Recent questions in Model Papers

Show that the vectors : $ \overrightarrow a = -4\hat i - 6\hat j - 2\hat k, \overrightarrow b = -\hat i + 4\hat j + 3\hat k \: and \: \overrightarrow c = -8\hat i - \hat j + 3\hat k $ are coplanar.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the sum of the vectors : $ \overrightarrow a = \hat i - 2\hat j + \hat k, \overrightarrow b = -2\hat i + 4\hat j + 5\hat k \: and \: \overrightarrow c = \hat i - 6\hat j - 7\hat k $

asked Jan 2, 2013 by thanvigandhi_1

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Find the equation of the plane passing through the point ( 1, -1, 2 ) having ( 2, 3, 2 ) as direction ratios of normal to the plane?

asked Jan 2, 2013 by thanvigandhi_1

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Let f : N $ \rightarrow $ N be defined by : $ f(n) = \left\{ \begin{array}{l l} \large \frac{n+1}{2} & \quad \text{if $n$ is odd}\\ \large \frac{n}{2} & \quad \text{ if $n$ is even} \end{array} \forall \: n \: \in\: N. \right.$ Find whether the function f is bijective.

asked Jan 2, 2013 by thanvigandhi_1

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Prove that : $ tan^{-1}\bigg(\frac{1}{3} \bigg)+tan^{-1}\bigg(\frac{1}{5}\bigg)+tan^{-1}\bigg(\frac{1}{7}\bigg)+tan^{-1}\bigg(\frac{1}{8}\bigg)=\large\frac{\pi}{4} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, show that $\begin{bmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{bmatrix} = (a + b + c )^3 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, show that $ \begin{bmatrix} b+c & a & b \\ c+a & c & a \\ a+b & b & c \end{bmatrix} = (a + b + c )(a-c)^2 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

If f(x) , defined by the following, is continuous at x = 0 , find the value of a, b and c. $f(x)=\left \{\begin{array}{1 1}\large\frac{\sin(a+1)x+\sin x}{x}, & if\;x< 0\\c, & if\;x= 0\\ \large\frac{\sqrt{x+bx^2}-\sqrt x}{bx^{\large\frac{3}{2}}}, & if\;x>0\end{array}\right.$

asked Jan 2, 2013 by thanvigandhi_1

0 answers

If $ x = a \bigg( \cos \theta +\ log \tan\large \frac{\theta}{2} \bigg)$ and $ y = a\sin\: \theta$, find the value of $ \large\frac{dy}{dx}$ at $ \theta = \large\frac{\pi}{4} $.

asked Jan 2, 2013 by thanvigandhi_1

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Find the intervals in which the function $ f(x) = \large\frac{4x^2+1}{x} $ is, (i) increasing, (ii) decreasing.

asked Jan 2, 2013 by thanvigandhi_1

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Find the equation of the tangent and the normal to the curve $ y = x^2+4x+1 $ at the point whose x - coordinate is 3.

asked Jan 2, 2013 by thanvigandhi_1

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Evaluate : \[ (i) \int \: (x-3) \sqrt {x^2+4x+3} dx \] \[ (ii) \int \: (3x+5) \sqrt {2x^2+3x+7} dx \] \[ (iii) \int \: (x-4) \sqrt {4+x-x^2} dx \] \[ (iv) \int \: (5x-1) \sqrt {6+5x-2x^2} dx \]

asked Jan 2, 2013 by thanvigandhi_1

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Evaluate : $ \int \: \large\frac{1}{\cos(x+a)\cos(x+b)}$$dx $

asked Jan 2, 2013 by thanvigandhi_1

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

asked Jan 2, 2013 by thanvigandhi_1

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For any three vectors : $ \overrightarrow a, \overrightarrow b \: and \: \overrightarrow c $ prove that $ \bigg[ \overrightarrow a + \overrightarrow b \: \: \overrightarrow b + \overrightarrow c \: \overrightarrow c + \overrightarrow a \bigg] = 2 \bigg[ \overrightarrow a \: \overrightarrow b \: \overrightarrow c \bigg]. $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the vector and cartesian equation of the sphere described on the join of the points (2, -3, 4 ) and (-5, 6, -7 ) as the extremities of a diameter.

asked Jan 2, 2013 by thanvigandhi_1

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Find the root of the perpendicular drawn from the point P(1, 6, 3) on the line $\large \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}. $ Also, find its distance from P.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

12 cards, numbered 1 to 12, are placed in a box, mixed up throughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system of equations : $ 2x-3y+5y = 11 ; 3x+2y-4z = -5 ; x+y-2z = -3 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is $ 8m^3$ . If building of tank cost Rs. 70 per sq. metre for the base and Rs. 45 per sq. metre for sides, what is the cost of least expensive tank?

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

asked Jan 2, 2013 by thanvigandhi_1

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

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Prove that $ x^2-y^2 = c ( x^2-y^2 )^2 $ is the general solution of differential equation $( x^3-3xy^2) dx = (y^3-3x^2y) dx$.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

From the point P(1, 2, 4 ) a perpendicular is drawn on the plane $ 2x+y-2z+3 = 0 $. Find the equation of the length and coordinates of the foot of the $ \bot $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

A farmer mixes two brands P and Q of cattle feed. Brand P costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C . The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units, respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Suppose, we have four boxes A, B, C and D containing coloured marbles as given below : One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A? box B? and box C?

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Relation R in the set of Z of all integers defined as R = {(x, y) : x-y is an integer }.

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Using principle value, evaluate the following : $ \cos^{-1} \bigg( \cos\large\frac{2\pi}{3} \bigg)$$ + \sin^{-1} \bigg( \sin\large\frac{2\pi}{3} \bigg) .$

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Find the value of x and y if : $2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

If matrix A = [ 1 2 3 ] Write AA' where A' is the transpose of matrix A.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Write the value of the determinant : $\begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{bmatrix} $

asked Jan 2, 2013 by thanvigandhi_1

1 answer

The radius of a circle is increasing uniformly at the rate of 3 cm per second. Find the rate in which the area of the circle is increasing when the radius is 10 cm.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

$ \int \: \large\frac{x^3+5x^2-4}{x^2}\normalsize dx. $

asked Jan 2, 2013 by thanvigandhi_1

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Find the value of the parallelopipped whose edges are $ \overrightarrow a = 2\hat i - 3\hat j + 4\hat k, \overrightarrow b = \hat i - 2\hat j - \hat k \: and \: \overrightarrow c = 2\hat i - \hat j + 2\hat k $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Find the scalar and vector components of the vector with initial point ( 2, 1 ) and terminal point ( -5, 7 ).

asked Jan 2, 2013 by thanvigandhi_1

1 answer

Write the direction cosines of a line equally inclined to the three coordinate axes.

asked Jan 2, 2013 by thanvigandhi_1

0 answers

(i) Is the binary operation * defined on set N, given a * b = $ \large\frac{a+b}{2} $ for all a, b $ \in$ N, comutative? (ii) Is the above binary operation * associative ?

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Write the following functions in the simplest form : \[ (i) tan^{-1} \bigg( \frac{\sqrt{1+x^2} - 1}{x} \bigg), x \neq 0 \] \[(ii) tan^{-1} \frac{1}{\sqrt{x^2-1}} ,| x | > 1 \]

asked Dec 31, 2012 by thanvigandhi_1

1 answer

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

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Prove that : $ (i) \large \frac{d}{dx} $$(a^x) = a^x log_e a. \; (ii) \large\frac{d}{dx}$$ (e^x) = e^x. $

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Find all the points of discontinuity of f, where $ f(x) = \left \{ \begin{array}{1 1} \large\frac{sinx}{x}, \quad & if\: x < 0 \\ x+1, \quad & if\: x \geq 0 \end{array}\right \}. $

asked Dec 31, 2012 by thanvigandhi_1

0 answers

A point source of light along a straight road is a height of 'a' metres. A boy 'b' metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of c metres per minute.?

asked Dec 31, 2012 by thanvigandhi_1

0 answers

The two equal sides of an isosceles triangles with fixed base b cm are decreasing at the rate of 3 cm/sec. How fast is the area decreasing when two equal sides are equal to the base.

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Evaluate : $ \int \: \bigg[ \large\frac{1}{log \: x} - \large\frac{1}{(log\: x )^2} \bigg]$$ dx. $

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Integrate : $\int \:\large \frac{x}{(x-1)^2(x+2)} $

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Integrate : $ \int \: \large\frac{3x+5}{x^2-x^2-x-1} $

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Integrate : $\int \: x(\ log x )^2dx.$

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Show that four points with position vectors $ 6\hat i - 7\hat j, 19 \hat j - 4\hat k, 3\hat j - 6\hat k \: and \: 2\hat i + 5\hat j + 10\hat k$ are not co-planar.

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the points ( 1, 2, 3 ) and (0, -1, 0 ) and parrallel to the line, $ \large\frac{x-1}{2} = \frac{y+2}{3} = \frac{z}{-3} $.

asked Dec 31, 2012 by thanvigandhi_1

1 answer

Find the image of the point ( 1, 2, 3 ) in the plane $ x + 2y + 4z = 38$.

asked Dec 31, 2012 by thanvigandhi_1

0 answers

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