
\(x \geq 3\) and \(x \leq -1\) 
C
\(x \geq 3\) or \(x < -1\) 
D
\(-1 \leq x < 3\) 
Explanation.
Share Answer

\(x^{2} - 2x \geq 3 \implies x^{2} - 2x - 3 \geq 0\)

\(x^{2} + x - 3x - 3 = (x + 1)(x - 3) \geq 0\)

\(x = -1 ; x = 3\)

Check: \(x = -1 : (-1)^{2} - 2(-1) = 1 + 2 \geq 3\) (satisfied)

\(-1 < x < 3 : 0^{2} - 2(0) = 0 \geq 3\) (not satisfied)

\(x < -1 : (-2)^{2} - 2(-2) = 4 + 4 = 8 \geq 3\) (satisfied)

\(x = 3 : 3^{2} - 2(3) = 9 - 6 = 3 \geq 3\) (satisfied)

\(x > 3 : 4^{2} - 2(4) = 16 - 8 = 8 \geq 3\) (satisfied)

\(\therefore x^{2} - 2x \geq\text{3 is satisfied in the region x}\leq \text{-1 and x} \geq 3\)


Correct Option:
B
Join Discuss Share Question Share to WhatsApp
Question 4
A
\(27\sqrt{2}\) 
B
\(27\sqrt{6}\) 
C
\(81\sqrt{2}\) 
D
\(81\sqrt{6}\) 
Explanation.
Share Answer

\(T_{n} = ar^{n - 1}\) (Geometric progression)

\(a = \sqrt{6}, r = \frac{T_{2}}{T_{1}} = \frac{3\sqrt{2}}{\sqrt{6}} \)

\(r = \frac{\sqrt{18}}{\sqrt{6}} = \sqrt{3}\)

\(\therefore T_{8} = (\sqrt{6})(\sqrt{3})^{8 - 1} \)

= \((\sqrt{6})(27\sqrt{3}) = 27\sqrt{18} = 81\sqrt{2}\)


Correct Option:
C
Join Discuss Share Question Share to WhatsApp
Question 5
A
300 
B
240 
C
120 
D
60 
Explanation.
Share Answer

In order to do this, simply find the option in the range where only the cos is +ve. This occurs in the range \(270 \leq x \leq 360\).

Check: \(\sin 300 = - \sin 60 = \frac{-\sqrt{3}}{2}\)

\(\cos 300 = \cos 60 = \frac{1}{2}\)


Correct Option:
A
Join Discuss Share Question Share to WhatsApp
Question 6
A

-3

 
B

0

 
C

\(\frac{5}{6}\)

 
D

1

 
Explanation.
Share Answer

\(\log_{10} (\frac{1}{3} + \frac{1}{4}) + 2\log_{10} 2 + \log_{10} (\frac{3}{7})\)

\(\frac{1}{3} + \frac{1}{4} = \frac{7}{12}\)

= \(\log_{10} (\frac{7}{12} \times 2^{2} \times \frac{3}{7})\)

= \(\log_{10} 1 = 0\)


Correct Option:
B
Join Discuss Share Question Share to WhatsApp
Question 7
A
8i + j 
B
2i - j 
C
-2i - 3j 
D
-8i - j 
Explanation.
Share Answer

\(\overrightarrow{SQ} = \overrightarrow{SR} + \overrightarrow{RQ}\)

\(\overrightarrow{RQ} = -\overrightarrow{QR} = - (3i + 2j) = -3i - 2j\)

\(\overrightarrow{SQ} = (-5i + 3j) - 3i - 2j = -8i + j\)


Correct Option:
A
Join Discuss Share Question Share to WhatsApp
Question 8
A
-8 
B
-4 
C
D
Explanation.
Share Answer

If (x + 1) is a factor, then f(-1) = 0.

\((-1)^{3} + p(-1)^{2} + (-1) + 6 = 0\)

\(-1 + p - 1 + 6 = 0 \implies p + 4 = 0\)

\(p = -4\)


Correct Option:
B
Join Discuss Share Question Share to WhatsApp
Question 9
A
-8 
B
-2 
C
D
Explanation.
Share Answer

Given: \(f(x + 1) = x^{3} + 3x^{2} - 4x + 2\).

\(f(2) = f(x + 1) \implies x + 1 = 2; x = 1\)

\(f(2) = 1^{3} + 3(1)^{2} - 4(1) + 2 = 1 + 3 - 4 + 2 = 2\)


Correct Option:
C
Join Discuss Share Question Share to WhatsApp
Question 10
A
B
C
D
Explanation.
Share Answer

The equation of a circle is given as: \((x - a)^{2} + (y - b)^{2} = r^{2}\)

Expanding, we have: \(x^{2} + y^{2} - 2ax - 2by + a^{2} + b^{2} = r^{2}\)

\(\implies x^{2} + y^{2} - 2ax - 2by = r^{2} - a^{2} - b^{2}\)

Comparing with the given equation: \(3x^{2} + 3y^{2} + 24x - 12y = 15\)

Making the coefficients of \(x^{2}\) and \(y^{2}\) = 1, we have

\(x^{2} + y^{2} + 8x - 4y = 5\)

\(2a = -8 \implies a = -4\)

\(2b = 4 \implies b = 2\)

\(r^{2} - a^{2} - b^{2} = 5 \implies r^{2} = 5 + (-4)^{2} + (2)^{2} = 5 + 16 + 4 = 25\)

\(\therefore r = 5\)


Correct Option:
D
Next Page
Question Map
Quiz link
Share Further Mathematics with your friends
Share to WhatsApp Share Quiz CBT mode Study mode copy link