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Recent questions tagged modelpaper

Simplify\( 2\tan^{-1} \bigg(\large \frac{1}{5} \bigg) + \sec^{-1} \bigg( \large\frac{5\sqrt 2}{7} \bigg) + 2\: tan^{-1} \bigg( \frac{1}{8} \bigg) \)

asked Dec 25, 2012 by thanvigandhi_1

1 answer

Two farmers Mukul and Vipul cultivates only three crops potato, tomato and wheat. The annual sale of 2010, 2011 are given below : \[ Sale 2010 = \begin{bmatrix} Potato & \quad Tomato & \quad Wheat \\ 20000 & \quad 15000 & \quad 80000 & \quad Mukul \\ 30000 & \quad 10000 & \quad 90000 & \quad Vipul \end{bmatrix} \] \[ Sale 2011 = \begin{bmatrix} Potato & \quad Tomato & \quad Wheat \\ 30000 & \quad 15000 & \quad 60000 & \quad Mukul \\ 20000 & \quad 20000 & \quad 50000 & \quad Vipul \end{bmatrix} \]

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the value of 'a' for which the function f defined as $ f(x) = \left\{ \begin{array} {l l} a\sin \large\frac{\pi}{{2}(n+1)}, & \quad {x \: \leq \: 0} \\ \large\frac{tanx - sinx}{x^3}, & \quad {x \: > 0} \end{array} \right .$ is continuous at x=0. Differentiate \(( x \: cos \: x )^x + ( x\sin\: x )^{\frac{1}{x}}\) w.r.t. \( x \).

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Verify Rolle's Theorem for the following in the given interval : \( f(x) = (x^2 - 4x + 3 ) e^{2x} \)in [ 1, 3 ]

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the intervals in which the function f given by $ f(x) = \large\frac{4sin x - 2x - x\cos\: x}{2+ cos\: x} $ is $\text { (i) increasing (ii) decreasing } $

asked Dec 25, 2012 by thanvigandhi_1

0 answers

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

asked Dec 25, 2012 by thanvigandhi_1

0 answers

Find the particular solution of the differential equation $\large \frac{dy}{dx} - \frac{y}{x} $$+ cosec \bigg( \large\frac{y}{x} \bigg) = 0 $

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point ( x, y ) is equal to the sum of the coordinates of the point

asked Dec 24, 2012 by thanvigandhi_1

1 answer

If a,b,c are the lengths of the sides BC, CA and AB of a triangle ABC respectively, prove that \( \large\frac{a}{sinA} = \frac{b}{sin B} = \frac{c}{sinC} \)

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Show that points ( -6, 3, 2 ) ( 3, -2, 4 ) (5, 7, 3 ) and ( -13, 17, -1 ) are coplaner.

asked Dec 24, 2012 by thanvigandhi_1

0 answers

In a hostel 50% of the student read Hindi magazine, 40% read English magazine and 20% read both Hindi and English magazine. A student is selected at random. (a) Find the probability that student read English magazine only. (b) Find the probability that student reads neither English nor Hindi magazines. (c) Explain the role of newspapers / magazines to aware a student.

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Using matrices, solve the following system of linear equations : $ \begin{array} xx + 2y - 3z = -4 \\ 2x + 3y + 2z = 2 \\ 3x - 3y - 4z = 11 \end{array} $

asked Dec 24, 2012 by thanvigandhi_1

0 answers

A catering agency has two kitchens to prepare food at two places A and B. From these place " Mid-day meal' is to be supplied to three different schools situated at P, Q, R. The monthly requirements of the schools are respectively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchen A and B are 60 and 70 packets per month respectively. The transportation cost per packet for the kitchen to school is given below. How many packets from each kitchen should be transported to school so that the cost of transportation is minimum? Also find the minimum cost.

asked Dec 24, 2012 by thanvigandhi_1

1 answer

The probability that a person will get an electric contract is \(\large \frac{2}{5}\) and the probability that he will not get plumbing contract is \( \frac{4}{7} \) . If the probability of getting atleast one contract is \( \large\frac{2}{3} \) . What is the probability that he will get both?

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Find the area of the smaller region bounded by ellipse $ \large\frac{x^2}{a^2} + \frac{y^2}{b^2} $$= 1$ and the line $\large\frac{x}{a} + \frac{y}{b}$$ = 1 $

asked Dec 24, 2012 by thanvigandhi_1

0 answers

A point on the hypotenuse of a triangle is at distance 'a' and 'b' fro the sides of the triangle. Show that the minimum length of the hypotenuse is $ \bigg( a^{\large\frac{2}{3}} + b^{\large\frac{2}{3}} \bigg)^{\frac{2}{3}}. $

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Evaluate : $ \int \:\large \frac{2\sin2\theta -\cos \theta}{6-\cos^\theta - 4\sin\theta}$$d\theta$

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Find the foot of the perpendicular from ( 0, 2, 3 ) on the line \(\large \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}\). Also find the length of perpendicular.

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Let * be a binary operation on set Q of rational numbers defined as ab = \( \large\frac{3ab}{7} \). Write the identity for , if any.

asked Dec 24, 2012 by thanvigandhi_1

1 answer

Let N be the set of natural numbers and R be the relation in N defined as \(R = {(a, b) : a = b – 2, b > 6}\). Then \[ \begin{array} ((A)\: (2, 4) ∈ R \quad & (B)\: (3, 8) ∈ R \\ (C) \: (6, 8) ∈ R \quad & (D)\: (8, 7) ∈ R. \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

1 answer

If \( sin^{–1} x = y\) , then \[ \begin{array} ((A)\: 0 ≤ y ≤ π \quad & (B) \: \frac{-\pi}{2} ≤ y ≤ \frac{\pi}{2} \\ (C)\: 0 < y < π \quad & (D) \: \frac{-\pi}{2} < y < \frac{\pi}{2} \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

\( sin \bigg (\frac{\pi}{3} - sin^{-1} (\frac{1}{2}) \bigg) \) is equal to \[ \begin{array} ((A) \: \frac{1}{2} \quad & (B)\: \frac{1}{3} \\ (C)\: \frac{1}{4} \quad & (D) \: 1 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

1 answer

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is \[ \begin{array} ((A)\: 27 \quad & (B)\: 18 \\ (C)\: 81 \quad & (D) \: 512 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Let A be a square matrix of order 3 × 3, then \( | kA | \) is equal to \[ \begin{array} ((A) \: k | A | \quad & (B)\: k^2 | A | \\ (C) \: k^3 | A | \quad & (D) \: 3k | A | \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

If \[ f(x) = \left\{ \begin{array}{l l} 1+x & \quad {when \: x\: \leq 2}\\ 5-x & \quad { when \: x\: \: \leq 3} \end{array} \right.\] , then \[ \begin{array} ((A)\: f\: is\: continuous \: at\: x = 2 \quad & (B)\: f\: is\: discontinuous \: at \: x = 2 \\(C) \: f\: is\: continuous\: at\: x = 3 \quad & (D)\: f\: is\: continuous\: at\: x = 3 \: and \: discontinuous \: at \: x=2 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

\( \frac{d}{dx} ( log\: tan\: x)\) is equal to \[ \begin{array} ((A)\: 2\: sec\: 2x \quad & (B)\: 2\: cosec\: 2 x \\ (C)\: sec \: 2 x \quad & (D)\: cosec \: 2x \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The rate of change of the area of a circle with respect to its radius r when r = 6 is \[ \begin{array} ((A) \: 10 π \quad & (B) \: 12 π \\ (C) \: 8 π \quad & (D)\: 11 π \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The interval in which \(y = x^2 e^{–x}\) is increasing is \[ \begin{array} ((A)\: (– ∞, ∞) \quad & (B)\: (–2, 0) \\ (C) \:(2, ∞) \quad & (D)\: (0, 2) \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

\( \int\: xsec^2xdx \) is equal to \[ \begin{array} ((A)\: tanx^2+C \quad & (B)\: tanx^2+C \\ (C)\: x\: tanx - log\: sin\: x \quad & (D)\: x\: tanx + log\: cos\: x+C \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

\( \int \: e^x \bigg(\frac{1}{x} - \frac{1}{x^2} \bigg) dx\) is equal to \[ \begin{array} ((A)\: \frac{-e^x}{x^2}+C \quad & (B)\: \frac{e^x}{x^2}+C \\ (C) \: \frac{-e^x}{x^2} \quad & (D) \: \frac{e^x}{x} \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

Area bounded by the curve \( y = sinx\: between\: x = 0\: and\: x = 2π\) is \[ \begin{array} (A) 2sq. units \quad & (B) 4 sq. units\\ (C) 8sq. units \quad & (D) 16sq. units \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \)is \[ \begin{array} (A)\: e^x+e^{-y}=C \quad & (B)\: e^x+e^y=C \\ (C)\: e^{-x}+e^y=C \quad & (D)\: e^{-x}+e^{-y}=C \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The integrating factor of the differential equation \( x \frac{dy}{dx} -y = 2x^2 \)is \[ \begin{array} (A)\: e^{-x} \quad & (B)\: e^{-y} \\ (C)\: \frac{1}{x} \quad & (D)\: x \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

The vector \( 2\hat i + \propto\hat j + \hat k \) is perpendicular to the vector \( 2\hat i - \hat j - \hat k \) if \[ \begin{array} (A) \: \propto = 5 \quad & (B) \: \propto = -5 \\ (C) \: \propto = -3 \quad & (D) \: \propto = 3 \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

If \( \overrightarrow u = 2\hat i + 2\hat j - \hat k \) and \( \overrightarrow v = 6\hat i - 3\hat j + 2\hat k \), then a unit vector perpendiculat to both \( \overrightarrow u \) and \( \overrightarrow v \) is \[ \begin{array} (A) \: \hat i - 10\hat j - 18\hat k \quad & (B) \: \frac{1}{\sqrt {17}} \bigg( \frac{1}{5}\hat i - 2\hat j -\frac{18}{5}\hat k \bigg) \\ (C)\: \frac{1}{\sqrt {473}} \bigg( 7\hat i - 10\hat j -18\hat k \bigg) \quad & (D)\: \frac{1}{\sqrt {425}} \bigg( \hat i - 10\hat j -18\hat k \bigg) \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

0 answers

: If the given lines \( \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} \) and \( \frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} \) are perpendicular, then k is \[ \begin{array} (A)\: -10 \quad & (B) \: \frac{10}{7} \\ (C)\: \frac{-10}{7} \quad & (D)\: \frac{-7}{10} \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

The angle between the planes \( \overrightarrow r .(3\hat i − 4\hat j +5\hat k) = 0 \: and\: \overrightarrow r .(2\hat i − \hat j − 2\hat k) \)is \[ \begin{array} (A)\: \frac{\pi}{3} \quad & (B) \: \frac{\pi}{2} \\ (C)\: \frac{\pi}{6} \quad & (D) \frac{\pi}{4} \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

The probability of obtaining an even prime numbr on each die, when a pair of dice is rolled is \[ \begin{array} (A)\: 0 \quad & (B) \: \frac{1}{3} \\ (C)\: \frac{1}{12} \quad & (D) \: \frac{1}{36} \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

If R is the set of real numbers and f : R\( \rightarrow\)R be the function defined by \(f (x) = 1 (3 – x^3 )^{\frac{1}{3}}\) , then (fof) (x) is equal to \[ \begin{array} (A) \: x^{\frac{1}{3}} \quad & (B)\: x^3 \\ (C) \: x \quad & (D)\: (3 – x^3) \end{array} \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

Let R be the set of real numbers, f : R\( \rightarrow \)R be the function defined by f (x) = 4x + 3, x ∈ R. Show that f is invertible. Find the inverse of f.

asked Dec 23, 2012 by thanvigandhi_1

0 answers

Prove that \[ tan^{-1} \bigg( \frac{\sqrt {1+x}-\sqrt {1-x}}{\sqrt {1+x}+\sqrt {1-x}} \bigg) = \frac{\pi}{4} - \frac{1}{2} cos^{-1}x , \frac{-1}{\sqrt 2} \leq x \: \leq 1 \]

asked Dec 23, 2012 by thanvigandhi_1

0 answers

If x, y, z are different numbers and \[ A = \begin{bmatrix} x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3 \end{bmatrix} = 0 \] , then show that 1 + x y z = 0

asked Dec 23, 2012 by thanvigandhi_1

0 answers

Is the function f defined by \[ f (x) = \left\{ \begin{array}{l l} x+5, & \quad {if\: x\: \leq 1}\\ x-5 & \quad \text{ if $x$ > 1} \end{array} \right.\] a continuous function? Justify your answer.

asked Dec 23, 2012 by thanvigandhi_1

0 answers

If \(x = a (cos\: t + t\: sin \: t), y = a (sin\: t – t \: cos \: t)\). Find \( \frac{d^2y}{dx^2} \)

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find \[ \int \mathrm \: ( sin^{-1} x)^2\: dx \]

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Evaluate \[ \int_{-2}^2 \mathrm \: | 1 - x^2 |dx \]

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find the solution of the differential equation \( (x^2 + y^2) dx = 2xy \: dy\)

asked Dec 22, 2012 by thanvigandhi_1

0 answers

If the magnitude of the vectors, \( \overline {a} , \overline{b} , \overline{c} \) are 3, 4, 5 respectively and \( \overline{a} \) and \( \overline{b} + \overline{c}, \overline{b} \) and \( ( \overline{c} + \overline{a} ), \overline{c}\: and\: ( \overline{a} + \overline{b} )\) are mutually perpendicular, then find the magnitude of \(( \overline{a} + \overline{b} + \overline{c} ) \)

asked Dec 22, 2012 by thanvigandhi_1

0 answers

Find the equation of the straight line passing through (1, 2, 3) and perpendicular to the plane \( x + 2y – 5z + 9 = 0\)

asked Dec 22, 2012 by thanvigandhi_1

0 answers

A man and his wife appear for an interview for two posts. The probability of husband’s selection is \( \frac{1}{7} \) and that of the wife’s selection is \( \frac{1}{5} \) . Find the probability that only one of them will be selected.

asked Dec 22, 2012 by thanvigandhi_1

1 answer

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