Find Plus2 Important Maths 5 Mark Questions Part 4 in this post. For more practice questions on Maths, refer to the other parts in the 'Related Post' section.
1) If $ A=\begin{bmatrix} 5\;\;\;\;\; 2 \\ 7\;\;\;\;\; 3 \\ \end{bmatrix} $ and $ B=\begin{bmatrix} 2\;\;\;\;\; -1 \\ -1\;\;\;\;\; 1 \\ \end{bmatrix} $ verify that
(i) $ (AB)^{-1}=B^{-1}A^{-1} $
(ii) $ (AB)^T=B^TA^T $
2) Solve: x + y+ 2x = 0; 3x + 2y +z = 0; 2x + y –z =0
3) Solve the following homogenous linear equations: x + 2y – 5z = 0; 3x + 4y + 6z = 0; x + y + z = 0
4) (i) For any vector $ \overrightarrow{r} $, prove that $ \overrightarrow{r}=(\overrightarrow{r}.\overrightarrow{i})\overrightarrow{i}+(\overrightarrow{r}.\overrightarrow{j})\overrightarrow{j}+(\overrightarrow{r}.\overrightarrow{k})\overrightarrow{k} $
(ii) Find the projection of the vector $ 7\overrightarrow{i}+\overrightarrow{j}-4\overrightarrow{k} $ on $ 2\overrightarrow{i}+6\overrightarrow{j}+3\overrightarrow{k} $
5) Angle in a semi-circle is a right angle. Prove by vector method.
6) Find the vectors of magnitude 6 which are perpendicular to both the vector $ 4\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k} $ and $ -2\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k} $
7) Show that the torque above the point A(3, –1, 3) of a force $ 4\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k} $ through the point B(5, 2, 4) is $ \overrightarrow{i}+2\overrightarrow{j}-8\overrightarrow{k} $
8) If $ arg(z-1) = \frac{\pi}{6} $ and $ arg(z+1) = 2\frac{\pi}{3} $, then prove |z|=1.
9) If $ cos\; \alpha+cos \;\beta+cos\; \gamma=0=sin\; \alpha+sin\; \beta+sin\; \gamma $, prove that
(i) $ cos\; 3\alpha+cos \; 3\beta+cos\; 3\gamma=3cos(\alpha+\beta+\gamma) $
(ii) $ sin\; 3\alpha+sin \; 3\beta+sin\; 3\gamma=3sin(\alpha+\beta+\gamma) $
10) Find all the values: $ (-\sqrt 3-i)^{\frac{2}{3}} $
11) Prove that if $ \omega^3=1 $, then $ \frac{1}{1+2\omega}-\frac{1}{1+\omega}+\frac{1}{2+\omega}=0 $
12) A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 9 mts. If the distance across (diameter) the top of the mirror is 160 cm, how deep is the mirror at the middle?
13) Find the equation of the ellipse whose vertices are (-1, 4) and (-7, 4) and eccentricity is $ \frac{1}{3} $
14) Find the equations of the tangent and normal to the hyperbola $ \frac{x^2}{9}-\frac{y^2}{12}=1 $ at $ \Theta=\frac{\pi}{6} $
15) Find the angle between the asymptotes of the hyperbola $ 4x^2-5y^2-16x+10y+31=0 $
OR
Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.
1) If $ A=\begin{bmatrix} 5\;\;\;\;\; 2 \\ 7\;\;\;\;\; 3 \\ \end{bmatrix} $ and $ B=\begin{bmatrix} 2\;\;\;\;\; -1 \\ -1\;\;\;\;\; 1 \\ \end{bmatrix} $ verify that
(i) $ (AB)^{-1}=B^{-1}A^{-1} $
(ii) $ (AB)^T=B^TA^T $
2) Solve: x + y+ 2x = 0; 3x + 2y +z = 0; 2x + y –z =0
3) Solve the following homogenous linear equations: x + 2y – 5z = 0; 3x + 4y + 6z = 0; x + y + z = 0
4) (i) For any vector $ \overrightarrow{r} $, prove that $ \overrightarrow{r}=(\overrightarrow{r}.\overrightarrow{i})\overrightarrow{i}+(\overrightarrow{r}.\overrightarrow{j})\overrightarrow{j}+(\overrightarrow{r}.\overrightarrow{k})\overrightarrow{k} $
(ii) Find the projection of the vector $ 7\overrightarrow{i}+\overrightarrow{j}-4\overrightarrow{k} $ on $ 2\overrightarrow{i}+6\overrightarrow{j}+3\overrightarrow{k} $
5) Angle in a semi-circle is a right angle. Prove by vector method.
6) Find the vectors of magnitude 6 which are perpendicular to both the vector $ 4\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k} $ and $ -2\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k} $
7) Show that the torque above the point A(3, –1, 3) of a force $ 4\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k} $ through the point B(5, 2, 4) is $ \overrightarrow{i}+2\overrightarrow{j}-8\overrightarrow{k} $
8) If $ arg(z-1) = \frac{\pi}{6} $ and $ arg(z+1) = 2\frac{\pi}{3} $, then prove |z|=1.
9) If $ cos\; \alpha+cos \;\beta+cos\; \gamma=0=sin\; \alpha+sin\; \beta+sin\; \gamma $, prove that
(i) $ cos\; 3\alpha+cos \; 3\beta+cos\; 3\gamma=3cos(\alpha+\beta+\gamma) $
(ii) $ sin\; 3\alpha+sin \; 3\beta+sin\; 3\gamma=3sin(\alpha+\beta+\gamma) $
10) Find all the values: $ (-\sqrt 3-i)^{\frac{2}{3}} $
11) Prove that if $ \omega^3=1 $, then $ \frac{1}{1+2\omega}-\frac{1}{1+\omega}+\frac{1}{2+\omega}=0 $
12) A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 9 mts. If the distance across (diameter) the top of the mirror is 160 cm, how deep is the mirror at the middle?
13) Find the equation of the ellipse whose vertices are (-1, 4) and (-7, 4) and eccentricity is $ \frac{1}{3} $
14) Find the equations of the tangent and normal to the hyperbola $ \frac{x^2}{9}-\frac{y^2}{12}=1 $ at $ \Theta=\frac{\pi}{6} $
15) Find the angle between the asymptotes of the hyperbola $ 4x^2-5y^2-16x+10y+31=0 $
OR
Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.
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