List of practice Questions

Two equal forces are acting at a point with an angle of 60$^{\circ}$ between them. If the resultant force is equal to $40\sqrt{3}$, the magnitude of each force is

From a balloon is moving Upwards with velocity $12\, ms^{-1}$. It releases a packet when it is at a height of $65\,m$ from the ground. How much time the packet will take to reach the ground $\left(g=10m/sec^2\right)$

Find all the points of discontinuity of f defined by f(x)=|x|-|x+1|.

Examine that sin|x| is a continuous function.

Show that the function defined by f(x)=|cosx| is a continuous function.

Show that the function defined by f(x)=cos(x²) is a continuous function.

Find the values of a and b such that the function defined by
$f(x)=\left\{\begin{matrix} 5, &if\,x\leq2 \\ ax+b,&if\,2<x<10 \\ 21,&if\,x\geq10 \end{matrix}\right.$

is a continuous function.

Find the values of k so that the function f is continuous at the indicated point.
$f(x)=\left\{\begin{matrix} kx+1, &if\, x\leq\pi \\ cos\,x,&if\,x>\pi \end{matrix}\right.\,at\,x=\pi$

Find the values of k so that the function f is continuous at the indicated point.
$f(x)=\left\{\begin{matrix} kx^2, &if\,x\leq2 \\ 3,&if\,x>2 \end{matrix}\right. \,at\,x=2$

Find the values of k so that the function f is continuous at the indicated point.
$f(x)=\left\{\begin{matrix} \frac{k\,cos\,x}{\pi-2x}, &if\,x\neq\frac{\pi}{2} \\ 3,&if\,x=\frac{\pi}{2} \end{matrix}\right.at\, x=\frac{\pi}{2}$

Examine the continuity of f, where f is defined by
$f(x)=\left\{\begin{matrix} sin\,x-cos\,x, &if\,x\neq0 \\ -1,& if\,x=0 \end{matrix}\right.$

Determine if f is defined by
$f(x)=\left\{\begin{matrix} x^2sin\frac{1}{x}, &if\,x\neq0 \\ 0,&if\,x=0 \end{matrix}\right.$
is a continuous function?

Find the points of discontinuity of f, where
$f(x)=\left\{\begin{matrix} \frac{sin\,x}{x} &if\,x<0 \\ x+1& if\,x\geq0 \end{matrix}\right.$

Discuss the continuity of the cosine,cosecant,secant and cotangent functions,

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 by
f(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.
$cos(\sqrt x)$