Recent questions tagged modelpaper

If the function f(x) given by $ f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if { x > 0} \\ 11, & \quad if { x = 1} \\ 5ax-2b, & \quad if { x < 1 } \end{array} \right. $ is continuous at x = 1, find the values of a and b.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove the following : $ \begin{vmatrix} x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3 \end{vmatrix} = xyz(x-y)(y-z)(z-x) $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $ \sin\bigg[ \large\frac{\pi}{3}$$-sin^{-1} \bigg( -\large\frac{1}{2} \bigg) \bigg]. $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

For a 2 x 2 matrix, A = $ [a_{ij} ], $ whose elements are given by $ a_{ij}=\large\frac{i}{j},$ write the value of $ a_{12}.$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

For what value of x, the matrix $ \begin{bmatrix} 5-x & x+1 \\ 2 & 4 \end{bmatrix} $ is singular?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

write $ A^{-1} $ for $ A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $ \int \sec\: x (\sec\: x+\tan \: x)dx $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $ \int \large\frac{dx}{x^2+16} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

For what value of 'a' the vectors $ 2\hat i - 3\hat j + 4\hat k $ and $ a\hat i + 6\hat j - 8\hat k $ are collinear?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the direction cosines of the vector $ -2\hat i + \hat j - 5\hat k $.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the intercept cut off by the plane $ 2x+y-z=5$ on x - axis.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Prove the following : $ \cot^{-1} \bigg[ \large\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}} \bigg] = \frac{x}{2}, x \in \bigg( 0, \frac{\pi}{4} \bigg). $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Find the value of $ \tan{-1} \bigg(\large \frac{x}{y} \bigg)-\tan^{-1} \bigg(\large \frac{x-y}{x+y} \bigg). $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove that $\begin{vmatrix} -a^2 & ab & ac \\ ba & -b^2 & bc \\ ca & cb & -c^2 \end{vmatrix} = 4a^2b^2c^2 $

asked Feb 4, 2013 by thanvigandhi_1

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Find the value of 'a' for which the function f defined as $ f(x) = \left\{ \begin{array}{l l}asin\frac{\pi}{2}(x+1), & \quad { x \leq 0} \\ \large \frac{\tan\: x-\sin x}{x^3}, & \quad { x > 0} \end{array} \right. $ is continuous at x = 0.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Differentiate $ x^{\large xcos\: x}+\large\frac{x^2+1}{x^2-1} $ w.r.t. $ x $.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

If $ x=a(\theta-\sin\theta),y=a(1+\cos\theta), $ find $ \large\frac{d^2y}{dx^2} $.

asked Feb 4, 2013 by thanvigandhi_1

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Sand is pouring from a pipe at the rate of $ 12 cm^3/s.$ The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

asked Feb 4, 2013 by thanvigandhi_1

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Find the points on the curve $ x^2+y^2-2x-3=0$ at which the tangents are parallel to x - axis.

asked Feb 4, 2013 by thanvigandhi_1

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Evaluate : $ \int \large\frac{5x+3}{\sqrt{x^2+4x+10}}$$dx $

asked Feb 4, 2013 by thanvigandhi_1

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

asked Feb 4, 2013 by thanvigandhi_1

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Solve the following differential equation: $e^x\tan\: y\: dx+(1-e^x)\sec^2ydy=0$

asked Feb 4, 2013 by thanvigandhi_1

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Solve the following differential equation : $ \cos^2x\frac{dy}{dx}+y=\tan \: x$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Find a unit vector perpendicular to each of the vectors $ \overrightarrow a+\overrightarrow b \: and \: \overrightarrow a-\overrightarrow b$, where $ \overrightarrow a=3\hat i+2\hat j +2 \hat k $ and $ \overrightarrow b = \hat i + 2\hat j -2\hat k.$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Find the angle between the following pair of lines : $\large\frac{-x+2}{-2}=\frac{y-1}{7}=\frac{z+3}{-3}$ and $\large\frac{x+2}{-1}=\frac{2y-8}{4}=\frac{z-5}{4} $ and check whether the lines are parallel or perpendicular.

asked Feb 4, 2013 by thanvigandhi_1

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Probabilities of solving a specific problem independently by A and B are $ \frac{1}{2} \: and \frac{1}{3} $ respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using matrix method, solve the following system of equations : $\large\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4,\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1, \frac{6}{x}+\frac{9}{y}-\frac{20}{z}$$=2; x,y,z \neq 0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

asked Feb 4, 2013 by thanvigandhi_1

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Using integration find the area of the triangular region whose sides have equations $ y=2x+1,y=3x+1 \: and \: x=4$.

asked Feb 4, 2013 by thanvigandhi_1

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

asked Feb 4, 2013 by thanvigandhi_1

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

asked Feb 4, 2013 by thanvigandhi_1

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Find the equation of the plane which contains the line of intersection of the planes $ \overrightarrow r.(\hat i + 2\hat j + 3\hat k)-4=0, \overrightarrow r. (2\hat i + \hat j - \hat k)+5=0$ and which is perpendicular to the plane $ \overrightarrow r. (5\hat i + 3\hat j - 6\hat k)+8=0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's time. If the profit on a racket and on a bat is Rs.20 and Rs.10 respectively, find the number of tennis rackets and crickets bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve it graphically.

asked Feb 4, 2013 by thanvigandhi_1

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Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

asked Feb 4, 2013 by thanvigandhi_1

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If $ f = \{(5,2),(6,3)\} g = \{(2,5),(3,6)\}$ write $ fog$

asked Feb 4, 2013 by thanvigandhi_1

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Write the range of one branch of $ \sin^{-1} x$ other than the pricipal branch.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Find the value of y, if $\begin{bmatrix} x-y & 2 \\ x & 5 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the adjoint of the following matrix $ \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Evaluate the determinant $ \begin{bmatrix} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{bmatrix} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $ \int e^{3\log\:x}.(x^4)dx $

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \large\frac{\sin^{-1}x}{\sqrt{1-x^2}}$$dx $

asked Feb 4, 2013 by thanvigandhi_1

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Find the angle between $ \overrightarrow a \: and \: \overrightarrow b$ with magnitude 1 and 2 respectively and when $ |\overrightarrow a$ x $ \overrightarrow b | = \sqrt 3 $

asked Feb 4, 2013 by thanvigandhi_1

0 answers

If $( \overrightarrow a \: and \: \overrightarrow b)$ are two vectors such that $( | \overrightarrow a. \overrightarrow b| = | \overrightarrow a x \overrightarrow b |)$, then what is the angle between $( \overrightarrow a \: and \: \overrightarrow b)?$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

If a line makes an angle of $ 30^{\circ}, 60^{\circ}, 90^{\circ}$ with the positive direction of x,y,z axes, respectively, then find its direction cosines.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Let * be binary operation on Q defined by $ ab=\large\frac{3ab}{5} $ show that is commutative as well as associative. Also find its identity element if it exists.

asked Feb 4, 2013 by thanvigandhi_1

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using properties of determinants, prove that $ \begin{vmatrix} -bc & b^2+bc & c^2+bc \\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{vmatrix} = (ab+bc+ca)^2 $

asked Feb 4, 2013 by thanvigandhi_1

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Using elements transformation find the inverse of $ \begin{bmatrix} 2 & -6 \\ 1 & -2 \end{bmatrix} $

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Show that the function f defined as follows, is continuous at x = 2 but not differentiable $f(x) = \left\{ \begin{array}{l l}3x-2, & \quad { 0 < x \leq 1} \\ 2x^2-x, & \quad { 1 < x \leq 2} \\ 5x-4, & \quad { x > 2 } \end{array} \right. $

asked Feb 4, 2013 by thanvigandhi_1

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