Recent questions tagged 2012

Find the area of the smaller part of the circle $ x^2+y^2=a^2$ cut off by the line $ x=\large\frac{a}{\sqrt 2}$

asked Jan 10, 2013 by thanvigandhi_1

2 answers

Form the differential equation of the family of circles in the first quadrant which touch the coordinates axes.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the equation of plane passing through the (-1, -1, 2) and $ \bot $ to each of the following planes : $ 2x+3y-3z=2 \: and \: 5x-4y+z=6 $.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

A farmer has a supply of chemical fertilizer of type I which contains 10% nitrogen and 6% phosphoric acid and type II fertilizer which contains 5% of nitrogen and 10% phosphoric acid. After testing the soil conditions of a field, it is found that atleast 14kg of nitrogen and 14 kg of phosphoric acid is required for a good crop. The fertilizer type I costs Rs.2/kg and the type II costs Rs. 3/kg. How many kilograms of each fertilizer should be used to meet the requirement and the cost be minimum?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

A can hit a target 4 times in 5 shots, B 3 times in 4 shots and C 2 times in 3 shots. Calculate the probability that \[\begin{array}{1 1} ((i) \: A,B,C\: all \: may \: hit. \\ (ii)\: Any\: two \: of \: A,B\: and \: C\: will\: hit \ the \: target. \\ (iii)\: None \: of\: them \: will \: hit \: the \: target.\\(iv)None \:of\:hem\:will\:hit\:the\:target. \end{array} \]

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Relation R in the set A = { 1, 2, 3, 4, 5, 6} defined as R = (x, y):y is divided by x}.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the value of the following: $ \tan^{-1}(1)+\cos^{-1} \bigg(\large -\frac{1}{2} \bigg)+$$\sin^{-1} \bigg(\large-\frac{1}{2} \bigg) $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Find the values of x and y make the following pairs of matrices equal : $\begin{bmatrix} 3x+6 & 5 \\ y+1 & 2-x \end{bmatrix} = \begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix} $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

What are the number of all possible matrices of 3 x 3 with each entry 0 or 1.

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Find the value of x if : $(i) \begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} = \begin{bmatrix} 2x & 4 \\ 6 & x \end{bmatrix} $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int (ax+b)^3dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int ( \sin2x-4e^{3x})dx$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ \overrightarrow a $ x $ \overrightarrow b = \overrightarrow c$ x $ \overrightarrow d$ and $ \overrightarrow a$ x $ \overrightarrow c = \overrightarrow b $ x $ \overrightarrow d $ show that $ \overrightarrow a - \overrightarrow d$ is parallel to $ \overrightarrow b-\overrightarrow c$ where $ \overrightarrow a \neq \overrightarrow d$ and $ \overrightarrow b \neq \overrightarrow c.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find a unit vector in the direction of $ \overrightarrow a = 3\hat i - 2\hat j + 6\hat k.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

For what value of $ \lambda $ are the vectors $ \overrightarrow a = 2\hat i + \lambda \hat j + \hat k \: and \: \overrightarrow b = \hat i - 2 \hat j + 3\hat k $ perpendicular to each other?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

In each of the follwing cases, state whether the function is one-one, onto or bijective. Justify your answer. $\begin{array}{1 1} (i) \: f: R \rightarrow R\: defined\: by \ f(x)=3-4x \\ (ii)\: f: R \rightarrow R\: defined\: by \ f(x)=1+x^2 .\end{array} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Solve for x : $ \tan^{-1} (2x)+\tan^{-1}(3x)=\large\frac{\pi}{4}. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants show that: $ \begin{bmatrix} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{bmatrix} = (x+y+z)(x-z)^2.. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ a, b, c$ are in A.P., show the following: $ \begin{bmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{bmatrix} = 0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that : $ \cot^{-1} \bigg[ \large\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \bigg]; $$0 < x <\large \frac{\pi}{2} = \large\frac{1}{2} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Differentiate $ (x^2-5x+8)(x^2+7x+9)$ in three ways mentioned : (i) by using product rule (ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation. Do they all give the same answer?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ u, v \: and \ w $ are function of $ x$ , then show that $ \large\frac{d}{dx}(u.v.w) = \frac{du}{dx}.v.w+u.\frac{dv}{dx}.w+u.v.\frac{dw}{dx}$In two ways first by repeated application or product rule, secondly by logatithmic differentiation.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate $\int\large \frac{3x-2}{(1+x^2)(x+3)}$$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate $\int\large \frac{\sin(x-\alpha)}{\sin(x+\alpha)}$$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Verify that $ y = e^x.\cos\: x$ is a solution of the differential equation : $\large \frac{d^2y}{dx^2}$$-2\large\frac{dy}{dx}$$+2y=0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation: $(x^2+3xy+y^2)dx-x^2\: dy = 0. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the points A, B and C with position vectors $ \overrightarrow a = 3\hat i - 4\hat j - 4\hat k, \overrightarrow b = 2\hat i - \hat j + 4\hat k \: and \: \overrightarrow c = \hat i - 3\hat j - 5\hat k,$ respectively, form the vertices of a right angled triangle.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Solve the equation $ 2x^2\large\frac{dy}{dx}$$- 2xy+y^2=0,y(e)=e.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the point on the line $ \large\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}$ at a distance $ 3\sqrt 2$ from the point (1, 2, 3).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the shortest distance between the following lines : $ \large\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1} $$ and $$ \large\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1} $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

A man is known to speak the truth 3 out of 4 times. He throws a dice and reports that it is a six. Find the probability that is actually a six.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}.$Prove that $ A^2-4A-51=0$. Hence find $ A^{-1} $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Using matrix method solve the following system of linear equations : $ x+y-z=1; 3x+y-2z=3 ;x-y-z = -1$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is $\large \frac{8}{27}$ of the volume of the sphere.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the equations of the tangent and normal to the curve $ 16x^2+9y^2 = 144$ at $ (x_1, y_1)$ where $ x_1=2\: and \: y_1 > 0.$ Also, find the points of intersection where both tangent and normal cut the $ x$ - axis.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Using the method of Integration, find the area of the region bounded by the lines $ 2x+y=4, 3x-2y=6,x-3y+5=0.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that : $\int_0^{\large\frac{\pi}{2}} ( \sqrt {cosx}+ \sqrt{cotx})dx = \sqrt 2 \pi. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the planes $ 2x+y+3x-2 = 0 \: and \: x-2y+5=0$ are perpendicular to each other.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of one kg food is given below : One kg of food x costs Rs.16 and one kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If $ f(x) = x+7 \: and \: g(x) = x-7, x\: \in \: R, $ find (fog) (7).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the principal value of the following : $ \sin^{-1} \bigg( -\frac{\large 1}{\large 2} \bigg). $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

For what value of x, is the following matrix singular? $ \begin{bmatrix} 3-2x & x+1 \\ 2 & 4 \end{bmatrix}.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements.

asked Jan 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \begin{bmatrix} a+ib & c+id \\ -c+id & a-ib \end{bmatrix} $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \large\frac{x^2}{1+x^3}$$dx.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int\large \frac{2\cos x}{3\sin^2x}$$dx.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the projection of $ \overrightarrow a $ on $ \overrightarrow b $ if $ \overrightarrow a . \overrightarrow b = 8 $ and $ \overrightarrow b = 2\hat i + 6\hat j + 3\hat k $.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Write the unit vector in the direction of $ \overrightarrow a = 2\hat i - 6\hat j + 3\hat k $.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Scalar product of the vector $ \hat i + \hat j + \hat k $ with the unit vector along the sum of vectors $ 2\hat i + 4\hat j -5\hat k $ and $ \lambda \hat i + 2\hat j + 3\hat k $ is equal to one, find the value of $ \lambda $.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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