Recent questions tagged misc

Choose the correct answer in the normal to the curve $x^2 = 4y$ passing $(1,2)$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the normal at the point $(1,1)$ on the curve $2y + x^2 = 3$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the line $y = mx + 1$ is a tangent to the curve $y^2 = 4x$ if the value of $m$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the slope of the tangent to the curve $x = t^2 + 3t – 8, y = 2t^2 – 2t – 5$ at the point $(2,– 1)$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in a cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi vertical angle $\alpha$ is one-third that of the cone and the greatest volume of cylinder is $\large \frac{4}{27}$$\pi h^3 \: tan^2 \: \alpha $

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\large\frac{2R}{\sqrt 3}$ . Also find the maximum volume.

asked Nov 28, 2012 by thanvigandhi_1

2 answers

Let $f$ be a function defined on $[a, b]$ such that $f' (x) > 0$, for all $x \: \in (a, b)$. Then prove that $f$ is an increasing function on $(a, b).$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $ \large\frac{4r}{3}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the absolute maximum and minimum values of the function $(f)$ given by $ f (x) = cos^2 x + sin\: x, x\: \in [0, \pi]$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the points at which the function $f$ given by $f (x) = (x – 2)^4 (x + 1)^3$ has a (i) local maxima, (ii) local minima and the (iii) point of inflexion.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

A point on the hypotenuse of a triangle is at distance $a$ and $b$ from the sides of the triangle.Show that the maximum length of the hypotenuse is $ (a^{\large\frac{2}{3}} + $$b^{\large\frac{2}{3}})^{\Large\frac{3}{2}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

The sum of the perimeter of a circle and square is $k$, where $k$ is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2 m$ and volume is $8 m^3$. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the maximum area of an isosceles triangle inscribed in the ellipse $\large\frac{x^2}{a^2} + \frac{y^2}{b^2} $$= 1 $ with its vertex at one end of the major axis.

asked Nov 28, 2012 by thanvigandhi_1

2 answers

Find the intervals in which the function $f$ given by $f(x) = x^3 + \large\frac{1}{x^3}$$, x \neq 0$ is $(i) \;increasing\qquad (ii) \;decreasing.$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the function $f$ given by $ f(x) =\large \frac{4sin\: x - 2x - xcos \: x}{2+cos\:x}$ is (i) increasing (ii) decreasing.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the normal at any point $\theta$ to the curve $ x = a \cos\theta + a \: \theta \sin\: \theta, y = a\sin\theta - a\theta \cos\theta$ is at a constant distance from the origin.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the equation of the normal to curve $y^2 = 4x$ which passes through the point $(1, 2).$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

The two equal sides of an isosceles triangle with fixed base $b$ are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the function given by $ f(x) =\large \frac{\log \: x}{x}$ has maximum at $x = e.$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Using differentials, find the approximate value of each of the following: $(33)^{-\large\frac{1}{5}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Using differentials, find the approximate value of each of the following: $ \left(\large\frac{17}{81}\right)^{\frac{1}{4}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Number of binary operations on the set $\{a, b\}$ are

asked Nov 27, 2012 by vaishali.a

1 answer

Let $f : R \to R$ be the Signum Function defined as \[ f(x) = \left \{ \begin {array} {1 1} 1, & \quad \text { x $>$ 0} \\ 0, & \quad \text { x $=$0} \\-1, & \quad \text { x $<$0} \\ \end {array} \right. \] and $g:R \to R$ be the greatest Integer Function given by $g(x)=[x]$ where $[x]$ is a greatest integer less thar or equal to $x$ Then, does $fog$ and $gof$ coincide in $(0,1]$?.

asked Nov 27, 2012 by vaishali.a

1 answer

Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is

asked Nov 27, 2012 by vaishali.a

1 answer

Let $ A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)\;$ and $\;(1, 3)$ which are reflexive and symmetric but not transitive is

asked Nov 27, 2012 by vaishali.a

1 answer

If A and B are any two events such that $P(A) + P(B) - P(A \text{and} B) = P(A)$, then choose the correct answer:

asked Nov 27, 2012 by balaji.thirumalai

1 answer

If P(A|B) > P(A), then which of the following is correct ?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

If A and B are two events such that P(A) $\neq$ 0 and P(B | A) = 1 . Then choose the correct answer.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: $ \text {P(A fails) = 0.2 P(B fails alone) = 0.15 P(A and B fail) = 0.15}$. Evaluate the following probabilities \[ \text {(i) P(A fails | B has failed) (ii) P(A fails alone)} \]

asked Nov 27, 2012 by balaji.thirumalai

1 answer

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$).

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Suppose we have four boxes A,B,C and D containing coloured marbles as given in the box below. One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C? \[\]$\begin{matrix} \text{Box} & &\text{Marble Colour} & \\ &\text{Red} &\text{White} & \text{Black}\\ A & 1 & 6 & 3\\ B & 6 & 2 &2 \\ C & 8 & 1 & 1\\ D & 0 & 6 & 4 \end{matrix}$

asked Nov 27, 2012 by balaji.thirumalai

1 answer

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins / loses.

asked Nov 27, 2012 by balaji.thirumalai

2 answers

How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6 . What is the probability that he will knock down fewer than 2 hurdles?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15 bear a mark 'Y'. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability for the following: (i) all will bear 'X' mark. (ii) not more than 2 will bear 'Y' mark. (iii) at least one ball will bear 'Y' mark. (iv) the number of balls with 'X' mark and 'Y' mark will be equal.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

A and B are two events such that P (A) $\neq$ 0. Find P(B|A), if (i) A is a subset of B (ii) A $\cap$ B = $\phi$

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Let $A=\{\text{-1,0,1,2}\}$ and $B=\{\text{-4,-2,0,2}\} and $f,g: A $\rightarrow B$ be functions defined by $f(x)=x^2-x, \;x \in A$ and $g(x)=2 |x- \frac {1} {2} | -1,\; x \in A$. Are $f$ and $g$ equal?

asked Nov 26, 2012 by vaishali.a

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

if $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$ prove that $A^n = \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix} , n \in N$.

asked Nov 23, 2012 by pady_1

1 answer

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