Recent questions tagged easy

Prove that the function $f$ given by $f (x) = \log \sin x$ is strictly increasing on$ \left(0, \: \large {\frac{\pi}{2}}\right)$ and strictly decreasing on $\left( \large {\frac{\pi}{2}}, \pi\right)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

On which of the following intervals is the function $f $ given by $f (x) = x^{100} + sin x –1$ strictly decreasing ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

For what values of $x$,
$\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix}$ = 0 ?

asked Nov 23, 2012 by pady_1

1 answer

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} $ show that $A^2 - 5A + 7I = 0$

asked Nov 23, 2012 by pady_1

1 answer

Prove that the function $f$ given by $f (x) = x^2 – x + 1$ is neither strictly increasing nor strictly decreasing on $(– 1, 1).$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that the logarithmic function is strictly increasing on $(0, \infty).$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that $ y =\large { \frac{4\sin\theta}{(2+\cos\theta)}}-\normalsize \theta$ is an increasing function of $\theta$ in $\left[0, \: \large {\frac{\pi}{2}}\right]$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the values of $x$ for which $y = [x(x – 2)]^2$ is an increasing function.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that $ y = log(1+x)-\large {\frac{2x}{2+x}}, \normalsize x > – \normalsize 1$, is an increasing function of $x$ throughout its domain.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the following functions are strictly increasing or decreasing: $(a) \;x^2 + 2x - 5$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the function f given by $f (x) = 2x^3 – 3x^2 – 36 x + 7$ is (a) strictly increasing and (b) strictly decreasing ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the function $f$ given by $f (x) = 2x^2 - 3x$ is $(a) $ strictly increasing

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = \sin x$ is neither increasing nor decreasing in $(0, \pi)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = \sin x$ is strictly decreasing in $(\large {\frac{\pi}{2}},$$ \pi)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = \sin \;x$ is strictly increasing in $\left(0, \: \large {\frac{\pi}{2}}\right)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f(x) = e^{2x}$ is strictly increasing on $R$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = 3x + 17$ is strictly increasing on $R$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the total revenue in Rupees received from the sale of $x$ units of a product is given by $Rx = 3x^2 +36x + 5$. The marginal revenue, when $x = 15$ is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ cm is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The total revenue in Rupees received from the sale of $x$ units of a product is given by $ R(x) = 13x^2 + 26x + 15.$ Find the marginal revenue when $x = 7$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $ C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000.$ Find the marginal cost when 17 units are produced.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Sand is pouring from a pipe at the rate of $12 cm^3/s$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical, has a variable diameter $\large\frac{3}{2}$$(2x+1)$. Find the rate of change of its volume with respect to $x.$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The radius of an air bubble is increasing at the rate of $ \large\frac{1}{2}$ cm/s. At what rate is the volume of the bubble increasing when the radius is $1\; cm$?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A particle moves along the curve $ 6y = x^3 +2$. Find the points on the curve at which the $y$-coordinate is changing $8$ times as fast as the $x$-coordinate.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10\; cm$.

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15\; cm$.

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The length $x$ of a rectangle is decreasing at the rate of $5\; cm/minute$ and the width y is increasing at the rate of $4\; cm/minute$. When $x = 8$ cm and $y = 6$ cm, find the rates of change of (a) the perimeter

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The radius of a circle is increasing at the rate of $0.7\; cm/s$. What is the rate of increase of its circumference?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A stone is dropped into a quiet lake and waves move in circles at the speed of $5 \;cm/s$. At the instant when the radius of the circular wave is $8 \;cm$, how fast is the enclosed area increasing?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

An edge of a variable cube is increasing at the rate of $3\; cm/s$. How fast is the volume of the cube increasing when the edge is $10\; cm$ long?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The radius of a circle is increasing uniformly at the rate of $3\; cm/s.$ Find the rate at which the area of the circle is increasing when the radius is $10\; cm$.

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A manufacturer produces three products $ x, y, z $ which he sells in two markets. Annual sales are indicated below: \[ \begin{array} { c c } \textbf{Market} & \textbf{Products} \\ I & 10,000 \quad 2,000 \quad 18,000 \\ II & 6,000 \quad 20,000 \quad 8,000 \end{array} \]
If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market

asked Nov 22, 2012 by pady_1

1 answer

Find the matrix $ X $ so that $ X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \: = \: \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix} $

asked Nov 22, 2012 by pady_1

1 answer

If the matrix A is both symmetric and skew symmetric, then

asked Nov 22, 2012 by pady_1

1 answer

If $A $ is a square matrix, such that $ A^2 = A $ , then $ (1 + A )^3 - 7A $ is equal to:

asked Nov 22, 2012 by pady_1

1 answer

Let $S=\{a,b,c\}\;$ and$ \;T = \{1,2,3\}$. Find the inverse of the following function $F$ from $S$ to $T$, if it exists - \[\;\; F=\{(a,3), (b,2), (c,1)\}\]

asked Nov 22, 2012 by vaishali.a

1 answer

Show that the function $f : R \to R$ given by $f (x) = x^3$ is injective.

asked Nov 21, 2012 by vaishali.a

1 answer

If $(f:R \to R)$ is defined by $f(x) = x^2$ - $3x+2$. Find $f(f(x))$:

asked Nov 21, 2012 by vaishali.a

1 answer

Let $f:R \to R$ be defined as $f(x)=10x+7.$Find the function $g:R \to R$ such that $g\;o\;f = f\;o\;g = I_R.$

asked Nov 21, 2012 by vaishali.a

1 answer

Consider a binary operation $\ast$ on $N$ defined as $a \ast b = a^3 + b^3$. Choose the correct answer:

asked Nov 21, 2012 by vaishali.a

1 answer

If $ x \sqrt {1+y} + y \sqrt{1+x} = 0$, for, $-1 < x < 1$, prove that $\large \frac{dy}{dx} = -\frac{1}{(1+x)^2} $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows: $\;\; a \ast b = a-b$. Find which of the binary operations are commutative and which are associative.

asked Nov 21, 2012 by vaishali.a

1 answer

Let $\ast$ be the binary operation on $N$ defined by $a \ast b=H.C.F. $ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on $N$?

asked Nov 21, 2012 by vaishali.a

1 answer

Find $\large \frac{dy}{dx} $, if $ y = \sin^{-1}x + \sin^{-1} \: \sqrt {1-x^2} , -1 \leq x \leq 1 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Find $ \large\frac{dy}{dx} $, if $ y = 12 ( 1 - \cos t ) , x = 10 ( t - \sin t ) , -\large\frac{\pi}{2}\normalsize < t <\large \frac{\pi}{2} $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Let $\ast '$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a \ast ' b = H.C.F$ of a and b. Is the operation same as the operation $\ast$ defined in the table below? $\begin{matrix} *&1&2&3&4&5 \\ 1&1&1&1&1&1 \\ 2&1&2&1&2&1 \\ 3&1&1&3&1&1 \\ 4&1&2&1&4&1 \\ 5&1&1&1&1&5 \end{matrix}$

asked Nov 21, 2012 by vaishali.a

1 answer

Differentiate w.r.t. $x$ the function in $ (\sin x - \cos x )^{\large(\sin x - \cos x )} , \large\frac {\pi}{4}\normalsize < x < \large\frac{3\pi}{4} $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $ \cos \: (a \cos x + b \sin x) $ , for some constant $a$ and $b$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $(5x)^{\large 3 \cos 2x} $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

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