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Recent questions tagged 2012

The cartesian equation of a line AB is \( \large\frac{2x-1}{\sqrt 3} = \frac{y+2}{2} = \frac{z-3}{3} \). Find the direction cosines of a line parallel to AB.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( \overrightarrow a = \hat i + 2\hat j - 3\hat k\), and \( \overrightarrow b = 2\hat i + 4\hat j - 5\hat k\). Find a unit vector parallel to \( \overrightarrow a + \overrightarrow b\).

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If matrix \( P = [ a_{ij}]_{m*m} \) is a diagonal matrix, then P is expressed as P = diagonal \( [ a_{11}, a_{22}, a_{33},....a_{mm} ].\) If A = diagonal [ 1, 4, -1 ]and B = diagonal [ 3, 2, 6], then write A+B.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Let A = set of boys of XII class in some school. Let a R b, if and only if a and b use unfair means in the exam, \( a, b \in A\) Is R a transitive relation?

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Solve for x : $ \begin{vmatrix} 3x-8 & 3 & 3 \\ 3 & 3x-8 & 3 \\ 3 & 3 & 3x-8 \end{vmatrix} = 0$

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove the following : $ \begin{vmatrix} 1 & a^2+bc & a^3 \\ 1 & b^2+ca & b^3 \\ 1 & c^2+ab & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a^2+b^2+c^2). $

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Solve for x : $ \tan^{-1}(x+1)+\tan^{-1}(x-1)=\tan^{-1} \bigg(\large\frac{8}{31} \bigg) $

asked Jan 25, 2013 by thanvigandhi_1

0 answers

Prove that $ 2\tan^{-1} \large\frac{1}{5}$$ + \tan^{-1} \large\frac{1}{8}$$ = \tan^{-1}\large \frac{4}{7} $

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Let $ f(x) = \left\{ \begin{array}{l l} \large\frac{1-\sin^3x}{3\cos^2x}, & \quad if { x < \large\frac{\pi}{2}}\\ a, & \quad if { x = \frac{\pi}{2}} \\ \large\frac{b(1-sin\: x)}{(\pi - 2x)^2}, & \quad if { x > \large\frac{\pi}{2}} \end{array}. \right. $ If f(x) be a continuous function at \( x =\large \frac{\pi}{2} \), find a and b.

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( y = [ \log (x+\sqrt{1+x^2} ]^2\), show that $ (1+x^2)\large\frac{d^2y}{dx^2}$$+x\large\frac{dy}{dx}$$-2=0 $

asked Jan 25, 2013 by thanvigandhi_1

0 answers

If \( y=x^x+(sin\: x)^x, \: find \: \frac{dy}{dx} \)

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the points on the curve \( 3x^2-y^2=8\), where the normal is parallel to line \( x+3y=4.\)

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^{\pi}\large \frac{x}{a^2\cos^2x+b^2\sin^2x}$$dx $

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^{\pi} \large\frac{x\tan x}{\sec x + \cos x}$$dx.$

asked Jan 24, 2013 by thanvigandhi_1

0 answers

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

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation $\cos x\large \frac{dy}{dx}$$+y = \sin x $.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the value of \( \lambda \), so that the two vectors \( 2\hat i + 4\hat j - 2\hat k \: and \: 3\hat i + 6\hat j - \lambda \hat k\) are (i) parallel (ii) perpendicular to each other.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the shortest distance between the lines $ \overrightarrow r = \hat i + 2\hat j + 3\hat k + \lambda(2\hat i + 3\hat j + 4 \hat k) \: and \: \overrightarrow r = 2\hat i + 4\hat j + 5\hat k + \mu(3\hat i + 4\hat j + 5 \hat k). $

asked Jan 24, 2013 by thanvigandhi_1

0 answers

A and B throws a pair of die turn by turn. The first to throw 9 is awarded by a prize. If A starts the game, show that the probability of A getting the prize is \(\large \frac{9}{17} \).

asked Jan 24, 2013 by thanvigandhi_1

1 answer

Using elementary transformations, find the inverse of the matrix : $\begin{bmatrix} 2 & -1 & 4 \\ 4 & 0 & 2 \\ 3 & -2 & 7 \end{bmatrix}$

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system of linear equations : $ 2x+y+z=2,x+3y-z=5, 3x+y-2z=6$

asked Jan 24, 2013 by thanvigandhi_1

1 answer

A window is in the form of rectangle above which there is a semi circle. If the perimeter of window is p units, show that the window will allow maximum light, when radius of the semicircle is \( \large\frac{p}{\pi + 4} \) units.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^{\Large\frac{\pi}{2}}\large \frac{dx}{5+4\sin x} $

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the points (1,2,3) and (0,-1,0) and parallel to the line $\large \frac{x-1}{2} = \frac{y+2}{3} = \frac{z}{-32}.$

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Using integration, find the area of triangle PQR, whose vertices are P(3, 0)Q(4, 5) and R(5, 1).

asked Jan 24, 2013 by thanvigandhi_1

0 answers

A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 6% phosphoric acid, and type B which contains 5% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, it was found that ateast 14 kg of nitrogen and 14 kg of phosphoric acid is required for producing a good crop. The fertilizer of type A costs Rs. 2 per kg and the type B costs Rs.3 per kg. How many kg of each type of the fertilizer should be used to meet the requirement at the minimum problem graphically.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

There are 20% chances for a worker of an industry to suffer from an occupational disease, 50 workers were selected at random and examined for the ocupational disease. Find the probabilty that (i) None is suffering from the disease. (ii) Only one is suffering from the disease.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

The probability of student E passing an examination is 0.6 and of student F is 0.2. Assuming that the two events 'E passes' and 'F passes' are independent. Find the probability of : (i) both the students passing the examination (ii) Only E passing the examination.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane through point (2, 0, -1) and perpendicular to the line joining the two points (1, 2, 3) and (3, -1, 6)

asked Jan 24, 2013 by thanvigandhi_1

1 answer

Find the projection of \( \overrightarrow a = 2\hat i - \hat j + \hat k \: on \: \overrightarrow b = \hat i - 2\hat j + \hat k \)

asked Jan 24, 2013 by thanvigandhi_1

0 answers

If $ |\overrightarrow a | = 3, |\overrightarrow b | = \large\frac{\sqrt 2}{3}$$ \: and \; \overrightarrow a \times \overrightarrow b$ is a unit vector, then find angle between $ \overrightarrow a \: and \overrightarrow b $.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \large\frac{1+\cos x}{1-\cos x}$$dx $

asked Jan 24, 2013 by thanvigandhi_1

0 answers

If the function \( f : Q \rightarrow Q\) defined by \( f(x) = 3x+5\) for all \( x \in Q \) is invertible, find \( f^{-1} \).

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Using principal values, evaluate \( tan^{-1} \bigg( tan\frac{7\pi}{6} \bigg). \)

asked Jan 24, 2013 by thanvigandhi_1

1 answer

Find a matrix X such that 2A + 2B + X = 0, where \( A = \begin{bmatrix} -1 & 3 \\ 2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 5 \\ 3 & 2 \end{bmatrix} \)

asked Jan 24, 2013 by thanvigandhi_1

1 answer

If \( A = \begin{bmatrix} 4 & 1 \\ 5 & 8 \end{bmatrix},\) show that A + A' is symmetric matrix.

asked Jan 24, 2013 by thanvigandhi_1

1 answer

If |A| = 5, where A is 3 x 3 matrix, find | adj A |.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \sec^2(5-7x)dx .$

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the differential equation of the family of ellipses having foci on y- axis and centre at the origin.

asked Jan 24, 2013 by thanvigandhi_1

1 answer

Solve the differential equation $\bigg( y-x\large\frac{dy}{dx} \bigg)$$ = a \bigg( y^2+\large\frac{dy}{dx} \bigg). $

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Let \( \overrightarrow a, \overrightarrow b, \overrightarrow c\) be three vectors of magnitude 3, 4, 2 respectively. If ech one is perpendicular to sum of other two vectors, prove that $ | \overrightarrow a + \overrightarrow b + \overrightarrow c | = \sqrt {29} .$

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Find the vector equation of the plane through the intersection of the plane : $ \overrightarrow r . (\hat i + \hat j + \hat k ) = 6 \: and \: \overrightarrow r . (2\hat i + 3\hat j + 4\hat k ) = -5$ and the point (1, 1, 1).

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Two cards are drawn one by one without replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

asked Jan 24, 2013 by thanvigandhi_1

2 answers

Find the probability of number of doublets in three throws of a pair of dice.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

If R is a relation in N x N, show that the relation R defined by : (a, b)R(c, d)if and only if ad = bc is an equivalence relation.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

Solve for x : $ \tan^{-1} \bigg(\large \frac{x-1}{x+1} \bigg) + \tan^{-1} \bigg( \frac{2x-1}{2x+1} \bigg) = \tan^{-1} \bigg( \frac{23}{36} \bigg). $

asked Jan 24, 2013 by thanvigandhi_1

1 answer

Prove that $ \tan^{-1} \large\frac{1}{7}$$ + 2\tan^{-1} \large\frac{1}{3} = \frac{\pi}{4} $

asked Jan 24, 2013 by thanvigandhi_1

1 answer

(i) Assume that on a diwali day Neeraj Gautam went to market with Vanya and Paurush to buy fireworks. Each of them selected fireworks of their choice. Neeraj gave Rs.60 for 3 rockets, 4 fountains and 2 bombs, while Vanya gave Rs. 76 for 6 fountains, 2 rockets and 3 bombs and Paurush gave Rs.80 for the cost of 2 fountains, 4 rockets and 6 bombs. If the cost of same item of firework is same. Find the cost of each item per unit by matrix method.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

If the function $ f(x) = \left\{ \begin{array}{l l} 3ax+b, & \quad if { x > 3}\\ 11, & \quad if { x=1} \\ 5ax-2b, & \quad if {x < 1} \end{array} \right. $ is continuous at x = 1 , find the values a and b.

asked Jan 24, 2013 by thanvigandhi_1

0 answers

If \( (cos\: x)^{\Large y }= (cos\: y)^{\Large x}\) find \( \large\frac{dy}{dx}\)

asked Jan 23, 2013 by thanvigandhi_1

0 answers

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