Trigonometry · JEE Main previous year questions

If ${\Delta }_1=\left|\begin{array}{ccc}x& \sin \theta & \cos \theta \\ -\sin \theta & -x& 1\\ \cos \theta & 1& x\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{ccc}x& \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x& 1\\ \cos 2\theta & 1& x\end{array}\right|,x\ne 0$ ; then for all $\theta \in \left(0,\frac{\pi }{2}\right)$ :

1. A.${\Delta }_1-{\Delta }_2=x(\cos 2\theta -\cos 4\theta )$
2. B.${\Delta }_1+{\Delta }_2=-2x^3$
3. C.${\Delta }_1+{\Delta }_2=-2(x^3+x-1)$
4. D.${\Delta }_1-{\Delta }_2=-2x^3$

Let A (3, 0, -1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then $\cos (\angle \mathrm{GOA})$ (O being the origin) is equal to :

1. A.$\frac{1}{\sqrt{15}}$
2. B.$\frac{1}{6\sqrt{10}}$
3. C.$\frac{1}{\sqrt{30}}$
4. D.$\frac{1}{2\sqrt{15}}$

ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are ${\cot }^{-1}(3\sqrt{2})$ and ${\mathrm{cosec}}^{-1}(2\sqrt{2})$ respectively, then the height of the tower (in metres) is :

1. A.$\frac{100}{3\sqrt{3}}$
2. B.25
3. C.20
4. D.$10\sqrt{5}$

The value of ${\cos }^{2}10^{\circ }-\cos 10^{\circ }\cos 50^{\circ }+{\cos }^{2}50^{\circ }$ is

1. A.$\frac{3}{2}+\cos 20^{\circ }$
2. B.$\frac{3}{4}$
3. C.$\frac{3}{2}(1+\cos 20^{\circ })$
4. D.$\frac{3}{2}$

The value of $\sin 10^{\circ }\sin 30^{\circ }\sin 50^{\circ }\sin 70^{\circ }$ is :-

1. A.$\frac{1}{36}$
2. B.$\frac{1}{16}$
3. C.$\frac{1}{32}$
4. D.$\frac{1}{18}$

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${\tan }^{-1}\left(\frac{1}{2}\right)$. Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :-

1. A.$\frac{1}{15\pi }$
2. B.$\frac{1}{5\pi }$
3. C.$\frac{1}{10\pi }$
4. D.$\frac{2}{\pi }$

If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}\sin \alpha \cos \beta$; $\alpha ,\beta \in [0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha -\beta )$ is equal to :

1. A.$-\sqrt{2}$
2. B.0
3. C.$-1$
4. D.$\sqrt{2}$

If the angle of elevation of a cloud from a point P which is 25 m above a lake be $30^{\circ }$ and the angle of depression of reflection of the cloud in the lake from P be $60^{\circ }$, then the height of the cloud (in meters) from the surface of the lake is :

1. A.45
2. B.42
3. C.50
4. D.60

All x satisfying the inequality $({\cot }^{-1}x{)}^2-7({\cot }^{-1}x)+10>0$, lie in the interval :

1. A.$(\cot 2,\infty )$
2. B.$(-\infty ,\cot 5)\cup (\cot 2,\infty )$
3. C.$(\cot 5,\cot 4)$
4. D.$(-\infty ,\cot 5)\cup (\cot 4,\cot 2)$

Given $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$ for a $\Delta \mathrm{ABC}$ with usual notation. If $\frac{\cos A}{\alpha }=\frac{\cos B}{\beta }=\frac{\cos C}{\gamma }$, then the ordered triad $(\alpha ,\beta ,\gamma )$ has a value :

1. A.$(19,7,25)$
2. B.$(7,19,25)$
3. C.$(5,12,13)$
4. D.$(3,4,5)$

If ${\cos }^{-1}\left(\frac{2}{3x}\right)+{\cos }^{-1}\left(\frac{3}{4x}\right)=\frac{\pi }{2}$ $\left(x>\frac{3}{4}\right)$, then x is equal to :

1. A.$\frac{\sqrt{145}}{10}$
2. B.$\frac{\sqrt{145}}{11}$
3. C.$\frac{\sqrt{145}}{12}$
4. D.$\frac{\sqrt{146}}{12}$

For any $\theta \in \left(\frac{\pi }{4},\frac{\pi }{2}\right)$, the expression $$3(\cos \theta -\sin \theta {)}^4+6(\sin \theta +\cos \theta {)}^2+4{\sin }^6\theta$$ equals :

1. A.$13-4{\cos }^2\theta +6{\sin }^2\theta {\cos }^2\theta$
2. B.$13-4{\cos }^6\theta$
3. C.$13-4{\cos }^2\theta +6{\cos }^4\theta$
4. D.$13-4{\cos }^4\theta +2{\sin }^2\theta {\cos }^2\theta$

If $\theta$ denotes the acute angle between the curves, $y=10-x^2$ and $y=2+x^2$ at a point of their intersection, the $|\tan \theta |$ is equal to :

1. A.$\frac{8}{15}$
2. B.$\frac{4}{9}$
3. C.$\frac{7}{17}$
4. D.$\frac{8}{17}$

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from $30^{\circ }$ to $45^{\circ }$ ; then after this, the time taken (in min.) by the car to reach the foot of the tower, is :

1. A.$9\left(1+\sqrt{3}\right)$
2. B.$18\left(1+\sqrt{3}\right)$
3. C.$18\left(\sqrt{3}-1\right)$
4. D.$\frac{9}{2}\left(\sqrt{3}-1\right)$

If sum of all the solutions of the equation $8\cos x\cdot \left(\cos \left(\frac{\pi }{6}+x\right)\cdot \cos \left(\frac{\pi }{6}-x\right)-\frac{1}{2}\right)=1$ in $[0,\pi ]$ is $\mathrm{k}\pi$, then k is equal to

1. A.$\frac{20}{9}$
2. B.$\frac{2}{3}$
3. C.$\frac{13}{9}$
4. D.$\frac{8}{9}$

PQR is a triangular park with $\mathrm{PQ}=\mathrm{PR}=200\mathrm{m}$. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively $45^{\circ }$, $30^{\circ }$ and $30^{\circ }$, then the height of the tower (in m) is :

1. A.$50\sqrt{2}$
2. B.$100$
3. C.$50$
4. D.$100\sqrt{3}$

A value of x satisfying the equation $\sin [{\cot }^{-1}(1+x)]=\cos [{\tan }^{-1}x]$, is :

1. A.$-\frac{1}{2}$
2. B.$-1$
3. C.$0$
4. D.$\frac{1}{2}$

If $$S=\left\{x\in [0,2\pi ]:\left|\begin{array}{ccc}0& \cos x& -\sin x\\ \sin x& 0& \cos x\\ \cos x& \sin x& 0\end{array}\right|=0\right\},$$ then ${\sum }_{x\in S}\tan \left(\frac{\pi }{3}+x\right)$ is equal to :

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

The value of ${\tan }^{-1}\left[\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right]$, $|x|<\frac{1}{\sqrt{2}}$, $x\ne 0$, is equal to :

1. A.$\frac{\pi }{4}+\frac{1}{2}{\cos }^{-1}{x}^2$
2. B.$\frac{\pi }{4}+{\cos }^{-1}{x}^2$
3. C.$\frac{\pi }{4}-\frac{1}{2}{\cos }^{-1}{x}^2$
4. D.$\frac{\pi }{4}-{\cos }^{-1}{x}^2$

A value of $\theta$ for which $\frac{2+3i\sin \theta }{1-2i\sin \theta }$ is purely imaginary, is :

1. A.${\sin }^{-1}\left(\frac{\sqrt{3}}{4}\right)$
2. B.${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{6}$

For $x\in R$, $f(x)=\left|\log 2-\sin x\right|$ and $g(x)=f(f(x))$, then :

1. A.$g\text{ is not differentiable at }x=0$
2. B.$g^{\prime }(0)=\cos (\log 2)$
3. C.$g^{\prime }(0)=-\cos (\log 2)$
4. D.$g\text{ is differentiable at }x=0\text{ and }{g}^{\prime }(0)=-\sin (\log 2)$

If $0\le x<2\pi$, then the number of real values of $x$, which satisfy the equation $\cos x+\cos 2x+\cos 3x+\cos 4x=0$ is:

1. A.$7$
2. B.$9$
3. C.$3$
4. D.$5$

Consider : $f(x)={\tan }^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right),x\in \left(0,\frac{\pi }{2}\right)$. A normal to $y=f(x)$ at $x=\frac{\pi }{6}$ also passes through the point:

1. A.$\left(\frac{\pi }{6},0\right)$
2. B.$\left(\frac{\pi }{4},0\right)$
3. C.$(0,0)$
4. D.$\left(0,\frac{2\pi }{3}\right)$

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is $30^{\circ }$. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is $60^{\circ }$. Then the time taken (in minutes) by him, from B to reach the pillar, is :

1. A.$6$
2. B.$10$
3. C.$20$
4. D.$5$

Let ${\tan }^{-1}y={\tan }^{-1}x+{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$, where $|x|<\frac{1}{\sqrt{3}}$. Then a value of $y$ is :

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