Recent questions in Model Papers

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

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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

0 answers

Find $\large\frac{dy}{dx},$ $if \: y=\sin^{-1} \bigg[ x \sqrt{1-x}-\sqrt x \sqrt{1-x^2} \bigg]$ .

asked Feb 4, 2013 by thanvigandhi_1

0 answers

If $y=\sqrt{x^2-1}-\log \bigg( \large\frac{1}{x}+\sqrt{1+\frac{1}{x^2}} \bigg)$ find $\large\frac{dy}{dx}$ .

asked Feb 4, 2013 by thanvigandhi_1

0 answers

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi-vertical angle is $\tan^{-1} \bigg(\large\frac{1}{2} \bigg)$ . Water is poured into it at a constant rate of 5 cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10 m.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int_0^{\pi}\large\frac{x\: dx}{a^2\cos^2x+b^2\sin^2x}$ $a,b>0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Show that the differential equation $2y.e^{\large\frac{x}{y}}dx+(y-2xe^{\large\frac{x}{y}})dy=0.$ is homogeneous and find its particular solution given that x = 0 when y = 1.

asked Feb 4, 2013 by thanvigandhi_1

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Find the particular solution of the differential equation : $\large\frac{dy}{dx}+y\cot x=2x+x^2\cot x, x \neq 0$ given that y = 0 when $x=\large\frac{\pi}{2}.$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Form a differential equation of the family of ellipse having foci on y - axis and centre at origin.

asked Feb 4, 2013 by thanvigandhi_1

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The scalar product of the vector $\hat i + 2\hat j + 4\hat k$ with a unit vector along the sum of vectors $\hat i + 2\hat j + 3\hat k$ and $\lambda \hat i + 4\hat j - 5\hat k$ is equal to one. Find the value of $\lambda$ .

asked Feb 4, 2013 by thanvigandhi_1

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Find a vector in the direction of vectors $5\hat i - \hat j + 2\hat k$ which has magnitude 8 units.

asked Feb 4, 2013 by thanvigandhi_1

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Find the value of p so that the lines $\large\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}$ and $\large\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}$ are right angles.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

A foot ball match may be either won, drawn or lost by the host country's team. So there are three ways of forecasting the result of any one match. One correct and two incorrect. Find the probability of forecasting at least three correct result for four matches.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

If a,b,c are positive and unequal, show that the following determinant is negative. $\Delta = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

using matrices, solve the following system of equations $x+2y-3z=-4;2x+3y+2z=2 ; 3x-3y-4z=11$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Show that the volume of the greatest cylinder that can be inscribed in a cone of height 'h' and semi-vertical angle $\alpha$ is $\large\frac{4}{27}\pi h^3\tan^2 \alpha$ .

asked Feb 4, 2013 by thanvigandhi_1

1 answer

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

asked Feb 4, 2013 by thanvigandhi_1

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Sketch the graph of $f(x) = \left\{ \begin{array}{l l}|x-2|+2, & \quad { x \leq 2} \\ x^2-2, & \quad { x > 2} \end{array} \right.$ Evaluate $\int_0^4 f(x) dx$ What does the value of this integral represent on the graph?

asked Feb 4, 2013 by thanvigandhi_1

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Find the equation of the plane passing through the point P(1,1,1) and containing the line $\overrightarrow r = (-3\hat i + \hat j + 5\hat k)+ \lambda (3\hat i - \hat j - 5\hat k).$ . Also show that the plane contains the line $\overrightarrow r = (-\hat i + 2\hat j + 5\hat k)+ H (\hat i - 2\hat j - 5\hat k).$

asked Feb 4, 2013 by thanvigandhi_1

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Two bags A and B contain 4 white and 3 black balls and 2 white,2 black balls respectively. From bag A, two balls are transferred to bag B. Find the probability of drawing (a) 2 white balls from Bag B? (b) 2 black balls from bag B? (c) 1 white and 1 black ball from bag B?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates. The corresponding values for rice are 0.05 g and .5 g respectively. Wheat costs Rs.4 per kg and rice Rs.6 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 5 g and 200 g respectively. In what qualities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of protiens and carbohydrates at minimum cost. Form a L.P.P and solve it graphically.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Write the principal value of $\cos^{-1} \bigg( -\large\frac{\sqrt 3}{2} \bigg).$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Write the range of the principal branch of $\sec^{-1}(x)$ defined on the domain R-(-1,1).

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Find x if $\begin{bmatrix} 3 & 4 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} 2x & 4 \\ -5 & 3 \end{bmatrix}$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

If A is a square matrix of order 3 such that |adj A|=64. Find |A'|.

asked Feb 3, 2013 by thanvigandhi_1

1 answer

If A is a square matrix satisfying $A^2=1$ , then what is the inverse of A?

asked Feb 3, 2013 by thanvigandhi_1

0 answers

If $f(x)=\sin x^{\circ},$ find $\large\frac{dy}{dx}.$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

What is the degree of the following differential equation? $y\large\frac{d^2y}{dx^2}+\bigg(\frac{dy}{dx} \bigg)^3$ $=x \bigg( \large\frac{d^3y}{dx^3} \bigg)^2$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

If $\overrightarrow a$ and $\overrightarrow b$ represent the two adjacent sides of a parallelogram, then write the area of parallelogram in terms of $\overrightarrow a \: and \: \overrightarrow b$

asked Feb 3, 2013 by thanvigandhi_1

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Find the angle between two vectors $\overrightarrow a \: and \: \overrightarrow b$ if $|\overrightarrow a |=3, |\overrightarrow b|=4\: and \: |\overrightarrow a * \overrightarrow b |=6$

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the direction cosines of a line, passing through origin and lying in the first octant, making equal angles with the three coordinates axes.

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Show that the relation R in the set $A = ( x ; x \in Z, 0 \leq x \leq 12)$ given by $R = \{(a,b) : |a-b|$ is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.

asked Feb 3, 2013 by thanvigandhi_1

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Solve for x : $2\tan^{-1}(\sin x)=\tan^{-1} (2\sec x),0 < x < \large\frac{\pi}{2}$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Show that : $\tan\large\frac{1}{2} \bigg[ \sin^{-1}\frac{2x}{1+x^2} + \cos^{-1}\large\frac{1-y^2}{1+y^2} \bigg] =\large \frac{x+y}{1-xy}$ $, |x| < 1, y > 0 \: xy < 1$

asked Feb 3, 2013 by thanvigandhi_1

1 answer

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