Topics

General Knowledge
General Science
General English
Aptitude
General Computer Science
General Intellingence and Reasoning
Current Affairs
Exams
Elementary Mathematics
English Literature

\( \Large log \cos x \sin x \ge 2 \) and \( \Large x \epsilon \left[ 0, 3 \pi \right] \), then \( \Large \sin x \) lines in the interval:

A) \( \Large \left[ 0, \frac{1}{2} \right] \)
B) \( \Large \left[\frac{\sqrt{5}-1}{2}, 1 \right] \)
C) \( \Large \left[ 0, \frac{\sqrt{5}-1}{2} \right] \)
D) none of these

Correct Answer:



C) \( \Large \left[ 0, \frac{\sqrt{5}-1}{2} \right] \)

Description for Correct answer:

\( \Large log \cos x \sin x \ge 2 => \sin x \le \cos^{2}x \)

=>\( \Large sinx \le 1 - sin^{2}x \)

=> \( \Large \sin^{2}x + \sin x - 1 \le 0 \)

=> \( \Large \left(\sin x + \frac{1}{2}\right)^{2} - \frac{5}{4} \le 0 \)

Also by definition of logarithm

\( \Large \sin x > 0, \cos x > 0, \cos x ≠ 1 \)

=> \( \Large \sin x + \frac{1}{2} \le \frac{\sqrt{5}}{2} \)

=> \( \Large 0 < \sin x \le \frac{\sqrt{5}-1}{2} \)

Part of solved Logarithms questions and answers : >> Elementary Mathematics >> Logarithms

Comments

Similar Questions

1). If \( \Large log_{2} x + log_{2} y \ge 6 \), then the least value of \( \Large \left(x + y\right) \) is

A). 4

B). 9

C). 16

D). 32

2). If \( \Large 4^{log_{3}\frac{31}{2}} + 9^{log2^{2}} = 10^{log_{x}83} \), (x belongs to R), then x is:

A). 4

B). 9

C). 10

D). none of these

3). Which of the following is not true?

A). \( \Large \frac{1}{log_{3} \pi } + \frac{1}{log_{4} \pi }>2 \)

B). \( \Large log_{3}5 \)

C). \( \Large \sqrt{8x} = \frac{10}{3} => x = 16 \)

D). \( \Large log_{x} \left(a^{2}+1\right)<0 \), (a?0) then 0

4). If \( \Large y = \frac{1}{a^{1-log ax}} \) and \( \Large z = \frac{1}{a^{1+log_{a}y}} \) then x is equal to:

A). \( \Large \frac{1}{a^{1+log_{a}z}} \)

B). \( \Large \frac{1}{a^{z+log_{a}z}} \)

C). \( \Large \frac{1}{a^{1-log_{a}z}} \)

D). none of these

5). The least value of n in order that the sum of first n terms of an infinite series \( \Large 1 + \frac{3}{4} + \left(\frac{3}{4}\right)^{2} + \left(\frac{3}{4}\right)^{3}+ .... \) should differ from the sum of the series by less then \( \Large 10^{-6} \) is
[\( \Large\ Given\ log_{10}2=0.30103, log_{10}3=0.47712 \)]

A). 14

B). 27

C). 53

D). 57

6). Solution of the equation \( \Large x log^{x^{2}} = log3 \left(x+y\right) \) and \( \Large x^{2}+y^{2} = 65 \) is:

A). x = 8, y = 1

B). x = 1, y = 8

C). (x=8, y=1); (x=1, y=8)

D). none of the above

7). The identity \( \Large log_{a}n\ log_{b}n\ +\ log_{b}n\ log_{c}n\ +\ log_{c}n\ log_{a}n \) ls:

A). \( \Large \frac{log_{a}n\ log_{b}n\ log_{c}n}{log_{abc}n} \)

B). \( \Large \frac{log_{abc}n}{log_{a}n} \)

C). \( \Large \frac{log_{b}n}{log_{abc}n} \)

D). none of these

8). The least value of the expression \( \Large 2\ log_{10}x\ -\ logx\ \left(0.01\right) \) for x > 1, is:

A). 10

B). 2

C). -0.01

D). none of the above

9). The angle of elevation of the top of a tower at a point on the ground is \( \Large 30 ^{\circ} \). If on walking 20m towards the tower the angle of elevation becomes \( \Large 60 ^{\circ} \) then the heights of the tower is:

A). 10 m

B). \( \Large \frac{10}{\sqrt{3}} m \)

C). \( \Large 10\sqrt{3} m \)

D). None of these

10). When the elevation of sun changes from \( \Large 45 ^{\circ} \) to \( \Large 30 ^{\circ} \) the shadow of a tower increases by 60 m, The height of the tower is:

A). \( \Large 30\sqrt{3} m \)

B). \( \Large 30 \left(\sqrt{2} + 1\right) m \)

C). \( \Large 30 \left(\sqrt{3} - 1\right) m \)

D). \( \Large 30 \left(\sqrt{3} + 1\right) m \)