Trigonometric Ratios

Q.

What does a cross product of $0$ mean?

Q. A $10$ inches long pencil $AB$ with mid point $C$ and a small eraser $P$ are placed on the horizontal top of a table such that $PC=\sqrt{5}$ inches and $\angle PCB={\tan }^{-1}\left(2\right).$
The acute angle through which the pencil must be rotated about $C$ so that the perpendicular distance between eraser and pencil becomes exactly $1$ inch is

1. ${\tan }^{-1}\left(\frac{1}{2}\right)$
2. ${\tan }^{-1}\left(\frac{3}{4}\right)$
3. ${\tan }^{-1}\left(1\right)$
4. ${\tan }^{-1}\left(\frac{4}{3}\right)$

Q. The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be ${45}^{\circ }$. After walking a distance of $80$ meters towards the top, up a slope inclined at an angle of ${30}^{\circ }$ to the horizontal plane, the angle of elevation of the top of the hill becomes ${75}^{\circ }$. Then the height of the hill (in meters) is

Q. If $P(x,y,z)$ is a point on the line segment joining $Q(2,3,4)$ and $R(3,5,6)$ such that the projections of the vector $\overrightarrow{OP}$ on the coordinate axes are $\frac{13}{5},\frac{21}{5},\frac{26}{5}$ respectively, where $O$ denotes the origin, then $P$ divides $QR$ in the ratio

1. $2:3$
2. $3:1$
3. $1:3$
4. $3:2$

Q. A trapezium $ABCD$ is inscribed into a semi-circle of radius $l$ so that the base $AD$ of the trapezium is diameter and the vertices $B$ and $C$ lie on the circumference. Then the value of base angle $\theta$ (in degree) of the trapezium $ABCD$ which has the greatest perimeter, is

Q. 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 ${30}^{\circ }$ from the ground, the angle of elevation of the summit is found to be ${60}^{\circ }$. Then the height (in km) of the summit from the ground is :

1. $\frac{1}{\sqrt{3}+1}$
   
2. $\frac{\sqrt{3}+1}{\sqrt{3}-1}$
   
3. $\frac{\sqrt{3}-1}{\sqrt{3}+1}$
   
4. $\frac{1}{\sqrt{3}-1}$
   

Q. Express the trigonometric ratios $\sin A,\sec A$ and $\tan A$ in terms of $\cot A$.

Q. State the converse of Pythagoras' Theorem and prove it.

Q.

Prove that: $(\text{cosec}\theta -\cot \theta {)}^2=\frac{1-\cos \theta }{1+\cos \theta }$

Q. Column Matching:

$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In a triangle}△XYZ,\text{let}a,b\text{and}c\text{be the}\\ & \text{lengths of the sides opposite to the angles}\\ & X,Y\text{and}Z,\text{respectively. If}2(a^2-b^2=c^2\\ & \text{and}\lambda =\frac{\sin (X-Y)}{\sin Z},\text{then possible values}\\ & \text{of}n\text{for which}\cos \left(n\pi \lambda \right)=0\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{In a triangle}△XYZ,\text{let}a,b\text{and}c\text{be}\\ & \text{the lengths of the sides opposite to the}\\ & \text{angles}X,Y\text{and}Z,\text{respectively. If}\\ & 1+\cos 2X-2\cos 2Y=2\sin X\sin Y,\text{then}\\ & \text{possible value(s) of}\frac{a}{b}\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{In}{R}^2,\text{let}\sqrt{3}\hat{i}+\hat{j},\hat{i}+\sqrt{3}\hat{j}\text{and}\beta \hat{i}+(1-\beta )\hat{j}\\ & \text{be the position vectors of}X,Y\text{and}Z\text{with}\\ & \text{respect to the origin}O,\text{respectively. If the}\\ & \text{distance of}Z\text{from the bisector of the acute}\\ & \text{angle of}\overrightarrow{OX}\text{with}\overrightarrow{OY}\text{is}\frac{3}{\sqrt{2}},\text{then possible}\\ & \text{value(s) of}|\beta |\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Suppose that}F\left(\alpha \right)\text{denotes the area of the}\\ & \text{region bounded by}x=0,x=2,y^2=4x\\ & \text{and}y=|\alpha x-1|+|\alpha x-2|+\alpha x,\text{where}\\ & \alpha \in \{0,1\}.\text{Then the value(s) of}F\left(\alpha \right)+\frac{8}{3}\sqrt{2}\text{,}\\ & \text{when}\alpha =0\text{and}\alpha =1,\text{is (are)}\end{array}& \text{(S) 5}\\ & \\ & \text{(T) 6}\\ & \end{array}$$
Option $\text{(D)}$ matches with which of the elements of right hand column?

1. $P$
2. $Q$
3. $R$
4. $S$
5. $T$

Q. Prove that: $\cot x\cot 2x-\cot 2x\cot 3x-\cot 3x\cot x=1$

Q. Evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos 2x}{3{\tan }^{2}x}$

Q. Solve: $\frac{tan\theta }{1-cot\theta }+\frac{cot\theta }{1-tan\theta }=1+sec\theta cosec\theta$

Q. In the figure given below, find the value of $\cos \theta$.

Q. period of cos[|sinx| - |cosx|]

Q. If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ and $|\overrightarrow{a}|=3,|\overrightarrow{b}|=5,|\overrightarrow{c}|=7$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\alpha$, then $\alpha =$_____

1. $\frac{2\pi }{3}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{3}$
4. $\frac{5\pi }{6}$

Q. Let $p$ and $q$ are real numbers such that $(\tan p-1{)}^3+2019(\tan p-1)=-1$ and $(1-\cot q{)}^3+2019(1–\cot q)=-1,$ $\tan p\ne \cot q$. Then number of possible values of $r$ which satisfy the equation $\tan p+\cot q+\sin r+\cos r=3,$ $r\in [-2\pi ,2\pi ]$ is

Q. $\frac{\tan {45}^{\circ }}{\text{cosec}{30}^{\circ }}+\frac{\sec {60}^{\circ }}{\cot {45}^{\circ }}-\frac{5\sin {90}^{\circ }}{2\cos 0^{\circ }}$

Q. Let $a=min\{x^2+2x+3,xϵR\}$ and $b=\underset{\theta \to 0}{lim}\frac{1-\cos \theta }{{\theta }^2}$.

1. $\frac{4^{n+1}-1}{{3.2}^n}$
2. $\frac{2^{n+1}-1}{{3.2}^n}$
3. none of these
4. $\frac{2^{n+1}+1}{{3.2}^n}$

Q. 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 is ${45}^{\circ }.$ When he moves $20\text{m}$ away from the bank, he finds the angle of elevation to be ${30}^{\circ }.$ The height of the tree is

1. $10(\sqrt{3}+1)\text{m}$
2. $15\sqrt{3}\text{m}$
3. $20(\sqrt{3}+1)\text{m}$
4. $10(\sqrt{3}-1)\text{m}$

Q. Prove that: $(1+\cot A+\tan A)\left(\sin A-\cos A\right)=\frac{\sec A}{{\csc }^{2}A}-\frac{\csc A}{\sec {}^{2}A}$

Q. If $\sec A+\tan A=m$, show that $\frac{m^2-1}{m^2+1}=\sin A$

Q. A man standing on a level plane observes the elevation of the top of a pole to be $\theta$. If he walks a distance equal to double the height of the pole towards the pole, the angle of elevation becomes $2\theta .$ Then the value of $\theta$ (in degrees) is

Q. Prove that $\frac{cos4x+cos3x+cos2x}{sin4x+sin3x+sin2x}=cot3x$

Q. If $\sec \theta +\tan \theta =x$, then $\tan \theta$ is :

1. $\frac{x^2+1}{2x}$
2. $\frac{x^2-1}{2x}$
3. $\frac{x^2+1}{x}$
4. $\frac{x^2-1}{x}$

Q. Solve: ${\tan }^2\theta +{\cot }^2\theta =2$

Q. The value of ${\int }_0^1\frac{{\log }_e(x+1)}{1+x^2}dx=\frac{\pi \ln a}{b}$, then $a^2+b^2$ equal to

1. $68$
2. $Noneofthese$
3. $20$
4. $40$

Q. Evaluate $\underset{x\to \frac{\pi }{2}}{lim}\left(\frac{\tan 2x}{x-\frac{\pi }{2}}\right)$.

Q. Find the integral of the function
$\frac{\cos 2x-\cos 2\alpha }{\cos x-\cos \alpha }$ with respect to x

Q. Solve :
$(cosec\theta -cot\theta )\sqrt{\frac{1+cos\theta }{1-cos\theta }}=1$