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

Using properties of determinants show that: \( \begin{bmatrix} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{bmatrix} = (x+y+z)(x-z)^2.. \)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If \( a, b, c\) are in A.P., show the following: \( \begin{bmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{bmatrix} = 0. \)

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Prove that : $ \cot^{-1} \bigg[ \large\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \bigg]; $$0 < x <\large \frac{\pi}{2} = \large\frac{1}{2} $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Differentiate \( (x^2-5x+8)(x^2+7x+9)\) in three ways mentioned : (i) by using product rule (ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation. Do they all give the same answer?

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If \( u, v \: and \ w \) are function of \( x\) , then show that $ \large\frac{d}{dx}(u.v.w) = \frac{du}{dx}.v.w+u.\frac{dv}{dx}.w+u.v.\frac{dw}{dx}$In two ways first by repeated application or product rule, secondly by logatithmic differentiation.

asked Jan 10, 2013 by thanvigandhi_1

0 answers

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

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Evaluate $\int\large \frac{\sin(x-\alpha)}{\sin(x+\alpha)}$$dx.$

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Verify that \( y = e^x.\cos\: x\) is a solution of the differential equation : $\large \frac{d^2y}{dx^2}$$-2\large\frac{dy}{dx}$$+2y=0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation: $(x^2+3xy+y^2)dx-x^2\: dy = 0. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the points A, B and C with position vectors \( \overrightarrow a = 3\hat i - 4\hat j - 4\hat k, \overrightarrow b = 2\hat i - \hat j + 4\hat k \: and \: \overrightarrow c = \hat i - 3\hat j - 5\hat k,\) respectively, form the vertices of a right angled triangle.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Solve the equation $ 2x^2\large\frac{dy}{dx}$$- 2xy+y^2=0,y(e)=e.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the point on the line \( \large\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}\) at a distance \( 3\sqrt 2\) from the point (1, 2, 3).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the shortest distance between the following lines : $ \large\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1} $$ and $$ \large\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1} $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}.$Prove that \( A^2-4A-51=0\). Hence find \( A^{-1} \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Using matrix method solve the following system of linear equations : $ x+y-z=1; 3x+y-2z=3 ;x-y-z = -1$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \(\large \frac{8}{27}\) of the volume of the sphere.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the equations of the tangent and normal to the curve \( 16x^2+9y^2 = 144\) at \( (x_1, y_1)\) where \( x_1=2\: and \: y_1 > 0.\) Also, find the points of intersection where both tangent and normal cut the \( x\) - axis.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Using the method of Integration, find the area of the region bounded by the lines \( 2x+y=4, 3x-2y=6,x-3y+5=0.\)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that : $\int_0^{\large\frac{\pi}{2}} ( \sqrt {cosx}+ \sqrt{cotx})dx = \sqrt 2 \pi. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the planes \( 2x+y+3x-2 = 0 \: and \: x-2y+5=0\) are perpendicular to each other.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of one kg food is given below : One kg of food x costs Rs.16 and one kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

An urn contains 25 balls of which 10 bear a mark 'X' and the remaining 15 mark 'Y'. A ball is drawn at random from the urn, its mark noted down and it is replaced. If 6 balls are drawn in this way, find the probability that (i) all will bear mark 'X' (ii) not more than two will bear 'Y' mark (iii) at least one ball will bear 'X' mark and Y mark will be equal.

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If \( f(x) = x+7 \: and \: g(x) = x-7, x\: \in \: R, \) find (fog) (7).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the principal value of the following : $ \sin^{-1} \bigg( -\frac{\large 1}{\large 2} \bigg). $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

For what value of x, is the following matrix singular? $ \begin{bmatrix} 3-2x & x+1 \\ 2 & 4 \end{bmatrix}.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements.

asked Jan 9, 2013 by thanvigandhi_1

1 answer

Evaluate : $ \begin{bmatrix} a+ib & c+id \\ -c+id & a-ib \end{bmatrix} $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int\large \frac{2\cos x}{3\sin^2x}$$dx.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

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 Jan 9, 2013 by thanvigandhi_1

0 answers

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

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Scalar product of the vector \( \hat i + \hat j + \hat k \) with the unit vector along the sum of vectors \( 2\hat i + 4\hat j -5\hat k \) and \( \lambda \hat i + 2\hat j + 3\hat k \) is equal to one, find the value of \( \lambda \).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that the relation R, defined in the set A of all triangles as \( R = (T_1, T_2):T_2\) is similar to \( T_2\)}, is equivalence relation. Consider three right triangles \( T_1\) , with sides 3, 4, 5; \( T_2\) with sides 5, 12, 13 and \( T_3\) with sides 6, 8, 10. Which triangle among \( T_1, T_2 \: and T_3\) are related?

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \sin \bigg[ \frac{\large \pi}{\large 3}-$$\sin^{-1} \bigg( -\frac{\large 1}{\large 2} \bigg) \bigg]. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Using properties of determinants prove that \( \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{bmatrix} = (a-b)(b-c)(c-a)(a+b+c). \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that \( \begin{bmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{bmatrix} = (5x+4)(4-x)^2. \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If \( y = \frac{\large sin-x}{\large \sqrt {1-x^2}}\) show that $ (1+x^2) \large\frac{ d^2y}{ dx^2}$$-3x \large\frac{ dy}{ dx}$$ - y=0. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

If \( y=x^x-2^{sinx}\) then find \( \frac{\large dy}{\large dx}.\)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the derivaties of the function given by \( f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)\) and hence find \( f(1) \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \frac{\large cos(x+a)}{\large sin(x+b)}dx$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Evaluate : $ \int \sin x \sqrt {1+\cos\: x} dx.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Verify that \( y=A\cos x - B\sin x\), is a solution of the differential equation \( \frac{\large d^2y}{\large dx^2}+y=0.\)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Solve the following differential equation : $(y^2-x^2)dy = 3xy\: dx.$

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Solve the differential equation : $\large \frac{dy}{dx}-\frac{y}{x}=\bigg( \frac{x-1}{x} \bigg)$$ e^x. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Show that each of the given three vectors is a unit vector : $\large \frac{1}{7}$$ (2\hat i + 3\hat j + 6\hat k),\large\frac{1}{7} $$(3\hat i - 6\hat j + 2\hat k),\large\frac{1}{7} $$(6\hat i + 2\hat j - 3\hat k)$ (Also show that they are mutually perpendicular to each other).

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the equation of the \( \bot \) drawn from point (2, 4, -1) to the line \( \frac{\large x+5}{\large 1}=\frac{\large y+3}{\large 4}=\frac{\large z-6}{\large -9}. \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Find the length and the foot of \( \bot \) drawn from point (2, -1, 5) to the line \( \frac{\large x-11}{\large 10}=\frac{\large y+2}{\large -4}=\frac{\large z+8}{\large -11}. \)

asked Jan 9, 2013 by thanvigandhi_1

0 answers

A speaks truth 8 times out of 10 times. A dice is tossed. He reports that it was 5. What is the probability that it was actually 5?

asked Jan 9, 2013 by thanvigandhi_1

1 answer

The cost of 4 kg onion , 3 kg wheat and 2 kg rice of Rs.60. The cost of 2 kg onion , 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find cost of each item per kg by matrix method.

asked Jan 9, 2013 by thanvigandhi_1

1 answer

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