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

Evaluate : $\int_1^2 \bigg( \large\frac{1}{x} - \frac{1}{x^2} \bigg)$ $e^x \: dx$

asked Feb 7, 2013 by thanvigandhi_1

1 answer

For two non zero vectors $\overrightarrow a \: and \: \overrightarrow b$ write when $|\overrightarrow a+\overrightarrow b | = | \overrightarrow a | + | \overrightarrow b |$

asked Feb 7, 2013 by thanvigandhi_1

1 answer

Find the value of p for which $p(\hat i + \hat j + \hat k)$ is a unit vector.

asked Feb 7, 2013 by thanvigandhi_1

0 answers

The cartesian equation of a line is $\large\frac{x+3}{2} = \large\frac{y-5}{4}=\large\frac{z+6}{2}$ Find vector equation of line.

asked Feb 7, 2013 by thanvigandhi_1

1 answer

Prove that $\tan^{-1} \large\frac{63}{16}=\sin^{-1}\bigg( \large\frac{5}{13} \bigg) + \cos^{-1} \bigg( \large\frac{3}{5} \bigg).$

asked Feb 7, 2013 by thanvigandhi_1

1 answer

Show that the relation R in the set $A = \{ x/x \in W, 0 \leq x \leq 12 \}$ given by $R = \{ (a,b):(a-b)$ is a multiple of 4 } is an equivalence relation. Also find the set of all element related to 2.

asked Feb 7, 2013 by thanvigandhi_1

0 answers

Examine which of the following is a binary operation.

asked Feb 7, 2013 by thanvigandhi_1

1 answer

If a,b and c are real number and $\begin{vmatrix} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{vmatrix}$ Show that either $a+b+c=0 \: or \: a=b=c$

asked Feb 7, 2013 by thanvigandhi_1

0 answers

Express the following matrix as the sum of a symmetric (B) and skew symmetric matrix (C) $\begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

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

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Show that the function $f(x) = |x+2 |$ is continuous at every $x \in R$ but fails to be differentiable at x = -2.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the equation of tangent lines to the curve $y=4x^3-3x+5$ which are perpendicular to the line $9y+x+3=0$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int \large\frac{\cos x\: dx}{(2+\sin x)(3+4\sin x) }$

asked Feb 6, 2013 by thanvigandhi_1

0 answers

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

asked Feb 6, 2013 by thanvigandhi_1

1 answer

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the area of triangle having the points as A(1,1,1), B(1,2,3) and C(2,3,1) as its vertices.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Two cards are drawn from a well - shuffled pack of 52 cards without replacement. What is the probability that one is a red queen and the other is a king of black colour.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Obtain the inverse of the following matrix, using elementary operations $A = \begin{bmatrix} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{bmatrix}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

A given quantity of metal is to be cast into a solid half cicular cylinder (i.e., with rectangular base and semicircular ends). Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its circular ends is $\pi : ( \pi + 2).$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find all the local maximum values and local minimum values of the function $f(x)=\sin 2x-x, -\large\frac{\pi}{2} < x < \large\frac{\pi}{2}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Prove that the image of the point (3,-2,1) in the plane $3x-y+4z=2$ lies on the plane $x+y+z+4=0$ .

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int \large\frac{x^4dx}{(x-1)(x^2+1)}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Two dice are thrown simultaneously. Let X denotes the number of sixes. Find the probability distribution of X. Also find the mean, variance and standard deviation of X using probability distribution table.

asked Feb 6, 2013 by thanvigandhi_1

0 answers

A catering agency has two kitchen to prepare food at two places A and B. From these places 'Mid-day Meal' is to be supplied to three different schools situated at P, Q , R. The monthly requirement of the school are respectively 40,40 and 70 food packets. A packet contains lunch for 1000 students. The capacity of kitchen A and B are 60 and 70 packets per month respectively. The transportation cost per packets from the kitchens 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 Feb 6, 2013 by thanvigandhi_1

1 answer

Write $( fog, \: if \: f : R \rightarrow R)$ are given by $( f(x) = 8x^3)$ and $( g(x)=x^{\large\frac{1}{3}} )$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int\large\frac{\cos \sqrt x}{\sqrt x}$ $dx$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Prove, using properties of determinants: $\begin{vmatrix} x+y+2z & x & y \\ z & y+z+2x & y \\ z & x & z+x+2y \end{vmatrix} = 2(x+y+z)^3$

asked Feb 6, 2013 by thanvigandhi_1

0 answers

For what value of $\lambda$ is the function $f(x) = \left\{ \begin{array}{l l}\lambda (x^2-2x), & \quad if { x \leq 0 } \\ 4x+1, & \quad if { x > 0 } \end{array} \right.$ is continuous at $x=0.$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $(1+x^2)\large\frac{dy}{dx}$ $+2xdy=\large\frac{1}{1+x^2},$ given y = 0 when x = 1.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes : $x+2y+3z-7=0 \: and \: 2x-3y+4z=0$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Write $fog,$ if $f : R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x) = | x| \: and \: g(x) = | 5x-2 |$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int\large \frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$ $dx$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Prove, using properties of determinants: $\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix} = (a+b+c)^3$

asked Feb 6, 2013 by thanvigandhi_1

0 answers

Find the value of k so that the function f, defined by $f(x) = \left\{ \begin{array}{l l}kx+1, & \quad if { x \leq \pi } \\ \cos x, & \quad if { x > \pi } \end{array} \right.$ is continuous at $x=\pi.$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Solve the following differential equation : $\large\frac{dy}{dx}$ $+2y\tan\: x = \sin\: x.$ given that $y=0, \: when\: x = \large\frac{\pi}{3}.$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the vector equation of the plane, passing through the points A(2,2,-1), B(3,4,2) and C(7,0,6). Also, find the cartesian equation of the plane.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Bag I contains 3 red and 4 black balls and bag II contains 4 red and 5 black balls. One ball is transferred from bag I to bag II and then a ball is drawn from bag II at random. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

If $f : R \rightarrow R$ is defined by $f(x)=3x+2$ , define $f[f(x)].$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

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

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Write the values of $x-y+z$ from the following equation : $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Write the order of the product matrix : $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

If $\begin{vmatrix} x & x \\ 1 & x \end{vmatrix} = \begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$ write the positive value of x.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int\limits_1^{\sqrt 3} \large\frac{dx}{1+x^2}$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Write the position vector of the mid-point of the vector joining the points P(2,3,4) and Q(4,1,-2).

asked Feb 6, 2013 by thanvigandhi_1

1 answer

If $\overrightarrow a.\overrightarrow a=0\: and \: \overrightarrow a.\overrightarrow b = 0$ , then what can be concluded about the vector $\overrightarrow b$ ?

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Consider $f : R_+ \rightarrow [ 4, \infty ]$ 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 Feb 6, 2013 by thanvigandhi_1

1 answer

Prove the following : $\large\frac{9\pi}{8}-\frac{9}{4}sin^{-1}\bigg(\large\frac{1}{3} \bigg) = \frac{9}{4}sin^{-1}\bigg(\large \frac{2\sqrt 2}{3} \bigg)$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Solve the following equation for x : $\tan^{-1} \bigg( \large\frac{1-x}{1+x} \bigg) = \large\frac{1}{2}\tan^{-1}(x), x > 0$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Prove, using properties of determinants $\begin{vmatrix} y+k & y & y \\ y & y+k & y \\ y & y & y+k \end{vmatrix} = k^2(3y+k)$

asked Feb 6, 2013 by thanvigandhi_1

1 answer

Find the points on the curve $y=x^3$ at which the slope of the tangent is equal to the y - coordinate of the point.

asked Feb 6, 2013 by thanvigandhi_1

1 answer

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