Recent questions in Model Papers

Assuming that petrol burnt per hour in driving a motorboat varies as the cube of its velocity. Show that the most economical speed when going against a current of p km/hour is $\large\frac{3p}{2}$ km/hour.

asked Jan 25, 2013 by thanvigandhi_1

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Evaluate : $\int_{-1}^{\large\frac{3}{2}} |x\sin\: \pi x |dx$

asked Jan 25, 2013 by thanvigandhi_1

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Evaluate : $\int_0^{\large\frac{\pi}{2}} \sin2x\tan^{-1}(\sin x)dx$

asked Jan 25, 2013 by thanvigandhi_1

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

asked Jan 25, 2013 by thanvigandhi_1

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Solve the following differential equation $(1+x^2)\large\frac{dy}{dx}$ $+y =\ tan^{-1}x$

asked Jan 25, 2013 by thanvigandhi_1

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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 25, 2013 by thanvigandhi_1

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Find the value of $\lambda$ so that the lines $\large\frac{1-x}{3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\large\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{6-z}{5}$ are perpendicular.

asked Jan 25, 2013 by thanvigandhi_1

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Three balls are drawn one by one without replacement from a bag containing 5 white and 4 black balls. Find the probability of the number of red balls.

asked Jan 25, 2013 by thanvigandhi_1

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A one hour routine check was made by security staff of MNC from $\large \frac{1}{2}$ hour before office opening time. Total 80 cars are checked and result was 9 cars carrierd no official, 20 cars carried one official, 16 cars carried 2 officials, 20 cars carried 3 officials and 15 cars carried 4 officials. (i) Using this information, write down a row matrix and a column whose product gives us the total number of official carried by the 80 cars. (ii) What was the average number of official per car?

asked Jan 25, 2013 by thanvigandhi_1

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A wire of length 30 cm 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 could be the lengths of two peices, so that the combined area of the square and the circle is minimum?

asked Jan 25, 2013 by thanvigandhi_1

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An open box with a square base is to be made out of given quantity of sheet of area $a^2$ . Show that maximum volume of the box is $\large\frac{a^3}{6\sqrt3}.$

asked Jan 25, 2013 by thanvigandhi_1

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Evalaute : $\int \sqrt{\large\frac{sin(x-\alpha)}{sin(x+\alpha)}}$ $dx$

asked Jan 25, 2013 by thanvigandhi_1

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Evalaute : $\int \large\frac{1}{\sin^2x+\sin2x}$ $dx.$

asked Jan 25, 2013 by thanvigandhi_1

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Find the area of the circle $4x^2+4y^2$ which is interior to the parabola $x^2=4y$ .

asked Jan 25, 2013 by thanvigandhi_1

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Find the image of (1, 6, 3) in the line $\large\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}.$

asked Jan 25, 2013 by thanvigandhi_1

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Akshay throw a die again and again until three fours are obtained. Find the probability of obtaining the third four in the sixth throw of die by Akshay.

asked Jan 25, 2013 by thanvigandhi_1

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A diet is to contain at least 80 units of minerals. Two foods $F_1 \: and F_2$ are available. Food $F_1$ costs Rs.4 per unit food and $F_2$ costs Rs.6 per unit. A unit of food $F_1$ contains atleast 3 units of vitamin A and 4 units of minerals. A unit of food $F_2$ contains atleast 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods, and also meets the minimal nutritional requirements.

asked Jan 25, 2013 by thanvigandhi_1

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If $f : R \rightarrow R$ is defined by $f(x) = x^2-3x+2$ , find $f (f(x)).$

asked Jan 25, 2013 by thanvigandhi_1

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Show that $sin^{-1} \bigg( 2x \sqrt{1-x^2} \bigg) = 2 sin^{-1}x.$

asked Jan 25, 2013 by thanvigandhi_1

1 answer

If $\begin{bmatrix} 2x+5 & 3 \\ 5x+2 & 9 \end{bmatrix} = 0$ , find x.

asked Jan 25, 2013 by thanvigandhi_1

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If $\begin{bmatrix} x+3 & 4 \\ y-4 & x+y \end{bmatrix} = \begin{bmatrix} 5 & 4 \\ 3 & 9 \end{bmatrix}$ , find x and y.

asked Jan 25, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int\large \frac{x^2+4x}{x^3+6x^2+5}$ $dx$

asked Jan 25, 2013 by thanvigandhi_1

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Evaluate : $\int_0^1 \large\frac{dx}{1+x^2}$

asked Jan 25, 2013 by thanvigandhi_1

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If $| \overrightarrow a | = 2, |\overrightarrow b | = \sqrt 3\: and \: \overrightarrow a.\overrightarrow b = \sqrt 3.$ Find the angle between $\overrightarrow a \: and \: \overrightarrow b.$

asked Jan 25, 2013 by thanvigandhi_1

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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

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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

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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

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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

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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

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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

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

asked Jan 24, 2013 by thanvigandhi_1

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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

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

asked Jan 24, 2013 by thanvigandhi_1

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

asked Jan 24, 2013 by thanvigandhi_1

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

asked Jan 24, 2013 by thanvigandhi_1

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

asked Jan 24, 2013 by thanvigandhi_1

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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

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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

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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

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

asked Jan 24, 2013 by thanvigandhi_1

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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

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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

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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

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