NEET Test Series from KOTA - 10 Papers In MS WORD
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Sets, Relation and Function
117357
The range of the function \(y=3\) sin \(\left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) is
1 \([0, \sqrt{3 / 2}]\)
2 \([0,1]\)
3 \([0,3 / \sqrt{2}]\)
4 \([0, \infty]\)
Explanation:
C Given, \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) \(\frac{\pi^2}{16}-x^2>0\) \(x^2\lt \frac{\pi^2}{16}\) \(x\lt \frac{\pi}{4}\) Hence, domain is \(\left[0, \frac{\pi}{4}\right]\) When, \(\mathrm{x}=0\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-0}\right)\) \(y=3 \sin \frac{\pi}{4}\) \(y=3 \times \frac{1}{\sqrt{2}}=\frac{3}{\sqrt{2}}\) When, \(x=\frac{\pi}{4}\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\left(\frac{\pi}{4}\right)^2}\right)\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\frac{\pi^2}{16}}\right)\) \(y=3 \sin 0\) \(y=0\)Hence, range of the function is \(\left[0, \frac{3}{\sqrt{2}}\right]\)
WB JEE-2014
Sets, Relation and Function
117358
For the function \(f(x)=\left[\frac{1}{[x]}\right]\), where \([x]\) denotes the greatest integer less than or equal to \(x\), which of the following statements are true?
A Given, \(\{x \in R:|\cos x| \geq \sin x\} \cap\left[0, \frac{3 \pi}{2}\right]\) When we draw the graphs of the function \(|\cos x|\) and \(\sin\) \(\mathrm{x}\), it is clear that - \(|\cos x| \geq \sin x \text { when, }\) \(x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\) \(\therefore \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right] \cap\left[0, \frac{3 \pi}{2}\right]\) \(\Rightarrow \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\)
WB JEE-2015
Sets, Relation and Function
117360
Given that, \(x\) is a real number satisfying \(\frac{5 x^2-26 x+5}{3 x^2-10 x+3}\lt 0\), then
1 \(\mathrm{x}\lt \frac{1}{5}\)
2 \(\frac{1}{5}\lt x\lt 3\)
3 \(x>5\)
4 \(\frac{1}{5}\lt x\lt \frac{1}{3}\) or \(3\lt x\lt 5\)
117357
The range of the function \(y=3\) sin \(\left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) is
1 \([0, \sqrt{3 / 2}]\)
2 \([0,1]\)
3 \([0,3 / \sqrt{2}]\)
4 \([0, \infty]\)
Explanation:
C Given, \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) \(\frac{\pi^2}{16}-x^2>0\) \(x^2\lt \frac{\pi^2}{16}\) \(x\lt \frac{\pi}{4}\) Hence, domain is \(\left[0, \frac{\pi}{4}\right]\) When, \(\mathrm{x}=0\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-0}\right)\) \(y=3 \sin \frac{\pi}{4}\) \(y=3 \times \frac{1}{\sqrt{2}}=\frac{3}{\sqrt{2}}\) When, \(x=\frac{\pi}{4}\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\left(\frac{\pi}{4}\right)^2}\right)\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\frac{\pi^2}{16}}\right)\) \(y=3 \sin 0\) \(y=0\)Hence, range of the function is \(\left[0, \frac{3}{\sqrt{2}}\right]\)
WB JEE-2014
Sets, Relation and Function
117358
For the function \(f(x)=\left[\frac{1}{[x]}\right]\), where \([x]\) denotes the greatest integer less than or equal to \(x\), which of the following statements are true?
A Given, \(\{x \in R:|\cos x| \geq \sin x\} \cap\left[0, \frac{3 \pi}{2}\right]\) When we draw the graphs of the function \(|\cos x|\) and \(\sin\) \(\mathrm{x}\), it is clear that - \(|\cos x| \geq \sin x \text { when, }\) \(x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\) \(\therefore \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right] \cap\left[0, \frac{3 \pi}{2}\right]\) \(\Rightarrow \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\)
WB JEE-2015
Sets, Relation and Function
117360
Given that, \(x\) is a real number satisfying \(\frac{5 x^2-26 x+5}{3 x^2-10 x+3}\lt 0\), then
1 \(\mathrm{x}\lt \frac{1}{5}\)
2 \(\frac{1}{5}\lt x\lt 3\)
3 \(x>5\)
4 \(\frac{1}{5}\lt x\lt \frac{1}{3}\) or \(3\lt x\lt 5\)
117357
The range of the function \(y=3\) sin \(\left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) is
1 \([0, \sqrt{3 / 2}]\)
2 \([0,1]\)
3 \([0,3 / \sqrt{2}]\)
4 \([0, \infty]\)
Explanation:
C Given, \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) \(\frac{\pi^2}{16}-x^2>0\) \(x^2\lt \frac{\pi^2}{16}\) \(x\lt \frac{\pi}{4}\) Hence, domain is \(\left[0, \frac{\pi}{4}\right]\) When, \(\mathrm{x}=0\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-0}\right)\) \(y=3 \sin \frac{\pi}{4}\) \(y=3 \times \frac{1}{\sqrt{2}}=\frac{3}{\sqrt{2}}\) When, \(x=\frac{\pi}{4}\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\left(\frac{\pi}{4}\right)^2}\right)\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\frac{\pi^2}{16}}\right)\) \(y=3 \sin 0\) \(y=0\)Hence, range of the function is \(\left[0, \frac{3}{\sqrt{2}}\right]\)
WB JEE-2014
Sets, Relation and Function
117358
For the function \(f(x)=\left[\frac{1}{[x]}\right]\), where \([x]\) denotes the greatest integer less than or equal to \(x\), which of the following statements are true?
A Given, \(\{x \in R:|\cos x| \geq \sin x\} \cap\left[0, \frac{3 \pi}{2}\right]\) When we draw the graphs of the function \(|\cos x|\) and \(\sin\) \(\mathrm{x}\), it is clear that - \(|\cos x| \geq \sin x \text { when, }\) \(x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\) \(\therefore \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right] \cap\left[0, \frac{3 \pi}{2}\right]\) \(\Rightarrow \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\)
WB JEE-2015
Sets, Relation and Function
117360
Given that, \(x\) is a real number satisfying \(\frac{5 x^2-26 x+5}{3 x^2-10 x+3}\lt 0\), then
1 \(\mathrm{x}\lt \frac{1}{5}\)
2 \(\frac{1}{5}\lt x\lt 3\)
3 \(x>5\)
4 \(\frac{1}{5}\lt x\lt \frac{1}{3}\) or \(3\lt x\lt 5\)
117357
The range of the function \(y=3\) sin \(\left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) is
1 \([0, \sqrt{3 / 2}]\)
2 \([0,1]\)
3 \([0,3 / \sqrt{2}]\)
4 \([0, \infty]\)
Explanation:
C Given, \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-x^2}\right)\) \(\frac{\pi^2}{16}-x^2>0\) \(x^2\lt \frac{\pi^2}{16}\) \(x\lt \frac{\pi}{4}\) Hence, domain is \(\left[0, \frac{\pi}{4}\right]\) When, \(\mathrm{x}=0\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-0}\right)\) \(y=3 \sin \frac{\pi}{4}\) \(y=3 \times \frac{1}{\sqrt{2}}=\frac{3}{\sqrt{2}}\) When, \(x=\frac{\pi}{4}\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\left(\frac{\pi}{4}\right)^2}\right)\) \(y=3 \sin \left(\sqrt{\frac{\pi^2}{16}-\frac{\pi^2}{16}}\right)\) \(y=3 \sin 0\) \(y=0\)Hence, range of the function is \(\left[0, \frac{3}{\sqrt{2}}\right]\)
WB JEE-2014
Sets, Relation and Function
117358
For the function \(f(x)=\left[\frac{1}{[x]}\right]\), where \([x]\) denotes the greatest integer less than or equal to \(x\), which of the following statements are true?
A Given, \(\{x \in R:|\cos x| \geq \sin x\} \cap\left[0, \frac{3 \pi}{2}\right]\) When we draw the graphs of the function \(|\cos x|\) and \(\sin\) \(\mathrm{x}\), it is clear that - \(|\cos x| \geq \sin x \text { when, }\) \(x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\) \(\therefore \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right] \cap\left[0, \frac{3 \pi}{2}\right]\) \(\Rightarrow \quad x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]\)
WB JEE-2015
Sets, Relation and Function
117360
Given that, \(x\) is a real number satisfying \(\frac{5 x^2-26 x+5}{3 x^2-10 x+3}\lt 0\), then
1 \(\mathrm{x}\lt \frac{1}{5}\)
2 \(\frac{1}{5}\lt x\lt 3\)
3 \(x>5\)
4 \(\frac{1}{5}\lt x\lt \frac{1}{3}\) or \(3\lt x\lt 5\)
