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Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 49

Warning: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead in /home/aarlxcom/sscradar.com/wp-content/plugins/latex/latex.php on line 47

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Trigonometry questions for SSC CGL and CHSL pdf download

Hieght and Distance questions for SSC cgl  and chsl pdf - Click Here to Download

1.If  tan θ + cot θ = 2, then value of θ is

  1. 45°
  2. 60°
  3. 90°
  4. 30°

Answer : 1. 45°

Solution:

tan θ + cotθ = 2

⇒ tan θ  + = 2

= 2

⇒ tan2 θ + 1= 2 tan θ

⇒ (tan2 θ - 2 tan θ + 1) = 0

⇒ (tan θ - 1)2 = 0

⇒ tan θ - 1 = 0

⇒ tan θ = 1

⇒ tan θ = tan 45°

⇒ θ = 45°

2. If Cos πx = x2 - x+ , then the value of x will be

  1. 0
  2. 1
  3. -1
  4. 2

Answer : 2. 1

solution :

⇒ Cos πx = x2 - x +

⇒ Cos πx = x2 - 2.x. + - +

⇒ Cos πx = (x - )2 + 1 > 1

3. The numerical value of - Sec2 27° + - Cosec2 27° is

  1. 1
  2. 2
  3. -1
  4. 0

Answer : 2. 0

Solution :

- Sec 2 27° + - Cosec2 27°

= 1 + tan2 63° - Sec2 27° +Cos2 63° - Cosec2 27°

= 1 + tan2 (90°-27°) - sec2 27° + Cosec2 (90°-27°) - Cosec2 27°

= 1 + Cot2 27° - Sec2 27° + Sec2 27° - Cosec2 27°

= Cosec2 27° - Cosec2 27°

= 0

4. If x = , then is equal to

  1. x - 1

Answer ; 2.

Solution:

⇒ x =

⇒ x = x

⇒ x = Cos θ

⇒ x = Cos θ     [  1-Sin2 θ = Cos2 θ ]

⇒ x =

=

5. If Sin = x2 - 2x = 2, then the value of x is

  1. 0
  2. 1
  3. -1
  4. None of these

Answer : 2. 1

Solution ;

Putting x = 1

⇒ Sin = 1 - 2+2

⇒ Sin = 1

6. The value of + - 8 Cos2 60° is

  1. 0
  2. 1
  3. 2
  4. -1

Answer :1. 0

Solution :

+ - 8 Cos2 60° = + - 8 Cos2 60°

+ - 8 x

⇒ 1 + 1 - 2 = 0

7. The  value of Cos2 30° + Sin2 60° + Tan2 45° + Sec2 60° + Cos 0° is

  1. 4
  2. 5
  3. 6
  4. 7

Answer : 4. 7

Solution : Cos2 30°+Sin2 60°+Tan2 45° + Sec2 60° + Cos 0°

= ()2+ ()2 + 1+ 22 + 1

= + + 1 + 4 +1

=

=

=

 

7. If Cos x + Cos2 x = 1, then Sin2 x + 2 Sin6 x + Sin4 x is equals to

  1. 0
  2. 3
  3. 2
  4. 1

Answer : 4. 1

solution :

⇒ Cos x + Cos2 x = 1

⇒ Cos x = 1-Cos 2 x

⇒ Cos x = Sin 2x

= Sin8 x + 2 Sin6 x + Sin  x

= (Sin 4 x + Sin2 x)2

= (Cos2 x + Sin2 x)

= 1

8. if x = p Cosec θ and y = q Cot θ then value of - is

  1. Sin2 θ
  2. 2 Tan θ
  3. 1
  4. 0

Answer : 3. 1

Solution :

= Cot2 θ

Now,

- = Cosec2 θ - Cot2 θ

- = 1

9. The value of Sin 22°+Sin2 68°+Cot2 30° is

  1. 4
  2. 3

Answer : 1. 4

Solution :

Sin2 22°+Sin2 68°+ Cot2 30°

=  Sin (90°-68°)+Sin2 68°+ Cot2 30°

=  Cos2 68°+Sin2 68°+ Cot2 30°

= 1+Cot2 30°

= 1+3 = 4

10.  If = 2, then value of Sin θ is

Answer : 2.

solution :

By Using Componendo and Dividendo,

=

= 2

  =

  =

⇒ Sin θ =

 

11. The value of the followng :

Cos 24°+Cos 55°+Cos 125°+Cos 204°+Cos 300°

is

  1. 2
  2. 1

Answer :   4.

Solution :

Cos 24°+Cos 55°+Cos 125°+Cos 204°+Cos 300°

= Cos 24°+Cos 55°+Cos (180°-55°)+Cos (180°-24°)+Cos (360°-60°) [ Cos (180°-θ) = -Cosθ, Cos (360°-θ) = Cosθ]

= Cos 24°+ Cos 55°- Cos 55°- Cos 24°+Cos 60°

= Cos 60° = 

12. If Tan A + Cot A = 2, then the value of Tan10 A + Cot10 A is

  1. 4
  2. 2
  3. 6
  4. 1

Answer : 2. 2

Solution : Tan A + Cot A = 2

⇒ TanA  = 2

= 2

⇒ Tan2 2A= 1 - 2Tan A = 0

⇒ (TanA - 1) 2 = 0

⇒ Tan A = 1 or Cot A =1

Thus Tan10 A + Cot10 A= 2

Trigonometry questions for ssc cgl pdf download


Height and Distances Important Questions for SSC CGL 

1. P and Q are two points observed from the top of Building with 10√3 m high. If the angle of depression of the point are complementary and PQ = 20 m, then distance of P from the building is

  1. 25 m
  2. 45 m
  3. 30 m
  4. 40 m

Answer: iii. 30 m

Solution :

Trigonometry questions for SSC CGL and CHSL pdf download
From ∆ ABQ

Tan θ = .............................(i)

From ∆ ABP

Tan (90°-θ) = ...............................(ii)

Now Multiplying eqn. (i) and eqn. (ii), we will get

Tan θ . Tan (90°-θ) = X

⇒ Tan θ . Cot θ =              [Tan θ . Cot θ = 1]  [Tan (90°-θ) = Cot θ ]

⇒ X2 + 20X = 300

⇒ X2 + 20X - 300 = 0

⇒ X (X+30) - 10 (X+30) = 0

⇒ (X-10) (X+30) = 0

⇒ X= 10 meters.

Total Distance = X+2 = 30 meters ...

2. From top of a cliff 90 m high, the angle of depression of the top and bottom of towers are observed to be 30° and 60° respectively. What is the height of tower ?

  1. 45 m
  2. 60 m
  3. 75 m
  4. 30 m

Answer : b. 60 m

Solution :

Trigonometry questions for SSC CGL and CHSL pdf download

From Δ ABD,

Tan 60° =

=

⇒ X =

⇒ X = 30√3 m

From Δ ACE,

Tan 30° =

=

⇒Y = 60 m ...

3. A car traveling on a straight road leading to a tower. From a point at a distance of 500 m from the tower, as seen by a driver, the angle of elevation of the top of tower is 30°. After driving toward the tower for 10 seconds, the angle of elevation of the top of the tower as seen by the driver is found to be 60°. Then speed of a car is

  1. 135 Km/hr
  2. 110 Km/hr
  3. 120 Km/hr
  4. 90  Km/hr

Answer : c. 120 Km/hr

Solution :
Trigonometry questions for SSC CGL and CHSL pdf download
From Δ ABD,

Tan 30° =

⇒ h =

From Δ ABC,

Tan 60° =

=

=

⇒ 3(500-X) = 500

⇒ 1500-3X = 500

⇒X = metres

⇒X = Km

Now Speed =

⇒ Speed =

=

= 120 Km/hr . . .

4. From an aeroplane just over a straight road , the angles of depression of two consecutive Km stone situated at the opposite side of the aeroplane were found to be 60° and 30° respectively. The height (in Kms) of the aeroplane from the road at that instant was : (Given √3 = 1.73)

  1. 0.433
  2. 8.66
  3. 4.33
  4. 0.866

Answer : d. 0.866

Solution :

Trigonometry questions for SSC CGL and CHSL pdf download

From Δ OAC,

Tan 60° =   [Tan 60°= √3]

=

⇒ h = √3 X ............(i)

From Δ OBC,

Tan 30° =      [Tan 30°= ]

⇒ (2-X) = √3 h

⇒ h = ............(ii)

From eqn. (i) and (ii)

√3 X =

⇒ X =

⇒ 3X = 2-X

⇒ 4X = 2

⇒ X=

From eqn. (i)

⇒ h = √3 X

Now putting the value of X= in above equation we will get,

h =

⇒ h =   (Given √3 = 1.73)

⇒ h = 0.86 (approx.)

5. Two towers A and B have length 45 m  and 15 m respectively. The angle of elevation from the bottom of the towers B to top of the tower A is 60°. If the angle of elevation from the bottom of the tower of A to the top of the tower B is θ then the value of sin θ is :

Answer :b.

Solution :

Trigonometry questions for SSC CGL and CHSL pdf download

From Δ DBA,

Tan 60° =

⇒ X = ..........(i)

From Δ BAC,

Tan θ =

⇒ X = ..........(ii)

Now from eqn .(i) and (ii), we will get

=

⇒ Tan θ =

⇒ Tan θ =

⇒ θ = 30°          [Tan 30° = ]

⇒ Sin θ = Sin 30° =

6. A kite is flying at the height of 75 m from the ground . The string makes an angle θ (Cot θ = ) with the level ground. Assuming that there is no stack in the string the length of the string is equal to :

  1. 85 m
  2. 65 m
  3. 75 m
  4. 40 m

Answer : a. 85 m

Solution : Given h = 75 m

Trigonometry questions for SSC CGL and CHSL pdf download

Cot θ =

We know

Cosec θ =

⇒ Cosec θ =

⇒ Cosec θ =

⇒ Cosec θ =

⇒ Cosec θ =

⇒ Sin θ = ...........(i)

From Δ ACB,

⇒ Sin θ =

⇒ Sin θ =

=   [ From eqn .(i) Sin θ = ]

=

⇒ X = 85 m

7. A person of height 6 ft. wants to pluck a fruit which is on a ft.  high tree. If the person is standing ft. away from the base of the tree, then at what angle should he throw a stone so that the stone hit the fruit ?

75°
30°
45°
60°

Answer : b. 30°

Solution :

Trigonometry questions for SSC CGL and CHSL pdf download

AB = CD = 6 Ft.

Now, DE = CE - DC

⇒ DE = { - 6} ft.

⇒ DE = ft.

From Δ DBE,

Tan θ =

⇒ Tan θ = = Tan 30°

⇒  θ = 30°

8. The angle of the top of a vertical tower situated perpendicular on a plane is observed as 60 from a point P on the same plane. From another point Q, 10 m vertically above the point P, the angle of depression of the foot of the tower is 30. The height of tower is :

  1. 15 m
  2. 30 m
  3. 20 m
  4. 25 m

Answer : b. 30 m

Solution :

Trigonometry questions for SSC CGL and CHSL pdf download

PQ = AC=10 m

From Δ QAP,

Tan 30° =

=

=

⇒ X = 10√3 m .........(i)

From Δ BAP,

Tan 60° =

⇒ √3 =

⇒ X = ...........(ii)

From (i) and (ii), we will get

10√3 =

⇒ Y = 30 - 10

⇒ Y = 20 m

Now, h = (10 + 20) m

⇒ h = 30 m

Trigonometry questions for ssc cgl pdf download