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

Prove that the relation R in the set A = {1, 2, 3, 4, 5}given by R = {(a, b):|a-b) is even }, is an equivalence relation.

asked Jan 6, 2013 by thanvigandhi_1

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Prove that : $\sin^{-1}\bigg(\large\frac{ 4}{ 5} \bigg)$ $+ \sin^{-1}\bigg(\large\frac{ 5}{ 13} \bigg)$ $+ \sin^{-1}\bigg(\frac{\large 16}{\large 65}\bigg) = \frac{\large \pi}{\large 2}.$

asked Jan 6, 2013 by thanvigandhi_1

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Solve for : $x : tan^{-1}3x+tan^{-1}2x=\frac{\large \pi}{\large 4}.$

asked Jan 6, 2013 by thanvigandhi_1

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Using properties of determinants, prove the following : $\begin{bmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{bmatrix} = 2abc(a+b+c)^33 - 3abc.$

asked Jan 6, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin{bmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{bmatrix} = 2abc(a+b+c)^3.$

asked Jan 6, 2013 by thanvigandhi_1

0 answers

If $y=3 \cos (log\: x) + 4\sin (\log x)$ then show that $x^2\frac{\large d^2y}{\large dx^2}+x\frac{\large dy}{\large dx}+y=0.$

asked Jan 6, 2013 by thanvigandhi_1

0 answers

Find $\large \frac{dy}{dx}$ if $(x^2+y^2)^2=xy$

asked Jan 6, 2013 by thanvigandhi_1

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Differentiate w.r.r. $x, \cos x . \cos 2x.\cos 3x$ .

asked Jan 6, 2013 by thanvigandhi_1

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If $y = \sqrt{\large\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$ then find $\large\frac{dy}{dx}$ .

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\large\frac{dx}{\sqrt {5-4x-2x^2}}$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int x\sin^{-1}x dx$

asked Jan 4, 2013 by thanvigandhi_1

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(i) Form the differential equation of the family of curves $y = a\: sin (x+b)$ , where a and b are arbitary constants. (ii) Solve the following differential equation : $2xy\: dx + (x^2+2y^2)dy = 0.$

asked Jan 4, 2013 by thanvigandhi_1

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Solve the following differential equation : $\large \frac{dy}{dx}$ $+ 2\tan x.y = \sin x.$

asked Jan 4, 2013 by thanvigandhi_1

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Find the position vector of a point R which divided the line joining the points whose positive vectors are $P(\hat i +2\hat j - \hat k ) and Q ( -\hat i + \hat j + \hat k )$ in the ratio 2 : 1. (i) internally (ii) externally.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Find the shortest distance between the following 2 lines - $\overrightarrow r = (1+2\lambda)\hat i + (1-\lambda)\hat j + \lambda\hat k ;\overrightarrow r = 2\hat i + \hat j - \hat k + \mu (3\hat i - 5\hat j + 2\hat k ) .$

asked Jan 4, 2013 by thanvigandhi_1

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An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

asked Jan 4, 2013 by thanvigandhi_1

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Using matrices, solve the following system of the linear equations : $\begin{array} xx - y = 4 \\ 2x + 3y + 4z = 17 \\ y +2z = 7 \end{array}$

asked Jan 4, 2013 by thanvigandhi_1

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Given : $A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}. B = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$ Find AB and use this result in solving the following system of equations : $\begin{array} xx - y + z = 4 \\ x - 2y - 2z = 9 \\ 2x + y +3z = 1 \end{array}$

asked Jan 4, 2013 by thanvigandhi_1

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Prove that the height of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius R is $\large\frac{2R}{\sqrt 3}.$ Also find the maximum volume.

asked Jan 4, 2013 by thanvigandhi_1

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Find the point on the curve $y^2= 4x$ which is nearest to the point ( 2, -8 ).

asked Jan 4, 2013 by thanvigandhi_1

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Find the area of the region included between the parabola $4y = 3x^2$ and the lines $3x-2y+12 = 0$ .

asked Jan 4, 2013 by thanvigandhi_1

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

asked Jan 4, 2013 by thanvigandhi_1

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If $l_1, m_1, n_1 \: and \: l_2, m_2, n_2$ are direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_1n_2-m_2n_1, n_1l_2-n_2l_1, l_1m_2-l_2m_1.$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

A toy company manufactures two types of dolls, A and B. Markets tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is almost half of that for dolls of type A. Further the production level of dolls of type A can exceed three times the production of dolls of other type by almost 600 units. If the company makes profit of Rs. 12 and Rs.16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

asked Jan 4, 2013 by thanvigandhi_1

0 answers

An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of an accident involving a scooter, a car and a truck are 0.02, 0.03 and 0.04, respectively. One of the insured vehicles meets with an accident. Find the probability that (i) it is a car (ii) it is a truck.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Find the value of a, b, c and d from the equation : $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$

asked Jan 4, 2013 by thanvigandhi_1

1 answer

Compute the following : $\begin{bmatrix} cos^2x & sin^2x \\ sin^2x & cos^2x \end{bmatrix} + \begin{bmatrix} sin^2x & cos^2x \\ cos^2x & sin^2x \end{bmatrix}$

asked Jan 4, 2013 by thanvigandhi_1

1 answer

Evaluate : $\begin{bmatrix} sin30^{\circ} & cos30^{\circ} \\ -sin60^{\circ} & cos60^{\circ} \end{bmatrix}$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Find the value of $k$ if the function $f(x) = \left\{ \begin{array}{l l} kx^2, & \quad { x \geq 1}\\ 4, & \quad \text{ $x$ < 1} \end{array}. \right.$ is continuous at $x = 1$ .

asked Jan 4, 2013 by thanvigandhi_1

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Prove that the logatithmic function is strictly increasing on $( 0, \infty )$ .

asked Jan 4, 2013 by thanvigandhi_1

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$\int \: (2x^2-3sinx+5\sqrt x)dx.$

asked Jan 4, 2013 by thanvigandhi_1

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Find the differential equation : $y-\cos y = x.$

asked Jan 4, 2013 by thanvigandhi_1

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Write two different vectors having same direction.

asked Jan 4, 2013 by thanvigandhi_1

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If P(A) = 0.8, P(B) = 0.5 and P(B/A) = 0.4. Find $(i)\: P(A \cap B ) (ii)\: P(A/B) (iii)\: P(A \cup B).$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and $P(E \cap F)$ . = 0.2. Find P(E/F) and P(F/E).

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Show that the relation R in the set A = {1, 2, 3, 4, 5} is given by R = {(a, b) : |a-b|is even }, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin{bmatrix} 1+a^2-b^2 & 2ab & -2b \\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2=b^2 \end{bmatrix} = (1 + a^2 + b^2)^3.$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin{bmatrix} 3a & -a+b & -a+c \\ a-b & 3b & c-b \\ a-c & b-c & 3c \end{bmatrix} = 3(a + b + c)(ab+bc+ca)$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

If $sin\: y = x\: sin (a+y)$ prove that $\large \frac{dy}{dx}=\frac{sin^2(a+y)}{sin\: a}$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

If $( cos \: x )^y = (sin\: y)^y$ find $\large\frac{dy}{dx}$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Prove that : $\large \frac{d}{dx}$ $(log_ex)=\large\frac{1}{x}.$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Find the intervals in which the function f given by $f(x) =\sin x + \cos x, 0 \leq x \leq 2\pi$ is strictly increasing or strictly decreasing.

asked Jan 4, 2013 by thanvigandhi_1

1 answer

Evaluate : $\int \large\frac{(x-4)e^x}{(x-2)^3}$ $dx$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int\large \frac{e^x}{\sqrt {5-4e^x-e^{2x}}}$ $dx$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int x^2\cot^{-1}xdx$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{1}{b^2sin^2x+a^2cos^2x}$ $dx$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

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

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation : $(x^2+1)\large \frac{dy}{dx}$ $+2xy = \sqrt {x^2+4}$

asked Jan 4, 2013 by thanvigandhi_1

0 answers

Find the direction cosines of the vector joining the points A(1, 2, -3 ) and B ( -1, -2, 1 ), directed from A to B.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

In a factory, which manufactures nuts, machines A, B and C manufacture respectively 25%, 35% and 40% of nuts. Of their outputs 5, 4 and 2 percent respectively are defective nuts. A nut is drawn at random from the product and is found to be defective. Find the probability that it is manufactured by machine B.

asked Jan 4, 2013 by thanvigandhi_1

0 answers

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