Trigonometry · JEE Main previous year questions

Let $f_k(x)=\frac{1}{k}({\sin }^{k}x+{\cos }^{k}x)$ where $x\in R$ and $k\ge 1$. Then $f_4(x)-f_6(x)$ equals :

1. A.$\frac{1}{4}$
2. B.$\frac{1}{12}$
3. C.$\frac{1}{6}$
4. D.$\frac{1}{3}$

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point $O$ on the ground is $45^{\circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to $30^{\circ }$. Then the speed (in m/s) of the bird is :

1. A.$20\sqrt{2}$
2. B.$20\left(\sqrt{3}-1\right)$
3. C.$40\left(\sqrt{2}-1\right)$
4. D.$40\left(\sqrt{3}-\sqrt{2}\right)$

The value of $\cos 255^{\circ }+\sin 195^{\circ }$ is

1. A.$\frac{\sqrt{3}-1}{2\sqrt{2}}$
2. B.$\frac{\sqrt{3}-1}{\sqrt{2}}$
3. C.$-\frac{\sqrt{3}-1}{\sqrt{2}}$
4. D.$\frac{\sqrt{3}+1}{\sqrt{2}}$

$f(x)=\int \frac{dx}{{\sin }^{6}x}$ is a polynomial of degree

1. A.$5\text{ in }\cot x$
2. B.$5\text{ in }\tan x$
3. C.$3\text{ in }\tan x$
4. D.$3\text{ in }\cot x$

If $f^{\prime }(x)=\sin (\log x)$ and $y=f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ equals

1. A.$\sin \left[\log \left(\frac{2x+3}{3-2x}\right)\right]$
2. B.$\frac{12}{(3-2x{)}^2}$
3. C.$\frac{12}{(3-2x{)}^2}\sin \left[\log \left(\frac{2x+3}{3-2x}\right)\right]$
4. D.$\frac{12}{(3-2x{)}^2}\cos \left[\log \left(\frac{2x+3}{3-2x}\right)\right]$

If in a triangle $ABC$, $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$, then $\cos A$ is equal to

1. A.$5/7$
2. B.$1/5$
3. C.$35/19$
4. D.$19/35$

Let $y(x)$ be a solution of $\frac{2+\sin x}{1+y}\frac{dy}{dx}=\cos x$. If $y(0)=2$, then $y\left(\frac{\pi }{2}\right)$ equals

1. A.$\frac{5}{2}$
2. B.2
3. C.$\frac{7}{2}$
4. D.3

Let $\cos (\alpha +\beta )=\frac{4}{5}$ and $\sin (\alpha -\beta )=\frac{5}{13}$, where $0\le \alpha ,\beta \le \frac{\pi }{4}$. Then $\tan 2\alpha =$

1. A.$\frac{56}{33}$
2. B.$\frac{19}{12}$
3. C.$\frac{20}{7}$
4. D.$\frac{25}{16}$

The area bounded by the curves $y=\cos x$ and $y=\sin x$ between the ordinates $x=0$ and $x=\frac{3\pi }{2}$ is

1. A.$4\sqrt{2}+2$
2. B.$4\sqrt{2}-1$
3. C.$4\sqrt{2}+1$
4. D.$4\sqrt{2}-2$

If $0<x<\pi$ and $\cos x+\sin x=\frac{1}{2}$, then $\tan x$ is :

1. A.$-\frac{(1-\sqrt{7})}{4}$
2. B.$-\frac{(4-\sqrt{7})}{3}$
3. C.$-\frac{(4+\sqrt{7})}{3}$
4. D.$-\frac{(1+\sqrt{7})}{4}$

The value of ${\sum }_{k=1}^{10}(\sin \frac{2k\pi }{11}+i\cos \frac{2k\pi }{11})$ is :

1. A.$i$
2. B.$1$
3. C.$-1$
4. D.$-i$

${\int }_0^{\pi }xf(\sin x)dx$ is equal to

1. A.$\pi {\int }_0^{\pi }f(\cos x)dx$
2. B.$\pi {\int }_0^{\pi }f(\sin x)dx$
3. C.$\frac{\pi }{2}{\int }_0^{\pi /2}f(\sin x)dx$
4. D.$\pi {\int }_0^{\pi /2}f(\cos x)dx$

The number of values of $x$ in the interval $[0,3\pi ]$ satisfying the equation $2{\sin }^{2}x+5\sin x-3=0$ is

1. A.4
2. B.6
3. C.1
4. D.2

In a triangle $PQR$, $\angle R=\frac{\pi }{2}$. If $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$ are the roots of $a{x}^2+bx+c=0,a\ne 0$ then

1. A.$a=b+c$
2. B.$c=a+b$
3. C.$b=c$
4. D.$b=a+c$

In a triangle $ABC$, let $\angle C=\frac{\pi }{2}$. If $r$ is the inradius and $R$ is the circumradius of the triangle $ABC$, then $2(r+R)$ equals :

1. A.$b+c$
2. B.$a+b$
3. C.$a+b+c$
4. D.$c+a$

If in a $\Delta ABC$, the altitudes from the vertices $A,B,C$ on opposite sides are in H.P, then $\sin A,\sin B,\sin C$ are in :

1. A.G. P.
2. B.A. P.
3. C.A.P.-G.P.
4. D.H. P

The normal to the curve $x=a(\cos \theta +\theta \sin \theta ),y=a(\sin \theta -\theta \cos \theta )$ at any point '$\theta$' is such that

1. A.it passes through the origin
2. B.it makes an angle $\frac{\pi }{2}+\theta$ with the $x$-axis
3. C.it passes through $\left(a\frac{\pi }{2},-a\right)$
4. D.it is at a constant distance from the origin

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.$-\frac{3}{\sqrt{130}}$

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.$\frac{2}{3}$

$\int \frac{dx}{\cos x-\sin x}$ 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.$\frac{1}{\sqrt{2}}\log \left|\tan \left(\frac{x}{2}-\frac{\pi }{8}\right)\right|+C$

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.$0$

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.$0$

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]

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.$60^{\circ }$

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.$20\mathrm{m}$