Recent questions tagged modelpaper

If A and B are independent events such that $ P(A) = 0.2, P(A \cup B)=0.5, \: find \: P(B).$

asked Feb 2, 2013 by thanvigandhi_1

1 answer

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

asked Feb 2, 2013 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}$, find $x$ and $y$ such that $ A^2-xA+yI=0. $

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove that : $ \begin{vmatrix} a+b+nc & (n-1)a & (n-1)b \\ (n-1)c & b+c+na & (n-1)b \\ (n-1)c & (n-1)a & c+a+nb \end{vmatrix} = n(a+b+c)^3$

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Assuming that petrol burnt per hour in driving a motor boat varies as the cube of its velocity. Show that the most economical speed when going against a current of k km/hour is $ \large\frac{3k}{2} $ km/hour.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

If $ x=a(\cos t+t\sin t), \: y=a(\sin t - t\cos t), \: find \; \large\frac{d^2y}{dx^2}$$ \: at \: t=\large\frac{\pi}{4}. $

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Show that the function $ f(x) = \left\{ \begin{array}{l l}-x^2, & \quad { x \leq 0} \\ x^2, & \quad { x \geq 0} \end{array} \right. $ is continuous at x = 0, also show that f is differentiable at x = 0.

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $ \int \large\frac{1}{1+x^4}$$ dx$

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $ \int\large \frac{dx}{2e^{2x}+3e^x+1} $

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $ \int_0^{\Large\frac{1}{\sqrt 2}} \large\frac{\sin^{-1}x}{(1-x^2)^{\frac{3}{2}}} $$dx.$

asked Jan 31, 2013 by thanvigandhi_1

0 answers

If $ \hat i+\hat j + \hat k, \: 2\hat i + 5\hat j,\: 3\hat i + 2\hat j - 3\hat k$ and $ \hat i - 6\hat j - \hat k$ respectively are the position vectors of the point A, B, C and D, then find the angle between AB and CD are parallel.

asked Jan 31, 2013 by thanvigandhi_1

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Using vectors, prove cosine formula $ \cos \: A = \large\frac{b^2+c^2-a^2}{2bc} $

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $ \int_0^3 f(x) \: dx $

asked Jan 31, 2013 by thanvigandhi_1

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Solve the differential equation : $ (x^2+xy)dy=(x^2+y^2)dx$

asked Jan 31, 2013 by thanvigandhi_1

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

asked Jan 31, 2013 by thanvigandhi_1

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In a meeting, 70% of the members favour a certain proposal, 30% being opposite. A member is selected at random and we take X = 0, if he opposed, and x = 1, if he is in favour. Find E(X) and variance (X).

asked Jan 31, 2013 by thanvigandhi_1

0 answers

In a family the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other.

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Let A = N x N and * be a binary operation on A defined by (a,b)(c,d)=(ac,bd), show that : (i) (A) is commutative. (ii) (A*) is associative (iii) Find the identity element, if any, in A.

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Show that volume of the greatest cylinder which can inscribed in a cone of height h and semi-vertical angle $ 30^{\circ} \: is \: \large\frac{4}{81} $$\: \pi h^3.$

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Find the area of greatest rectangle which can be inscribed in a circle of radius 10 cm.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Use definite integration, find the area of triangle whose vertices are (-1,6), (1,2) and (5,4).

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Find the area of the smaller region bounded by the ellipse $ \large\frac{x^2}{9}+\frac{y^2}{4}$$=1$ and the straight line $\large \frac{x}{3}+\frac{y}{2}=1$.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 5% phosphoric acid, and type B which contains 6% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, it was found that atleast 14 kg of nitrogen and 14 kg of phosphoric acid is required for producing a good crop. The fertilizer of type A costs Rs.5 per kg and the type B costs Rs.3 per kg. How many kg of each type of the fertilizer should be used to meet the requirement at the minimum possible cost? Using L.P.P. solve the above problem graphically.

asked Jan 31, 2013 by thanvigandhi_1

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Solve by matrix method : $ x+y+z=20, \: 2x+y-z=23, \: 3x+y+z=46 $

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Show that line whose vector equation is $ \overrightarrow r = 2\hat i - 2\hat j + 3\hat k + \lambda (\hat i - \hat j + 4\hat k)$ is parallel to the plane $ \overrightarrow r .(\hat i + 5\hat j + \hat k)=5$ and find the distance between them.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

An insurance company insured 2000 scooters and 3000 motor cycles. The probability of an accident involving a scooter is 0.01 and that of a motor cycle is 0.02. An insured vehicle met with an accident. Find the probability that the accidental vehicle was a motor cycle.

asked Jan 31, 2013 by thanvigandhi_1

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Show that the function $ f : R \rightarrow R $ defined by $ f(x)=3x+7$ is one-one.

asked Jan 31, 2013 by thanvigandhi_1

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Prove that $ \tan^{-1}\large\frac{1}{3}$$+\tan^{-1}\large\frac{1}{2}=\frac{\pi}{4} $

asked Jan 31, 2013 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} cos\: \alpha & sin\: \alpha \\ -sin\: \alpha & cos\: \alpha \end{bmatrix} $, then show that AA' = I.

asked Jan 31, 2013 by thanvigandhi_1

1 answer

If $ A = [a_{ij}]_{m*n}$ is a skew-symmetric, what is the value of $ a_{ij}\: \forall \: i$?

asked Jan 31, 2013 by thanvigandhi_1

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Write a value of $ \int \large\frac{1-\tan x}{x+\log\cos\: x}$$ dx $

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $ \int e^{2-5x}\: dx.$

asked Jan 31, 2013 by thanvigandhi_1

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Without expanding, show that : $ \begin{vmatrix} 42 & 1 & 6 \\ 28 & 7 & 4 \\ 14 & 3 & 2 \end{vmatrix} = 0 $

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Show that the vectors $ \hat i + \hat j + \hat k $ is equally inclined to the axes.

asked Jan 31, 2013 by thanvigandhi_1

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If vectors $ 2\hat i - 5\hat j + b\hat k \: and \: 2\hat i + a\hat j+4\hat k $ are parallel, find the values of a and b.

asked Jan 31, 2013 by thanvigandhi_1

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A line makes an angles $ 30^{\circ}, 90^{\circ}, 60^{\circ}$ with co-ordinate axes. Find its direction cosines.

asked Jan 31, 2013 by thanvigandhi_1

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Given relation R in the set Z of all integers is given by $ R = {(a,b) \in Z * Z\: a-b \: is\: an\: integer }$. Find whether R is an equivalence relation or not.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Let * be binary operation of N x N defined by (a,b)(c,d)=(a+c, b+d). Show that is commutative as well as associative. Find the identity element for the operation * on N x N, if any.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Solve for x : $ 2\: tan^{-1}(\cos x)=tan^{-1}(2 cosec x) $

asked Jan 31, 2013 by thanvigandhi_1

1 answer

There are two families P and Q. There are 3 men, 3 women and 12 children in family P and 2 men, 2 women and 4 children in family Q. The recommended daily allowance for calories is : Man : 2400, woman : 2000, child: 1400 and man : 60 g for proteins is : woman: 40 g and child: 35 g. (i) Represent the above information by matrices. Using matrix multiplication, calculate the total requirement of calories and proteins of each of the two families.

asked Jan 31, 2013 by thanvigandhi_1

1 answer

If the function $ f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if\: { x > 1} \\ 11, & \quad if \: { x = 1} \\ 5ax-2b, & \quad if \: { x < 1} \end{array} \right. $ is continuous at x = 1, find the values of a and b.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Find $ \large\frac{dy}{dx}, \: if \: x^y+y^x=a^b.$

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Show that equation of the tangent to the ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$ at $ (x_0,y_0) $ is $\large\frac{xx_0}{a^2}+\frac{yy_0}{b^2}$$=1 $

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{5x+3}{\sqrt{x^2+4x+16}}$$dx.$

asked Jan 31, 2013 by thanvigandhi_1

0 answers

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

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Prove that an angle in a semi-circle is a right angle (using vector only).

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Find p, when the scalar projection of $ \overrightarrow a = p\hat i + \hat j + 4\hat k \: on \: \overrightarrow b = 2\hat i + 6\hat j + 3\hat k$ is 4 units.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Show that the plane through (1, 1, 1), (1, -1, 1) and (-7, 3, 5) is perpendicular to XZ-plane.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

A card from the pack of 52 cards is lost from the remaining cards of the pack, two cards are drawn and are found to be both diamonds. (i) Find the probability of lost card being a diamond.(ii)While playing in the ground you found a purse containing rupees and some documents,what will you do?

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Find the equation of a curve passing through the point (0,2) given that the sum of the coordinate of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

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