Applications of Cross Product

Q. Let $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$ be three vectors in three-dimensional space, where $\overrightarrow{u}$ and $\overrightarrow{v}$ are unit vectors which are not perpendicular to each other and $\overrightarrow{u}\cdot \overrightarrow{w}=1,\overrightarrow{v}\cdot \overrightarrow{w}=1,\overrightarrow{w}\cdot \overrightarrow{w}=4.$ If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$ is $\sqrt{2}$, then the value of $|3\overrightarrow{u}+5\overrightarrow{v}|$ is

Q.

If the vectors $ai+j+k,i+bj+k,i+j+ck(a,b,c\ne 1)$ are coplanar, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is?

1. $-1$

2. $0$

3. $1$

4. $\frac{1}{3}$

Q.

Let $\alpha$ and $\beta$ be the roots of the equation, $5x^2+6x-2=0$. If $S_n={\alpha }^n+{\beta }^n$, $n=1,2,3,..$ then

1. $5S_6+6S_5+2S_4=0$

2. $6S_6+5S_5=2S_4$

3. $6S_6+5S_5+2S_4=0$

4. $5S_6+6S_5=2S_4$

Q.

A possible value of $\tan \left(\frac{1}{4}{\sin }^{-1}\left(\frac{\sqrt{63}}{8}\right)\right)$ is:

1. $\frac{1}{2\sqrt{2}}$

2. $\frac{1}{\sqrt{7}}$

3. $\sqrt{7}-1$

4. $2\sqrt{2}-1$

Q.

The value of $\sin \left(\frac{\mathrm{\pi }}{16}\right)\sin \left(\frac{3\mathrm{\pi }}{16}\right)\sin \left(\frac{5\mathrm{\pi }}{16}\right)\sin \left(\frac{7\mathrm{\pi }}{16}\right)$ is

1. $\frac{1}{16}$

2. $\frac{\sqrt{2}}{16}$

3. $\frac{1}{8}$

4. $\frac{\sqrt{2}}{8}$

Q.

The imaginary part of $i^i$ is

1. $0$

2. $1$

3. $2$

4. $-1$

Q.

A vector $\overrightarrow{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}\left(\alpha ,\beta \in R\right)$ lies in the plane of the vectors, $\overrightarrow{b}=\hat{i}+\hat{j}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\overrightarrow{a}$ bisects the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$, then

1. $\overrightarrow{a}·\hat{i}+3=0$

2. $\overrightarrow{a}·\hat{k}+4=0$

3. $\overrightarrow{a}·\hat{i}+1=0$

4. $\overrightarrow{a}·\hat{k}+2=0$

Q.

How do you find a unit vector parallel to another vector?

Q. Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2).$ Let a vector $\overrightarrow{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ be such that $\overrightarrow{a}$ is parallel to the plane $P$, perpendicular to $(\hat{i}+2\hat{j}+3\hat{k})$ and $\overrightarrow{a}\cdot (\hat{i}+\hat{j}+2\hat{k})=2.$ Then $(\alpha -\beta +\gamma {)}^2$ equals

Q.

Find the area of a rhombus if its vertices are $\left(3,0\right)$, $\left(4,5\right)$, $\left(-1,4\right)$ and $\left(-2,-1\right)$ taken in order. [Hint:$\mathrm{Area}\mathrm{of}\mathrm{Rhombus}=\frac{\mathrm{Product}\mathrm{of}\mathrm{Diagonals}}{2}$]

Q.

Let$k$ be an integer such that the triangle with vertices $(k,-3k),(5,k)$and $(-k,2)$ has area $28\mathrm{sq}.\mathrm{units}$. Then the orthocenter of this triangle is at the point

1. $\left(1,\frac{3}{4}\right)$

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

3. $\left(2,\frac{1}{2}\right)$

4. $\left(2,\frac{-1}{2}\right)$

Q.

If $\sin \left(\mathrm{\theta }\right)=\frac{2}{3}$ find all other trigonometric ratios?

Q. Out of the four alternatives, choose the one which expresses the sentence below in passive voice.

Do you understand what I mean?

1. What I mean is that understood by you?
2. Was what I mean understood by you?
3. Is what I mean understood by you?
4. What I mean is understood by you?

Q.

If $f\left(x\right)={\cos }^{-1}\left(\frac{1-{\left(\log x\right)}^2}{1+{\left(\log x\right)}^2}\right)$, then $f\left(e\right)$ is equal to

1. $\frac{2}{e}$

2. $\frac{1}{e}$

3. $\frac{3}{e}$

4. None of these

Q. Let $P$ be a matrix of order $3\times 3$ such that all the entries in $P$ are from the set $\{-1,0,1\}.$Then the maximum possible value of the determinant of$P$ is

Q.

Moving along the X-Axis are two points with $x=10+6t$ and $x=3+t^2$ The Speed with which they are reaching from each other at the time of encounter is ( $x$ is in $cm$ and $t$ is in $sec$ )

1. $16cm.se{c}^{-1}$

2. $20cm.se{c}^{-1}$

3. $8cm.se{c}^{-1}$

4. $12cm.se{c}^{-1}$

Q.

What is the solution to the differential equation $\frac{dy}{dx}=5y^2$ with the initial condition $y\left(0\right)=3$?

Q. If $\alpha ,\beta ,\gamma$ be the direction angles of a vector and $cos\alpha =\frac{14}{15},cos\beta =\frac{1}{3}thencos\gamma =$

1. 
   

2. None of these

3. 
   

4. 
   

Q.

If the scalar and vector products of two vectors A and B are equal in magnitude, then the angle between the two vectors is

1. $45°$

2. $90°$

3. $180°$

4. $360°$

Q.

Angle between vectors $a$ and $b$ where $a,b\text{and}c$ are unit vectors satisfying $a+b+\sqrt{3}c=0$ is

Q.

The solution of the differential equation $\frac{dy}{dx}+\frac{2y}{x}=0$ with $y\left(1\right)=1$is given by

1. $y=\frac{1}{x^2}$

2. $y=\frac{1}{y^2}$

3. $x=\frac{1}{y}$

4. $x=y$

Q.

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$and $\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$. A vector coplanar to $\overrightarrow{a}$ and $\overrightarrow{b}$ has a projection along $\overrightarrow{c}$ of magnitude $\frac{1}{\sqrt{3}}$, then the vector is?

1. $4\hat{i}-\hat{j}+4\hat{k}$

2. $4\hat{i}+\hat{j}-4\hat{k}$

3. $2\hat{i}+\hat{j}+\hat{k}$

4. None of these

Q. If $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$, and $\overrightarrow{c}$ is unit vector perpendicular to the vector $\overrightarrow{a}$ and coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$then a unit vector $\overrightarrow{d}$ perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{c}$, is

1. $\frac{1}{\sqrt{6}}(2\hat{i}-\hat{j}+\hat{k})$
2. $\frac{1}{\sqrt{2}}(\hat{j}+\hat{k})$
3. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
4. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{k})$

Q.

The value of $\sum _{k=1}^6\left(\sin \left(\frac{2\mathrm{\pi k}}{7}\right)-i\cos \left(\frac{2\mathrm{\pi k}}{7}\right)\right)$

1. $-1$

2. $0$

3. $-i$

4. $i$

Q.

The non - zero vectors $\stackrel{\rightharpoonup }{a},\stackrel{\rightharpoonup }{b}$ and $\stackrel{\rightharpoonup }{c}$ are related by $\stackrel{\rightharpoonup }{a}=8\stackrel{\rightharpoonup }{b}$ and $\stackrel{\rightharpoonup }{c}=-7\stackrel{\rightharpoonup }{b}$. Then, the angle between $\stackrel{\rightharpoonup }{a}$ and $\stackrel{\rightharpoonup }{c}$ is?

1. $\mathrm{\pi }$

2. $0$

3. $\frac{\mathrm{\pi }}{4}$

4. $\frac{\mathrm{\pi }}{2}$

Q. Dividing $f\left(z\right)byz-i$, we obtain the remainder i and dividing it by $z+i$, we get the remainder $1+i$. The remainder upon the division of
$f\left(z\right)by{z}^2+1$ is

1. $\frac{1}{2}(z+1)+i$
2. $\frac{1}{2}(iz+1)+i$
3. $\frac{1}{2}(iz-1)+i$
4. $\frac{1}{2}(z+1)+i$

Q.

Find $\mathrm{a}$ and $\mathrm{b}$ if $(\mathrm{a}+\mathrm{ib})(1+\mathrm{i})=2+\mathrm{i}$.

Q.

The number of distinct real values of $\lambda$, for which the vectors $-{\lambda }^{2}i+j+k$, $i-{\lambda }^{2}j+k$ and $i+j-{\lambda }^{2}k$ are coplanar is$?$

1. Zero

2. One

3. Two

4. Three

Q. $\underset{\log 2}{\overset{a}{\int }}\frac{e^x}{\sqrt{e^x-1}}dx=2,$ then $a=$

1. $2\log 2$
2. $\log 5$
3. $\log 2$
4. $0$

Q. Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^2=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this parabola, such that the area of $\Delta PXQ$ is maximum. Then this maximum area (in sq. units) is:

1. $\frac{125}{2}$
2. $\frac{625}{4}$
3. $\frac{125}{4}$
4. $\frac{75}{2}$