Integral Calculus & Area · JEE Main previous year questions

2005 4 questions

1. Q262005 · 28 Apr 2005 · Shift 1

   If $a^2+b^2+c^2=-2$ and $$f(x)=\left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& 1+c^{2}x\end{array}\right|,$$ then $f(x)$ is a polynomial of degree :

   1. A.$1$
   2. B.$0$
   3. C.$3$
   4. D.$2$
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   Answer: D

2. Q282005 · 28 Apr 2005 · Shift 1

   $\int {\left\{\frac{\log x-1}{1+(\log x{)}^2}\right\}}^{2}dx$ is equal to

   1. A.$\frac{\log x}{(\log x{)}^2+1}+C$
   2. B.$\frac{x}{x^2+1}+C$
   3. C.$\frac{x{e}^x}{1+x^2}+C$
   4. D.$\frac{x}{(\log x{)}^2+1}+C$
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   Answer: D

3. Q302005 · 28 Apr 2005 · Shift 1

   The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is :

   1. A.$1$
   2. B.$2$
   3. C.$3$
   4. D.$4$
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   Answer: A

4. Q332005 · 28 Apr 2005 · Shift 1

   The value of ${\int }_{-\pi }^{\pi }\frac{{\cos }^{2}x}{1+a^x}dx$, $a>0$, is

   1. A.$a\pi$
   2. B.$\frac{\pi }{2}$
   3. C.$\frac{\pi }{a}$
   4. D.$2\pi$
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   Answer: B