Recent questions tagged modelpaper-2014-chennai

Evaluate : \[\] $ \int \large\frac{1}{\sin^4x+\sin^2x \cos^2x + \cos^4x}$$dx$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is $60^{\circ}$.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award Rs.x each, Rs.y each and Rs.z each for the three respective values to 3,2 and 1 students respectively with a total award money of Rs. 1,600. School B wants to spend Rs. 2,300 to award its 4, 1 and 3 students on the respective values ( by giving the same award money to the three values as before ). If the total amount of award for one prize on each value is Rs. 900 using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Two numbers are selected at random ( without replacement ) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

There are three coins. One is a two-headed coin ( having head on both faces ), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

asked Mar 25, 2014 by thanvigandhi_1

2 answers

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours of fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs. 80 on each piece of type A and Rs. 120 on each piece of type B. How many pieces of type A and B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 2) ( 1, 5) and ( 3, 4)

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Find the distance of the point (2, 12, 5) from the point of intersection of the line $ \overrightarrow r = 2\hat i - 4\hat j + 2 \hat k + \lambda ( 3\hat i + 4 \hat j + 2 \hat k ) $ and the plane $ \overrightarrow r. ( \hat i - 2 \hat j + \hat k ) =0.$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Find the equation of the plane through the line of intersection of the planes $ x+y+z=1$ and $2x+3y+4z=5$ which is perpendicular to the plane $x-y+z=0$. Also find the distance of the plane obtained above, from the origin.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Find the value of p, so that the lines $ l_1 : \large\frac{1-x}{3} $$= \large\frac{7y-14}{p} $$= \large\frac{z-3}{2}$ and $l_2 : \large\frac{7-7x}{3p} $$= \large\frac{y-5}{1} $$ = \large\frac{6-z}{5}$ are perpendicular to each other. Also find the equations of a line passing through a point ( 3, 2, -4 ) and parallel to line $l_1$.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Find the particular solution of the differential equation $ log \bigg( \large\frac{dy}{dx} \bigg) =3x+4y,$ given that $ y=0$ when $x=0$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

If $ x = \cos\: t ( 3-2 \: \cos^2 t)\: and \: y = \sin \: t ( 3-2 \sin^2\: t)$ find the value of $ \large\frac{dy}{dx} $ at $ t = \large\frac{\pi}{4}$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Using properties of determinants, prove that, \[\] $\begin{vmatrix} 1+a & 1 & 1 \\[0.3em] 1 & 1+b & 1 \\[0.3em] 1 & 1 & 1+c \end{vmatrix} = abc +bc+ca+ab$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ y = Pe^{ax}+Qe^{bx}$, show that \[\] $\large\frac{d^2y}{dx^2}$$-(a+b) \large\frac{dy}{dx}$$+aby=0$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least three successes

asked Mar 25, 2014 by thanvigandhi_1

1 answer

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

asked Mar 25, 2014 by thanvigandhi_1

0 answers

prove that : \[\] $ \tan^{-1} \bigg[ \large\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}} \bigg]$$ = \large\frac{\pi}{4}$$- \large\frac{1}{2}$$ \cos^{-1}x , \large\frac{-1}{\sqrt 2}$$ \leq x \leq 1$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If the function $ f : R \rightarrow R $ be given by $ f(x) = x^2+2$ and $ g : R \rightarrow R $ be given by $ g(x) = \large\frac{x}{x-1} , x \neq 1$, find fog and gof and hence find fog (2) and gof (-3)

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Find the equations of the tangent and normal to the curve $ \large\frac{x^2}{a^2}$$ - \large\frac{y^2}{b^2} = 1$ at the point $ ( \sqrt 2a, b)$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Find the value(s) of $x$ for which $y = [x(x-2)]^2$ is an increasing function.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Evaluate : \[\] $\int \large\frac{x+2}{\sqrt{x^2+5x+6}} dx$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Evaluate : \[\] $ \int_0^{\pi} \large\frac{4x \sin x}{1+ \cos^2x} dx$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

The scalar product of the vector $ \overrightarrow a = \hat i + \hat j + \hat k$ with a unit vector along the sums of vectors $ \overrightarrow b = 2 \hat i + 4\hat j - 5 \hat k$ and $ \overrightarrow c = \lambda \hat i + 2 \hat j + 3 \hat k\;$ is equal to one. Find the value of $ \lambda $ and hence find the unit vector along $ \overrightarrow b + \overrightarrow c$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Show that the four points A, B, C and D with position vectors $ 4\hat i + 5 \hat j + \hat k, \: \: -\hat j - \hat k, \: \: 3\hat i + 9 \hat j + 4\hat k,$ and $ 4(-\hat i + \hat j + \hat k) $ respectively are coplanar.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Solve the differential equation $(1+x^2) \large\frac{dy}{dx}$$ +y = e^{ tan^{-1}}x$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \overrightarrow a$ and $ \overrightarrow b$ are perpendicular vectors $ | \overrightarrow a + \overrightarrow b | = 13$ and $ | \overrightarrow a |=5$, find the value of $ | \overrightarrow b|.$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \int_0^a \large\frac{1}{4+x^2}$$ dx = \large\frac{\pi}{8}$, find the value of $a$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If the cartesian equation of the lines are $ \large\frac{3-x}{5} $$= \large\frac{y+4}{7}$$=\large\frac{2z-6}{4}$, write the vector equation of the line.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ R = \{ (x, y) : x + 2y =8\}$ is a relation on N. write the range of R

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Find the value of 'P' for which the vectors $ 3 \hat i + 2 \hat j + 2 \hat k$ and $ \hat i - 2P \hat j + 3 \hat k $ are parallel.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

If $ f(x) =\int_0^x t\: \sin\: t\: dt$, then write the value of $f'(x)$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \begin{bmatrix} 3x & 7 \\[0.3em] -2 & 4 \end{bmatrix} = \begin{bmatrix} 8 & 7 \\[0.3em] 6 & 4 \end{bmatrix}$, find the value of $x$.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $\tan^{-1}x+ \tan^{-1}y = \large\frac{\pi}{4},$$ xy < 1$, then the write the value of $x+y+xy$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \begin{bmatrix} x-y & z \\[0.3em] 2x-y & w \end{bmatrix} =\begin{bmatrix} -1 & 4 \\[0.3em] 0 & 5 \end{bmatrix} $, find the value of $X$ and $Y$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $A$ is a square matrix such that $A^2=A$, then write the value of $7A -(I+A)^3$ where $I$ is an identity matrix.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

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