Applications Scalar Triple Product

Q.

Solve $8x+5y=9,3x+2y=4$by any method.

Q.

$x+7y=10,3x-2y=7;$solve the following.

Q.

$R$ is a relation on $N$ given by$R=\left\{(x,y):4x+3y=20\right\}$. Which of the following belongs to $R$?

1. $(-4,12)$

2. $(5,0)$

3. $(3,4)$

4. $(2,4)$

Q.

If the lines $ax+2y+1=0,bx+3y+1=0$ and $cx+4y+1=0$ are concurrent, then $a,b,c$ are in:

1. A.P.

2. G.P.

3. H.P.

4. None of these

Q.

Solve the following system of equations using the cross multiplication method.

$$\begin{gathered} 21x=13y+21, \\ 4x+7y-20=0 \end{gathered}$$

Q.

If $\overrightarrow{a}=2\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}$and a unit vector $\stackrel{\rightharpoonup }{c}$is coplanar. If $\overrightarrow{c}$is perpendicular to$\overrightarrow{a}$, then $\overrightarrow{c}$is equal to?

1. $\pm \frac{1}{\sqrt{2}}\left(-\hat{j}+\hat{k}\right)$

2. $\pm \frac{1}{\sqrt{3}}\left(-\hat{i}-\hat{j}+\hat{k}\right)$

3. $\pm \frac{1}{\sqrt{5}}\left(\hat{i}-2\hat{j}\right)$

4. $\pm \frac{1}{\sqrt{3}}\left(\hat{i}-\hat{j}-\hat{k}\right)$

Q.

If $\mathrm{t}$ is a parameter then $\mathrm{x}=\mathrm{a}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)$ and $\mathrm{y}=\mathrm{b}\left(\mathrm{t}-\frac{1}{\mathrm{t}}\right)$ represents

Q.

If the volume of parallelepiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k},\overrightarrow{v}=\hat{i}+\hat{j}+3\hat{k},\overrightarrow{w}=2\hat{i}+\hat{j}+\hat{k}$ be $1$ cubic unit. If $\theta$ be the angle between the edges $\overrightarrow{u},\overrightarrow{w}$ then, $\cos \theta$ can be:

1. $\frac{7}{6\sqrt{6}}$

2. $\frac{5}{7}$

3. $\frac{7}{6\sqrt{3}}$

4. $\frac{5}{3\sqrt{3}}$

Q.

What multiple of equation $2$ should be subtracted from equation $1$?
$$\begin{array}{rl}& 2x-4y=6\\ & -x+5y=0\end{array}$$
After this elimination step, solve the triangular system.

If the right-hand side changes to $(-6,0)$, what is the new solution?

Q.

If the lines of regression are $3x+12y=19$ and $3y+9x=46$, then $r$ will be:

1. $0.289$

2. $-0.289$

3. $0.209$

4. None of these

Q.

Choose the correct answer for the given question:

Which rule represents the relation in the table $?$

$x$	$2$	$3$	$4$	$5$	$6$
$y$	$1$	$3$	$5$	$7$	$9$

1. $y=x-1$

2. $y=x+3$

3. $y=2x-3$

4. $y=2x=3$

Q.

Match the linear equations in Column-I with their solution is Column-II.

COLUMN I	COLUMN II
(A) $2\mathrm{x}+3\mathrm{y}=5\sqrt{2}$	(p) $(\sqrt{2},0)$
(B) $\sqrt{2}\mathrm{x}+4\mathrm{y}=6$	(q) $(\sqrt{2},\sqrt{2})$
(C) $3\mathrm{x}+\frac{\mathrm{y}}{2}=\frac{1}{\sqrt{2}}$	(r) $(\sqrt{2},1)$
(D) $\sqrt{2}\mathrm{x}+5\mathrm{y}=2$	(s) $(0,\sqrt{2})$

1. $\left(A\right)\to p;\left(B\right)\to r;\left(C\right)\to q;\left(D\right)\to s$

2. $\left(A\right)\to r;\left(B\right)\to p;\left(C\right)\to q;\left(D\right)\to s$

3. $\left(A\right)\to q;\left(B\right)\to r;\left(C\right)\to s;\left(D\right)\to p$

4. $\left(A\right)\to r;\left(B\right)\to s;\left(C\right)\to p;\left(D\right)\to q$

Q.

If the curve $ay+x^2=7$, and $x^3=y$ cut orthogonally at $\left(1,1\right)$, then the value of $a$ is

1. $1$

2. $0$

3. $6$

4. 6

Q.

Solve the following pairs of equations by reducing them to a pair of linear equations:

$$\begin{gathered} \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2 \\ \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1 \end{gathered}$$

Q.

How do you convert the parametric equations into a Cartesian equation by eliminating the parameter $r$ : $x=r^2+r,y=r^2-r$ ?

Q.

Solve the following system of equations using the cross multiplication method.

$$\begin{gathered} 0.4x+0.3y=1.7 \\ 7x-2y=8 \end{gathered}$$

Q. If $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+0\hat{j}+\hat{k}$ then the vectors $\overrightarrow{c}$ satisfying the conditions, (i) that it is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$ (ii) that it is perpendicular to $\overrightarrow{b}$, and (iii) that $\overrightarrow{a}.\overrightarrow{c}=7$, is

1. $-\frac{3}{2}\hat{i}+\frac{5}{2}\hat{j}+3\hat{k}$
2. $-3\hat{i}+5\hat{j}+6\hat{k}$
3. $-6\hat{i}+0\hat{j}+5\hat{k}$
4. $-\hat{i}+2\hat{j}+2\hat{k}$

Q.

Solve: $5(3+\mathrm{y})=2\mathrm{y}$

Q. The volume of the tetrahedron with vertices at (1, 2, 3), (4, 3, 2), (5, 2, 7), (6, 4, 8) is

1. $\frac{22}{3}$
2. $\frac{11}{3}$
3. $\frac{1}{3}$
4. $\frac{16}{3}$

Q.

If $y$ varies directly with $x$ and $y=12$ when $x=0.5$, then what is $y$ when $x=3$?

Q. The volume of the parallellopiped whose coterminal edges are $2\hat{i}-3\hat{j}+4\hat{k},\hat{i}+2\hat{j}-2\hat{k},3\hat{i}-\hat{j}+\hat{k}$ is

1. 5
2. 6
3. 7
4. 8

Q.

Solve the following system of equations using the cross multiplication method.

Q. Let $\overrightarrow{a}=(2+\sin \theta )\hat{i}+\cos \theta \hat{j}+\sin 2\theta \hat{k}$, $\overrightarrow{b}=\sin (\theta +\frac{2\pi }{3})\hat{i}+\cos (\theta +\frac{2\pi }{3})\hat{j}+\sin (2\theta +\frac{4\pi }{3})\hat{k}$ and $\overrightarrow{c}=\sin (\theta -\frac{2\pi }{3})\hat{i}+\cos (\theta -\frac{2\pi }{3})\hat{j}+\sin (2\theta -\frac{4\pi }{3})\hat{k}$ be three vectors where $\theta \in (0,\frac{\pi }{2})$. The maximum volume of the tetrahedron whose coterminous edges are given by the vectors $2\overrightarrow{b}\times \overrightarrow{c},3\overrightarrow{c}\times \overrightarrow{a}$ and $\overrightarrow{a}\times 4\overrightarrow{b}$ is

Q.

Solve the system:

$$\begin{gathered} -4\mathrm{x}+3\mathrm{y}=14 \\ \mathrm{y}=2\mathrm{x}+8 \end{gathered}$$

Q.

Solve it :-

$1.2y=2.4$

Q.

Find the value of $\mathrm{x}$ for given value of $\mathrm{y}=5\mathrm{x}-7$when $\mathrm{y}=-22$.

Q.

Solve the following system of equations using the cross multiplication method.

$$\begin{gathered} 7x-y=33 \\ 12y-x=19 \end{gathered}$$

Q.

Solve the system:

$$\begin{gathered} \mathrm{x}+\mathrm{y}=-4 \\ 6\mathrm{x}+2\mathrm{y}=4 \end{gathered}$$

Q.

Solve $8\mathrm{x}+2\mathrm{y}=-12,5\mathrm{x}-\mathrm{y}=4$