If a,b, and c are real numbers and determinant \(\Delta = \begin{vmatrix} b+c &c+a &a+b \\ c+a&a+b &b+c \\ a+b&b+c &c+a \end{vmatrix}\)Show that either a+b+c=0 or a=b=c.
Prove that the determinant \(\begin{vmatrix} x &sin\theta &cos\theta \\ -sin\theta&-x &1 \\ cos\theta&1 &x \end{vmatrix}\) is independent of θ.
Compute the magnitude of the following vectors:\(\overrightarrow{a}\)=\(\hat{i}\)+\(\hat j+\hat k\);\(\overrightarrow{b}\)=2\(\hat{i}\)-7\(\hat{j}\)-3\(\hat{k}\); \(\overrightarrow{c}\)= \(\frac{1}{\sqrt 3}\hat i+\frac{1}{\sqrt 3}\hat j-\frac{1}{\sqrt 3}\hat k\)
Answer the following as true or false. (i)a→and -a→are collinear. (ii)Two collinear vectors are always equal in magnitude. (iii)Two vectors having same magnitude are collinear. (iv)Two collinear vectors having the same magnitude are equal.
In figure, identify the following vectors.
(i)Coinitial (ii)Equal (iii)Collinear but not equal
Classify the following as scalar and vector quantities. (i)Time period (ii)Distance (iii)Force (iv)Velocity (v)Work done
Represent graphically a displacement of 40km,30°east of north.
If A is an invertible matrix of order 2,then det(A-1) is equal to
Write Minors and Cofactors of the elements of following determinants: I. \(\begin{vmatrix}2&-4\\0&3\end{vmatrix}\)
II. \(\begin{vmatrix}a&c\\b&d\end{vmatrix}\)
Find the area of the region bounded by the parabola y=x2 and y=|x|
The area between x=y2 and x=4 is divided into two equal parts by the line x=a, find the value of a.
Find the area of the smaller part of the circle x2+y2=a2 cut off by the line x=a/√2
Find the area of the region in the first quadrant enclosed by x-axis,line x=√3y and the circle x2+y2=4
Verify A(adj A)=(adj A)A=\(\mid A \mid I\).
\(\begin{bmatrix}1&-1&2\\3&0&-2\\1&0&3\end{bmatrix}\)
By using properties of determinants, show that:
\(\begin{vmatrix}a^2+1&ab&ac\\ab&b^2+1&bc\\ca&cb&c^2+1\end{vmatrix}\)=1+a2+b2+c2
Show that points A (a,b+c),B (b,c+a),C (c,a+b) are collinear
Find area of the triangle with vertices at the point given in each of the following:I. (1,0),(6,0),(4,3)II. (2,7),(1,1),(10,8)III. (−2,−3),(3,2),(−1,−8)
If A=\(\begin{bmatrix}3&-2\\4&-2\end{bmatrix}\) and I=\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\),find k so that A2=kA-2I
Show that (i)\(\begin{bmatrix}5&-1\\6&7\end{bmatrix}\)\(\begin{bmatrix}2&1\\3&4\end{bmatrix}\)\(\neq \begin{bmatrix}2&1\\3&4\end{bmatrix}\begin{bmatrix}5&-1\\6&7\end{bmatrix}\)
(ii)\(\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}\)\(\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}\)\(\neq \) \(\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}\)\(\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}\)
If F(x)= \(\begin{bmatrix}\cos x&\sin x&0\\\sin x&cos x&0\\0&0&1\end{bmatrix}\)and F(y)=\(\begin{bmatrix}\cos y&-\sin y&0\\\sin y&cos y&0\\0&0&1\end{bmatrix}\),show that F(x)+F(y)=F(x+y)
If x \(\begin{bmatrix}2\\3\end{bmatrix}\)+y \(\begin{bmatrix}-1\\1\end{bmatrix}\)=\(\begin{bmatrix}10\\5\end{bmatrix}\),find values of x and y.
Solve the equation for x,y,z and t if 2\(\begin{bmatrix}x&y\\y&t\end{bmatrix}\)+3\(\begin{bmatrix}1&-1\\0&2\end{bmatrix}\)=3\(\begin{bmatrix}3&5\\4&6\end{bmatrix}\)
Find x and y, if 2\(\begin{bmatrix}1&3\\0&x\end{bmatrix}\)+\(\begin{bmatrix}y&0\\1&2\end{bmatrix}\)=\(\begin{bmatrix}5&6\\1&8\end{bmatrix}\)
Simplify \(\cos\theta\) \(\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\)+\(\sin\theta\) \(\begin{bmatrix}\sin\theta&-\cos\theta\\\cos\theta&\sin\theta\end{bmatrix}\)
