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Questions >> CBSE XII >> Math >> Model Papers

Show that the right-circular cone of least curved surface and given volume has an altitude equal to $\sqrt 2$ times the radius of the base.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int_{\Large\frac{\pi}{6}}^{\Large\frac{\pi}{3}}\large \frac{dx}{1+\sqrt{tan \: x}}$

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{6x+7}{\sqrt{(x-5)(x-4)}}$ $dx$

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Stetch the graph of y = | x + 3 | and evaluate the area under the curve y = | x + 3 | above x - axis and between x = -6 to x = 0.

asked Feb 5, 2013 by thanvigandhi_1

0 answers

Find the distance of the point (-1,-5,-10) from the point of intersection of the line $\overrightarrow r = (2\hat i - \hat j + 2\hat k)+\lambda (3\hat i + 4\hat j + 2\hat k)$ and the plane $\overrightarrow r = (\hat i - \hat j + \hat k)=5.$

asked Feb 5, 2013 by thanvigandhi_1

1 answer

Given three identical boxes I, II and III each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\cos^{-1} \bigg( \cos\large\frac{7\pi}{6} \bigg).$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\int \large\frac{2-3sin \: x}{cos^2x}$ $dx$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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 4, 2013 by thanvigandhi_1

1 answer

Find the value of a and b such that the following function f(x) is a continuous functions : $f(x) = \left\{ \begin{array}{l l}5, & \quad { x \leq 2} \\ ax+b, & \quad { 2 < x < 10} \\ 21, & \quad { x \geq 10 } \end{array} \right.$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation: $(1+y^2)(1+\log x)dx+xdy=0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

If two vectors $\overrightarrow a \: and \: \overrightarrow b$ are such that $|\overrightarrow a|=2,|\overrightarrow b|=1\: and \: \overrightarrow a.\overrightarrow b=1$ , then find the value of $(3\overrightarrow a-5\overrightarrow b).(2\overrightarrow a+7\overrightarrow b).$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Show that of all the rectangles of given area, the square has the smallest perimeter.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\tan^{-1} \bigg[\tan\large \frac{3\pi}{4} \bigg]$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Form the differential equation of the family of parabolas having vertex at the origin and axis along positive y - axis.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\overrightarrow a = 2\hat i + 3\hat j - \hat k \: and \: \overrightarrow b = \hat i - 2\hat j + \hat k$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

If the function f(x) given by $f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if { x > 0} \\ 11, & \quad if { x = 1} \\ 5ax-2b, & \quad if { x < 1 } \end{array} \right.$ is continuous at x = 1, find the values of a and b.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove the following : $\begin{vmatrix} x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3 \end{vmatrix} = xyz(x-y)(y-z)(z-x)$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\sin\bigg[ \large\frac{\pi}{3}$ $-sin^{-1} \bigg( -\large\frac{1}{2} \bigg) \bigg].$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

For a 2 x 2 matrix, A = $[a_{ij} ],$ whose elements are given by $a_{ij}=\large\frac{i}{j},$ write the value of $a_{12}.$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

For what value of x, the matrix $\begin{bmatrix} 5-x & x+1 \\ 2 & 4 \end{bmatrix}$ is singular?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

write $A^{-1}$ for $A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\int \sec\: x (\sec\: x+\tan \: x)dx$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the value of $\int \large\frac{dx}{x^2+16}$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

For what value of 'a' the vectors $2\hat i - 3\hat j + 4\hat k$ and $a\hat i + 6\hat j - 8\hat k$ are collinear?

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the direction cosines of the vector $-2\hat i + \hat j - 5\hat k$ .

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Write the intercept cut off by the plane $2x+y-z=5$ on x - axis.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Prove the following : $\cot^{-1} \bigg[ \large\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}} \bigg] = \frac{x}{2}, x \in \bigg( 0, \frac{\pi}{4} \bigg).$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Find the value of $\tan{-1} \bigg(\large \frac{x}{y} \bigg)-\tan^{-1} \bigg(\large \frac{x-y}{x+y} \bigg).$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using properties of determinants, prove that $\begin{vmatrix} -a^2 & ab & ac \\ ba & -b^2 & bc \\ ca & cb & -c^2 \end{vmatrix} = 4a^2b^2c^2$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Find the value of 'a' for which the function f defined as $f(x) = \left\{ \begin{array}{l l}asin\frac{\pi}{2}(x+1), & \quad { x \leq 0} \\ \large \frac{\tan\: x-\sin x}{x^3}, & \quad { x > 0} \end{array} \right.$ is continuous at x = 0.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Differentiate $x^{\large xcos\: x}+\large\frac{x^2+1}{x^2-1}$ w.r.t. $x$ .

asked Feb 4, 2013 by thanvigandhi_1

0 answers

If $x=a(\theta-\sin\theta),y=a(1+\cos\theta),$ find $\large\frac{d^2y}{dx^2}$ .

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Sand is pouring from a pipe at the rate of $12 cm^3/s.$ The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Find the points on the curve $x^2+y^2-2x-3=0$ at which the tangents are parallel to x - axis.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{5x+3}{\sqrt{x^2+4x+10}}$ $dx$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{2x}{(x^2+1)(x^2+3)}$ $dx$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation: $e^x\tan\: y\: dx+(1-e^x)\sec^2ydy=0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation : $\cos^2x\frac{dy}{dx}+y=\tan \: x$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Find the angle between the following pair of lines : $\large\frac{-x+2}{-2}=\frac{y-1}{7}=\frac{z+3}{-3}$ and $\large\frac{x+2}{-1}=\frac{2y-8}{4}=\frac{z-5}{4}$ and check whether the lines are parallel or perpendicular.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Probabilities of solving a specific problem independently by A and B are $\frac{1}{2} \: and \frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Using matrix method, solve the following system of equations : $\large\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4,\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1, \frac{6}{x}+\frac{9}{y}-\frac{20}{z}$ $=2; x,y,z \neq 0$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Using elementary transformations, find the inverse of the matrix $\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$

asked Feb 4, 2013 by thanvigandhi_1

1 answer

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Using integration find the area of the triangular region whose sides have equations $y=2x+1,y=3x+1 \: and \: x=4$ .

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int_0^{\large\frac{\pi}{2}} 2\sin x\cos x\tan^{-1}(\sin x)dx$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int_0^{\Large\frac{\pi}{2}}\large \frac{xsin\: x\cos x}{\sin^4x+cos^4x}$ $dx$

asked Feb 4, 2013 by thanvigandhi_1

0 answers

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