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Recent questions tagged modelpaper

Find the values of a and b, so that the function f given by $ f(x) = \left\{ \begin{array}{l l} 1, & \quad if { x \leq 3}\\ ax+b, & \quad if { 3<x<5} \\ 7, & \quad if {x \geq 5} \end{array} \right. $ is continuous at x = 3 and x = 5.

asked Jan 23, 2013 by thanvigandhi_1

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Differentiate \( sin(x^x)\) w.r.t. $ \tan^{-1} \sqrt{ \large\frac{1-x^2}{1+x^2}} $

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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

asked Jan 23, 2013 by thanvigandhi_1

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Using properties of definite integrals, evaluate : $ \int_0^1\large \frac{\log \: x}{\sqrt{1-x^2}} $$dx.$

asked Jan 23, 2013 by thanvigandhi_1

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If \( \overrightarrow a = 4\hat i + 5\hat j - \hat k, \overrightarrow b = \hat i - 4\hat j + 5\hat k, \overrightarrow c = 3\hat i + \hat j - \hat k, \) find a vector \( \overrightarrow d\) , which is perpendicular to both \( \overrightarrow a \: and \: \overrightarrow b \: and \: \overrightarrow d.\overrightarrow c = 21 \).

asked Jan 22, 2013 by thanvigandhi_1

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Find the vector equation of the line parallel to the line \( \large\frac{x-1}{5} = \frac{3-y}{2} = \frac{z+1}{4}\) and passing through (3, 0, -4). Also, find the distance between these two lines.

asked Jan 22, 2013 by thanvigandhi_1

0 answers

Prove that the image of the point (3, -2, 1) in the point \( 3x-y+4z=2\) lies in the plane \( x+y+z+4=0\).

asked Jan 22, 2013 by thanvigandhi_1

0 answers

In a school, 15 N.S.S. volunteer selected for cultural activities of which 3 are extras 3 volunteers are selected at random out of them. Let X denote the number of volunteer as extras. Find the probability distribution of X.

asked Jan 22, 2013 by thanvigandhi_1

0 answers

Use product $ \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix} $ to solve the system of equations : $ x-y+2z=1; 2y-3z=1; 3x-2y+4z = 2$

asked Jan 22, 2013 by thanvigandhi_1

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A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of hypotenuse is \( \bigg[ a^{\large\frac{2}{3}} + b^{\large\frac{2}{3}} \bigg]^{\large\frac{3}{2}} \).

asked Jan 22, 2013 by thanvigandhi_1

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Show that the right circular cone of least curved surface and given volume has an altitude equal to \( \sqrt 2\) times the radius of the base.

asked Jan 22, 2013 by thanvigandhi_1

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Find the area of the circle \( 4x^2+4y^2=9\) which is interior to the parabola \( x^2 = 4y\)

asked Jan 22, 2013 by thanvigandhi_1

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Find the particular solution of the differential equation $ (x^2+1)\large\frac{dy}{dx}$$-2xy=(x^4+2x^2+1)\cos \: x ; y=0, \: when\: x = 0.$

asked Jan 22, 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 shortest distance

asked Jan 22, 2013 by thanvigandhi_1

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A furniture firm manufactures chairs and tables, each requiring the use of three machines A, B and C. Production of each chair requires 2 hours on machine A, 1 hour on machine B and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximise profit? Formulate the problem as LPP and solve it graphically.

asked Jan 22, 2013 by thanvigandhi_1

1 answer

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both hearts. Find the probability of the lost card being a heart.

asked Jan 22, 2013 by thanvigandhi_1

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Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

asked Jan 22, 2013 by thanvigandhi_1

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Find the principal value of \( \sec^{-1}(2). \)

asked Jan 22, 2013 by thanvigandhi_1

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Evaluate : $ \sin \bigg[ \large\frac{\pi}{3} - $$\sin^{-1} \bigg( -\large\frac{1}{2} \bigg) \bigg] $

asked Jan 22, 2013 by thanvigandhi_1

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Evaluate : $ \begin{bmatrix} a+ib & c+id \\ -c+id & a-ib \end{bmatrix} $

asked Jan 22, 2013 by thanvigandhi_1

1 answer

If A is a square matrix of order 3 such that | adj A | = \( 8I_2\), find |A|.

asked Jan 22, 2013 by thanvigandhi_1

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If $ y = \sin^{-1}x+\cos^{-1}x, $ then find $\large \frac{dy}{dx} $.

asked Jan 22, 2013 by thanvigandhi_1

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Find the value of a and b, if $\begin{bmatrix} 2 & -2 \\ 3 & 7 \end{bmatrix} = \begin{bmatrix} 2 & a \\ 3 & 2a+b \end{bmatrix} $

asked Jan 22, 2013 by thanvigandhi_1

1 answer

Find the order and degree of the differential equation $\bigg[ 1+ \bigg(\large \frac{dy}{dx} \bigg)^2 \bigg]^{\large\frac{3}{2}} = \frac{d^2y}{dx^2} $

asked Jan 22, 2013 by thanvigandhi_1

1 answer

Find the value of \( \lambda \) so that the vectors \( \overrightarrow a = 2\hat i + \lambda\hat j + \lambda\hat k \: and \: \overrightarrow b = \hat i - 2\hat j + 3\hat k\) are perpendicular.

asked Jan 22, 2013 by thanvigandhi_1

0 answers

For any two vectors \( \overrightarrow a \: and \: \overrightarrow b\) , prove that \( | \overrightarrow a + \overrightarrow b | = |\overrightarrow a - \overrightarrow b|, \: if \: \overrightarrow a \bot \overrightarrow b\).

asked Jan 22, 2013 by thanvigandhi_1

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If the equation of the line AB is \( \large\frac{x-3}{1} = \frac{y+2}{-2} = \frac{z-5}{4} ,\) find the direction ratios of a line parallel to AB.

asked Jan 22, 2013 by thanvigandhi_1

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For each binary operation * defined below, determine whether * is (i) commutative (ii) associative \( a*b = \large\frac{ab}{2}\) on Q?

asked Jan 22, 2013 by thanvigandhi_1

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If $ tan^{-1} \bigg(\large \frac{x-1}{x-2} \bigg) $$+ \tan^{-1} \bigg(\large \frac{x+1}{x+2} \bigg) =\large \frac{\pi}{4}, $ find the value of x.

asked Jan 22, 2013 by thanvigandhi_1

1 answer

Prove that : \[ sin^{-1}\frac{3}{5}+sin^{-1}\frac{8}{17}=cos^{-1}\frac{36}{85} \]

asked Jan 21, 2013 by thanvigandhi_1

1 answer

Principal and staff of a school decided to award prizes to their students for three values Honesty (x) cleanliness (y), Punctuality (z) in two groups. Group A ( upto class VIII) and group B ( IX - XII). It is decided that Rs. 12000 for three values (x, y, z) for 3, 4, 5 students of Group A and Rs. 11040 for 5, 4, 3 student of group B respectively. If all the three prizes together amount to Rs. 2880, then (i) Represent above situation by a matrix equation and form linear equation using matrices. (ii) Is it possible to solve the system of linear equations so obtained using matrices? (iii) Which value you prefer to be rewarded most and why?

asked Jan 21, 2013 by thanvigandhi_1

0 answers

If \( y = x^{\large\sin x}+(\sin x)^{\large x}, \: find\: \large\frac{dy}{dx}.\)

asked Jan 21, 2013 by thanvigandhi_1

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show that the function \( f(x) = \left\{ \begin{array}{l l} \large \frac{3}{2}-x, & \quad {\large\frac{1}{2}< x < 1}\\ \frac{3}{2}, & \quad { x = 1} \\ \large\frac{3}{2}+x, & \quad {1 \leq x < 2} \end{array} \right. \) is continuous at x = 1.

asked Jan 21, 2013 by thanvigandhi_1

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Differentiate \( tan^{-1} \bigg[ \large\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}} \bigg] \) w.r.t. \( sin^{-1} \bigg( \large\frac{2x}{1+x^2} \bigg). \)

asked Jan 21, 2013 by thanvigandhi_1

0 answers

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

asked Jan 21, 2013 by thanvigandhi_1

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

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Find the intervals in which the following function is strictly increasing or decreasing \( f(x)=8+36x+3x^2-2x^3\)

asked Jan 21, 2013 by thanvigandhi_1

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Evaluate : $ \int_{\large\frac{\pi}{6}}^{\large\frac{\pi}{4}}\large \frac{1}{1+\sqrt{\tan x}}$$ dx.$

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Prove by vector method \( \cos(A+B) = \cos A \cos B - \sin A \sin B\)

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the points (0,-1,-1), (4,5,1) and (3,9,4).

asked Jan 21, 2013 by thanvigandhi_1

0 answers

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

asked Jan 21, 2013 by thanvigandhi_1

0 answers

(i) Jayant contested election for the posts A and B; selection for which is independent. The probability his winning for post A is \( \frac{1}{5} \) and for post B is \( \frac{1}{8} \). What is the probability that Jayant got success in at least one of the post. (ii) How you define ' democracy'?

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Given that \( A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 1 & 2 \end{bmatrix}\) and \( B = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 -1 & 5 \end{bmatrix}\). Find AB. Hence, using this to solve the following system of linear equations. \( x-y=3;\: 2x+3y+4z=17; \: y+2z=7\)

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Show that semi-vertical angle of the cone of maximum volume and of given slant height is \( \tan^{-1} \sqrt 2\).

asked Jan 21, 2013 by thanvigandhi_1

1 answer

Prove that the area of right angled triangle of given hypotenuse is maximum when the triangle is isoceles.

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Calculate the area of the region enclosed between the circles \( x^2+y^2=9 \: and \: (x-3)^2+y^2=9.\)

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Find the particular solution of the differential equation .$ (xdy-ydx)y. \sin \bigg(\large\frac{y}{x} \bigg) =$$ (ydx-xdy) x \cos \bigg(\large \frac{y}{x} \bigg)$ given that \( y= \pi, \: when \: x=3\)

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Find the distance of the point (-2, 3, -4) from the line \( \large \frac{x+2}{3} = \frac{2y+3}{4} = \frac{3z+4}{5} \) measured parallel to the plane \( 4x+12y-3z+1=0.\)

asked Jan 21, 2013 by thanvigandhi_1

0 answers

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