Question 2 :
$A$ and $B$ are two points and $C$ is any point collinear with $A$ and $B$. IF $AB=10$, $BC=5$, then $AC$ is equal to:
Question 3 :
The number of line segments possible with three collinear points is ________.
Question 4 :
If $-9$ is a root of the equation $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix}=0$, then the other two roots are
Question 5 :
If the area of the triangle with vertices $(2, 5), (7, k)$ and $(3, 1)$ is $10$, then find the value of $k$.<br>
Question 7 :
The value of k for which $kx+3y-k+3=0$ and $12x+ky=k$, have infinite solutions, is?
Question 8 :
If the area of the triangle formed by $ (0,0), (a,0) $ and $ \left( \dfrac{1}{2} , a \right) $ is equal to $ \dfrac {1}{2} $ sq unit, then the values of $a$ are :
Question 9 :
The system of equations which can be solved by matrix inversion method have_______.
Question 10 :
Which of the given values of $x$ and $y$ make the following pair of matrices equal.<br>$\displaystyle \begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}=\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$
Question 12 :
If the points $(a, 1), (2, -1)$ and $\left(\dfrac{1}{2}, 2\right)$ are collinear, then $a$ is equal to:
Question 13 :
The points (2, -3), (4,3) and (5, k/2) are on the same straight line. The value(s) of k is (are):
Question 14 :
Two points $(a, 0)$ and $(0, b)$ are joined by a straight line. Another point on this line is
Question 15 :
If $(3, 2)$, $\left (x, \dfrac {22}{5}\right), (8, 8)$ lie on a line, then $x$ is equal to
Question 16 :
If the points $(a,\,b),\,(3,\,-5)$ and $(-5,\,-2)$ are collinear. Then find the value of $3a+8b$
Question 17 : The system of equations<div>$\displaystyle  <br/>\begin{matrix}kx +y+z=1&  & \\ <br/> x+ky+z=k&  & \\ <br/> x+y+kz=k^{2}&  & <br/>\end{matrix}$<br/>have no solution,if k equals ?<br/></div>
Question 18 :
The value of $\lambda$ and$\mu$ for which the simultaneous equation<br/>$x+y+z=6$, $x+2y+3z=10$ and $x+2y+\lambda z=\mu$ have a unique solution are
Question 19 :
The equations $\displaystyle \:x+y+z=6, x+2y+3z= 10, x+2y+mz = n$ give infinite number of values of the triplet $(x,y,z)$ if
Question 20 :
Points $(1, 5), (2, 3)$ and $(-2, -11) $ are ____
Question 21 :
If the lines $p_{ 1 }x+q_{ 1 }y=1, p_{ 2 }x+q_{ 2 }y=1$ and $p_{3}x+q_{3}y=1$ be concurrent, show that the points $(p_{1},q_{1}), (p_{2}, q_{2})$ and $ (p_{3}, q_{3})$ are collinear.
Question 22 :
Solve the system of equations<br>$\quad x+y+z = 6 \\ \quad x+2y+3z = 14 \\ \quad x+4y+7z = 30$
Question 23 :
Value of p for which the points (-5, 1), (1, p) and (4, -2) are collinear is
Question 24 :
If the following system of equations possess a non-trivial solution over the set of rationals<br>$x + ky + 3z = 0$<br>$3x + ky - 2z = 0$<br>$2x + 3y - 4z = 0$,<br>then x,y,z are in the ratio
Question 25 :
If the lines $\mathrm{x}+\mathrm{p}\mathrm{y}+\mathrm{p}=0,\ \mathrm{q}\mathrm{x}+\mathrm{y}+\mathrm{q}=0$ and $\mathrm{r}\mathrm{x}+\mathrm{r}\mathrm{y}+1 =0 (\mathrm{p},\mathrm{q}, \mathrm{r}$ being distinct and $ \neq$ 1) are concurrent, then the value of<br/>$\displaystyle \frac{p}{p-1}+\frac{q}{q-1}+\frac{r}{r-1}=$<br/>
Question 26 :
If the lines ${x}+{a}{y}+{a}=0,\ {b}{x}+{y}+{b}=0,\ {c}{x}+{c}{y}+1 =0 ({a}\neq{b}\neq {c}\neq1)\ $ are concurrent, then the value of $\displaystyle \frac{{a}}{{a}-1}+\frac{{b}}{{b}-1}+\frac{{c}}{{c}-1}$, is<br/>
Question 27 : <div><span>${ tan }^{ -1 }x+{ tan }^{ -1 }y={ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy<1$</span><br/></div><div>                                    $=\pi +{ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy>1$.</div><div><br/></div><div><span> Evaluate:  ${ tan }^{ -1 }\dfrac { 3sin2\alpha  }{ 5+3cos2\alpha  } +{ tan }^{ -1 }\left( \dfrac { tan\alpha  }{ 4 }  \right) $</span><br/></div><div>                                  where $-\dfrac { \pi  }{ 2 } <\alpha <\dfrac { \pi  }{ 2 } $</div>
Question 30 :
If $n - 1\sum\limits_{}^\infty {{{\cot }^{ - 1}}\left( {{{{n^2}} \over 8}} \right) = \pi .} $ where ${a \over b}$ is rational number in its lowest, then correct option is/are
Question 31 :
The value of $\sin^{-1} \left( \dfrac{2 \sqrt 2}{3} \right ) + \sin^{-1} \left( \dfrac{1}{3}\right )$ is equal to
Question 33 :
The value of $\sin \left( \cos ^{ -1 } \left( -\cfrac { 1 }{ 7 }  \right) +\sin ^{ -1 }\left( -\cfrac { 1 }{ 7 }  \right) \right)=$ ____
Question 36 :
If $\sin^{-1}\dfrac{1}{3} + \sin^{-1}\dfrac{2}{3} = \sin^{-1}x$, then $x$ is equal to-
Question 37 :
The value of $\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$ is 
Question 39 :
If $x,y,z \in [-1,1]$ such that $\cos^{-1}x +\cos^{-1}y +\cos^{-1}z=0$, find $x+y+z$.
Question 40 :
${\tan ^{ - 1}}2 + {\tan ^{ - 1}}3$ is equal to
Question 42 :
Simplify ${\cot ^{ - 1}}\dfrac{1}{{\sqrt {{x^2} - 1} }}$ for $x <  - 1$
Question 43 :
$\cos ^{-1}\left ( \cos \left ( \frac{5\pi}{4} \right ) \right )$ is given by
Question 44 :
The number of real values of x satisfying the equation $\tan^{-1}\left(\dfrac{x}{1-x^2}\right)+\tan^{-1}\left(\dfrac{1}{x^3}\right)=\dfrac{3\pi}{4}$, is?
Question 47 :
Consider the following statements:<br/>1. $\tan^{-1} 1+ \tan^{-1} (0.5) = \dfrac {\pi}2$<br/>2. $\sin^{-1}{\cfrac{1}{3} }+ \cos^{-1}{\cfrac{1}{3}} =\cfrac{\pi}{2}$<br/>Which of the above statements is/are correct ? 
Question 48 :
$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {\dfrac{{2r}}{{1 - {r^2} + {r^4}}}} \right)} $ is equal to
Question 49 :
The value of ${ sin }^{ -1 }(sin12)+{ cos }^{ -1 }(cos12)$ is equal to :
Question 50 :
Choose the correct answer <br>$ \cos ^{-1} ( \cos \dfrac{7\pi }{6}) $ is equal to
Question 52 :
Number of triplets $\left ( x, y, z \right )$ satisfying $\sin ^{-1}x+\sin ^{-1}y+\cos ^{-1}z=2\pi$ is<br>
Question 53 :
If $sin^{-1}$ a is the acute angle between the curves $x^2+y^2=4x$ and $x^2+y^2=8$ at (2, 2), then a= _____<br>
Question 55 :
If $\cos ^{ -1 }{ x } -\cos ^{ -1 }{ \cfrac { y }{ 2 } } =\alpha $ where $-1-1\le x\le 1,-2\le y\le 2,x\le \cfrac { y }{ 2 } $ then for all $4{ x }^{ 2 }-4xy\cos { \alpha } +{ y }^{ 2 }$ is equal to
Question 61 :
The value of $\sin [2 \tan ^{-1} (.75)]$ is equal to
Question 63 :
If $x \in \left ( \frac{3\pi}{2}, 2\pi \right )$ then the value of the expression $sin^{-1}[cos({cos^{-1}(cos \, x)}+sin^{-1}(sin \, x))]$, is 
Question 64 :
If $\tan^{-1} (-x) + \cos^{-1}\left (\dfrac {-1}{2}\right ) = \dfrac {\pi}{2}$, them the value of $x$ is
Question 68 :
If ${ sin }^{ -1 }\left( \dfrac { \sqrt { x } }{ 2 } \right) +{ sin }^{ -1 }\left( \sqrt { 1-\dfrac { x }{ 4 } } \right) +tan^{ -1 }y=\dfrac { 2\pi }{ 3 } $ then
Question 69 :
The value of $sin^{-1} x + cos^{-1} x (|x| \geq 1)$ is
Question 70 :
If $A=\tan {-1}(\frac {x\sqrt 3}{2k-x})$ and $B=\tan ^{-1}(\frac {2x-k}{k\sqrt 3})$, then the value of A-B is
Question 71 :
If $x={ sin }^{ -1 }(sin10)$ and $y={ cos }^{ -1 }(cos10)$, then the value of (y - x) is
Question 73 :
$\displaystyle\int_0^{\pi /2} {\dfrac{{\sin x}}{{\sqrt {1 + {\mathop{\rm cosx}\nolimits} } }}} dx = $
Question 74 :
$ \int_{0}^{\frac{\pi}{2}} \sin \theta \cdot \sin 2 \theta d \theta $
Question 76 :
$ \int_0^{ \pi /2 } \sin^5 x \cos ^6 x dx = $
Question 81 :
$\displaystyle \int _{ 0 }^{ 1 }{ \tan ^{ -1 }{ \left( \dfrac { x }{ \sqrt { 1-{ x }^{ 2 } } } \right) } dx }$
Question 82 :
$\displaystyle \int _{ 0 }^{ 1 }{ \tan ^{ -1 }{ \left( \dfrac { 2x }{ 1-{ x }^{ 2 } }  \right) dx }  } =\dfrac{\pi}{a}-\ln a$. Find $a$.
Question 85 :
$\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ x }.x }{ { \left( x+1 \right) }^{ 2 } } dx= }$
Question 87 :
The quadratic polynomial $p(x)$ has the following properties: $p(x)\ge 0$ for all real number, $p(1)=$ and $p(2)=2$ value of $p(0)+p(3)$ is equal to
Question 89 :
The value of the definite integral $\displaystyle \int_{0}^{\pi/2}sinx\space sin2x\space sin3xdx$ is equal to
Question 91 :
If $\int _{ 0 }^{ 1 }{ \cfrac { \tan ^{ -1 }{ x } }{ x } } dx=k\int _{ 0 }^{ \pi /2 }{ \cfrac { x }{ \sin { x } } } dx$ then the value of $k$ is
Question 97 :
$\displaystyle \int _{ 0 }^{ { \pi  }/{ 8 } }{ { \cos }^{ 3 } } 4\theta d\theta $ is equal to:
Question 99 :
Obtain $ \displaystyle \int _{ 0 }^{ \pi  }{ \sqrt { 1+\cos { 2x }  } dx } $
Question 104 : Find proper substitution<div>$\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ -x } }{ 1+{ e }^{ -x } } dx }$</div>
Question 112 : <div>The value of $\displaystyle \int _0^{\pi/2} \sin x \cos x dx $</div><div><br/></div>
Question 114 :
If $\displaystyle \int_{0}^{\dfrac{\pi}{3}}\dfrac{\tan \theta}{\sqrt{2k \sec \theta}}d\theta=1-\dfrac{1}{\sqrt{2}},(k>0),$ then the value of k is :
Question 116 :
Integrate $\displaystyle \int _{ 0 }^{ 2\pi  }{ \sqrt {1+\sin x }}dx$
Question 117 :
If $ \int _0^1 xdx = \dfrac {\pi}{4} - \dfrac {1}{2} ln 2 $ then the value of definite integral $ \int _0^1 \tan^{-1} (1-x+x^2) dx $ equals :
Question 119 :
The value of the integral $\displaystyle \int_{-\pi/2}^{\pi/2} \left(x^{2}+\log \dfrac{\pi-x}{\pi+x}\right) \cos x dx $ is 
Question 123 :
Evaluate: $\int _{ 1/3 }^{ 1 }{ \cfrac { { \left( x-{ x }^{ 3 } \right)  }^{ 1/3 } }{ { x }^{ 4 } }  } dx=$
Question 124 :
Let $I = \int\limits_1^3 {\sqrt {{x^4} + {x^2}} dx,} $then I= 
Question 125 :
<br/>$\displaystyle \int_{-1}^{3}\left( \tan^{ -1 }\frac { x }{ x^{ 2 }+1 } +\tan ^{ -1 } \frac { x^{ 2 }+1 }{ x }  \right) dx=$<br/>
Question 126 :
The value of the definite integral $\displaystyle \int_{0}^{\pi / 2} \dfrac{\sin 5 x}{\sin x} d x$ is
Question 127 :
$\int _{ 0 }^{ \pi /2 }{ \sin { 2x } .\sin { x } } dx=.....$
Question 128 :
The value of $\int_{0}^{[x]} (x-[x])dx$, where $[x]$ is the greatest integer $|le x$ is equal to
Question 130 :
The area of the region bounded by the lines $x = 1, x = 2$, and the curves $x(y - e^x) = \sin x$ and $2xy = 2 \sin x + x^3$ is
Question 131 :
The value of $\int _{ 1/e }^{ \tan { x } }{ \dfrac { t }{ 1+{ t }^{ 2 } } dt+ } \int _{ 1/e }^{ \cot { x } }{ \dfrac { t }{ t\left( 1+{ t }^{ 2 } \right) } dt } $, where $x\in \left( \pi /6,\ \pi /3 \right) $, is equal to :
Question 135 :
If $\vec a=\hat i+2\hat j$ and $\vec b = 3\hat j$, then $\vec a\cdot\vec b=$<br/>
Question 136 :
If $\left| \overrightarrow { a } \right| =7,\ \left| \overrightarrow { b } \right| =11,\ \left| \overrightarrow { a } +\overrightarrow { b } \right| = 10\sqrt 3$, then $\left| \overrightarrow { a } -\overrightarrow { b } \right| =$
Question 137 :
The values of a, for which the points A,B,C with position vectors $\displaystyle 2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $\displaystyle a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right angled triangle with C=$\displaystyle \frac{\pi}{2} $ are
Question 138 :
For $O$ being the origin and $3$ points $P,Q$ and $R$ lie on a plane. If $\displaystyle \vec{PO}+\vec{OQ}=\vec{QO}+\vec{OR}$, then $P, Q, R$ are <br/>
Question 139 :
If $\overline{a}$ and $\overline{b}$ are position vectors of $A$ and $B$ respectively, then the position vector of a point $C$ in $AB$ produced such that $\overline{AC}=3\overline{AB}$ is<br/>
Question 140 :
The vector which can give unit along the x-axis is <span> $\overrightarrow { A } =2\hat { i } -4\hat { j } +7\hat { k } , \overrightarrow { B } =7\hat { i } +2\hat { j } -5\hat { k } $. Find C?</span>
Question 141 :
If $G$ is the centroid of the triangle $ABC$ then $\vec{GA}+\vec{GB}+\vec{GC}$ is equal to <br/>
Question 143 :
The work done by the force $\vec { F } = 2 \hat { i } - \hat { j } - \hat { \mathbf { k } }$ in moung an object along the vector $3 \hat { i } + 2 j - 5 \hat { k }$ <span>is</span>
Question 146 :
The set of values of $c$ for which the angle between the vectors $cx\hat{i}-6\hat{j}+3\hat{k}$ and $x\hat{i}-2\hat{j}+2cx\hat{k}$ is acute for every $x\in R$ is
Question 147 :
If $\mid \vec a \mid = 2$ and $\mid \vec b \mid = 3$ and $\vec a \cdot \vec d = 0$, then $(\vec a \times (\vec a \times (\vec a \times (\vec a \times \vec b ))))$ is equal to
Question 148 :
If A and B are the points $(2,1,-2),(3,-4,5)$, then the angle that $OA$ makes with $OB$ is:
Question 149 :
If $\vec {a} . \hat {i} = \vec {a} . (\hat {i} + \hat {j}) = \vec {a} (\hat {i} + \hat {j} + \hat {k})$, thus $\vec {a}=$
Question 150 :
The vector<br/>$\vec a + \vec b,\vec a - k\vec b$ where $k$ scalar are collinear, for
Question 151 :
Position vectors of mid point of the vector joining the points $P(2, 3, 4)$ and $Q (4, 1, -2)$ is<br/>
Question 152 :
$P, Q, R, S$ have position vectors $\overline{p},\overline{q},\overline{r},\overline{s}$ respectively such that $\overline{p}-\overline{q}=2(\overline{s}-\overline{r})$, then which of the following is correct<br/>
Question 153 :
If $\vec{a} = (2, 1, -1), \vec{b} = (1,-1,0), \vec{c} = (5, -1, 1) $ , then what is the unit vector parallel to $ \vec{a} + \vec{b} - \vec{c} $ in the opposite direction ?
Question 154 :
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}$ is a vector such that $\vec{a}.\vec{b}=0$ and $\vec{a}\times \vec{b}=0$. Then which of following is correct?
Question 155 :
If $A(\overline{a})$ , $B(\overline{b})$ and $C(\overline{c})$ be the vertices of a triangle $ABC$ whose circumcentre is the origin then orthocentre is given by<br/>
Question 156 :
The position vectors of the points $A$ and $B$ are respectively $3\hat { i } -5\hat { j } +2\hat { k } $ and $\hat { i } +\hat { j } -\hat { k } $. What is the length of $AB$?
Question 157 :
If $a=\lambda \hat { i } +2\hat { j } +2\hat { k } $ and $b=2\hat { i } +2\hat { j } +\lambda \hat { k } $ are at right angle, then the value of $\left| a+b \right| -\left| a-b \right| $ is
Question 158 :
Let $\vec{A}=\hat{i}+2\hat{j}{+}3\hat{k},\ \vec{B}=4\hat{i}+2\hat{j},\ \vec{C}=2\hat{i}+2\hat{j}{+}2\hat{k}$. Then the ratio in which $C$ divides $AB$ is<br/>
Question 159 :
Consider two vectors $\vec{F_{1}}=2\hat{i}+5\hat{k}$ and $\vec{F_{2}}=3\hat{j}+4\hat{k}$. The magnitude of the scalar product of these vectors is
Question 160 :
The projection of the vector $\vec a = 4\hat i - 3\hat j + 2\hat k$ on the vector making equal angles (acute) with coordinate axes having magnitude $\sqrt{3}$ is
Question 161 :
If $\bar{a}$ is collinear with $\bar{b}=3\bar{i}+6\bar{j}+6\bar{k}$ and $\bar{a}\cdot\bar{b}=27$. Then $\bar{a}$ is equal to?
Question 162 :
If $\vec a$ is parallel to $\vec b \times \vec c$, then $(\vec a \times \vec b) \cdot (\vec a \times \vec c)$ is equal to
Question 163 :
<span>Given that u is a vector of length $2$, v is a vector of length $3$ and the angle between them when placed tail to tail is $\displaystyle 45^{\circ} $, which option is closest to the exact value of $\vec u\cdot\vec v$ ?</span>
Question 164 :
Let $\vec{OB} =\hat { i } +2\hat { j } +2\hat { k }$ and $\vec{OA} =4\hat { i } +2\hat { j } +2\hat { k } $. The distance of the point $B$ from the straight line passing through $A$ and parallel to the vector $2\hat { i } +3\hat { j } +6\hat { k } $ is
Question 165 :
If $O$ and $O'$ are circumcenter and orthocenter of a triangle $ABC$ then $\left( OA+OB+OC \right) $ equals
Question 166 :
If $(2, -1, 2)$ is the centroid of tetrahedron $OABC$ and $G_{1}$ is the centroid of $\Delta ABC$ then $|\overline{OG}_{1}|=$<br/>
Question 167 :
In a tetrahedron if two pairs of opposite edges are at a right angles then the third pair is inclined at an angle of<br>
Question 168 :
If $\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =4$, if $\left( \overrightarrow { a } +\lambda \overrightarrow { b } \right) $ is perpendicular to $\left( \overrightarrow { a } -\lambda \overrightarrow { b } \right) $ then $\lambda =$
Question 169 :
The magnitude of resultant of three vectors of magnitude $1,2$ and $3$ whose direction are those of the sides of an equilateral triangle taken in order is:
Question 170 :
In a parallelogram $ABCD,\left| AB \right| =a,\left| AD \right| =b$ and $\left| AC \right| =c.$ Then, $DB.AB$ has the value
Question 171 :
If $\overline { a } $ and $\overline { b } $ include an angle of ${120}^{o}$ and their magnitudes are $2$ and $\sqrt{3}$ then $\overline { a } .\overline { b } $ is
Question 172 :
If a vector is multiplied by a real number, then which of the following statements is incorrect?
Question 173 :
If $S$ is the circumcenter, $O$ is the orthocenter of $\triangle ABC$, then $\displaystyle \vec{SA}+\vec{SB}+\vec{SC}= $<br/>
Question 174 :
If $\bar a, \bar b, \bar c$  are unit vector and $\bar c=\bar a+\bar b$, then $|\bar a-\bar b|$ is
Question 176 :
Let $\vec {p}$ is the position vector of the orthocentre & $\vec {g}$ is the position vector of the centroid of the triangle $ABC$ where circumcentre is the origin. If $\vec {p}= K\vec{g},$ then $K=$
Question 177 :
If $a.b=0$ and $a+b$ makes an angle of ${ 60 }^{ o }$ with $b$, then $\left| a \right| $ is equal to
Question 178 :
If $\overline {a} + \overline {b} + \overline {c} = \overline {0}$ and $|\overline {a}| = 3, |\overline {b}| = 5, |\overline {c}| = 7$ and $(\overline {a}, \overline {b}) = \alpha$, then $\alpha =$ ________.
Question 179 : <div>The position vectors of points $\vec{A},\vec{B},\vec{C}$ are respectively $\vec{a},\vec{b},\vec{c}$. If $P$ divides $\vec{AB}$ in the ratio $3:4$ and $Q$ divides $\vec{BC}$ in the ratio $2:1$ both externally then $\vec{PQ}$ is</div><div><br/></div>
Question 180 :
If $\displaystyle \left ( \bar A+\bar B \right )$ is perpendicular to$\bar B$ and If $\displaystyle \left ( \bar A+2\bar B \right )$ is perpendicular to $\bar A$, then<br>
Question 181 :
lf $\vec{a}$ and $\vec{b}$ are two non-parallel unit vectors and the vector $\alpha\vec{a}+\vec{b}$ bisects the internal angle between $\vec{a}$ and $\vec{b}$, then $\alpha$ is<br/>
Question 182 :
Let $\overrightarrow a$ and  $\overrightarrow b$ be two unit vectors if the vectors $\overrightarrow c = \overrightarrow a + 2\overrightarrow b$ and $\overrightarrow d = 2\overrightarrow a - 4\overrightarrow b$ are perpendicular to each other. Then the angle between $\overrightarrow a$ and $\overrightarrow b$ is:
Question 183 :
If $A, B$ are two points on the curve $y=x^{2}$ in the $x-y$ plane satisfying $\vec{OA}.\hat{i}=1$ and $\vec{OB}.\hat{i}=-2$ then the length of the vectors $2\vec{OA}-3\vec{OB}$ is<br/>
Question 184 :
Let us define, the length of a vector $a\overline{i}+b\overline{j}+c\overline{k}$ as $|{a}|+|{b}|+|{c}|$. This definition coincides with the usual definition of the length of a vector $a\overline{i}+b\overline{j}+c\overline{k}$ if<br/>
Question 185 :
If $\overrightarrow a$ and $\overrightarrow b$ are unit vectors, then angle between $\overrightarrow a$ and $\overrightarrow b$ for $\sqrt 3 \overrightarrow a - \overrightarrow b$ to be unit vector is
Question 186 :
If $\vec \alpha | =4$ and $ -3 \le \lambda \le 2$, then the range of $ | \lambda \vec \alpha |$ is
Question 187 :
Given that $\vec a, \vec b, \vec p, \vec q$ are four vectors such that $\vec a + \vec b = \mu \vec p, \vec b \cdot \vec q = 0$ and $(\vec b)^2 = 1,$ where $\mu$ is scalar. Then $\mid (\vec a \cdot \vec q) \vec p - (\vec p \cdot \vec q)\vec a \mid$ is equal to
Question 188 :
Let $A\left( \vec { a } \right) B\left( \vec { b } \right)$ and $C\left( \vec { c } \right)$ be the vertices of $\triangle ABC$ and $D\left( \vec { d } \right)$ be a point such that $2\left( \vec { d } \right) \left( \vec { a } -\vec { b } \right) -{ \left| \vec { a } \right| }^{ 2 }-{ \left| \vec { b } \right| }^{ 2 }$ and $2\vec { d } .\left( \vec { b } -\vec { c } \right) -{ \left| \vec { b } \right| }^{ 2 }-{ \left| \vec { c } \right| }^{ 2 }$ then in $\triangle ABC,D$ must be
Question 189 :
If C is the mid-point of AB and P is any point outside AB , then
Question 190 :
$\mathrm{l}\mathrm{n}$ a triangle O$\mathrm{A}\mathrm{B},\ \mathrm{E}$ is the mid-point of $\mathrm{O}\mathrm{B}$ and $\mathrm{D}$ is a point on $\mathrm{A}\mathrm{B}$ such that $\mathrm{A}\mathrm{D}$: $\mathrm{D}\mathrm{B}=2: 1$. lf $\mathrm{O}\mathrm{D}$ and $\mathrm{A}\mathrm{E}$ interesect at $\mathrm{P}$, then the ratio $\displaystyle\frac{OP}{PD}$ is<br/>
Question 191 : <div><br/></div><div>Let two non-collinear unit vectors $\hat{{a}}$ and $\hat{{b}}$ form an acute angle. ${A}$ point ${P}$ moves so that at any time ${t}$ the position vector $\vec{{O}{P}}$ (where $O$ is the origin) is given by $\hat{a}\cos {t}+\hat{{b}} \sin t$. When ${P}$ is farthest from origin $O$, let ${M}$ be the length of $\vec{{O}{P}}$ and $\hat{{u}}$ be the unit vector along $\vec{{O}{P}}$. Then,</div>
Question 192 :
The values of $\lambda$ such that $(x, y, z) \neq (0, 0, 0)$ and $(\hat{i} + \hat{j} + 3\hat{k})x + (3\hat{i} - 3\hat{j} + \hat{k})y + (-4\hat{i} + 5\hat{j})z = \lambda(x\hat{i} + y\hat{j} + z\hat{k})$ are
Question 193 :
If $\vec{A} = i-j, \vec{B} = i+j+k$ are two vectors and $\vec{C}$ is another vectors such that $\vec{A} \times \vec{C} + \vec{B} =\vec{0}$ and $\vec{A} . \vec{C} =0$, then $|\vec{C}|^2$ =
Question 195 :
If $A=\left[\begin{array}{lll}<br/>1 & -2 & 3\\<br/>-4 & 2 & 5<br/>\end{array}\right]$ and $B=\left[\begin{array}{ll}<br/>2 & 3\\<br/>4 & 5\\<br/>2 & 1<br/>\end{array}\right],$ then <br/>
Question 196 :
If $A = \left[ {\begin{array}{*{20}{c}}  2&{ - 3} \\   { - 4}&1 \end{array}} \right]$, then $\left[ {3{A^2} + 12A} \right]$ is equal to 
Question 197 :
If $A = \begin{bmatrix}3 & 1 \\ -1 & 2\end{bmatrix}$, Then $A^2$ =
Question 198 :
If $A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{bmatrix}$, then $A^2$ is equal to
Question 199 :
If $A=[1\ \  2\ \  3\ \  4]$ and $AB = [3 \ \ 4\ \  -1],$ then the order of<br/>matrix $B$ is 
Question 200 :
If $A= \begin{bmatrix}<br/>1 & 2 & 3\\ <br/>4 & 5 & 6<br/>\end{bmatrix}$ and $B= \begin{bmatrix}<br/>1\\ <br/>0\\ <br/><br/>5\end{bmatrix},$ then $AB = $
Question 201 :
iF $A=\begin{bmatrix} 1& -1\\ -1& 1\end{bmatrix}$, then the expression $A^3-2A^2$ is
Question 202 :
If $A = \begin{bmatrix}a & b\end{bmatrix},\space B = \begin{bmatrix}-b & -a \end{bmatrix}$ and $C = \begin{bmatrix}a \\ -a\end{bmatrix}$, then the correct statement is
Question 203 :
Given $A= \begin{bmatrix}  3&4  \\ 4&-3 \end{bmatrix}$ and $B = \begin{bmatrix} 24 \\ 7\end{bmatrix},$ find the matrix $X$<b> </b>such that $AX=B$.
Question 204 :
What is $\begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} a& h & g\\ h & b & f\\ g & f & c\end{bmatrix}$ equal to?
Question 205 :
If $A = \begin{bmatrix} 2& 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\end{bmatrix}$, then $A^6 =$
Question 207 :
If $A=\begin{bmatrix} 2 & x-3 & x-2 \\ 3 & -2 & -1 \\ 4 & -1 & -5 \end{bmatrix}$ is a symmetric matrices then $x=$
Question 208 :
If $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$, then $BA =$
Question 209 :
If $A=\begin{bmatrix} -1 & 0 \\ 0 & 2 \end{bmatrix} $, then $ A^{3}-A^{2}=$
Question 210 :
If $[2\ 3\ 4] \begin{bmatrix}1 & x &3 \\ 2 & 4 & 5\\ 3 & 2 &x \end{bmatrix} \begin{bmatrix} x\\ 2 \\ 0 \end{bmatrix} = 0$, then $x =$ ________.<br>
Question 211 :
If $A=\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 4 \end{bmatrix}$, then $AB$ equal to 
Question 212 : <div>Find the value in place of question mark in the following:</div>$A_{6 \times 2}\times B_{2\times 6} = C_{?\times6}$?<br/>
Question 213 :
Find the output order for the following matrix multiplication $A_{4 \times 2}\times B_{2\times4}$?<br/>
Question 214 :
$[A]_{n\times m}, [B]_{m\times m},$ are the two matrices. If multiplication AB exist, then<br/>
Question 215 :
What is the output for the following matrix multiplication $A_{3 \times 2}\times B_{2\times 3}$?<br>
Question 216 :
If $\begin{bmatrix}3 & -1 \\ 2 & 5\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix} = \begin{bmatrix}4 \\ -3\end{bmatrix},$ find $x$ and $y$
Question 217 :
If $A=\displaystyle \left[ \begin{matrix} 1 & -6 & 2 \\ 0 & -1 & 5 \end{matrix} \right] $ and $\displaystyle B=\left[ \begin{matrix} 2 \\ 1 \end{matrix} \right] $, then $AB$ equals
Question 218 :
If the matrices $A=\begin{bmatrix}2 & 1 & 3 \\4 & 1 & 0\end{bmatrix}$ and $B=\begin{bmatrix}1 & -1\\ 0 & 2 \\5 & 0\end{bmatrix}$, then AB will be
Question 219 :
$A=\begin{bmatrix} 1 & 2 \\ 3 & 2 \\ -1 & 0 \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 3 & 2 \\ 4 & -1 & 3 \end{bmatrix}, $then $AB$ can be defined of order 
Question 220 :
If $A$ is of order $3\times 4$ and $B$ is of order $4\times 3$ , then the order of $BA$ is :
Question 221 :
IF $A=\begin{bmatrix} -1 & 0 & 2 \\ 3 & 1 & 2 \end{bmatrix}$ and $B=\begin{bmatrix} -1 & 5 \\ 2 & 7 \\ 3 & 10 \end{bmatrix},$ then
Question 222 :
If $\mathrm{A}=\left[\begin{array}{lll}<br/>1 & -3 & -4\\<br/>-1 & 3 & 4\\<br/>1 & -3 & -4<br/>\end{array}\right]$, then $\mathrm{A}^{2}=$<br/>
Question 223 :
If $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{bmatrix}$ then $A^ {2}$ is equal to
Question 224 :
lf $\mathrm{A}= \left[\begin{array}{lll}<br/>o & c & -b\\<br/>-c & o & a\\<br/>b & -a & o<br/>\end{array}\right]\mathrm{a}\mathrm{n}\mathrm{d}$ $ \mathrm{B}=\left[\begin{array}{lll}<br/>a^{2} & ab & ac\\<br/>ab & b^{2} & bc\\<br/>ac & bc & c^{2}<br/>\end{array}\right],$ then $\mathrm{A}\mathrm{B}=$<br/>
Question 225 :
If $A=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, then $A^{4}=$<span><br/></span>
Question 226 :
What is the output order for the following matrix multiplication $A_{2 \times 1}\times B_{1\times 2}$?<br/>
Question 227 :
If $A$ and $B$ are two matrices such that $A + B$ and $AB$ are both defined, then 
Question 229 :
$I$ $A=\left[\begin{array}{ll}<br>0 & 1\\ 1 & 0 \end{array}\right]$, $A^{4}=$ <br>($I$ is an identity matrix.)<br>
Question 231 :
Consider the following statements:<br>1. The product of two non-zero matrices can never be identity matrix.<br>2. The product of two non-zero matrices can never be zero matrix.<br>Which of the above statements is/are correct?
Question 232 :
If $A$ is any matrix, then the product $AA$ is defined only when A is a matrix of order $m \times n$ where : <span><br/></span>
Question 233 :
lIf $\mathrm{A} =\left[\begin{array}{ll}<br/>a & 0\\<br/>a & 0<br/>\end{array}\right],\ \mathrm{B}=\left[\begin{array}{ll}<br/>0 & 0\\<br/>b & b<br/>\end{array}\right],$ then $\mathrm{A}\mathrm{B}=$ <br/>
Question 234 :
If $A=\begin{bmatrix} 1&2 \\ 2 &3\\3 & 4\end{bmatrix}$ and $B=\begin{bmatrix} 1 &  2\\ 2 &  1\end{bmatrix},$ then which one of the following is correct?
Question 235 :
If $I = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix},$ then find $I^3$<br>
Question 236 :
If $\begin{bmatrix} 1 & 2 & 3   \end{bmatrix}   B=\begin{bmatrix}  3 & 4   \end{bmatrix}$, then the order of the matrix $B$ is
Question 237 :
If $A$ is matrix of order $\displaystyle m\times n$ and $B$ is a matrix of order $\displaystyle n\times p,$ then the order of $AB$ is 
Question 238 :
If a matrix $A$ is of order $3\times 4$ and a matrix $B$ is of order $4\times 3$, then the order of $BA$ is
Question 240 :
If $\displaystyle \begin{bmatrix} 1&2&3\end{bmatrix} A=\begin{bmatrix} 4&5 \end{bmatrix}, $ then what is the order of matrix $A$?
Question 241 :
The order of $[\mathrm{x} \space \mathrm{y} \space\mathrm{z}] \left[\begin{array}{lll}<br/>\mathrm{a} & \mathrm{h} & \mathrm{g}\\<br/>\mathrm{h} & \mathrm{b} & \mathrm{f}\\<br/>\mathrm{g} & \mathrm{f} & \mathrm{c}<br/>\end{array}\right]\left[\begin{array}{l}<br/>\mathrm{x}\\<br/>\mathrm{y}\\<br/>\mathrm{z}<br/>\end{array}\right]$ is<br/>
Question 242 :
Given $A$ is a matrix of order $3\times 2$. If order of $AB$ is $3\times 3$, then order of $B$ will be 
Question 243 :
if A=$\left[ \begin{matrix} 2 & 3 \\ 5 & -7 \end{matrix} \right] then\quad \left( { A }^{ '} \right) ^{ 2 }=$
Question 244 :
If $\begin{bmatrix} 3 & 2 & -1 \\ 4 & 9 & 2 \\ 5 & 0 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} 0 \\ 7 \\ 2 \end{bmatrix}$, then $(x,\ y,\ z)=$
Question 245 :
If $\displaystyle \begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix} \: \begin{bmatrix} 2 \\ -2 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$ then $x-y$ is equal to
Question 247 :
If $A=\begin{bmatrix} 2 & 1 & 3 \\ 4 & 5 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} 2 & 3 \\ 4 & 2 \\ 1 & 5 \end{bmatrix}$, then
Question 248 :
Let $\displaystyle A = \begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}$. The only correct statement about the matrix $A$ is<br>
Question 249 :
Given $A, B, C$ are three matrices such that <br/>$A = \begin{bmatrix}x & y & z\end{bmatrix}$, $B = \begin{bmatrix} a  & h & g \\ h & b & f \\ g & f & c\end{bmatrix}$, $C = \begin{bmatrix}x \\ y \\ z\end{bmatrix}.$ Evaluate $ABC$.<br/>
Question 250 :
$\begin{bmatrix} 10 & 20 & 30 \\ 20 & 45 & 80 \\ 30 & 80 & 171 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 4 & 1 \end{bmatrix}\begin{bmatrix} x & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix}$ then ${x}=$<br>
Question 251 :
If $A =\begin{pmatrix} -2& 2\\ 2 & -2\end{pmatrix}$, then which one of the following is correct?
Question 252 :
<span>Multiplication of two matrices $A$ and $ B$ i.e. $AB,$ is possible if and only if</span>
Question 253 :
If $[1\ 2\ 3] B = [3\ 4],$ then order of the matrix $B$ is
Question 254 :
If $[1\,x\,1]  \begin{bmatrix} 1&3&2 \\ 0&5&1\\0&2&0 \end{bmatrix}$ $\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix}=0$, then the values of $x$ are:
Question 255 :
If $\displaystyle A\times \begin{bmatrix} 1 & 2 &3  \\ 4 & 5 & 6  \end{bmatrix}=\begin{bmatrix} 1 & 2 &3  \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{bmatrix}$ then the order of A is _______
Question 256 :
If $A = \left[ \begin{array} { c c } { a b } & { b ^ { 2 } } \\ { - a ^ { 2 } } & { - a b } \end{array} \right]$ and $A ^ { n } = 0$ then the minimum value of $n$ is
Question 257 :
If $A =\begin{bmatrix} ab&b^2 \\-a^2 &-ab \end{bmatrix}$, then $A^2$ is equal<br>
Question 258 :
If P = $ \begin{bmatrix}<br>1\\ <br>3\\ <br><br>4\end{bmatrix}$ , Q = $\begin{bmatrix}<br>2 & -1&5 <br>\end{bmatrix}$ then PQ =
Question 259 :
$\left[\begin{array}{lll}<br/>x & 0 & 0\\<br/>y & \mathrm{z} & 0\\<br/>l & m & n<br/>\end{array}\right]\left[\begin{array}{lll}<br/>a & 0 & 0\\<br/>0 & b & 0\\<br/>0 & 0 & c<br/>\end{array}\right] =$<br/>
Question 260 :
If $I =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & \quad \quad 0 & \quad 1 \end{pmatrix},$ $P=\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & \quad \quad 0 & \quad -2 \end{pmatrix},$ then the matrix $P^3+2P^2$ is equal to 
Question 261 :
If $A=\begin{bmatrix}x&y&z\end{bmatrix},$ $ B=\begin{bmatrix}a&h&g\\h&b&f\\g&f&c\end{bmatrix}, C=\begin{bmatrix}\alpha & \beta & \gamma \end{bmatrix}^{T}$ then $ABC$ is
Question 262 :
If $A=\begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}$, then $(A-2I)(A-3I)=$ 
Question 263 :
The  matrix equation satisfied by $A$ is ______$, $ if $A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}.$ 
Question 264 :
If the matrix A is such that $\begin{bmatrix} 1 & 3\\ 0 & 1\end{bmatrix} A=\begin{bmatrix} 1 & 1 \\ 0 & -1\end{bmatrix}$, then what is equal to A?<br/>
Question 266 :
If $A=\begin{bmatrix} 1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix},  B=\begin{bmatrix} 2 & 3 & 4 \\ 1 & 2 & 3 \\  -1 & 1 & 2 \end{bmatrix}$  then  $AB=$
Question 267 :
If $\left[ \begin{matrix} 2 & -3 \\ 1 & \lambda  \end{matrix} \right] \times \left[ \begin{matrix} 1 & 5 & \mu  \\ 0 & 2 & -3 \end{matrix} \right] =\left[ \begin{matrix} 2 & 4 & 1 \\ 1 & -1 & 13 \end{matrix} \right],$ then
Question 268 :
If $A=\begin{bmatrix} 1 & -2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$ and $B=\begin{bmatrix} 2 & 1 \\ 3 & 2 \\ 1 & 1 \end{bmatrix}$, then ${ \left( AB \right) }^{ T }$ is equal to
Question 270 :
Let $A = \begin{bmatrix}x + y & y\\ 2x & x - y\end{bmatrix}, B = \begin{bmatrix} 2& -1\end{bmatrix}$ and $C = \begin{bmatrix} 3& 2\end{bmatrix}.$ If $AB = C$, then $A^{2}$ is equal to<br/>
Question 271 :
If $A+I=\begin{bmatrix} 3 & -2 \\ 4 & 1 \end{bmatrix}$, then $\left( A+I \right) \cdot \left( A-I \right) $ is equal to
Question 272 :
If $\begin{bmatrix} 1 & 1\\ -1 & 1\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}=\begin{bmatrix} 2 \\ 4\end{bmatrix}$, then the values of $x $ and $y $ respectively are?
Question 273 :
Let $\left[\begin{array}{ll}<br/>2 & -2\\-2 & \ 5<br/>\end{array}\right]=\left[\begin{array}{ll}<br/>1 & 0\\<br/>-1 & 1<br/>\end{array}\right]\left[\begin{array}{ll}<br/>2 & 0\\<br/>0 & x<br/>\end{array}\right]\left[\begin{array}{ll}<br/>1 & -1\\<br/>0 & 1<br/>\end{array}\right]$, then the value of $x$ is<br/>
Question 274 :
If the matrix $\displaystyle A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ then $\displaystyle A^2$ is
Question 275 :
If $\mathrm{A}=\left[\begin{array}{ll}<br/>1 & 2\\<br/>0 & 3<br/>\end{array}\right]$ and $\mathrm{B}=[3 \space-1]$, then $\mathrm{B}\mathrm{A}=$<br/>
Question 276 :
Let  $A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\   and\  B=\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix},  a,b\in N.$ Then:
Question 277 :
If $A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix}$ and $kA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix}$, then the value of $k, a, b $, are respectively. 
Question 278 :
If $\displaystyle A=\left[ \begin{matrix} 3 & 1 \\ -1 & 2 \end{matrix} \right] $ and $\displaystyle I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] $, then the correct statement is:
Question 279 :
If $\displaystyle A = \begin{bmatrix} 4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3 \end{bmatrix}$ then $\displaystyle A^2$ is equal to
Question 280 : <div><span>$B=A+A^{2}+A^{3}+A^{4}$ </span><br/></div><div><span>If order of $A$ is $3$ then order of $B$ is </span></div>
Question 281 :
If $A$ and $B$ are square matrices of the same order, then $(A+B)(A-B)$ is equal to
Question 282 :
If $A=\begin{bmatrix} 0 & 0 \\ 0 & 5 \end{bmatrix}$, then ${ A }^{ 12 }$ is
Question 283 :
If $O\left( A \right) =2\times 3,$ $O\left( B \right) =3\times 2$ and $O\left( C \right) =3\times 3$, which one of the following is not defined?
Question 284 :
If commutativity is not true in any multiplication of $ 3 \times 3 $ matrices $ A $ and $ B( $ or their powers) then the <span>number of distinct terms in the expansion of $ (A+B)^{3} $ must be:</span>
Question 285 :
If $A=<br/>\begin{bmatrix}<br/>2 & -1 \\<br/>-1 & 2<br/>\end{bmatrix}$ and $I$ is the unit matrix of order $2$, then $A^2$equals
Question 286 :
If the matrix $A = \begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2\end{bmatrix}$, then $A^n=\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ b & 0 & a\end{bmatrix}. n \in N$ where
Question 288 :
A vector is equally inclined to the $x$-axis, $y$-axis and $z$-axis respectively, its direction cosines are
Question 289 :
The direction angles of the line $x = 4z + 3, y = 2 - 3z$ are $\alpha, \beta$ and $\gamma$, then $\cos \alpha + \cos \beta + \cos \gamma =$ ________.
Question 291 :
The direction cosines of a line which is equally inclined to axes, is given by
Question 292 :
The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the $3$ concurrent edges of the cube are coordinate axes)<br/>
Question 293 :
If the $d.c's$ of two lines are connected by the equations $l + m + n = 0, l^2 + m^2 - n^2 = 0$, then angle between the lines is
Question 294 : <span>From the point $P(3, -1, 11)$, a perpendicular is drawn on the line $L$ given by the equation $\dfrac {x}{2} = \dfrac {y - 2}{3} = \dfrac {z - 3}{4}$. Let $Q$ be the foot of the perpendicular.</span><div>What are the direction ratios of the line segment $PQ$?</div>
Question 295 :
A line passes through the points (6, -7, -1) and (2, -3, 1). What are the direction ratios of the line ? <span><br></span>
Question 296 :
Cosine of the angle between two diagonals of a cube is equal to
Question 297 :
$l = m =n = 1$ represents the direction cosines of <br/><br/>
Question 298 :
The projection of the join of the two points $(1,4,5), (6,7,2)$ on the line whose d.s's are $(4,5,6)$ is
Question 299 : <table class="table table-bordered"><tbody><tr><td> List I</td><td>List II </td></tr><tr><td><span>1) d.c's of $x -$ axis</span></td><td><span>a) $(1,1,1)$</span> </td></tr><tr><td><span>2) d.c's of $y -$ axis</span></td><td><span>b)$\left(\displaystyle \frac{]}{\sqrt{3}}\frac{]}{\sqrt{3}},\frac{]}{\sqrt{3}}\right)$</span></td></tr><tr><td><span>3) d.c's of $z -$ axis</span></td><td><span>c) $(1,0,0)$</span><br/></td></tr><tr><td><span>4) d.c's of a line makes </span><span>equal angles with axes</span></td><td><span>d) $(0,1,0)$</span></td></tr><tr><td> </td><td><span>e) $(0,0,1)$</span></td></tr></tbody></table>The correct order for 1, 2, 3, 4 is
Question 300 :
If a line has the direction ratio $18, 12, 4 $, then its direction cosines are:<br/>
Question 301 :
If a line makes the angles $ \alpha , \beta$ and $\gamma$ with the axes, then what is the value of $1+\cos 2\alpha +\cos 2\beta+\cos 2\gamma$<span> equal to ?</span>
Question 302 :
A line passes through the points $(6, -7, -1)$ and $(2, -3, 1)$. The direction cosines of the line so directed that the angle made by it with the positive direction of x-axis is acute, is?
Question 303 : The projections of a directed line segment on the coordinate axes are $12, 4, 3$ respectively.<div>What are the direction cosines of the line segment?</div>
Question 304 :
Can $\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}, \dfrac{-2}{\sqrt{3}}$ be the direction cosines of any directed line?
Question 305 :
Direction cosines $l, m, n$ of two lines are connected by the equation $l-5m+3n=0$ and $7l^{2}+5m^{2}-3n^{2}=0$. The direction cosines of one of the lines are
Question 306 :
Direction cosines of the line $\cfrac { x+2 }{ 2 } =\cfrac { 2y-5 }{ 3 } ,z=-1$ are ____
Question 307 :
What are the DR's of vector parallel to $\left( 2,-1,1 \right) $ and $\left( 3,4,-1 \right) $?
Question 308 :
The direction cosines of the vectors $2\vec {i} + \vec {j} - 2\vec {k}$ is equal to
Question 309 :
If $P(x, y, z)$ moves such that $x=0, z=0$, then the locus of $P$ is the line whose d.cs are<br/>
Question 311 :
If the dr's the line are $(1+\lambda, 1-\lambda, 2)$ and it makes an angle ${60}^{o}$ with the Y-axis then $\lambda$ is
Question 312 :
Direction cosines of ray from $P(1, -2, 4)$ to $Q(-1, 1, -2)$ are
Question 313 :
The projection of the line segment joining $(0, 0, 0)$ and $(5, 2, 4)$ on the line whose direction ratios are $2, -3, 6$ is<br/>
Question 314 :
The direction ratios of the line $6x - 2 = 3y + 1 = 2z - 2$ are 
Question 315 :
If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is
Question 316 :
ABC is a triangle where $A(2,3,5), B(-1,3,2)$ and $C(\lambda , 5, \mu)$. Let the median through $A$ is equally inclined to the axes.<br/>The value of $\mu - \lambda$ is equal to:
Question 317 :
A line makes the same angle $\theta $ with each of the $x$ and $z$-axes. If the angle $\beta$, which it makes with $y$-axis, is such that $\sin ^{ 2 }{ \beta } =3\sin ^{ 2 }{ \theta } $, then $\cos ^{ 2 }{ \theta } $ is equal to
Question 318 :
Let the direction - cosines of the line which is equally inclined to the axis be $\displaystyle \pm \frac{1}{\sqrt{k}}$. Find $k$ ?
Question 319 : The direction ratios of the line perpendicular to the lines<br/><br/><div>$\dfrac {x - 7}{2} = \dfrac {y + 17}{-3} = \dfrac {z - 6}{1}$ and, $\dfrac {x + 5}{1} = \dfrac {y + 3}{2} = \dfrac {z - 4}{-2}$ are proportional to</div>
Question 320 :
If a line makes angles $\alpha, \beta, \gamma$ and $\delta$ with the diagonals of a cube, Then, $\cos^2\alpha +\cos^2\beta +\cos^2\gamma +\cos^2\delta =\dfrac {a}{b}$, where $a$ and $b$ are in lowest form, find $a+b$
Question 321 :
Which of the following triplets give the direction cosines of a line ?
Question 323 :
Direction ratio of two lines are $l_{1}, m_{1},n_{1}$ and $l_{2},m_{2},n_{2}$ then direction ratios of the line perpendicular to both the lines are
Question 324 :
If $\alpha,\beta,\gamma\in[0,2\pi]$, then the sum of all possible values of $\alpha, \beta,\gamma$ if $\sin \alpha=-\dfrac{1}{\sqrt{2}}$, $\cos \beta=-\dfrac{1}{2}$, $\tan \gamma=-\sqrt{3}$, is
Question 325 :
If direction cosines of two lines are proportional to $(2,3,-6)$ and $(3,-4,5)$, then the acute angle between them is
Question 326 :
If a ray makes angles $\alpha, \beta, \gamma$ and $\delta$ with the four diagonals of a cube and<br>$\mathrm{A}:\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+\cos^{2}\delta$<br>$\mathrm{B}:\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma+\sin^{2}\delta$<br>$\mathrm{C}:\cos 2\alpha+\cos 2\beta+\cos 2\gamma+\cos 2\delta$<br>Arrange $A,B,C$ in descending order<br>
Question 327 :
The projection of a directed line segment on the co-ordinate axes are $12, 4, 3$, the DC's of the line are<br/>
Question 328 :
A particle moves along a straight line according to the law $s=16-2t+3t^3$, where $s$ metres is the distance of the particle from a fixed point at the end of $t$ second. The acceleration of the particle at the end of $2$ s is
Question 329 :
If $y = 6x -x^3$ and $x$ increases at the rate of $5$ units per second, the rate of change of slope when $x = 3$ is
Question 330 :
A particle moves along a curve so that its coordinates at time $t$ are $\displaystyle x = t, y = \frac{1}{2} t^{2}, z =\frac{1}{3}t^{3}$ acceleration at $ t=1 $ is<br>
Question 331 :
The sides of two squares are $x$ and $y$ respectively, such that $y = x + x^{2}$. The rate of change of area of <span>second square with respect to area of first square is ________.</span>
Question 332 :
The radius of a circular plate is increased at $ 0.01 \text {cm/sec}.$ If the area is increased at the rate of $\frac{\pi }{{10}}$. Then its radius is 
Question 334 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?
Question 335 :
What is the rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ $cm$.
Question 336 :
The radius of a circle is uniformly increasing at the rate of $3cm/s$. What is the rate of increase in area, when the radius is $10cm$?
Question 337 :
If the displacement of a particle moving in straight line is given by $x=3t^2+2t+1$ at time $t$ then  the acceleration of the particle at time $t=3$ is
Question 338 :
The rate of change of surface area of a sphere of radius $r$ when the radius is increasing at the rate of $2 cm/sec$ is proportional to
Question 339 :
The area of an equilateral triangle of side $'a'$ feet is increasing at the rate of $4 sq.ft./sec$. The rate at which the perimeter is increasing is
Question 340 :
The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+15$ is
Question 341 :
Consider the following statements:<br/>1. $\dfrac {dy}{dx}$ at a point on the curve gives slope of the tangent at that point.<br/>2. If $a(t)$ denotes acceleration of a particle, then $\displaystyle \int a(t) dt + c$ give velocity of the particle.<br/>3. If $s(t)$ gives displacement of a particle at time $t$, then $\dfrac {ds}{dt}$ gives its acceleration at that instant.<br/>Which of the above statements is/ are correct?
Question 342 :
A ladder, 5 meter long, standing on a horizontal floor, leans against vertical wall. If the top of the ladder slides downwards at the rate of 10cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is:
Question 343 :
The rate of change of the area of a circle with respect to its radius $r$ at $r = 6 cm$ is.
Question 344 :
If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is
Question 345 :
A stone is dropped into a quiet lake and waves move in circles at the speed of $5$ cm/s At the instant when the radius of the circular wave is $8$ cm how fast is the enclosed area increasing?<br/>
Question 346 :
The position of a particle is given by $s={ t }^{ 3 }-6{ t }^{ 2 }-15t$ where $s$ in metres, $t$ is in seconds. If the particle is at rest, then time $t=.....$
Question 347 :
The side of a square sheet is increasing at the rate of $4 cm$ per minute. The rate by which the area increasing when the side is $8 cm$ long is.
Question 348 :
If the radius of a sphere is measured as $8\ cm$ with a error of $0.03\ cm$, then the approximate error calculate its volume is
Question 349 :
Excluding stoppages, the speed of a bus is $72\ kmph$ and including stoppages, it is $60\ kmph$. For how many minutes does the bus stop per hour?
Question 350 :
If the distance $s$ travelled by a particle in time $t$ is given by $s=t^2-2t+5$, then its acceleration is
Question 351 :
If the rate of increase of the population of the city is $5\%$ per year. In t time population of city is P, then expression of P in terms of t is _________.
Question 352 :
A balloon which always remain spherical has a variable diameter $\cfrac { 3 }{ 2 } \left( 2x+3 \right) $. The rate of change of its volume w.r.t $x$, is
Question 353 :
If $x\sqrt{1+y} + y\sqrt{1+x} = 0$ and $x \neq y $ then $\dfrac{dy}{dx} = $
Question 354 :
Find the rate of change of the area of a circle with respect to its radius $r$ when<br>(i) $r = 3$ cm<br>(ii) $r=4$ cm
Question 355 :
If a circular iron sheet of radius $30 cm$ is heated such that its area increases at the uniform rate of $6 \pi \:cm^2 / hr$, then the rate (in mm / hr) at which the <span>radius of the circular sheet increases is.</span>
Question 356 :
A point on the parabola $y^2 = 18x$ at which the ordinate increase at twice the rate of the abscissa is _______, $\left(\dfrac{dx}{dt} \neq 0\right)$.
Question 357 :
An object stars from rest at $t=0$ and accelerates at a rate given by $a=6t$. What i) its velocity and ii) its displacement at any time $t$?
Question 358 :
If a ball is thrown vertically upwards and the height 's' reached in time 't' is given by $s = 22 t - 11 t^2$, then the total distance travelled by the ball is
Question 359 :
The equation of motion of a particle moving along a straight line is $s=2t^3-9t^2+12t$, where the units of s and t are centimetre and second. The acceleration of the particle will be zero after.
Question 360 :
Water runs into an inverted conical tent at the rate of $20\:{ft}^3/min$ and leaks out at the rate of $5\:{ft}^3/min$. The height of the water is three times the radius of the water's surface. The radius of the water surface is increasing when the radius is $5\:ft$, is
Question 361 :
If metallic circular plate of radius $50 cm$ is heated so that its radius increases at the rate of $1$ mm per hour, then the rate at which the area of the plate increases (in $cm^2/hr$) is.
Question 362 :
The approximate change in the volume of a cube of side $x$ metres caused by increasing <span>the side by $3$% is :</span>
Question 363 :
A cylindrical tank of radius $10m$ is being filled with wheat at the rate of $314$ cubic metre per hour. then the depth of the wheat is increasing at the rate of
Question 364 :
Find an angle $\theta, 0 < \theta < \dfrac{\pi}{2}$, which increases twice as fast as its sine.
Question 365 :
The total revenue received from the sale of $x$ units is given by $R(x)=10{ x }^{ 2 }+20x+1500$. The marginal revenue, when $x=2015$, is _______
Question 366 :
The relation between the time $t$ and distance '$x$' is given by $t=p{ x }^{ 2 }+qx$, where $p$ and $q$ are constants. The relation between velocity $v$ and acceleration $f$ is
Question 367 :
If $s=ae^t+b{e}^{-t}$ is the equation of motion of a particle, then its acceleration is equal to
Question 368 :
<span>At time $t > 0$ the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At $t = 0$ the radius of the sphere is $1$ unit and at $t = 15$ the radius is $2$ units. Find the radius of the sphere as a function of time $t$.</span>
Question 369 :
A point on the parabola $y^2=18x$ at which the ordinate increase at twice the rate of the abscissa is<br>
Question 370 :
If the distance '$s$' metres transversed by a particle in $t$ seconds is given by $\displaystyle s={ t }^{ 3 }-3{ t }^{ 2 }$, then the velocity of the particle when the acceleration is zero, in metre/sec is
Question 371 :
The volume of metal in a hollow sphere is constant.If the inner radius is increasing at the rate of $1 cm/sec$, then the rate of increase of the outer radius when the radii are $4 cm$ and $8 cm$ <span>respectively is.</span>
Question 372 :
The $x $ and $y$ coordinate of a particle at any time t are given by $x=7t+4{ t }^{ 2 }$ and $y=5t$, where $x$ and $y$ are in meter and $t$ in seconds. The acceleration of particle at $t=5s$ is
Question 373 :
For what values of $ x$ is the rate of increase of $x^3 - 5x^2 + 5x +8$ is twice the rate of increase of $x$?
Question 374 :
If a particle moving along a line follows the law $s = \sqrt{1+t}$, then the accelertion is proportional to
Question 375 :
The rate of change of the surface area of the sphere of radius $r$ when the radius is increasing at the rate of $2$ cm/sec is proportional to :
Question 376 :
An open cylindrical can of given capacity is to be made from a metal sheet of uniform thickness. If no allowance is to be made for waste material, what will be the most economical ratio of the radius to the height of the can ?
Question 377 :
The side of a square sheet is increasing at the rate of $4\ cm$ per minute. The rate by which the area increasing when the side is $8\ cm$ long is-
Question 378 :
A stone is dropped into a quiet lake and waves move in a circle at a speed of $3.5 cm/sec$. At the instant when the radius of the circular wave is $7.5 cm$. The enclosed area increases as fastly as.
Question 379 :
The volume of a sphere is increasing the rate of $1200\ c.cm/sec$. The rate of increase in its surface area when the radius is $10\ cm$ is
Question 380 :
Bacteria multiply at a rate proportional to the number present. If the original number $N$ double in $3$ hours, the number of the bacteria will be $4N$ is (in hours).
Question 381 :
If the radius of a circle is doubled, its area is increased by :
Question 382 :
If the volume of a spherical ball is increasing at the rate of $4\pi\ cc/sec$, then the rate of increase of its radius (in cm/sec), when the volume is $288\pi\ cc$, is 
Question 383 :
A particle moves along the curve $y = x^2 + 2x$. Then the points on the curve are the x and y coordinates of the particle changing at the same rate, are.
Question 384 :
The point on the curve $6y={ x }^{ 3 }+2$ at which y-co-ordinate is changing 8 times as fast as x-coordinate is ................
Question 385 :
The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is $1cm$ the altitude is $6 cm$. When the radius is $6cm$, then volume is increasing at the rate of $1$ Cu $cm/sec$. When the radius is $36cm$, the volume is increasing at a rate of $n$ cu. $cm/sec$. The value of '$n$' is equal to:
Question 386 :
The rate of change of area of a circle with respect to its radius at $r=2 \ cm$ is
Question 387 :
The temperature at $12$ noon was ${10}^{o}C$ above zero. If it decreases at the rate of ${2}^{o}C$ per hour until midnight, at what time would the temperature be ${8}^{o}C$ below zero? What would be the temperature at midnight?
Question 388 :
A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermost ripple increases at the rate of $5$ cm/sec. Then rate of change of area after $2$ seconds is _______.
Question 389 :
The sides of an equilateral triangle are increasing at the rate of $2{ cm }/{ s }$. The rate at which the area increases when the side is $10 cm$, is
Question 390 :
The motion of a particle along a straight line is described by equation<br/><br/>$x = 8 + 12t t^{3}$<br/><br/>where x is in metre and t in second. The retardation of the particle when its velocity becomes zero is<br/>
Question 391 :
A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey, it move with constant acceleration $f$ and in the seconds part with constant retardation $r$. The value of $t$ given by
Question 392 :
Two particle start simultaneously from the same point and move along two straight lines, one with uniform velocity $\vec {u}$ and the other from rest with uniform acceleration $\vec {f}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle with respect to the first least after a time
Question 393 :
A particle moves along the curve $\displaystyle y=x^{3/2}$ in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units per second. The value of when x = 3 is
Question 394 :
A spherical balloon is filled with $4500\ \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\ \pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49\ minutes$ after the leakage began is:<br>
Question 395 :
A balloon which always remains spherical is being inflated by pumping in $10$ cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is $15\ cms$
