Trigonometry · JEE Main previous year questions

2004 9 questions

1. Q12004 · 24 Apr 2004 · Shift 1

   Let $\alpha ,\beta$ be such that $\pi <\alpha -\beta <3\pi$. If $\sin \alpha +\sin \beta =-\frac{21}{65}$ and $\cos \alpha +\cos \beta =-\frac{27}{65}$ then the value of $\cos \frac{\alpha -\beta }{2}$ :

   1. A.$\frac{-6}{65}$
   2. B.$\frac{3}{\sqrt{130}}$
   3. C.$\frac{6}{65}$
   4. D.$-\frac{3}{\sqrt{130}}$
   Show answer

   Answer: D

2. Q32004 · 24 Apr 2004 · Shift 1

   A line makes the same angle $\theta$, with each of the $x$ and $z$ axes. If the angle $\beta$, which it makes with y-axis, is such that ${\sin }^2\beta =3{\sin }^2\theta$, then ${\cos }^2\theta$ equals :

   1. A.$\frac{2}{5}$
   2. B.$\frac{1}{5}$
   3. C.$\frac{3}{5}$
   4. D.$\frac{2}{3}$
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   Answer: C

3. Q282004 · 24 Apr 2004 · Shift 1

   $\int \frac{dx}{\cos x-\sin x}$ is equal to

   1. A.$\frac{1}{\sqrt{2}}\log \left|\tan \left(\frac{x}{2}+\frac{3\pi }{8}\right)\right|+C$
   2. B.$\frac{1}{\sqrt{2}}\log \left|\cot \left(\frac{x}{2}\right)\right|+C$
   3. C.$\frac{1}{\sqrt{2}}\log \left|\tan \left(\frac{x}{2}-\frac{3\pi }{8}\right)\right|+C$
   4. D.$\frac{1}{\sqrt{2}}\log \left|\tan \left(\frac{x}{2}-\frac{\pi }{8}\right)\right|+C$
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   Answer: A

4. Q302004 · 24 Apr 2004 · Shift 1

   The value of $I={\int }_0^{\pi /2}\frac{(\sin x+\cos x{)}^2}{\sqrt{1+\sin 2x}}dx$ is

   1. A.$3$
   2. B.$1$
   3. C.$2$
   4. D.$0$
   Show answer

   Answer: C

5. Q312004 · 24 Apr 2004 · Shift 1

   If ${\int }_0^{\pi }xf(\sin x)dx=A{\int }_0^{\pi /2}f(\sin x)dx$, then $A$ is

   1. A.$2\pi$
   2. B.$\pi$
   3. C.$\frac{\pi }{4}$
   4. D.$0$
   Show answer

   Answer: B

6. Q432004 · 24 Apr 2004 · Shift 1

   If $f:R\to S$, defined by $f(x)=\sin x-\sqrt{3}\cos x+1$, is onto, then the interval of $S$ is

   1. A.[-1, 3]
   2. B.[-1, 1]
   3. C.[0, 1]
   4. D.[0, 3]
   Show answer

   Answer: A

7. Q582004 · 24 Apr 2004 · Shift 1

   The sides of a triangle are $\sin \alpha ,\cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}$. Then the greatest angle of the triangle is :

   1. A.$150^{\circ }$
   2. B.$90^{\circ }$
   3. C.$120^{\circ }$
   4. D.$60^{\circ }$
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   Answer: C

8. Q592004 · 24 Apr 2004 · Shift 1

   A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $60^{\circ }$ and when he retires $40$ meters away from the tree the angle of elevation becomes $30^{\circ }$. The breadth of the river is :

   1. A.$60\mathrm{m}$
   2. B.$30\mathrm{m}$
   3. C.$40\mathrm{m}$
   4. D.$20\mathrm{m}$
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   Answer: D

9. Q612004 · 24 Apr 2004 · Shift 1

   The normal to the curve $x=a(1+\cos \theta ),y=a\sin \theta$ at ${}^{\prime }{\theta }^{\prime }$ always passes through the fixed point

   1. A.$(a,a)$
   2. B.$(0,a)$
   3. C.$(0,0)$
   4. D.$(a,0)$
   Show answer

   Answer: D