Recent questions tagged exemplar

The vectors $\overrightarrow{a}=3\hat i+2\hat j+2\hat k$ and $\overrightarrow{b}=-\hat i+2\hat k$ are the adjacent sides of a parallelogram .The acute angle between its diagonals is__________.

asked Jan 17, 2013 by sreemathi.v

1 answer

If $\overrightarrow{r}.\overrightarrow{a}=0,\overrightarrow{r}.\overrightarrow{b}=0$ and $\overrightarrow{r}.\overrightarrow{c}=0$ for some non-zero vector $\overrightarrow{r}$,then the value of $\overrightarrow{a}.(\overrightarrow{b}\times\overrightarrow{c})$ is___________.

asked Jan 17, 2013 by sreemathi.v

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The vector $\overrightarrow{a}+\overrightarrow{b}$ bisects the angle between the non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ if__________.

asked Jan 17, 2013 by sreemathi.v

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The number of vectors of unit length perpendicular to the vectors $\overrightarrow{a}=2\hat i+\hat j+2\hat k$ and $\overrightarrow{b}=\hat j+\hat k$ is

asked Jan 17, 2013 by sreemathi.v

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If $\mid\overrightarrow{a}\mid=4$ and $-3\leq\:\lambda\leq 2$,then the range of $\mid\lambda\overrightarrow{a}\mid$ is

asked Jan 17, 2013 by sreemathi.v

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If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three vectors such that $\mid\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\mid=0$ and $\mid\overrightarrow{a}\mid=.2, \mid\overrightarrow{b}\mid=3,\mid\overrightarrow{c}\mid=5$,then value of $\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}$ is

asked Jan 17, 2013 by sreemathi.v

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Projection vector of $\overrightarrow{a}$ on $\overrightarrow{b}$ is

asked Jan 17, 2013 by sreemathi.v

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If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors such that$\mid\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\mid=0$,then the value of $\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}$ is

asked Jan 17, 2013 by sreemathi.v

1 answer

The vector $\hat i+\lambda\hat j+\hat k,\hat i+\hat j-\hat k$ and $3\hat i+6\hat j-5\hat k$ are coplanar if

asked Jan 17, 2013 by sreemathi.v

1 answer

If $\mid\overrightarrow a\mid=10,\mid\overrightarrow b\mid=2$ and $\overrightarrow a.\overrightarrow b=12$,then value of $\mid\overrightarrow a\times\overrightarrow b\mid$ is

asked Jan 17, 2013 by sreemathi.v

1 answer

For any vector $\overrightarrow{a}$,the value of $(\overrightarrow{a}.\hat i)^2+ (\overrightarrow{a}.\hat j)^2+ (\overrightarrow{a}.\hat k)^2 $ is equal to

asked Jan 17, 2013 by sreemathi.v

1 answer

The vector from origin to the points $A$ and $B$ are $\overrightarrow{a}=2\hat i-3\hat j-2\hat k$ and $\overrightarrow{b}=2\hat i+3\hat j-\hat k,$ respectively,then the areqa of triangle $OAB$ is

asked Jan 17, 2013 by sreemathi.v

1 answer

The value of $\lambda$ for which the vectors $3\hat i+6\hat j+\hat k $ and $ 2\hat i+4\hat j+\lambda\hat k$ are parallel is

asked Jan 17, 2013 by sreemathi.v

1 answer

Find the value of $\lambda$ such that the vectors $\overrightarrow{a}=2\hat i+\lambda\hat j+\hat k$ and $\overrightarrow{b}=\hat i+2\hat j+3\hat k$ are orthogonal

asked Jan 17, 2013 by sreemathi.v

1 answer

The angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $\sqrt 3$ and 4,respectively,and $\overrightarrow{a}.\overrightarrow{b}=2\sqrt 3$ is

asked Jan 17, 2013 by sreemathi.v

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The vector having initial and terminal points as $(2,5,0)$ and $(-3,7,4)$ respectively is

asked Jan 17, 2013 by sreemathi.v

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The position vector of the point which divides the join of points $2\overrightarrow{a}-3\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ in the ratio 3:1 is

asked Jan 17, 2013 by sreemathi.v

1 answer

The vector in the direction of the vector $\hat i+2\hat j+2\hat k$ that has magnitude $9$ is

asked Jan 17, 2013 by sreemathi.v

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If $\overrightarrow{a}=\hat i+\hat j+\hat k$ and $\overrightarrow{b}=\hat j-\hat k$,find a vector $\overrightarrow{c}$ such that $\overrightarrow{a}\times\overrightarrow{c}=\overrightarrow{b}$ and $\overrightarrow{a}.\overrightarrow{c}=3$.

asked Jan 17, 2013 by sreemathi.v

1 answer

Show that the area of the parallelogram whose diagonals are given by $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\mid \overrightarrow{a} \times \overrightarrow{b}\mid}{2}$.Also find the area of the parallelogram whose diagonals are $2\hat i+\hat j+\hat k$ and $\hat i+3\hat j+\hat k$.

asked Jan 17, 2013 by sreemathi.v

1 answer

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ represent the vertices of a triangle,then show that $\frac{1}{2}\mid\overrightarrow{b}\times\overrightarrow{c}+\overrightarrow{c}\times\overrightarrow{a}+\overrightarrow{a}\times\overrightarrow{b}\mid$ gives the vector area of the triangle .Hence deduce the condition that the three points $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are collinear.Also find the unit vector normal to the plane of the triangle.

asked Jan 17, 2013 by sreemathi.v

1 answer

Prove that in any triangle ABC,$ cos A=\large \frac{b^2+c^2-a^2}{2bc}$,$where \overrightarrow{a},\overrightarrow{b} and \overrightarrow{c}$ are the vectors of sides BC, CA and AB respectively.

asked Jan 17, 2013 by sreemathi.v

1 answer

Using vectors,prove that the parallelogram on the same base and between the same parallels are equal in area.

asked Jan 17, 2013 by sreemathi.v

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Using vectors,find the area of the triangle ABC with vertices $A(1,2,3),B(2,-1,4)$ and $C(4,5,-1).$

asked Jan 17, 2013 by sreemathi.v

1 answer

If $A,B,C,D$ are the points with position vectors $\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k},2\overrightarrow{i}+\overrightarrow{j}+3\overrightarrow{k},2\overrightarrow{i}-3\overrightarrow{k},3\overrightarrow{i}+2\overrightarrow{i}+\overrightarrow{k},$ respectively ,find the projection of $\overrightarrow{AB}$ along $\overrightarrow{CD}.$

asked Jan 17, 2013 by sreemathi.v

1 answer

Find the sine of the angle between the vectors $\overrightarrow{a}=3\hat i+\hat j+2\hat k$ and $\overrightarrow{b}=-2\hat i+2\hat j+4\hat k.$

asked Jan 17, 2013 by sreemathi.v

1 answer

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,$show that $\overrightarrow{a} \times \overrightarrow{b}=\overrightarrow{b} \times \overrightarrow{c} =\overrightarrow{c} \times \overrightarrow{a}$.Interpret the result geometrically?

asked Jan 17, 2013 by sreemathi.v

1 answer

Find the angle between the vectors $2\hat i\;-\hat j\;-\hat k$ and $3\hat i\;+4\hat j\;+\hat k$.

asked Jan 17, 2013 by sreemathi.v

1 answer

FInd a vector of magnitude 6 ,which is perpendicular to both the vectors $2\hat i-\hat j+2\hat k$ and $4\hat i-\hat j+3\hat k.$

asked Jan 17, 2013 by sreemathi.v

1 answer

A vector $\overrightarrow{r}$ has magnitude 14 and direction ratios 2,3,-6.Find the direction cosines and components of $\overrightarrow{r}$,given that $\overrightarrow{r}$ makes an acute angle with x-axis.

asked Jan 17, 2013 by sreemathi.v

1 answer

A vector $\overrightarrow{r}$ is inclined at equal angles to the three axes.If the magnitude of $\overrightarrow{r}$ is $2\sqrt 3$ units,find $\overrightarrow{r}.$

asked Jan 17, 2013 by sreemathi.v

1 answer

Using vectors,find the value of k such that the points $(k,-10,3),(1,-1,3)$ and $(3,5.3)$ are collinear.

asked Jan 17, 2013 by sreemathi.v

1 answer

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the position vectors of $A$ and $B$,respectively,find the position vector of a point $C$ in $BA$ produced such that $BC=1.5BA$.

asked Jan 17, 2013 by sreemathi.v

1 answer

Find a unit vector in the direction of $\overrightarrow{PQ},$where $P$ and $Q$ have co-ordinates $(5,0,8)$ and $(3,3,2)$,respectively.

asked Jan 17, 2013 by sreemathi.v

1 answer

If $\overrightarrow{a}=\hat i-\hat j+2\hat k\;and\;\overrightarrow{b}=2\hat i-\hat j-2\hat k,$then find the unit vector in the direction of $ 6 \overrightarrow{b} $

asked Jan 17, 2013 by sreemathi.v

1 answer

Find the unit vector in the direction of sum of vectors $\overrightarrow{a}=2\hat i+\hat j+\hat k\;and\;\overrightarrow{b}=2\hat j-\hat k$

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the curve $x=2y+3$ and the y-lines $y=1$ and $y=-1$ is \[(A)\;4 sq.units\quad(B)\;\frac{3}{2} sq. units\quad(C)\;6 sq.units\quad(D)8 sq.units\]

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the curve $y=x+1$ and the lines $x=2$ and $x=3$ is \[(A)\frac{7}{2} sq.units\quad(B)\;\frac{9}{2}sq. units\quad(C)\frac{11} {2}sq.units\quad(D)\frac{13}{2} sq.units\]

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the circle $x^2+y^2=1$ is \[(A)\;2\pi sq.units\quad(B)\;\pi sq. units\quad(C)\;3\pi sq.units\quad(D)4\pi sq.units\]

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the ellipse $\large\frac{x^2}{25}+\frac{y^2}{16}=1$ is \begin{array}{1 1}(A)\;2 \pi sq.units & (B)\;20{\pi}^2 sq.units\\(C)\;16{\pi}^2 sq.units & (D)\;25\pi sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the curve $y=sin x$ between the ordinates $x=0$ and $x=\large\frac{\pi}{2}$ and the x-axis is

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by parabola $y^2=x$ and the straight line $2y=x$ is \begin{array}{1 1}(A)\;\frac{4}{3} sq.units & (B)\;1 sq.units\\(C)\;\frac{2}{3} sq.units & (D)\;\frac{1}{3}sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the curve $y=\cos x$ between $x=0$ and $x=\pi$ is \begin{array}{1 1}(A)\;2 sq.units & (B)\;4 sq.units\\(C)\;3 sq.units & (D)\;1sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region in the first quadrant enclosed by the x-axis,the line y=x and the circle $x^2+y^2=32$ is \begin{array}{1 1}(A)\;16\pi\; sq.units & (B)\;4\pi \;sq.units\\(C)\;32\pi\; sq.units & (D)\;24\pi\;sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

2 answers

The area of the region bounded by the curve $y=\sqrt {16-x^2}$ and x-axis is \begin{array}{1 1}(A)\;8\pi \;sq.units & (B)\;20\pi\; sq.units\\(C)\;16\pi\; sq.units & (D)\;256\pi\; sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is \begin{array}{1 1}(A)\;\frac{3}{8}sq.units & (B)\;\frac{5}{8}sq.units\\(C)\;\frac{7}{8}sq.units & (D)\;\frac{9}{8} sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

1 answer

The area of the region bounded by the y-axis ,$y=\cos x$ and $y=\sin x,0\leq x\leq \large\frac{\pi}{2}$ is\begin{array}{1 1}(A)\;1\sqrt 2sq.units & (B)\;(\sqrt 2+1)sq.units\\(C)\;(\sqrt 2-1)sq.units & (D)\;(2\sqrt 2-1)sq.units \end{array}

asked Jan 16, 2013 by sreemathi.v

1 answer

Draw a rough sketch of the given curve $y=1+|x+1|,x=-3,x=3,y=0$ and find the area of the region bounded by them,using integration.

asked Jan 16, 2013 by sreemathi.v

1 answer

Find the area bounded by the curve $y=2\cos x$ and the $x$-axis from $x=0$ to x=2$\pi$.

asked Jan 16, 2013 by sreemathi.v

1 answer

Find the area bounded by the lines $y=4x+5,y=5-x$ and $4y=x+5.$

asked Jan 16, 2013 by sreemathi.v

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