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Trigonometry

Trigonometric Ratio & Identities: JEE Main & Advanced PYQs

381 previous year questions on Trigonometric Ratio & Identities from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 101-150 of 381

  1. A vertical pole of height $10\sqrt{3}$ meters is observed from three points $A$, $B$ and $C$ in the same horizontal line passing through the foot '$O$' of the pole. The …
  2. The angle of elevation of the summit of a mountain from a point on the ground is $45^\circ$. After climbing up one km towards the summit at an inclination of $30^\circ$ …
  3. If $\tan A$ is equal to the integral solution of the inequality $4x^{2}-24x+35<0$ and $\sin B$ is equal to the slope of the angle bisector of the second quadrant, then …
  4. If $0<x<\pi$ and $\cos x+\sin x=\frac12$, then $\tan x$ is :
  5. If $A=\sin^{2}x+\cos^{4}x$, then for all real $x$:
  6. Which one is not periodic?
  7. Let $\mathbf A$ and $\mathbf B$ denote the statements $\mathbf A$: $\cos\alpha+\cos\beta+\cos\gamma=0$ $\mathbf B$: $\sin\alpha+\sin\beta+\sin\gamma=0$ If …
  8. Let $f_k(x)=\frac{1}{k}\left(\sin^{k}x+\cos^{k}x\right)$ where $x\in R$ and $k\ge1$. Then $f_4(x)-f_6(x)$ equals :
  9. 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 …
  10. If $u=\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}+\sqrt{a^{2}\sin^{2}\theta+b^{2}\cos^{2}\theta}$ then the difference between the maximum and minimum values of …
  11. If $\operatorname{cosec}\theta=\frac{p+q}{p-q}\ (p\neq q,\ p\neq0)$, then $\left|\cot\left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right|$ is equal to :
  12. The number of values of $\alpha$ in $[0,2\pi]$ for which $2\sin^{3}\alpha-7\sin^{2}\alpha+7\sin\alpha=2$, is :
  13. The period of $\sin^{2}\theta$ is
  14. The number of solutions of the equation $\sin2x-2\cos x+4\sin x=4$ in the interval $[0,5\pi]$ is :
  15. The expression $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$ can be written as :
  16. In a $\Delta ABC$, $\frac ab=2+\sqrt3$, and $\angle C=60^\circ$. Then the ordered pair $(\angle A,\angle B)$ is equal to :
  17. Given $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$ for a $\Delta ABC$ with usual notation. If $\frac{\cos A}{\alpha}=\frac{\cos B}{\beta}=\frac{\cos C}{\gamma}$, then …
  18. The sum of all values of $\theta\in\left(0,\frac\pi2\right)$ satisfying $\sin^{2}2\theta+\cos^{4}2\theta=\frac34$ is
  19. Let $f_k(x)=\frac1k\left(\sin^{k}x+\cos^{k}x\right)$ for $k=1,2,3,\ldots$ Then for all $x\in R$, the value of $f_4(x)-f_6(x)$ is equal to :
  20. Let $f(\theta)=3\left(\sin^{4}\left(\frac{3\pi}{2}-\theta\right)+\sin^{4}(3\pi+\theta)\right)-2\left(1-\sin^{2}2\theta\right)$ and …
  21. The value of $\cos^{2}10^\circ-\cos10^\circ\cos50^\circ+\cos^{2}50^\circ$ is
  22. If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is :
  23. The value of $\cos\frac{\pi}{2^2}\cdot\cos\frac{\pi}{2^3}\cdot\ldots\cdot\cos\frac{\pi}{2^{10}}\cdot\sin\frac{\pi}{2^{10}}$ is :
  24. The number of solutions of the equation $1+\sin^4x=\cos^23x$, $x\in\left[-\frac{5\pi}2,\frac{5\pi}2\right]$ is:
  25. Let a vertical tower $AB$ have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that $AP=2AB$. If …
  26. Let $P=\left\{\theta:\sin\theta-\cos\theta=\sqrt2\cos\theta\right\}$ and $Q=\left\{\theta:\sin\theta+\cos\theta=\sqrt2\sin\theta\right\}$, be two sets. Then
  27. An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt3$ km above it, is observed at an elevation of $60^{\circ}$ from a point on the ground. …
  28. Let $S=\left\{\theta\in[-2\pi,2\pi]:2\cos^{2}\theta+3\sin\theta=0\right\}$. Then the sum of the elements of $S$ is :
  29. 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 …
  30. Two vertical poles of height $20$ m and $80$ m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of …
  31. $PQR$ is a triangular park with $PQ=PR=200$ 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 …
  32. If the sum of all the solutions of the equation $8\cos x\cdot\left(\cos\left(\frac\pi6+x\right)\cdot\cos\left(\frac\pi6-x\right)-\frac12\right)=1$ in $[0,\pi]$ is …
  33. For any $\theta\in\left(\frac\pi4,\frac\pi2\right)$, the expression $3(\sin\theta-\cos\theta)^4+6(\sin\theta+\cos\theta)^2+4\sin^6\theta$ equals:
  34. The number of solutions of $\sin3x=\cos2x$, in the interval $\left(\frac\pi2,\pi\right)$ is
  35. If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3x^{2}-10x-25=0$, then the value of $3\sin^{2}(A+B)-10\sin(A+B)\cdot\cos(A+B)-25\cos^{2}(A+B)$ is
  36. The equation $y=\sin x\sin(x+2)-\sin^2(x+1)$ represents a straight line lying in:
  37. If $\cos\alpha+\cos\beta=\frac32$ and $\sin\alpha+\sin\beta=\frac12$ and $\theta$ is the arithmetic mean of $\alpha$ and $\beta$, then $\sin2\theta+\cos2\theta$ is equal …
  38. Let $10$ vertical poles standing at equal distances on a straight line, subtend the same angle of elevation $\alpha$ at a point $O$ on this line and all the poles are on …
  39. The angle of elevation of the top of a vertical tower from a point A, due east of it is $45^{\circ}$. The angle of elevation of the top of the same tower from a point B …
  40. The lengths of two adjacent sides of a cyclic quadrilateral are $2$ units and $5$ units and the angle between them is $60^\circ$. If the area of the quadrilateral is …
  41. If $A>0,\ B>0$ and $A+B=\frac\pi6$, then the minimum value of $(\tan A+\tan B)$ is :
  42. If in a triangle $ABC$, $AB=5$ units, $\angle B=\cos^{-1}\left(\frac35\right)$ and radius of circumcircle of $\Delta ABC$ is $5$ units, then the area (in sq. units) of …
  43. With the usual notation in $\triangle ABC$, if $\angle A+\angle B=120^\circ$, $a=\sqrt3+1$ units and $b=\sqrt3-1$ units, then the ratio $\angle A:\angle B$ is :
  44. Consider a triangular plot $ABC$ with sides $AB=7$ m, $BC=5$ m and $CA=6$ m. A vertical lamp-post at the mid-point $D$ of $AC$ subtends an angle $30^{\circ}$ at $B$. The …
  45. If $0\le x<2\pi$, then the number of real values of $x$, which satisfy the equation $\cos x+\cos2x+\cos3x+\cos4x=0$, is
  46. Two poles standing on a horizontal ground are of heights $5$ m and $10$ m respectively. The line joining their tops makes an angle of $15^\circ$ with the ground. Then …
  47. If $2\tan^2\theta-5\sec\theta=1$ has exactly $7$ solutions in the interval $\left[0,\frac{n\pi}{2}\right]$, for the least value of $n\in\mathbb N$, then …
  48. The angle of elevation of the top of a vertical tower from a point $P$ on the horizontal ground was observed to be $\alpha$. After moving a distance $2$ metres from $P$ …
  49. If $n$ is the number of solutions of the equation $2\cos x\left(4\sin\left(\frac\pi4+x\right)\sin\left(\frac\pi4-x\right)-1\right)=1$, $x\in[0,\pi]$ and $S$ is the sum …
  50. The set of all values of $\lambda$ for which the equation $\cos^{2}2x-2\sin^{4}x-2\cos^{2}x=\lambda$ has a real solution $x$, is

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