This is the final part of the Question paper series AA1 on Mathematics. We present some 5 mark questions for solving in this post. If you wish to refer to previous sections, the navigation links are provided below.
Part 1- Questions
Part 2- Questions
Part 3- Questions
Part 1- Questions
Part 2- Questions
Part 3- Questions
46) (i) Show the points whose position vectors $ 4\overrightarrow{i}-3\overrightarrow{j}+\overrightarrow{k} $, $ 2\overrightarrow{i}-4\overrightarrow{j}+5\overrightarrow{k} $, $ \overrightarrow{i}-\overrightarrow{j}$ form a right-angled triangle.
(ii) A force given by $ 3\overrightarrow{i}+2\overrightarrow{j}-4\overrightarrow{k} $ is applied at the point (1, –1, 2). Find the moment of force about the point (2, –1, 3).
47) (i) Show that the following two lines are skew lines and find the distance between them.
$ \overrightarrow{r}=(3\overrightarrow{i}+5\overrightarrow{j}+7\overrightarrow{k}) + t(\overrightarrow{i}-2\overrightarrow{j}+\overrightarrow{k}) $ and $ \overrightarrow{r}=(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}) + s(7\overrightarrow{i}+6\overrightarrow{j}+7\overrightarrow{k}) $
48) (i) Find the area of parallelogram whose diagonals are represented by $ 2\overrightarrow{i}+3\overrightarrow{j}+6\overrightarrow{k} $ and $ 3\overrightarrow{i}-6\overrightarrow{j}+2\overrightarrow{k} $.
(ii) Find the angle between the line $ \frac{x+2}{3}=\frac{y+1}{-1}=\frac{z-3}{-2} $ and the plane $ 3x+4y+z+5=0. $
49) If $ \begin{bmatrix}-1\;\;\;\;\; 2\;\;\;\;\; -2 \\ 4\;\;\;\;\; -3\;\;\;\;\; 4 \\ 4\;\;\;\;\; -4\;\;\;\;\; 5 \end{bmatrix} $, show that $ A=A^{-1} $.,
50) Solve the following system of linear equation by determinant method: 2x+3y=8; 4x+6y=16.
51) If $ \overrightarrow{a}=2\overrightarrow{i}+3\overrightarrow{j}-5\overrightarrow{k},\; \overrightarrow{b}=-\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k},\;\overrightarrow{c}=4\overrightarrow{i}-2\overrightarrow{j}+3\overrightarrow{k} $, show that $ (\overrightarrow{a} \times \overrightarrow{b}) \times \overrightarrow{c} \neq \overrightarrow{a} \times (\overrightarrow{b} \times \overrightarrow{c}) $
52) Find the shortest distance between the skew lines $ \frac{x-6}{3}=\frac{y-7}{-1}=\frac{z-4}{1} $ and $ \frac{x}{-3}=\frac{y+9}{2}=\frac{z-2}{4} $.
53) Find the vector and cartesian equation of the sphere whose centre is (1, 2, 3) and which passes through the point (5, 5, 3).
54) If p represents the complex number z, find the locus of P, if |2z-3|=2.
55) (i) Express $ -1+i\sqrt3 $ in polar form.
(ii) Find the real and imaginary parts of $ \frac{1}{1+i} $.
56) Find the locus of a point which moves so that the sum of its distances from (3, 0) and (-3, 0) is 9.
57) The tangent at any point of the rectangular hyperbola $ xy=c^2 $ makes intercepts a, b and the normal at the point makes intercepts p, q on the axes, prove that ap+bq=0.
58) Find the equation of the ellipse if the foci are $ (\pm3, 0) $ and the vertices are $ (\pm5, 0) $
59) Obtain the Maclaurin’s series expansion for: $ \frac{1}{1+x} $
60) Find the intervals of Concavity, and the pints of inflection of the function $ f(x) = 2x^3+5x^2-4x $
61) Evaluate: $ lim_{x\rightarrow 1} \; x^{\frac{1}{x-1}} $
62) Find $ \frac{\partial w}{\partial u} $ and $ \frac{\partial w}{\partial v} $, if $ w=sin^{-1}xy $, where x=u+v, y=u-v.
63) (i) Prove that the points representing the complex numbers 2i, 1+i, 4+4i and 3+5i on the Argand plane are the vertices of a rectangle.
(or)
(ii) Find the two positive numbers whose product is 100 and whose sum is minimum.
Ten mark questions:
64) If $ \overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}, \overrightarrow{b}=2\overrightarrow{i}+\overrightarrow{k}, \overrightarrow{c}=2\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ and $ \overrightarrow{d}=\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k} $, then verify $ (\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=[\overrightarrow{a} \overrightarrow{b}\overrightarrow{d}]\overrightarrow{c}-[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]\overrightarrow{d} $
65) (i) Prove by vector method: Sin (A+B) = SinA CosB + CosA SinB
(ii) Prove that the altitudes of a triangle are concurrent.
66) Find the vector and cartesian equations of a plane containing the line $ \frac{x-2}{2}=\frac{y-2}{3}=\frac{z-1}{-2} $ and passing through the point (-1, 1, –1)
67) A bag contains 3 types of coins namely Re. 1, Rs. 2, Rs. 5. There are 30 coins amounting to Rs. 100 in total. Find the number of coins in each category.
68) Solve the non-homogenous equations of three unknowns by using determinants 2x+2y+z=5; x-y+z=1; 3x+y+2z=4. If $ A=\begin{bmatrix} -1\;\;\;\;\; 2 \\ 1\;\;\;\;\; -4 \\ \end{bmatrix} $, verify the result $ A(adj A) = (adj A)A = |A|I_2 $.
69) Find the vector and cartesian equations of the plane passing through the points (-1, 1, 1) and (1, –1, 1) and perpendicular to the plane x+2y+2z=5.
70) Show that the lines $ \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z}{3} $ and $ \frac{x-2}{1}=\frac{y-1}{2}=\frac{-z-1}{1} $ intersect and find their point of intersection.
71) Find the vector and cartesian equations of the plane passing through the points (2, 2, –1), (3, 4, 2) and (7, 0, 6).
72) Solve the equation: $ x^7+x^4+x^3+1=0 $
73) If $ a=cos\;2\alpha +i\;sin\;2\alpha,\; b=cos\;2\beta +i\;sin\;2\beta,\; c=cos\;2\gamma +i\;sin\;2\gamma $, prove that:
(i) $ \sqrt{abc}\;+\;\frac{1}{\sqrt{abc}}=2\; cos(\alpha+\beta+\gamma) $
(ii) $ \frac{a^2b^2+c^2}{abc}=2\;cos\;2(\alpha+\beta-\gamma) $
74) Find the equation of the hyperbola if its asymptotes are parallel to x+2y-12=0 and x-2y+8=0, (2, 4) is the centre of the hyperbola and it passes through (2, 0).
75) Find the eccentricity, vertices, focus and centre of the ellipse $ 9x^2+25y^2-18x-100y-116=0 $.
76) Prove that the line 5x+12y=9 touches the hyperbola $ x^2-9y^2=9 $ and find its point of contact.
77) Verify $ \frac{\partial^2 u}{\partial x \partial y} = \frac{\partial^2 u}{\partial y \partial x} $ for the function $ u=tan^{-1}(\frac{x}{y}) $.
78) A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of $ 2\;m^3 /min $, find the rate at which the water level is rising when the water is 3m deep.
79) Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
80) (i) Prove by vector method that Cos (A+B) = Cos A Cos B – Sin A Sin B.
(or)
(ii) The girder of a railway bridge is in the parabolic form with span 100 ft and the highest point in the arch is 10 ft above the bridge. Find the height of the bridge at 10 ft to the left or right from the mid point of the bridge.
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