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Recent questions tagged 2012

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

0 answers

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

asked Jan 22, 2013 by thanvigandhi_1

0 answers

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

0 answers

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

0 answers

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

0 answers

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

0 answers

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

0 answers

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

0 answers

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

0 answers

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

A fruit grower can use two types of fertilzer in his garden, brand P and brand Q. The amounts ( in kg ) of nitrogen , phosphoric, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs atleast 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should he use? What is the minimum amount of nitrogen added in the garden? What steps do you advice to a farmer to avoid overuse of fertilizers.

asked Jan 21, 2013 by thanvigandhi_1

0 answers

In a class having 70% boys, 20% of the boys, and 10% of the girls are players. A student is selected at random from the class and found to be a player. Find the probability that the selected is a girl. Explain the contribution of sports in inculcating human values.

asked Jan 21, 2013 by thanvigandhi_1

0 answers

A student is given a test with 8 questions of true-false type. If he gets 6 or more questions correct, he is declared pass, given that he guesses the answer to each question, compute the probability that student will pass in the exam. Evaluation is necessary in student learning, Explain.

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Give an example of relation which is reflexive but neither symmetric nor transitive.

asked Jan 21, 2013 by thanvigandhi_1

1 answer

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

asked Jan 17, 2013 by thanvigandhi_1

1 answer

If \( \begin{bmatrix} x & 2 \\ 18 & x \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 18 & 6 \end{bmatrix} \), then find x?

asked Jan 17, 2013 by thanvigandhi_1

1 answer

For what value of k, the matrix \( \begin{bmatrix} k & 2 \\ 3 & 4 \end{bmatrix} \) has no inverse?

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Using properties of determinant, prove that \( \begin{bmatrix} 1 & x & y+z \\ 1 & y & z+x \\ 1 & z & x+y \end{bmatrix} = 0 \)

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int_0^1\large \frac{dx}{\sqrt{1-x^2}} $

asked Jan 17, 2013 by thanvigandhi_1

0 answers

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

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Find a vector in the direction of vector \( 5\hat i - \hat j + 2\hat k \) which has magnitude 8 units.

asked Jan 17, 2013 by thanvigandhi_1

0 answers

If |\( \overrightarrow a \) | =1 , | \( \overrightarrow b \) | = 2 and \( \overrightarrow a . \overrightarrow b\) = 1, find the angle between \( \overrightarrow a \: and \overrightarrow b.\)

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Find the direction cosine of the line joining the points (-2, 4, -5) and (1,2,3).

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Show that the relation R : N \( \rightarrow \) N defined by (a,b)R(c,d) \( \iff \) a+d = b+c for all (a,b), (c,d) \( \in \) N x N is an equivalence relation.

asked Jan 17, 2013 by thanvigandhi_1

0 answers

If \( cos^{-1}x+cos^{-1}y+cos^{-1}z=\pi\) prove that \( x^2+y^2+z^2+2xyz=1\)

asked Jan 17, 2013 by thanvigandhi_1

1 answer

Solve: $ \tan^{-1}(x+1)+\tan^{-1}(x-1)=\tan^{-1}\large\frac{8}{31}$$;x > 0.$

asked Jan 17, 2013 by thanvigandhi_1

0 answers

(i) The cost of 5 kg of sugar, 5kg of wheat and 2kg of rice is Rs. 340. The cost of 4 kg of rice, 2kg of sugar and 5kg of wheat is Rs. 270. Rs. 264 is the cost of 3kg of wheat, 2kg of rice and 4 kg of sugar. Represent the above problem by a system of linear equations and find the cost of each item per kg by matix method. (ii) How increase in the prices of essential commodities affect economic weaker section of society?

asked Jan 17, 2013 by thanvigandhi_1

0 answers

If the function \( f(x) = \left\{ \begin{array}{l l} x^2+3x+a, & \quad { x \leq 1}\\ bx+2, & \quad { x > 1} \end{array}. \right. \) is differentiable for all x. Find a and b.

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Differentiate : $\large \frac{x}{a} $$\sqrt {x^2-a^2}+\large\frac{a^2}{2}$$\sin^{-1}\large\frac{x}{a}$

asked Jan 17, 2013 by thanvigandhi_1

0 answers

If \( y=x^{x^{x....^{\infty}}} \), then prove that \( \large\frac{dy}{dx}=\frac{y^2}{x(1-ylog\: x )}\)

asked Jan 17, 2013 by thanvigandhi_1

0 answers

Show that \( y = \log (1+x)-\large\frac{2x}{2+x}\) is an increasing function of x for all value of x > -1.

asked Jan 17, 2013 by thanvigandhi_1

0 answers

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