Recent questions tagged modelpaper

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

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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

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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

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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

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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

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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

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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

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

asked Jan 9, 2013 by thanvigandhi_1

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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

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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

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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

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

asked Jan 9, 2013 by thanvigandhi_1

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

asked Jan 9, 2013 by thanvigandhi_1

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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

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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

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

asked Jan 9, 2013 by thanvigandhi_1

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

asked Jan 9, 2013 by thanvigandhi_1

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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

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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

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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

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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

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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

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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

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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

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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

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

asked Jan 9, 2013 by thanvigandhi_1

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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

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

asked Jan 9, 2013 by thanvigandhi_1

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

asked Jan 9, 2013 by thanvigandhi_1

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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

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

asked Jan 9, 2013 by thanvigandhi_1

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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

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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

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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

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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

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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

Using matrix method solve the following system of linear equations : $x+2y+z=7; x+3z=11 ; 2x-3y = 1$

asked Jan 9, 2013 by thanvigandhi_1

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Find the equation of the line through the point P(3, 4) which cuts from the first quadrant a triangle of the minimum area. Also find the area of the triangle.

asked Jan 9, 2013 by thanvigandhi_1

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Find the area of the region bounbed by $ y^2=4x, \: x=1, \: x=4$ and $ x$ - axis in the first quadrant.

asked Jan 9, 2013 by thanvigandhi_1

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Evaluate : $ \int_0^2 (x^2+x+1)dx$ as limit of a sum.

asked Jan 9, 2013 by thanvigandhi_1

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Evaluate : $ \int_0^{\pi} \large\frac{dx}{5+4cos\: x}. $

asked Jan 9, 2013 by thanvigandhi_1

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Show that the line joining the origin to the point (2, 1, 1) is perpendicular the line determined by the point (3, 5, -1),(4, 3, -1).

asked Jan 8, 2013 by thanvigandhi_1

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There are two types of fertilisers $ F_1 and F_2. F_1$ contain 10% nitrogen and 6% phosphoric acid and $ F_2$ consists of 5% of nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14kg of phosphoric acid for the crop. If $ F_1 $ costs Rs.6/kg and $ F_2 $ costs Rs. 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

asked Jan 8, 2013 by thanvigandhi_1

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An electronic assembly consists of two subsystems say A and B. From the previous testing procedures, the following probabilities are assumed to be known : P(A fails)=0.2, P(B fails alone)=0.15. Evaluate the following probabilities : (i) P(A fails / B has failed) (ii) P(A fails alone).

asked Jan 8, 2013 by thanvigandhi_1

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Let * be a operation on N given by ab=H.C.F, (a, b), a,b $ \in $ N. Write the value of 224.

asked Jan 8, 2013 by thanvigandhi_1

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Using principal value, calculate the following : $ \sin^{-1} \bigg( \sin \large\frac{3\pi}{5} \bigg). $

asked Jan 8, 2013 by thanvigandhi_1

1 answer

Find the value of x if $ \begin{bmatrix} 3x+y & -y \\ 2y-x & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ -5 & 3 \end{bmatrix} $

asked Jan 8, 2013 by thanvigandhi_1

1 answer

If A is an invertible matrix of order 3 and |A|=5 then find [adj.A].

asked Jan 8, 2013 by thanvigandhi_1

1 answer

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