Discuss the continuity of the following functions.(a) f(x)=sinx+cosx(b) f(x)=sinx−cosx(c) f(x)=sinx\(\times\)cosx
Is the function defined by \(f(x)=x^2-sin\,x+5\) continuous at \(x=p\) ?
Show that the function defined by \(g(x)=x-[x]\) is discontinuous at all integral points. Here \([x]\) denotes the greatest integer less than or equal to \(x\).
For what value of λ is the function defined by\(f(x)=\left\{\begin{matrix} \lambda (x^2-2x) &if\,x\leq0 \\ 4x+1&if\,x>0 \end{matrix}\right.\)continuous at x=0? What about continuity at x=1?
Find the relationship between a and b so that the function f is defined byf(x)=\(\left\{\begin{matrix} ax+1 &if\,x\leq3 \\ bx+3&if\,x>3 \end{matrix}\right.\) is continuous at x=3.
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)\(\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}\)\(\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}\)\(\neq \) \(\begin{bmatrix}-1&1&0\\0&-1&1\\ 2&3&4\end{bmatrix}\)\(\begin{bmatrix}1&2&3\\0&1&0\\ 1&1&0\end{bmatrix}\)
Prove that the function f given by \(f(x)=|x+1|\), x∈R is not differentiable at x=1.
Differentiate the functions with respect to x.\(cos(\sqrt x)\)
Differentiate the functions with respect to x.\(2\sqrt{cot\ (x^2)}\)
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)
Differentiate the functions with respect to x.\(sin\ (ax+b)\)
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 \(\cos\theta\) \(\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\)+\(\sin\theta\) \(\begin{bmatrix}\sin\theta&-\cos\theta\\\cos\theta&\sin\theta\end{bmatrix}\)
Discuss the continuity of the function f, where f is defined by\(f(x)=\left\{\begin{matrix} 2x, &if\,f(x)<0 \\ 0,&if\,0\leq x\leq 1\\ 4x,&if\,x>1 \end{matrix}\right.\)
Verify A(adj A)=(adj A)A=IAII. \(\begin{bmatrix}2&3\\-4&-6\end{bmatrix}\)