Recent questions tagged modelpaper

If $ \overrightarrow a, \overrightarrow b $ and $ \overrightarrow c $ are three unit vectors such that $ \overrightarrow a.\overrightarrow b = \overrightarrow a. \overrightarrow c = 0$ and angle between $ \overrightarrow b \: and \: \overrightarrow c $ is $ \large\frac{\pi}{6}, $ prove that $ \overrightarrow a = \pm\: 2 ( \overrightarrow b$ x $ \overrightarrow c ) $.

asked Dec 28, 2012 by thanvigandhi_1

0 answers

Show that the four points ( 0, -1, -1 ), ( 4, 5, 1 ) and (-4, 4, 4 ) are coplanar. Also find the equation of the plane containing them.

asked Dec 28, 2012 by thanvigandhi_1

1 answer

A coin is based so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, find the probabilty distribution of number of tails.

asked Dec 28, 2012 by thanvigandhi_1

0 answers

Using properties of determinants, show that $ \Delta = \begin{bmatrix} (a+b)^2 & ab & ca \\ ab & (a+b)^2 & bc \\ ac & bc & (a+b)^2 \end{bmatrix} = 2abc\: (a + b + c )^3 $

asked Dec 28, 2012 by thanvigandhi_1

0 answers

The sum of the perimeter of a circle and a 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 Dec 28, 2012 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{1}{\sin x (5-4\cos x )}$$ dx$

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Using integration, find the area of the region $ \{ (x,y) : | x - 1 | \leq y \leq \sqrt {5-x^2} \} $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Show that the lines $ \large\frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5} $ and $\large \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{5} $ are coplanar. Also find the equation of the plane.

asked Dec 26, 2012 by thanvigandhi_1

1 answer

From a pack of 52 cards, a card is lost. Fro the remaining 51 cards, two cards are drawn at random ( without replacement ) and are found to be both diamonds. What is the probability that the lost card was a card of heart?

asked Dec 26, 2012 by thanvigandhi_1

0 answers

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods X and Y are available at a cost of Rs.4 and Rs.3 per unit respectively. One unit of food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas 1 unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of foods X and Y should be used to have atleast cost, satisfying the requirements. Make it an LPPand solve it graphically.

asked Dec 26, 2012 by thanvigandhi_1

0 answers

A fair die is rolled up. If 1 turns up, a ball is picked up at random from bag A. If 2 0r 3 turns up, a ball is picked up at random from bag B, otherwise a ball is picked up from bag C. Bag A contains 3 red and 2 white balls, bag B contains 3 red and 5 white balls and bag C contains 4 red and 5 white balls. The die is rolled, a bag is picked up and a ball is drawn from it. If the ball drawn is red, what is the probability that bag B was picked up?

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Give an example to show that the relation R in the set of natural numbers, defined by $ R = ( (x,y), x, y \in\: N, x \leq \: y^2) $ is not transitive.

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Write the principal value of $ cos^{-1} \bigg( cos \frac{5\pi}{3} \bigg) $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Find x, if $ \begin{bmatrix} 5 & 3x \\ 2y & z \end{bmatrix} $ = $ \begin{bmatrix} 5 & 4 \\ 12 & 6 \end{bmatrix}^T $

asked Dec 26, 2012 by thanvigandhi_1

1 answer

For what value of a, $ \begin{bmatrix} 2a & -1 \\ -8 & 3 \end{bmatrix} $ is a singular matrix?

asked Dec 26, 2012 by thanvigandhi_1

0 answers

A square matrix A, of order 3, has | A | = 5, find | A.adj.A |.

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Evaluate : $\int \: 5^x dx $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Write the value of $ \int_{\Large\frac{-\pi}{2}}^{\Large\frac{\pi}{2}} \sin^5 x dx $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Find the position vector of the midpoint of the line segment joining the points $ A ( 5\hat i + 3\hat j )\: and \: B ( 3\hat i - \hat j ) $.

asked Dec 26, 2012 by thanvigandhi_1

0 answers

If $ \overrightarrow a = 2\hat i - \hat j + 3\hat k \: and \: \overrightarrow b = (6\hat i + \lambda\hat j + 9\hat k )$ and $ \overrightarrow a | | \overrightarrow b, $ find the value of $ \lambda $.

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Find the distance of the point (a, b, c) from x - axis.

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Let N be the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d) if ad = bc. Show that R is an equivalence relation.

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Prove that $ tan^{-1} \bigg(\frac{\large 1}{\large 4} \bigg) + tan^{-1} \bigg( \frac{\large 2}{\large 9} \bigg ) = \frac{\large 1}{\large 2} cos^{-1} \bigg( \frac{3}{5} \bigg). $

asked Dec 26, 2012 by thanvigandhi_1

1 answer

Using properties of determinants, prove that : $ \begin{bmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{bmatrix} = (1+a^2+b^2+c^2) $

asked Dec 26, 2012 by thanvigandhi_1

0 answers