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Recent questions tagged modelpaper

Find the equation of the plane passing through the point (1,1,1) and containing the line \( \overrightarrow r = (3\hat i + \hat j + 5\hat k) + \lambda (3\hat i - \hat j - 5\hat k)\). Also, show that the plane contains the line \( \overrightarrow r = (-\hat i + 2\hat j + 5\hat k) + \lambda (\hat i - 2\hat j - 5\hat k)\).

asked Feb 3, 2013 by thanvigandhi_1

0 answers

A company sells two different products A and B. The products are produced in a common production process which has total capacity of 500 man hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The demand in the market shows that the maximum number of units of A that can sold is 70 and that of B is 125. Profit on each unit of A is Rs. 20 and on B is Rs.15. How many units of A and B should be produced to maximise the profit? Form an L.P.P and solve it graphically.

asked Feb 3, 2013 by thanvigandhi_1

1 answer

In an examination, 10 questions of true-false type are asked. A student toses a fair coin to determine his answer to each question. If the coin falls heads, he answers 'true' and if it falls tails, he answers 'false'. Show that the probability that he answers at most 7 questions correctly is \( \frac{121}{128} \).

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Two bags A and B contain 4 white and 3 black balls and 2 white and 2 black balls respectively. From bag A, two balls are drawn at random and then transferred to bag B, A ball is then drawn from bag B and is found to be a black ball. What is the probability that the transferred balls were 1 white and 1 black?

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Using principal value, write the value of $ \cos^{-1} \bigg( \large\frac{1}{2} \bigg)$$ + 2\sin^{-1} \bigg(\large \frac{1}{2} \bigg) $

asked Feb 3, 2013 by thanvigandhi_1

1 answer

If $ \sin\: \{ \sin^{-1}\large\frac{1}{5}$$+\cos^{-1}x \} = 1 $, then find the value of x.

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Find the value of x, if : $ \begin{bmatrix} 1 & x & 1 \end{bmatrix} \; \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5 \end{bmatrix} \; \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix} = 0 $

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Find the values of x, if $\begin{bmatrix} 2 & 4 \\ 5 & 1 \end{bmatrix} = \begin{bmatrix} 2x & 4 \\ 6 & x \end{bmatrix} $

asked Feb 3, 2013 by thanvigandhi_1

1 answer

Differentiate \( e^{\large \sin\: x} \) w.r.t. \( x \).

asked Feb 3, 2013 by thanvigandhi_1

0 answers

If \( A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \) and A + A' = I, then find the value of \( \theta \)

asked Feb 3, 2013 by thanvigandhi_1

0 answers

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

asked Feb 3, 2013 by thanvigandhi_1

0 answers

Find the magnitude of two vectors \( \overrightarrow a \: and \: \overrightarrow b\) having the same magnitude and such that angle between them is \( 30^{\circ} \) and their scalar product is 3.

asked Feb 3, 2013 by thanvigandhi_1

0 answers

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 Feb 3, 2013 by thanvigandhi_1

0 answers

Find the distance between the parallel planes \( x+y-z=-4\) and \( 2x+2y-2z+10=0\)

asked Feb 3, 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 Feb 2, 2013 by thanvigandhi_1

0 answers

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

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Prove that : $ 2\tan^{-1} \frac{1}{2} + \tan^{-1}\large \frac{1}{7} $$= \tan^{-1} \large\frac{31}{17} $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Simplify : $ \cos^{-1} \bigg(\large \frac{3}{5}$$ \cos \theta +\large \frac{4}{5}$$ sin \theta \bigg) $

asked Feb 2, 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) and 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 students 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.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Find the point (s) of discountinuity of the function 'f' given by $ f(x) = \left\{ \begin{array}{l l}|x|+3, & \quad { x \leq -3} \\ -2x, & \quad {-3 < x < 3} \\ 6x+2, & \quad { x > 3 } \end{array} \right. $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

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

asked Feb 2, 2013 by thanvigandhi_1

0 answers

If $x= \bigg( \cos\theta + \log \tan\large\frac{\theta}{2} \bigg), $$y=\sin\: \theta, \large\frac{d^2y}{dx^2}$$\: at \: \theta=\large\frac{\pi}{4} $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

If \( y=log \bigg( x+\sqrt{x^2+1} \bigg)\), then prove that $(x^2+4)\large\frac{d^2y}{dx^2}$$+x\large\frac{dy}{dx}=0$.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

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

asked Feb 2, 2013 by thanvigandhi_1

0 answers

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

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \large\frac{\sin x\cos x}{a^2\sin^2x+b^2+\cos^2x}$$dx $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate the following integral using limit of sums $\int_1^4 \: x^2-xdx $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^a \large\frac{\sqrt x}{\sqrt x+\sqrt{a-x} }$$dx $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Show that each of the given three vectors is a unit vector. \( \frac{1}{7} (2\hat i + 3\hat j + 6\hat k), \frac{1}{7} (3\hat i - 6\hat j + 2\hat k), \frac{1}{7} (6\hat i + 2\hat j - 36\hat k). \)Also, show that they are mutually perpendicular to each other.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

Find the equation of a line passing through the point (-1,-1,2) and perpendicular to the planes \( 3x+2y-3z=1 \: and \: 5x-4y+z=5\)

asked Feb 2, 2013 by thanvigandhi_1

0 answers

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

asked Feb 2, 2013 by thanvigandhi_1

1 answer

In a group of student, selection of four students for cultural activity from 3 boys and 3 girls has to be done. Find the probability that either 3 boys and 1 girl or 3 girls and one boy are selected.

asked Feb 2, 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 Feb 2, 2013 by thanvigandhi_1

0 answers

Show that semi vertical angle of right circular cone of given surface area and maximum volume is $ \sin^{-1} \bigg( \large\frac{1}{3} \bigg). $

asked Feb 2, 2013 by thanvigandhi_1

0 answers

A wire of length 36 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum.

asked Feb 2, 2013 by thanvigandhi_1

1 answer

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

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Solve the differential equation : $(x+y)dy + (x-y) dx=0$

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane passing through the line intersection of the plane \( x-2y+z=1\) and \( 2x+y+z=8\) and parallel to the line with direction ratios 1,2,1. Also, find the perpendicular distance of the point P(3,1,2) from this plane.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

A retired person has 70000 to invest and two types of bonds are available in the market for investment. First type of bond yields an annual income of 8% on the amount invested and the second type of bond yields 10% per annum. As per norms, he has to invest minimum of 10000 in the first type and not more than 30000 in the second type. How should he plan his investment, so as to get maximum return after one year of investment. Make it an L.P.P. and solve the above problem.

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Probability of solving a mathematical problem by A, B, and C respectively are \( \frac{1}{3}, \frac{1}{2} \: and \: \frac{1}{4}\). If they try to solve the problem indepently. Find the probability that : (i) The problem is solved. (ii) exactly one of them solve the problem (iii) exactly two of them solve the problem.

asked Feb 2, 2013 by thanvigandhi_1

0 answers

A company has two plants to manufacture bicycles. Plant I manufactures 70% of the bicycles and plant II manufactures 30%. At plant I 80% of the bicycles are rated as of standard quality and a plant II 90% of the bicycles are rated as of standard quality. A bicycle is choosen at random and is found to be of standard quality. What is the probability that it has come from plant II?

asked Feb 2, 2013 by thanvigandhi_1

1 answer

If A is square matrix of order 3 such that | adj.A |=81, find | A |.

asked Feb 2, 2013 by thanvigandhi_1

2 answers

If a matrix has 12 elements, what are the possible order it can have?

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Differentiate \( \cos^{-1}x^4\) w.r.t. \( x.\)

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Find the product of matrix \( A = \begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix} \) and its transpose.

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Find a, for which \( f(x) = x^2-2ax+6\) is strictly increasing when x > 0.

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_0^{1.5} [x] \: dx $

asked Feb 2, 2013 by thanvigandhi_1

1 answer

Find the differential equation that will represent the family of straight lines \( y=mx+c\), where m,c are parameters.

asked Feb 2, 2013 by thanvigandhi_1

1 answer

For what values of p, $ \overrightarrow a = (2\hat i + 6\hat j + 3\hat k)$ p is a unit vector.

asked Feb 2, 2013 by thanvigandhi_1

1 answer

A coin is tossed twice, where E : Tail appear on one coin and F : one coin shows head. Find P(E/F).

asked Feb 2, 2013 by thanvigandhi_1

0 answers

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