Recent questions tagged modelpaper

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

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane which is perpendicular to the plane $ 5x+3y+6z+8=0$ and which contains the line of intersection of the planes $ x+2y+3z-4=0 \: and \: 2x+y-z+3=0.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 10, 2013 by thanvigandhi_1

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Prove that : $ \cot^{-1} \bigg[\large \frac{\sqrt{1+\sin\: x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \bigg] = \large \frac{x}{2}; $$x \in \bigg( 0, \large\frac{\pi}{2} \bigg). $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $ \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{bmatrix} = (a-b)(b-c)(c-a). $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $ \begin{bmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{bmatrix} = 2(a+b+c)^3. $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Find $\large \frac{dy}{dx} $ of following : $ y=(\sin \: x)^x+\sin^{-1} \sqrt x. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using Mathematical Induction prove that $ \large\frac{d}{dx}(x^n)$$=nx^{n-1}, \forall \: n \: \in \: N$ for all positive integers $ n $.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the intervals in which the function f given by $ f(x)=2x^3-3x^2-36x+7$ is : \[ \begin{array} ((i)\: strictly\: increasing \quad & (ii)\: strictly\: decreasing \end{array} \]

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that the curves $ x=y^2$ and $ xy=k$ cut at right angles if $ 8k^2 = 1. $

asked Jan 10, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{2\sin\: x+3\cos \: x}{3\sin \: x + 4 \cos \: x} $$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \ (x+3) \sqrt {3-4x-x^2} dx. $

asked Jan 10, 2013 by thanvigandhi_1

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Evaluate : $\int\large \frac{e^x}{\sqrt {5-4e^x-e^{2x}}}$$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int\large \frac{1}{5cos\: x-12sin\: x}$$dx$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find $ | \overrightarrow a | \: and \: | \overrightarrow b |, $ if $ ( \overrightarrow a + \overrightarrow b).( \overrightarrow a - \overrightarrow b)=8 \: and \: | \overrightarrow a |=8| \overrightarrow b|.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the points (2, 3, 2), (2, -2, -1) and (7, 0, 6).

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 10, 2013 by thanvigandhi_1

0 answers

There are two bags I and II. Bag I contains 3 white and 4 black balls and Bag II contains 5 white and 6 black balls. One ball is drawn at random from one of the bags and is found to be white. Find the probability that it was drawn from bag I.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using matrix solve the following system of linear equations : $ x+y+z=4;2x-y+z=-1;2x+y-3z = -9$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ A = \begin{bmatrix} 3 & 1 \\ 7 & 5 \end{bmatrix} $, find x and y that $ A^2+x1 = yA.$ Hence find $ A^{-1} $.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $ \large\frac{1}{3}h.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the area of the smaller part of the circle $ x^2+y^2=a^2$ cut off by the line $ x=\large\frac{a}{\sqrt 2}$

asked Jan 10, 2013 by thanvigandhi_1

2 answers

Form the differential equation of the family of circles in the first quadrant which touch the coordinates axes.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the equation of plane passing through the (-1, -1, 2) and $ \bot $ to each of the following planes : $ 2x+3y-3z=2 \: and \: 5x-4y+z=6 $.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

A farmer has a supply of chemical fertilizer of type I which contains 10% nitrogen and 6% phosphoric acid and type II fertilizer which contains 5% of nitrogen and 10% phosphoric acid. After testing the soil conditions of a field, it is found that atleast 14kg of nitrogen and 14 kg of phosphoric acid is required for a good crop. The fertilizer type I costs Rs.2/kg and the type II costs Rs. 3/kg. How many kilograms of each fertilizer should be used to meet the requirement and the cost be minimum?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

A can hit a target 4 times in 5 shots, B 3 times in 4 shots and C 2 times in 3 shots. Calculate the probability that \[\begin{array}{1 1} ((i) \: A,B,C\: all \: may \: hit. \\ (ii)\: Any\: two \: of \: A,B\: and \: C\: will\: hit \ the \: target. \\ (iii)\: None \: of\: them \: will \: hit \: the \: target.\\(iv)None \:of\:hem\:will\:hit\:the\:target. \end{array} \]

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Relation R in the set A = { 1, 2, 3, 4, 5, 6} defined as R = (x, y):y is divided by x}.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Find the value of the following: $ \tan^{-1}(1)+\cos^{-1} \bigg(\large -\frac{1}{2} \bigg)+$$\sin^{-1} \bigg(\large-\frac{1}{2} \bigg) $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Find the values of x and y make the following pairs of matrices equal : $\begin{bmatrix} 3x+6 & 5 \\ y+1 & 2-x \end{bmatrix} = \begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix} $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

What are the number of all possible matrices of 3 x 3 with each entry 0 or 1.

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Find the value of x if : $(i) \begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} = \begin{bmatrix} 2x & 4 \\ 6 & x \end{bmatrix} $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int (ax+b)^3dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int ( \sin2x-4e^{3x})dx$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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 10, 2013 by thanvigandhi_1

0 answers

Find a unit vector in the direction of $ \overrightarrow a = 3\hat i - 2\hat j + 6\hat k.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

For what value of $ \lambda $ are the vectors $ \overrightarrow a = 2\hat i + \lambda \hat j + \hat k \: and \: \overrightarrow b = \hat i - 2 \hat j + 3\hat k $ perpendicular to each other?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

In each of the follwing cases, state whether the function is one-one, onto or bijective. Justify your answer. $\begin{array}{1 1} (i) \: f: R \rightarrow R\: defined\: by \ f(x)=3-4x \\ (ii)\: f: R \rightarrow R\: defined\: by \ f(x)=1+x^2 .\end{array} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Solve for x : $ \tan^{-1} (2x)+\tan^{-1}(3x)=\large\frac{\pi}{4}. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants show that: $ \begin{bmatrix} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{bmatrix} = (x+y+z)(x-z)^2.. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ a, b, c$ are in A.P., show the following: $ \begin{bmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{bmatrix} = 0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that : $ \cot^{-1} \bigg[ \large\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \bigg]; $$0 < x <\large \frac{\pi}{2} = \large\frac{1}{2} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Differentiate $ (x^2-5x+8)(x^2+7x+9)$ in three ways mentioned : (i) by using product rule (ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation. Do they all give the same answer?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ u, v \: and \ w $ are function of $ x$ , then show that $ \large\frac{d}{dx}(u.v.w) = \frac{du}{dx}.v.w+u.\frac{dv}{dx}.w+u.v.\frac{dw}{dx}$In two ways first by repeated application or product rule, secondly by logatithmic differentiation.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate $\int\large \frac{\sin(x-\alpha)}{\sin(x+\alpha)}$$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Verify that $ y = e^x.\cos\: x$ is a solution of the differential equation : $\large \frac{d^2y}{dx^2}$$-2\large\frac{dy}{dx}$$+2y=0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the points A, B and C with position vectors $ \overrightarrow a = 3\hat i - 4\hat j - 4\hat k, \overrightarrow b = 2\hat i - \hat j + 4\hat k \: and \: \overrightarrow c = \hat i - 3\hat j - 5\hat k,$ respectively, form the vertices of a right angled triangle.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the point on the line $ \large\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}$ at a distance $ 3\sqrt 2$ from the point (1, 2, 3).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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