Recent questions in Model Papers

Find the particular solution of the differential equation $ log \bigg( \large\frac{dy}{dx} \bigg) =3x+4y,$ given that $ y=0$ when $x=0$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

If $ x = \cos\: t ( 3-2 \: \cos^2 t)\: and \: y = \sin \: t ( 3-2 \sin^2\: t)$ find the value of $ \large\frac{dy}{dx} $ at $ t = \large\frac{\pi}{4}$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Using properties of determinants, prove that, \[\] $\begin{vmatrix} 1+a & 1 & 1 \\[0.3em] 1 & 1+b & 1 \\[0.3em] 1 & 1 & 1+c \end{vmatrix} = abc +bc+ca+ab$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ y = Pe^{ax}+Qe^{bx}$, show that \[\] $\large\frac{d^2y}{dx^2}$$-(a+b) \large\frac{dy}{dx}$$+aby=0$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least three successes

asked Mar 25, 2014 by thanvigandhi_1

1 answer

If $ \tan^{-1} \bigg( \large\frac{x-2}{x-4} \bigg) $$+ \tan^{-1} \bigg( \large\frac{x+2}{x+4} \bigg) $$ = \large\frac{\pi}{4}$ find the value of x.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

prove that : \[\] $ \tan^{-1} \bigg[ \large\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}} \bigg]$$ = \large\frac{\pi}{4}$$- \large\frac{1}{2}$$ \cos^{-1}x , \large\frac{-1}{\sqrt 2}$$ \leq x \leq 1$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If the function $ f : R \rightarrow R $ be given by $ f(x) = x^2+2$ and $ g : R \rightarrow R $ be given by $ g(x) = \large\frac{x}{x-1} , x \neq 1$, find fog and gof and hence find fog (2) and gof (-3)

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Find the equations of the tangent and normal to the curve $ \large\frac{x^2}{a^2}$$ - \large\frac{y^2}{b^2} = 1$ at the point $ ( \sqrt 2a, b)$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Find the value(s) of $x$ for which $y = [x(x-2)]^2$ is an increasing function.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Evaluate : \[\] $\int \large\frac{x+2}{\sqrt{x^2+5x+6}} dx$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

Evaluate : \[\] $ \int_0^{\pi} \large\frac{4x \sin x}{1+ \cos^2x} dx$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

The scalar product of the vector $ \overrightarrow a = \hat i + \hat j + \hat k$ with a unit vector along the sums of vectors $ \overrightarrow b = 2 \hat i + 4\hat j - 5 \hat k$ and $ \overrightarrow c = \lambda \hat i + 2 \hat j + 3 \hat k\;$ is equal to one. Find the value of $ \lambda $ and hence find the unit vector along $ \overrightarrow b + \overrightarrow c$

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Show that the four points A, B, C and D with position vectors $ 4\hat i + 5 \hat j + \hat k, \: \: -\hat j - \hat k, \: \: 3\hat i + 9 \hat j + 4\hat k,$ and $ 4(-\hat i + \hat j + \hat k) $ respectively are coplanar.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

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

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \overrightarrow a$ and $ \overrightarrow b$ are perpendicular vectors $ | \overrightarrow a + \overrightarrow b | = 13$ and $ | \overrightarrow a |=5$, find the value of $ | \overrightarrow b|.$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \int_0^a \large\frac{1}{4+x^2}$$ dx = \large\frac{\pi}{8}$, find the value of $a$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If the cartesian equation of the lines are $ \large\frac{3-x}{5} $$= \large\frac{y+4}{7}$$=\large\frac{2z-6}{4}$, write the vector equation of the line.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ R = \{ (x, y) : x + 2y =8\}$ is a relation on N. write the range of R

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Find the value of 'P' for which the vectors $ 3 \hat i + 2 \hat j + 2 \hat k$ and $ \hat i - 2P \hat j + 3 \hat k $ are parallel.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

If $ f(x) =\int_0^x t\: \sin\: t\: dt$, then write the value of $f'(x)$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \begin{bmatrix} 3x & 7 \\[0.3em] -2 & 4 \end{bmatrix} = \begin{bmatrix} 8 & 7 \\[0.3em] 6 & 4 \end{bmatrix}$, find the value of $x$.

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $\tan^{-1}x+ \tan^{-1}y = \large\frac{\pi}{4},$$ xy < 1$, then the write the value of $x+y+xy$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $ \begin{bmatrix} x-y & z \\[0.3em] 2x-y & w \end{bmatrix} =\begin{bmatrix} -1 & 4 \\[0.3em] 0 & 5 \end{bmatrix} $, find the value of $X$ and $Y$

asked Mar 25, 2014 by thanvigandhi_1

0 answers

If $A$ is a square matrix such that $A^2=A$, then write the value of $7A -(I+A)^3$ where $I$ is an identity matrix.

asked Mar 25, 2014 by thanvigandhi_1

1 answer

Two schools $P$ and $Q$ want to award their selected students on the values of Tolerance,Kindness and Leadership. The school $P$ wants to award $Rs.x,\:Rs.y\:and\:\:Rs.z$ each for the three respective values to $3,2\:and\:1$ students respectively with a total award money of $ Rs.2,200.$ School $Q$ wants to spend $Rs.3,100$ to award its 4,1 and 3 students on the respective values (by giving the same award money to the three values as school $P$). If the total amount of award for one prize on each value is $Rs.1,200,$ using matrices, find the award money for each value.

asked Mar 23, 2014 by rvidyagovindarajan_1

1 answer

If $x=a\:cos\theta+b\:sin\theta$ and $y=a\:sin\theta-b\:cos\theta$, then show that $y^2.\large\frac{d^2y}{dx^2}$$-x\large\frac{dy}{dx}$$+y=0$

asked Mar 23, 2014 by rvidyagovindarajan_1

1 answer

Three cards are drawn at random (without replacement) from a well shuffled pack of 52 playing cards. Find the probability distribution of number of red cards. Hence find the mean of the distribution.

asked Mar 23, 2014 by rvidyagovindarajan_1

1 answer

Show that $cot^{-1}7+cos^{-1}8+cot^{-1}18=cot^{-1}3$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Solve: $cos(tan^{-1}x)=sin(cot^{-1}(\large\frac{3}{4}))$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Solve the differential equation $(x^2-yx^2)dy+(y^2+x^2y^2)dx=0$ Given $y=1$ when $x=1$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Find a unit vector perpendicular to each of the vector $ \overrightarrow a + \overrightarrow b$ and $ \overrightarrow a - \overrightarrow b$, where $ \overrightarrow a = \hat i + \hat j + \hat k$ and $ \overrightarrow b = \hat i + 2\hat j +3\hat k$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Find the distance of the point $P(-1,-5,-10)$ from the point of intersection of the line joining the points $A(2,-1,2)$ and $B(5,3,4)$with the plane $x-y+z=5$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Find the equation of the plane that contains the point $(1,-1,2)$ and is $\perp$ to the planes $2x+3y-2z=5$ and $x+2y-3z=8$. Hence find the distance of the point $(-2,5,5)$ from the plane obtained above.

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Evaluate: $\large \int \limits_0^1$$ x\;e^{x^2}\; dx$

asked Mar 21, 2014 by balaji

1 answer

Find the distance between the two lines $l_1$ and $l_2$ where $l_1: \overrightarrow r=(\hat i+2\hat j-4\hat k)+\lambda(2\hat i+3\hat j+6\hat k)$ and $l_2: \overrightarrow r=(3\hat i+3\hat j-5\hat k)+\mu(4\hat i+6\hat j+12\hat k)$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Using the property of determinants and without expanding, prove that \[ \begin{vmatrix} a^2&bc&ac+c^2 \\ a^2+ab&b^2&ac \\ ab&b^2+bc&c^2 \end{vmatrix} = 4a^2b^2c^2\]

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Find the value of $\hat i\times (\hat j+\hat k)+\hat j\times (\hat k+\hat i)+\hat k\times (\hat i+\hat j)$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Find the value of $a-2b$ if $\begin{bmatrix}a+4 & 3b\\8 & -6\end{bmatrix}=\begin{bmatrix}2a+2 & b+2\\8 & a-8b\end{bmatrix}\qquad$

asked Mar 21, 2014 by rvidyagovindarajan_1

1 answer

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are diamodns? (ii) only 3 cards are diamonds? (iii) none is a diamonds?

asked Mar 20, 2014 by balaji.thirumalai

1 answer

Evlauate$ \Large \int_0^{\pi/4}$$ \large \frac{\sin x+\cos x}{9+16\sin 2x}$$dx$

asked Mar 20, 2014 by balaji.thirumalai

1 answer

Find the projection of the vector $ \hat i + \hat j + \hat k$ on the vector $\hat j$.

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

Find the value of the following:\[cos^{-1} \big(- \frac {1} {2} \big)+2 sin^{-1}\frac {1} {2} \]

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

Let $R=\{(a,a^3):\:a\: is\: a\: prime\:number\: less\: than\: 5\}$ be a relation. Find the Range of $R$.

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

If $A$ is a $3\times 3$ matrix, and $ |A|\neq0$ and $|3A|=k.|A|,$ then write the value of $k$

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

Find the angle between the pairs of lines: $ \overrightarrow r = 2\hat i -5 \hat j +\hat k + \lambda (3\hat i +2 \hat j +6\hat k)\: and \: \overrightarrow r = 7\hat i - 6\hat k + \mu (\hat i +2\hat j +2\hat k) $

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

Use elementary column operation $C_2\rightarrow C_2-2C_1$ in the matrix equation $\begin{bmatrix}4 & 2\\3 & 3\end{bmatrix}=\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}\:\begin{bmatrix}2 & 0\\1 & 1\end{bmatrix}$

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

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

asked Mar 20, 2014 by rvidyagovindarajan_1

1 answer

Show that of all the rectangles with a given perimeter,the square has the largest area.

asked Jul 25, 2013 by sreemathi.v

0 answers

Differentiate $( x \cos \: x )^x + ( x\sin\: x )^{\Large\frac{1}{x}}$ w.r.t. $ x $.

asked Jul 25, 2013 by sreemathi.v

1 answer

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