Recent questions tagged medium

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

asked Nov 27, 2012 by thanvigandhi_1

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Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

asked Nov 27, 2012 by thanvigandhi_1

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Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

asked Nov 27, 2012 by thanvigandhi_1

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Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

asked Nov 27, 2012 by thanvigandhi_1

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Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum.

asked Nov 27, 2012 by thanvigandhi_1

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What is the maximum value of the function $\sin \: x + \cos x$?

asked Nov 27, 2012 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} \frac{2}{3} & 1 & \frac{5}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{7}{3} & 2 & \frac{2}{3} \end{bmatrix} \text{ and } B = \begin{bmatrix} \frac{2}{5} & \frac{3}{5} & 1 \\ \frac{1}{5} & \frac{2}{5} & \frac{4}{5} \\ \frac{7}{5} & \frac{6}{5} & \frac{2}{5} \end{bmatrix} \text{ then compute } 3A - 5B $

asked Nov 26, 2012 by pady_1

1 answer

The approximate change in the volume of a cube of side $x$ metres caused by increasing the side by $3\%$ is

asked Nov 26, 2012 by thanvigandhi_1

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If $f(x) = 3x^2 + 15x + 5$, then the approximate value of $f (3.02)$ is

asked Nov 26, 2012 by thanvigandhi_1

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If the radius of a sphere is measured as $7 \: m$ with an error of $0.02 \: m$, then find the approximate error in calculating its volume.

asked Nov 26, 2012 by thanvigandhi_1

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Find the approximate value of $f (5.001)$, where $f (x) = x^3 – 7x^2 + 15.$

asked Nov 26, 2012 by thanvigandhi_1

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Find the approximate value of $ f (2.01)$, where $f (x) = 4x^2 + 5x + 2$.

asked Nov 26, 2012 by thanvigandhi_1

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if $ A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} ,$ prove that $ A^3 - 6A^2 + 7A + 2I = 0 $

asked Nov 25, 2012 by pady_1

1 answer

If $ A = \begin{bmatrix} 0 & -tan \frac{\alpha}{2} \\ tan\frac{\alpha}{2} & 0 \end{bmatrix} $ and $I$ is the identity matrix of order $2$, show that $ I + A = ( I - A ) \begin{bmatrix} cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix} $

asked Nov 25, 2012 by pady_1

1 answer

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

asked Nov 25, 2012 by pady_1

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$ (i)$ Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.

asked Nov 25, 2012 by pady_1

1 answer

Find $ \frac{1}{2}(A + A')$ and $\frac{1}{2}(A - A')$ , when $ A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix} $

asked Nov 25, 2012 by pady_1

1 answer

Express the following matrices as the sum of a symmetric and a skew symmetric matrix: $ \quad \begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$

asked Nov 25, 2012 by pady_1

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Find the equation of the tangent to the curve $y = \sqrt {3x - 2}$ which is parallel to the line $4x - 2y + 5 = 0 $.

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Find the equations of the tangent and normal to the hyperbola $\large {\frac{x^2}{a^2}} -\large { \frac{y^2}{b^2}} =\normalsize 1$ at the $ (x_0, \: y_0)$.

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Prove that the curves $x = y^2$ and $xy = k$ cut at right angles* if $8k^2 = 1.$

asked Nov 24, 2012 by thanvigandhi_1

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Find the equations of the tangent and normal to the parabola $y^2 = 4ax$ at the point $(at^2, 2at)$.

asked Nov 24, 2012 by thanvigandhi_1

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Find the equation of the normals to the curve $y = x^3 + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$.

asked Nov 24, 2012 by thanvigandhi_1

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For the curve $y = 4x^3 – 2x^5$, find all the points at which the tangent passes through the origin.

asked Nov 24, 2012 by thanvigandhi_1

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Find the points on the curve $y = x^3$ at which the slope of the tangent is equal to the $y$ - coordinate of the point.

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 3 & 10 \\ 2 & 7 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Using elementary transformations, find the inverse of the matrix if it exists - $ \begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix} $

asked Nov 23, 2012 by pady_1

1 answer

Let $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $, show that $ (aI + bA)^n = a^nI + na^{n-1}bA $, where $\;I\;$ is the identity matrix of order 2 and $n \in N$.

asked Nov 23, 2012 by pady_1

1 answer

Find the equation of all lines having slope $2$ that are tangents to the curve $ y= \large\frac{1}{x-3}, $$\: x \neq 3$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find points at which the tangent to the curve $y = x^3 - 3x^2 - 9x + 7$ is parallel to the $x$ - axis.

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

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

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

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

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

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

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

Differentiate w.r.t. x the function $cot^{-1} \begin{bmatrix} \frac{{\sqrt {1+sin \: x}} + {\sqrt{1-sin \: x}}} {{\sqrt {1+sin \: x}} - {\sqrt{1-sin \: x}}} \\[0.3em] \end{bmatrix}$, where $0 < x < \frac{\pi}{2}$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

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