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Recent questions tagged modelpaper

if A is an invertible matrix of order 3 and |A|=5, then find |adj.A|.

asked Feb 17, 2013 by thanvigandhi_1

1 answer

If \( \overrightarrow a \times \overrightarrow b=\overrightarrow c \times \overrightarrow d\: and \: \overrightarrow a \times \overrightarrow c=\overrightarrow b \times \overrightarrow d\) show that \( \overrightarrow a-\overrightarrow d\) is parallel to \( \overrightarrow b-\overrightarrow c\) where \( \overrightarrow a \neq \overrightarrow d\: and \overrightarrow b \neq\overrightarrow c\)

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Prove that : $ \sin^{-1} \bigg(\large \frac{4}{5} \bigg)$$+\sin^{-1} \bigg(\large \frac{5}{13} \bigg) $$+ \sin^{-1} \bigg( \large\frac{16}{65} \bigg) = \large\frac{\pi}{2} $

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Solve for x : \( \tan^{-1}3x+\tan^{-1}2x=\large\frac{\pi}{4} \)

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Find the value of \( \lambda \) so that the line $\large \frac{1-x}{3}=\frac{7y-14}{2\lambda}=\frac{5z-10}{11} $$\: and \; \large\frac{7-7x}{3\lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are perpendicular to each other.

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $\large \frac{dy}{dx}$$+y=\cos\: x-\sin\: x$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Find the particular solution, satisfying the given condition, for the following differential equation : $\large \frac{dy}{dx}-\frac{y}{x}$$+cosec \bigg( \large\frac{y}{x} \bigg)$$=0; y=0\: when \: x = 1$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

By using properties of determinants, prove the following: \[ \begin{vmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{vmatrix}=(5x+4)(4-x)^2 \]

asked Feb 17, 2013 by thanvigandhi_1

1 answer

A die is thrown again and again untill three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Differentiate the following function w.r.t \(x\) $ x^{\large \sin x}+(\sin\: x)^{\large \cos x}$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Evaluate : \[ \frac{e^x}{\sqrt{5-4e^x-e^{2x}}}dx\]

asked Feb 15, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \large\frac{(x-4)e^x}{(x-2)^3}$$dx $

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( y=3\cos(\log\: x)+4\sin(\log\: x),\) then show that $ x^2.\large\frac{d^2y}{dx^2}$$+x\large\frac{dy}{dx}$$+y=0$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find $\large \frac{dy}{dx}$ if $( (x^2+y^2)^2=xy)$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find the equation of the tangent to the curve $ y= \sqrt{3x-2}$ which is parallel to the line $4x-2y+5=0$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find the intervals in which the function f given by \( f(x)=x^3+\frac{1}{x^3}, x \neq 0\) is (i) increasing (ii) decreasing.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is \( 8m^3\). If building of tank costs Rs. 70 per sq. metre for the base and Rs.45 per sq. metre for sides, what is the cost of least expensive tank?

asked Feb 15, 2013 by thanvigandhi_1

0 answers

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods \( F_1 \: and F_2\) are available . Food \( F_1 \) costs Rs. 4 perunit and \( F_2\) costs Rs. 6 per unit. One unit of food \( F_1 \) contains 3 units of vitamin A and 4 units of minerals. One unit of food \( F_2 \) contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

asked Feb 15, 2013 by thanvigandhi_1

0 answers

Three bags contain balls as shown in the table below. A bag is chosen at random and two balls are drawn from it. They happen to be white and red. What is the probability that they came from the III bag?

asked Feb 15, 2013 by thanvigandhi_1

0 answers

Using matrices, solve the following system of equations : \[ 2x-3y+5z=11\] \[ 3x+2y-4z=-5\]\[x+y-2z=-3\]

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Evaluate :$ \int_0^{\pi}\large \frac{e^{\large \cos x}}{e^{\large \cos x}+e^{\large-\cos x}}$$dx$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int_0^{\large\frac{\pi}{2}} (2\log\sin\: x-\log\sin 2x)dx$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Using the method of integration, find the area of the region bounded by the lines\( 2x+y=4, 3x-2y=6 \:and \: x-3y+5=0\)

asked Feb 15, 2013 by thanvigandhi_1

0 answers

Find the equation of the plane passing through the point (-1,3,2) and perpendicular to each of the planes \( x+2y+3z=5\: and \: 3x+3y+z=0\)

asked Feb 15, 2013 by thanvigandhi_1

0 answers

Evaluate : \[ tan^{-1} \bigg( sin \bigg(\frac{-\pi}{2} \bigg) \bigg).\]

asked Feb 15, 2013 by thanvigandhi_1

0 answers

Solve for x : \[ tan^{-1}\frac{1-x}{1+x}=\frac{1}{2}tan^{-1}x:x > 0.\]

asked Feb 15, 2013 by thanvigandhi_1

0 answers

If \( \begin{bmatrix} x+3y & y \\ 7-x & 4 \end{bmatrix}=\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix} \), find the values of x and y.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( \begin{bmatrix} 2x & 5 \\ 8 & x \end{bmatrix}=\begin{bmatrix} 6 & 5 \\ 8 & 3 \end{bmatrix} \), then find x.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Differentiate with respect to x. $( \sec(\tan(\sqrt x)))$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find order and degree ( if defined ) of following differential equation : $ \bigg( \large\frac{d^2y}{dx^2} \bigg)^2$$+\cos \bigg( \large\frac{dy}{dx} \bigg)=0$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( |\overrightarrow a|=\sqrt 3, |\overrightarrow b|=2\) and the angle between \( \overrightarrow a \: and \: \overrightarrow b\) is \( 60^{\circ}\), find \( \overrightarrow a.\overrightarrow b\)

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find a vector in the direction of vector \( \overrightarrow a = \hat i - 2\hat j\), whose magnitude is 7.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

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 Feb 15, 2013 by thanvigandhi_1

1 answer

if \( \begin{bmatrix} x+2 & 3 \\ x+5 & 4 \end{bmatrix}=3,\) find the value of x.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Let T be the set of all triangles in a plane with R as relation in T given by \( R = {(T_1,T_2): T_1 \cong T_2}\). Show that R is an equivalence relation.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Solve : \[ tan^{-1}(x+1)+tan^{-1}(x-1)=tan^{-1}\frac{8}{31}\]

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Prove the following : \[ tan^{-1} \bigg( \frac{x}{y} \bigg)- tan^{-1} \bigg( \frac{x-y}{x+y} \bigg)= \frac{\pi}{4},y \neq 0. \]

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Using the properties of determinants, prove the following : \[ \begin{vmatrix} b+c & c+a & a+b \\ c+a & a+b & c+a \\ a+b & b+c & c+a \end{vmatrix} = 2(a+b+c)(ab+bc+ca-a^2-b^2-c^2).\]

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Discuss the continuity of the following function at x = 0: $ f(x) = \left\{ \begin{array}{l l}\large\frac{x^4+2x^3+x^2}{\tan^{-1}x}, & \quad if { x \; \neq 2 } \\ 0, & \quad if { x\; = 0 } \end{array} \right. $

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Verify Lagrange's mean value theorem for the following function: $ f(x)=x^2-2x+3,\: for \: [4,6]$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If $ f(x)=\sqrt{\large\frac{\sec x-1}{\sec x+1}}$$\: find\: f'(x). $ Also find $( f' \bigg( \large\frac{\pi}{2}\bigg).)$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( x\sqrt{1+y}+y\sqrt{1+x}=0\:, find \: \large\frac{dy}{dx}\)

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Show that $ \int_0^{\large\frac{\pi}{2}} \sqrt{\tan\: x}+\sqrt{\cot x}=\sqrt {2\pi} $

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( y=e^{\large \tan\: x},\) prove that $ \cos^2x\large\frac{d^2y}{dx}$$-(1+\sin\: 2x)\large\frac{dy}{dx}$$=0$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( y=(x)^x+(\cos x)^{2x} \: find \: \large\frac{dy}{dx}\)

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Evaluate : $\begin{align} \int \limits_{0}^{2}(x^2+3)dx \end{align}$ as limit of sum.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{8\: dx}{(x+2)(x^2+4)}$

asked Feb 15, 2013 by thanvigandhi_1

1 answer

If \( \hat i+ \hat j+\hat k,\: 2\hat i+5\hat j,\: 3\hat i+2\hat j=3\hat k\: and \: \hat i-6\hat j-\hat k\) are the position vectorsof the points A,B,C and D, find the angle between \( \overrightarrow{AB}\: and \: \overrightarrow{CD}\). Deduce that \( \overrightarrow{AB}\: and \: \overrightarrow{CD}\) are collinear.

asked Feb 15, 2013 by thanvigandhi_1

1 answer

Find the equation of the line passing through the point P(4,6,2) and the point of intersection of the line \( \frac{x-1}{3}=\frac{y}{2}=\frac{z+1}{7} \) and the plane \( x+y-z=8\).

asked Feb 11, 2013 by thanvigandhi_1

0 answers

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