Recent questions in Model Papers

Evaluate:$\large \int\frac{x+2}{\sqrt{x^2+2x+3}}$dx

asked Mar 20, 2013 by sreemathi.v

1 answer

Evaluate: $\int\large \frac{cos 2x-cos 2\alpha}{cos x-cos\alpha}$$dx$

asked Mar 20, 2013 by sreemathi.v

1 answer

If x=$acos^3\theta$ and y=$asin^3\theta,$then find the value of $\Large \frac{d^2y}{dx^2}$ at $ \theta=\Large \frac{\pi}{6}$.

asked Mar 20, 2013 by sreemathi.v

1 answer

Find the value of k, for which $f(x) =\left\{\begin{array}{1 1}\frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}, & if\;-1\leq x<0\\\frac{2x+1}{x-1}, & if\;0\leq x<1\end{array}\right.$ is continuous at x=0.

asked Mar 20, 2013 by sreemathi.v

1 answer

Differentiate the following with respect to $x: \sin^{-1}\bigg(\large \frac{2^{x+1}.3^x}{1+(36)^x}\bigg)$

asked Mar 20, 2013 by sreemathi.v

1 answer

If $y^x=e^{y-x}$,prove that $\large \frac{dy}{dx}=\frac{(1+log y)^2}{log y}$

asked Mar 20, 2013 by sreemathi.v

1 answer

Using properties of determinants,prove the following $\begin{vmatrix}x& x+y & x+2y\\x+2y & x & x+y\\x+y & x+2y & x\end{vmatrix}=9y^2(x+y)$

asked Mar 20, 2013 by sreemathi.v

1 answer

Solve the following equation :$\cos\big(tan^{-1}x\big)=sin\bigg(cot^{-1}\frac{3}{4}\bigg)$

asked Mar 20, 2013 by sreemathi.v

1 answer

Show that: $tan\bigg(\frac{1}{2}sin^{-1}\frac{3}{4}\bigg)=\frac{4-\sqrt 7}{3}$

asked Mar 20, 2013 by sreemathi.v

1 answer

Consider $( f:R_+ \to [4,\infty)$ given by $(f(x)=x^2 +4)$. Show that $(f)$ is invertible with the inverse $(f^{-1})$ of $(f)$ given by $(f^{-1}(y) = \sqrt {y-4})$,where $R_+$ is the set of all non-negative real numbers.

asked Mar 20, 2013 by sreemathi.v

1 answer

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue).If the total revenue(in rupees) received from the sale of Xunits of a product is given by R(x)=$3x^2+36x+5$,find the marginal revenue,when x=5,and write which value does the question indicate.

asked Mar 20, 2013 by sreemathi.v

1 answer

Find the length of the perpendicular drawn from the origin to the plane 2X-3Y+6Z+21=0.

asked Mar 20, 2013 by sreemathi.v

1 answer

Find $\mid\overrightarrow{x}\mid$,if for a unit vector $\overrightarrow{a},(\overrightarrow{x}-\overrightarrow{a}).(\overrightarrow{x}+\overrightarrow{a})=15.$

asked Mar 20, 2013 by sreemathi.v

1 answer

P and Q are two points with position vectors $3\overrightarrow{a}-2\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ respectively.Write the position vector of a point R which divides the line segment PQ in the ratio 2 : 1 externally.

asked Mar 20, 2013 by sreemathi.v

1 answer

If $A_{ij}$ is the cofactor of the element $a_{ij}$ of the determinant $\begin{vmatrix}2 & -3 & 5\\6 & 0 & 4\\1 & 5 & -7\end{vmatrix}$,then write the value of $a_{32}.A_{32}$.

asked Mar 20, 2013 by sreemathi.v

1 answer

Write the differential equation representing the family of curves y=mx,where m is an arbitrary constant.

asked Mar 20, 2013 by sreemathi.v

1 answer

For what value of x, is the matrix A=$\begin{bmatrix}0 & 1 & -2\\-1 & 0 & 3\\x & -3 & 0\end{bmatrix}$ a skew-symmetric matrix?

asked Mar 20, 2013 by sreemathi.v

1 answer

Write the value of $\tan^{-1}\begin{bmatrix}2\sin\bigg(2\cos^{-1}\frac{\sqrt 3}{2}\bigg)\end{bmatrix}$

asked Mar 20, 2013 by sreemathi.v

1 answer

Write the principal value of $\tan^{-1}(\sqrt 3)-cot^{-1}(-\sqrt 3).$

asked Mar 20, 2013 by sreemathi.v

1 answer

Write a unit vector in the direction of $ \overrightarrow a = 2\hat i - 6\hat j + 3\hat k.$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Find the projection of $ \overrightarrow a \: on \: \overrightarrow b\: if \: \overrightarrow a.\overrightarrow b=8\: and \: \overrightarrow b = 2\hat i + 6\hat j + 3\hat k$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Find the direction cosines of the line passing through the points $(-2,4,-5)$ and $(1,2,3)$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

If matrix A = (1,2,3), write AA' where A' is the transpose of matrix A.

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Write the value of the determinants \[ \begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix} \]

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Using principal value, evaluate the following : \[ sin^{-1} \bigg( sin \frac{3\pi}{5} \bigg) \]

asked Feb 17, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \int \large\frac{\sec^2x}{3+\tan\:x}$$dx $

asked Feb 17, 2013 by thanvigandhi_1

1 answer

If $ \int_0^1 (3x^2+2x+k)dx=0,$ find the value of k

asked Feb 17, 2013 by thanvigandhi_1

1 answer

If the binary operation * on the set of integers Z, is defined by $ab=( a+3b^2)$ then find the value of $24$

asked Feb 17, 2013 by thanvigandhi_1

1 answer

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 : $ \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 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

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

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