Recent questions in Model Papers

Find the area of the smaller region bounded by the ellipse $\large\frac{x^2}{a^2}+\large\frac{y^2}{b^2}$ $=1$ and the straight line $\large\frac{x}{a}+\frac{y}{b}$ $=1$ .

asked Jan 29, 2013 by thanvigandhi_1

0 answers

Find the area of the region $\{(x,y); x^2+y^2 \leq 1 \leq x+y \}.$

asked Jan 28, 2013 by thanvigandhi_1

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

asked Jan 28, 2013 by thanvigandhi_1

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Evaluate : $\int_0^{\pi} \cos 2x\log\sin\:x\: dx$

asked Jan 28, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{\sin x}{\sin 4x}$ $dx$

asked Jan 28, 2013 by thanvigandhi_1

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Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to each of the lines $\large\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ and $\large\frac{x+2}{-3} = \frac{y-1}{2} = \frac{z+1}{5}$ . Also, find its vector equation.

asked Jan 28, 2013 by thanvigandhi_1

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A company sells two different products A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 70 and that of B is 125. If the profit is Rs.20 per unit for the product A and Rs.15 per unit for the product B, how many units for each product should be sold to maximize profit.

asked Jan 28, 2013 by thanvigandhi_1

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Find the area bounded by the curve $x^2 = 4y$ and the straight line 4y - 2 = x

asked Jan 28, 2013 by fingeazy

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Express the vector $\overset \rightharpoonup {a} = 5\hat{i} - 2\hat{j} + 5\hat{k}$ as sum of two vectors such that one is parallel to the vector $\overset \rightharpoonup {b} = 3\hat{i} + \hat{k}$ and the other is perpendicular to $\overset \rightharpoonup {b}$

asked Jan 28, 2013 by fingeazy

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If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$ . $\hat{i} + \hat{k}$ and $c\hat{i} + c\hat{j} + b\hat{k}$ be coplanar, show that $c^2$ = ab

asked Jan 28, 2013 by fingeazy

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A car, traveling with a uniform acceleration, has a velocity of 18 km/hour at a certain time and 54 km/hour after covering a distance of 500m. How much further will it travel to attain a velocity of 72 km/hour?

asked Jan 28, 2013 by fingeazy

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A body falls freely from the top of a tower; It covers $\large\frac{5}{9}$ th second. Find the height of the tower and the total by the body to fall down. of the whole distance in the last

asked Jan 28, 2013 by fingeazy

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A cricket ball is project with a velocity of 29.4 m/sec. Find (a) The greatest range on the horizontal plane; and (b) The angle of projection to give a range of 44.10 m.

asked Jan 28, 2013 by fingeazy

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Find the co-ordinates of the foot of the perpendicular drawn from the point A(1,8,4,) to the line joining the points B(0,-1,3) and C(2,-3,-1).

asked Jan 28, 2013 by fingeazy

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$\overset \rightharpoonup {P}$ and $\overset \rightharpoonup {Q}$ are two unlike parallel forces. When the magnitude of $\overset \rightharpoonup {P}$ is doubled, it is found that the line of action of $\overset \rightharpoonup {Q}$ is midway between the lines of action of the new and the original resultants, find the ratio of P and Q.

asked Jan 28, 2013 by fingeazy

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There are 40 workers in a factory who are participating in strike for increase in wages. Out of them 5 workers are sincere to management and believe that management know each thing and do the needful. Management selects 2 workers at random out of them. Write the probability distribution for the selected worker who are sincere. Also find the mean of the distribution.

asked Jan 28, 2013 by thanvigandhi_1

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Let * be binary operation on N given by $ab = 1.c.m.(a,b),a,b \in N.$ Find (23)*6.

asked Jan 28, 2013 by thanvigandhi_1

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Prove that $\cos^{-1}(4x^3-3x)=3\cos^{-1}x.$

asked Jan 28, 2013 by thanvigandhi_1

1 answer

Find the values of x,y and z, if $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$

asked Jan 28, 2013 by thanvigandhi_1

1 answer

If $A = \begin{bmatrix} 2 & 5 \\ 6 & -3 \end{bmatrix}$ then show that : |5A|=25|A|.

asked Jan 28, 2013 by thanvigandhi_1

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Find the co-factor of $a_{13}$ in the following : $\begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}$

asked Jan 28, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int \large\frac{\sin(\tan^{-1}x)}{1+x^2}$ $dx$

asked Jan 28, 2013 by thanvigandhi_1

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Evaluate : $\int_{-\pi}^{\pi} x^8\sin^5x dx.$

asked Jan 28, 2013 by thanvigandhi_1

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Find a unit vector parallel to the sum of the vectors $\overrightarrow a = 2\hat i + 4\hat j - 5\hat k \: and \: \overrightarrow b = \hat i + 2\hat j - 3\hat k$

asked Jan 28, 2013 by thanvigandhi_1

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Show that the vectors $2\hat i - 3\hat j + 4\hat k\: and \: -4\hat i + 6\hat j - 8\hat k$ are collinear.

asked Jan 28, 2013 by thanvigandhi_1

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Find the area of a parallelogram whose adjacent sides are given by $\overrightarrow a = 3\hat i + \hat j + 4\hat k \: and \: \overrightarrow b = \hat i - \hat j + \hat k$

asked Jan 28, 2013 by thanvigandhi_1

0 answers

Let A = set of workers in a publication house. Let a R b, if and only if a and b tell lie; $a,b \in A$ (i) Is R symmetric relation? (ii) Is this relation an equivalence relation?

asked Jan 28, 2013 by thanvigandhi_1

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Prove that $\large \frac{9\pi}{8}-\frac{9}{4}$ $\sin^{-1}\large\frac{1}{3}=\large\frac{9}{4}$ $\sin^{-1}\large\frac{2\sqrt2}{3}$

asked Jan 28, 2013 by thanvigandhi_1

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Express $\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$ as the sum of a symmetic and skew-symmetric matrix.

asked Jan 28, 2013 by thanvigandhi_1

1 answer

If $A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix},$ show that $A^2-7A+10I=0.$ Hence , find $A^{-1}$

asked Jan 28, 2013 by thanvigandhi_1

1 answer

For what value of k, is the following continuous at x = 0? $f(x) = \left\{ \begin{array}{l l}\large\frac{\sqrt{1+x}-\sqrt{1-x}}{sin^{-1}x}, & \quad { x \neq 0}\\ k, & \quad { x = 0} \\ \end{array}. \right.$

asked Jan 28, 2013 by thanvigandhi_1

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Differentiate $\sin(x^x)+(\sin x)^x,$ with respect to x.

asked Jan 28, 2013 by thanvigandhi_1

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Find the equations of tangent and normal to curve $y=x-\sin x\cos x$ at $x=\large\frac{\pi}{2}.$

asked Jan 28, 2013 by thanvigandhi_1

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

asked Jan 28, 2013 by thanvigandhi_1

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

asked Jan 28, 2013 by thanvigandhi_1

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Show that $x\cos \: \bigg(\large\frac{y}{x} \bigg) \large\frac{dy}{dx}=$ $y\cos \bigg(\large \frac{y}{x} \bigg) +x$ is homogeneous and solve it.

asked Jan 28, 2013 by thanvigandhi_1

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

asked Jan 28, 2013 by thanvigandhi_1

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Solve the differential equation: $\large \frac{dy}{dx}=\frac{y}{x}$ $+\log x$

asked Jan 28, 2013 by thanvigandhi_1

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Let $\overrightarrow a = \hat i + \hat j,\overrightarrow b = \hat i - \hat j \: and \: \overrightarrow c = \hat i + 2\hat j + 3\hat k,$ if $n$ is unit vector such that $\overrightarrow a.\overrightarrow n=0 \: and \: \overrightarrow b.\overrightarrow n=0,$ then show that $\overrightarrow c.\overrightarrow n=3.$

asked Jan 28, 2013 by thanvigandhi_1

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Show that the points with position vectors are given by $2\hat i - \hat j + \hat k, \hat i - 3\hat j - 5\hat k$ and $3\hat i - 4\hat j - 4\hat k$ from a right-angled triangle.

asked Jan 28, 2013 by thanvigandhi_1

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Find the equation of plane through the points (2,2,1), (9,3,6) and perpendicular to the plane $2x+6y+6z=9.$

asked Jan 28, 2013 by thanvigandhi_1

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In a village of 50 people, 20 people have their fast on 2nd october. Two persons are selected at random out of them. Write the probability distribution for selected persond who are on fast. Also, find the mean of the distribution.

asked Jan 28, 2013 by thanvigandhi_1

0 answers

For the matrix $A= \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}$ , show that $A^3-6A^2+5A+11I=0.$ Hence, find $A^{-1}.$

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Prove that the area of a right-angled triangle of given hypotenuse is maximum when it is isosceles.

asked Jan 27, 2013 by thanvigandhi_1

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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 Jan 27, 2013 by thanvigandhi_1

0 answers

Find the area of the region bounded by the parabola $y=x^2+1$ and the lines $y=x,x=0 \: and \: x=2.$

asked Jan 27, 2013 by thanvigandhi_1

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Prove that $\int_0^{\large\frac{\pi}{2}} ( \sqrt{\tan\:x}+ \sqrt{\cot\:x})dx= \sqrt2 \pi$

asked Jan 27, 2013 by thanvigandhi_1

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Find the equation of plane through the intersection of the plane $x+y+z=9 \: and \: 2x+3y+4z+5=0$ and passing through the point (1,1,1).

asked Jan 27, 2013 by thanvigandhi_1

0 answers

Find the shortest distance between the lines $\large \frac{x-8}{3} = \frac{y+9}{-16}=\frac{z-10}{7}$ and $\large\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$ Also, find equation of the line of shortest distance.

asked Jan 27, 2013 by thanvigandhi_1

0 answers

If a young man rides his motorcycle at 25 km/hour, he had to spend Rs.2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs.5 per km. He has Rs.100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as L.P.P. and solve it graphically.

asked Jan 27, 2013 by thanvigandhi_1

1 answer

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