Jee Math Test Series Quiz-8

Let \(a, b, c \in \mathbf{R}\) be such that \(a^2+b^2+c^2=1\). If \(a \cos \theta=b \cos \left(\theta+\frac{2 \pi}{3}\right)=c \cos \left(\theta+\frac{4 \pi}{3}\right)\), where \(\theta=\frac{\pi}{9}\), then the angle between the vectors \(a \hat{i}+b \hat{j}+c \hat{k}\) and \(b \hat{i}+c \hat{j}+a \hat{k}\) is :

Let \(\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}\) be two vectors. If a vector perpendicular to both the vectors \(\vec{a}+\vec{b}\) and \(\vec{a}-\vec{b}\) has the magnitude 12 then one such vector is :

If a unit vector \(\vec{a}\) makes angles \(\pi / 3[latex] with [latex]\hat{i}, \pi / 4\) with \(\hat{j}\) and \(\theta \in(0, \pi)\) with \(\hat{k}\), then a value of, is:

If the position vectors of the vertices \(A, B\) and \(C\) of a \(\triangle \mathrm{ABC}\) are respectively \(4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(2 \hat{i}+5 \hat{j}+7 \hat{k}\), then the position vector of the point, where the bisector of \(\angle A\) meets \(B C\) is \(\quad\)

Let \(\overrightarrow{\mathrm{u}}\) be a vector coplanar with the vectors \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}\) and \(\vec{b}=\hat{j}+\hat{k}\). If \(\vec{u}\) is perpendicular to \(\vec{a}\) and \(\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{b}}-24\), then \(|\overrightarrow{\mathrm{u}}|^2\) is equal to:

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=\hat{j}-\hat{k}\) and a vector \(\vec{b}\) be such that \(\vec{a} \times \vec{b}=\vec{c}\) and \(\vec{a} \cdot \vec{b}=3\). Then \(|\vec{b}|\) equals?

If \(\vec{a}, \vec{b}\), and \(\overrightarrow{\mathrm{c}}\) are unit vectors such that \(\vec{a}+2 \vec{b}+2 \overrightarrow{\mathrm{c}}=\overrightarrow{0}\), then \(|\vec{a} \times \overrightarrow{\mathrm{c}}|\) is equal to

Let the volume of a parallelepiped whose coterminous edges are given by \(\vec{u}=\hat{i}+\hat{j}+\lambda \hat{k}, \vec{v}=\hat{i}+\hat{j}+3 \hat{k}\), and \(\vec{w}=2 \hat{i}+\hat{j}+\hat{k}\), be 1 cu.unit. If \(\theta\) be the angle between the edges \(\vec{u}\) and \(\vec{w}\), then \(\cos \theta\) can be:

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

Let \(\overrightarrow{\mathrm{a}}=2 \hat{i}+\lambda_1 \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=4 \hat{i}+\left(3-\lambda_2\right) \hat{j}+6 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=3 \hat{i}+6 \hat{j}+\left(\lambda_3-1\right) \hat{k}\) be three vectors such that \(\overrightarrow{\mathrm{b}}=2 \overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{a}}\) is perpendicular to \(\overrightarrow{\mathrm{c}}\) Then a possible value of \(\left(1_1, 1_2, 1_3\right)\) is:

Let \(\overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\sqrt{2} \hat{k}, \overrightarrow{\mathrm{b}}=\mathrm{b}_1 \hat{i}+\mathrm{b}_2 \hat{j}+\sqrt{2} \hat{k}\) and \(\overrightarrow{\mathrm{c}}=5 \hat{i}+\hat{j}+\sqrt{2} \hat{k}\) be three vectors such that the projection vector of \(\overrightarrow{\mathrm{b}}\) on \(\vec{a}\) is \(\vec{a}\).
If \(\vec{a}+\vec{b}\) is perpendicular to \(\vec{c}\), then \(|\vec{b}|\) is equal to:

If \(|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=3\) and \(|2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|=5\), then \(|2 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|\) equals:

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}-\hat{k}\) be three vectors. A vector \(\vec{v}\) in the plane of \(\vec{a}\) and \(\vec{b}\), whose projection on \(\vec{c}\) is \(\frac{1}{\sqrt{3}}\), is given by

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 \(\overrightarrow{\mathrm{OP}}\) (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 \(\overrightarrow{O P}\) and \(\hat{u}\) be the unit vector along \(\overrightarrow{O P}\). Then,

If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero, non-coplanar vectors and \(\overrightarrow{b_1}=\vec{b}-\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a}, \overrightarrow{b_2}=\vec{b}+\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a}\)
\(
\overrightarrow{c_1}=\vec{c}-\frac{\vec{c} \cdot \vec{a}}{|\vec{a}|^2} \vec{a}+\frac{\vec{b} \cdot \vec{c}}{|\vec{c}|^2} \overrightarrow{b_1},
\)
\(
\overrightarrow{c_2}=\vec{c}-\frac{\vec{c} \cdot \vec{a}}{|\vec{a}|^2} \vec{a}-\frac{\overrightarrow{b_1} \cdot \vec{c}}{\left|\overrightarrow{b_1}\right|^2} \overrightarrow{b_1},
\)
\(
\overrightarrow{c_3}=\vec{c}-\frac{\vec{c} \cdot \vec{a}}{|\vec{c}|^2} \vec{a}+\frac{\vec{b} \cdot \vec{c}}{|\vec{c}|^2} \overrightarrow{b_1}, \quad \overrightarrow{c_4}=\vec{c}-\frac{\vec{c} \cdot \vec{a}}{|\vec{c}|^2} \vec{a}=\frac{\vec{b} \cdot \vec{c}}{|\vec{b}|^2} \overrightarrow{b_1},
\)
\(
\text { then the set of orthogonal vectors is }
\)

If \(\vec{a}\) and \(\vec{b}\) are two unit vectors such that \(\vec{a}+2 \vec{b}\) and \(5 \vec{a}-4 \vec{b}\) are perpendicular to each other then the angle between \(\vec{a}\) and \(\vec{b}\) is

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors, then \(|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2\) does NOT exceed

Let \(\vec{u}, \vec{v}\) and \(\vec{w}\) be vectors such that \(\vec{u}+\vec{v}+\vec{w}=0\). If \(|\vec{u}|=3,|\vec{v}|=4\) and \(|\vec{w}|=5\), then \(\vec{u} \cdot \vec{v}+\vec{v} \cdot \vec{w}+\vec{w} \cdot \vec{u}\) is

Let \(\vec{p}\) and \(\vec{q}\) be the position vectors of \(P\) and \(Q\) respectively, with respect to \(O\) and \(|\vec{p}|=p,|\vec{q}|=q\). The points \(R\) and \(S\) divide \(P Q\) internally and externally in the ratio \(2: 3\) respectively. If \(O R\) and \(O S\) are perpendicular then

If the vectors \(\overrightarrow{\mathrm{AB}}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is

Let \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\) be the points on the plane with position vectors \(-2 \hat{i}-\hat{j}, 4 \hat{i}, 3 \hat{i}+3 \hat{j}\) and \(-3 \hat{i}+2 \hat{j}\) respectively. The quadrilateral PQRS must be a

Let \(\alpha, \beta, \gamma\) be distinct real numbers. The points with position vectors \(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}, \beta \hat{i}+\gamma \hat{j}+\alpha \hat{k}, \gamma \hat{i}+\alpha \hat{j}+\beta \hat{k}\)

The points with position vectors \(60 i+3 j, 40 i-8 j, a i-52 \mathrm{j}\) are collinear if

If \(\vec{a}\) and \(\vec{b}\) are non-collinear vectors, then the value of \(\alpha\) for which the vectors \(\vec{u}=(\alpha-2) \vec{a}+\vec{b}\) and \(\vec{v}=(2+3 \alpha) \vec{a}-3 \vec{b}\) are collinear is :

If \(\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}+(2 r-1) \hat{k}\) are three vectors such that \(\vec{c}\) is parallel to the plane of \(\vec{a}\) and \(\vec{b}\), then \(r\) is equal to

Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors which are pairwise non-collinear. If \(\vec{a}+3 \vec{b}\) is collinear with \(\vec{c}\) and \(\vec{b}+2 \vec{c}\) is collinear with \(\vec{a}\), then \(\vec{a}+3 \vec{b}+6 \vec{c}\) is:

If the \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}(p \neq q \neq r \neq 1)\) vector are coplanar, then the value of \(p q r-(p+q+r)\) is

The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\hat{j}+\hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then which one of the following gives possible values of \(\alpha\) and \(\beta\)?

\(A B C\) is a triangle, right angled at \(A\). The resultant of the forces acting along \(\overline{A B}, \overline{B C}\) with magnitudes \(\frac{1}{A B}\) and \(\frac{1}{A C}\) respectively is the force along \(\overline{A D}\), where \(D\) is the foot of the perpendicular from \(A\) onto \(B C\). The magnitude of the resultant is

If \(C\) is the mid point of \(A B\) and \(P\) is any point outside \(A B\), then

Let \(a, b\) and \(c\) be distinct non-negative numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) lie in a plane, then \(c\) is

If \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(\bar{a}+2 \bar{b}+3 \bar{c}, \lambda \bar{b}+4 \bar{c}\) and \((2 \lambda-1) \bar{c}\) are non coplanar for

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that no two of these are collinear. If the vector \(\vec{a}+2 \vec{b}\) is collinear with \(\vec{c}\) and \(\vec{b}+3 \vec{c}\) is collinear with \(\vec{a}\) ( \(\lambda\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals

Consider points \(A, B, C\) and \(D\) with position vectors \(7 \hat{i}-4 \hat{j}+7 \hat{k}, \hat{i}-6 \hat{j}+10 \hat{k},-\hat{i}-3 \hat{j}+4 \hat{k} \quad\) and \(\quad 5 \hat{i}-\hat{j}+5 \hat{k}\)
respectively. Then \(A B C D\) is a

If \(\left|\begin{array}{lll}a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3\end{array}\right|=0\) and vectors \(\left(1, a, a^2\right)\), \(\left(1, b, b^2\right)\) and \(\left(1, c, c^2\right)\) are non-coplanar, then the product abc equals

The vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k} \quad \& \quad \overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(A B C\). The length of the median through \(A\) is

In a triangle \(A B C\), right angled at the vertex \(A\), if the position vectors of \(A, B\) and \(C\) are respectively \(3 \hat{i}+\hat{j}-\hat{k},-\hat{i}+3 \hat{j}+p \hat{k}\) and \(5 \hat{i}+q \hat{j}-4 \hat{k}\), then the point \((p, q)\) lies on a line:

In a parallelogram \(\mathrm{ABD},|\overrightarrow{A B}|=a,|\overrightarrow{A D}| \Rightarrow b\) and \(|\overrightarrow{A C}|=c\), then \(\overrightarrow{D A} \cdot \overrightarrow{A B}\) has the value :

If \(\hat{x}, \hat{y}\) and \(\hat{z}\) are three unit vectors in three-dimensional space, then the minimum value of
\(
|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2
\)

If \(|\vec{a}|=2,|\vec{b}|=3\) and \(|2 \vec{a}-\vec{b}|=5\), then \(|2 \vec{a}+\vec{b}|\) equals:

If \(\hat{a}, \hat{b}\) and \(\hat{c}\) are unit vectors satisfying \(\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}\), then the angle between the vectors \(\hat{a}\) and \(\hat{c}\) is :

Let \(\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}-2 \hat{k}\) be three vectors. A vector of the type \(\vec{b}+\lambda \vec{c}\) for some scalar \(\lambda\), whose projection on \(\vec{a}\) is of magnitude \(\sqrt{\frac{2}{3}}\) is :

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\vec{p}\) and \(\angle B A D\) be an acute angle. If \(\vec{r}\) is the vector that coincide with the altitude directed from the vertex \(\mathrm{B}\) to the side \(A D\), then \(\vec{r}\) is given by:

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors. If the vectors \(\vec{c}=\hat{a}+2 \hat{b}\) and \(\vec{d}=5 \hat{a}-4 \hat{b}\) are perpendicular to each other, then the angle between \(\hat{a}\) and \(\hat{b}\) is :

If \(a+b+c=0,|\vec{a}|=3,|\vec{b}|=5\) and \(|\vec{c}|=7\), then the angle between \(\vec{a}\) and \(\vec{b}\) is

A unit vector which is perpendicular to the vector \(2 \hat{i}-\hat{j}+2 \hat{k}\) and is coplanar with the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(2 \hat{i}+2 \hat{j}-\hat{k}\) is

\(A B C D\) is parallelogram. The position vectors of \(A\) and \(C\) are respectively, \(3 \hat{i}+3 \hat{j}+5 \hat{k}\) and \(\hat{i}-5 \hat{j}-5 \hat{k}\). If \(M\) is the midpoint of the diagonal \(D B\), then the magnitude of the projection of \(\overrightarrow{O M}\) on \(\overrightarrow{O C}\), where \(O\) is the origin, is

If the vectors \(\vec{a}=\hat{i}-\hat{j}+2 \hat{k}, \quad \vec{b}=2 \hat{i}+4 \hat{j}+\tilde{\hat{k}}\) and \(\vec{c}=\hat{\lambda}+\hat{j}+\hat{\mu}\) are mutually orthogonal, then \((\lambda, \mu)=\)

The non-zero vectors are \(\vec{a}, \vec{b}\) and \(\vec{c}\) are related by \(\vec{a}=8 \vec{b}\) and \(\vec{c}=-7 \vec{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is

The values of a, for which points A, B, C with position vectors \(2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}\) and \(a \hat{i}-3 \hat{j}+\hat{k}\) respectively are the vertices of a right angled triangle with \(C=\frac{\pi}{2}\) are