Recent questions tagged sec-b

Find the equation of all lines having slope $–1$ that are tangents to the curve $ y= \large\frac{1}{x-1}, $$ \: x \neq 1$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find a point on the curve $y = (x – 2)^2 $ at which the tangent is parallel to the chord joining the points $(2, 0)$ and $(4, 4)$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the normal to the curve $ x = a \cos^3\theta, \: y = a \sin^3\theta$ at $ \theta =\Large {\frac{\pi}{4}}$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ then prove that $ A^n = \begin{bmatrix} 1+2n & -4n \\ n & 1 - 2n \end{bmatrix} $ , where $n$ is any positive integer.

asked Nov 23, 2012 by pady_1

1 answer

The interval in which $y = x^2 e ^{- x}$ is increasing is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

If $A$ and $B$ are symmetric matrices, prove that $AB - BA$ is a skew symmetric matrix.

asked Nov 23, 2012 by pady_1

1 answer

Show that the matrix $B'AB$ is symmetric or skew symmetric according as A is symmetric or skew symmetric.

asked Nov 23, 2012 by pady_1

1 answer

Find the values of $x, y, z $ if the matrix $ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} $ satisfy the equation $A'A = I $

asked Nov 23, 2012 by pady_1

1 answer

Let $I$ be any interval disjoint from $[–1, 1]$. Prove that the function $f$ given by $ f(x) = x + \frac{1}{\large x}$ is strictly increasing on $I$.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the least value of a such that the function $f$ given by $f (x) = x^2 + ax + 1$ is strictly increasing on $(1, 2).$

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

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

Find $x$, if $ \begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = 0 $

asked Nov 23, 2012 by pady_1

1 answer

Which of the following functions are strictly decreasing on $\left(0, \: \large {\frac{\pi}{2}}\right)$

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

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

asked Nov 23, 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

Define a binary operation $\ast$ on the set $\{0, 1, 2, 3, 4, 5\}$ as \[ a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a$+$b $<$ 6} \\ a+b-6, & \quad \text{ if a+b $\geq$ 6} \\ \end{array} \right. \] Show that zero is the identity for this operation and each element $a\neq0$ of the set is invertible with $6-a$ being the inverse of $a$.

asked Nov 22, 2012 by vaishali.a

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

Given a non-empty set $ X,$ let $\ast :\; P(X)\; \times\; P(X) \to P(X) $ be defined as $A \ast B = \; ( A-B)\; \cup \; (B-A),\; \forall A, B \in \; P(X).$. Show that the empty set $\emptyset $ is the identity for the operation $\ast$ and all the elemnets $A$ of $ P(X) $ are invertible with $ A^{-1} \;= A$.

asked Nov 22, 2012 by vaishali.a

1 answer

Consider the binary operation $\ast :\; R \times R \rightarrow R$ and $o :\; R \times R \rightarrow R$ defined as $a \ast b = | a \text{-b}|$ and $\;a\;o\;b=a, \forall a,\;b \in R.$ Show that $\ast$ is commutative but not associative, $o$ is associative but not commutative. Further, show that $\forall\; a,\; b,\; c \in R,\; a\; \ast\; (b\; o\; c) = (a \ast b) \;o\; (a \ast c)$. [If it is so, we say that the operation $\ast$ distributes over $o$]. Does $o$ distribute over? Justify your answer.

asked Nov 22, 2012 by vaishali.a

1 answer

Let $f:W \to W$ be defined as $f(n)=n$ - $1$, if $n\;is\;odd\;and\; f(n)=n+1,\;if\;n\;is\; even.$ Show that $f$ is invertible. Find the inverse of $f$. Here, $W$ is the set of all whole numbers.

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

If $( x-a)^2 + (y-b)^2 = c^2$, for some $ c > 0$, prove that $\Large\frac{\begin{bmatrix} 1 + \left(\frac{dy}{dx}\right)^2 \\[0.3em] \end{bmatrix}^{\frac{\Large 3}{\Large 2}}}{\Large\frac{d^2y}{dx^2}}$is a constant independent of $a$ and $b$.

asked Nov 21, 2012 by thanvigandhi_1

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 $(A=N \times N \,and\, * )$ be the binary operation on $(A)$ defined by $( (a, b) * (c, d) = (a + c, b + d))$. Show that * is commutative and associative. Find the identity element for * on $( A )$, if any.

asked Nov 21, 2012 by vaishali.a

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 $N$ given by $a\ast b=L.C.M\,.of\,a\,and\,b$. Find $\begin{array}{1 1}(i)\;\; 5 \ast7,\; 20 \ast 16 & (ii)\;\; Is\; \ast \;commutative?\\(iii)\;\; Is\; \ast \;associative? & (iv)\;\; Find\, the\,identity \, of \ast \, in N\\(v)\;\; Which \, elements\, of \, N\, are\, invertible\, for\, the\, operaation\,\ast ? & \;\end{array}$

asked Nov 21, 2012 by vaishali.a

2 answers

Differentiate w.r.t. $x$ the function in $ x^{\large x} + x^{\large a} + a^{\large x} + a^{\large a} $, for some fixed $ a > 0 $ and $ x > 0 $

asked Nov 21, 2012 by thanvigandhi_1

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 $( \log x )^{\large\log x}, x > 1 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $\large\frac{\cos^{ -1}\Large\frac{ x}{ 2}}{\sqrt {2x + 7}}, \normalsize-2 < x < 2 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $ sin^{-1}(x \: \sqrt x ), 0 \leq x \leq 1 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $ (3x^2 - 9x + 5)^9 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Verify Mean Value Theorem, if $ f (x) = x^3 - 5x^2 - 3x $ in the interval $ [a,b]$, where $ a = 1$ and $ b = 3$. Find all $ c \in (1,3) $ for which $ f'(c) = 0$.

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Verify Mean Value Theorem, if $ f (x) = x^2 - 4x - 3 $ in the interval $ [a,b]$, where $ a = 1 $ and $b = 4 $.

asked Nov 21, 2012 by thanvigandhi_1

1 answer

If $ f : [ -5,5] \rightarrow R $ is a differentiable function and if $ f' (x) $ does not vanish anywhere, then prove that $ f (-5) \: \neq \: f (5)$.

asked Nov 21, 2012 by thanvigandhi_1

1 answer

For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,ab\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,ab\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,ab\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, ab= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,ab\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, ab= \frac {a} {b+1} & \;\end{array}$

asked Nov 20, 2012 by vaishali.a

5 answers

If $ e^y ( x + 1 ) = 1$, show that $ \large\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $ y = Ae^{mx} + Be^{nx}$, show that $ \large\frac{d^2y}{dx^2} -\normalsize ( m + n ) \large\frac{dy}{dx} +\normalsize mny = 0 $

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $ y =\cos^{-1}x $ , find $ \large\frac{d^2y}{dx^2}$ in terms of $ y $ alone.

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $x$ and $y$ are connected parametrically by the equations given in $x = a \big(\cos t +\ log \tan \large\frac{t}{2}\big) $ and $ y = a \sin t $ without eliminating the parameter, Find $ \frac{\large dy}{\large dx}$

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $x$ and $y$ are connected parametrically by the equations given in $x = a ( \theta - \sin \theta ), y = a ( 1 +\cos \theta) $ without eliminating the parameter, Find $\large\frac{dy}{dx}$

asked Nov 20, 2012 by thanvigandhi_1

1 answer

Let $f : R - \{ - \frac {4} {3} \} \to R $ be a function defined as $f(x)= \frac {4x} {3x+4} $ .The inverse of $f$ is the map $g$: Range $f \to R - \{ - \frac {4} {3} \} $ given by

asked Nov 19, 2012 by vaishali.a

1 answer

If $x$ and $y$ are connected parametrically by the equations given in $ x = \cos \theta -\ cos 2\theta, y = \sin \theta -\ sin 2\theta $ without eliminating the parameter, Find $\large\frac{dy}{dx}$

asked Nov 19, 2012 by thanvigandhi_1

1 answer

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