+2 Maths Important Questions AA5- Part 1

In this post, we provide the readers with some six mark questions in Mathematics for solving. There are totally 15 questions for solving,which are considered important from an examination angle. Take these questions are practice questions only, and prepare them well. You should also write the steps clearly to solve every question and must not jump steps in between. Share your answers with us in the comments section.

1) Solve the matrix inversion method of the following system of linear equation:

2x-y+z=7;        3x+y-5z=13;        x+y+z=5

2) Forces $ 2\overrightarrow{i}+7\overrightarrow{j}, \:2\overrightarrow{i}-5\overrightarrow{j}+6\overrightarrow{k}, \:-\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} $ act at a point P whose position vector is$ 4\overrightarrow{i}-3\overrightarrow{j}-2\overrightarrow{k} $. 
Find the moment of the resultant of three forces acting at P about the point Q whose position vector is $ 6\overrightarrow{i}+\overrightarrow{j}-3\overrightarrow{k} $

3) If $ \overrightarrow{a}=2\overrightarrow{i}+3\overrightarrow{j}-\overrightarrow{k}, \: \overrightarrow{b}=-2\overrightarrow{i}+5\overrightarrow{k}, \:\overrightarrow{c}=\overrightarrow{j}-3\overrightarrow{k} $. Verify that $ \overrightarrow{a} \times (\overrightarrow{b} \times \overrightarrow{c})=(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c} $

4) Find the modulus and argument of the following complex numbers:
a) i) i
   ii) $ -\sqrt2+i\sqrt2 $

b) If $ z^2=(0, 1) $ find z.

5) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from the centre of the reflector.

6) Find the equation of the ellipse whose vertices are (-1, 4) and (-7, 4) and eccentricity is 1/3.

7) A water tank has the shape of an inverted circular cone with base radius 2 meters and height 4 meters. 
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.

8) Find the intervals of concavity and the points of inflection of the function

$ f(x)=(x-1)^{\frac{1}{3}} $

9) a) Using Euler’s theorem prove that $ u=xy^2 \: sin(\frac{x}{y}) $, show that

$ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=3u $

    b) If U=(x-y) (y-z) (z-x), then show that $ U_x+U_y+U_z=0 $

10) Evaluate:

i) $ \int_{0}^{\frac{\pi}{2}}cos^9 \:x \: dx $

ii) $ \int_{0}^{1}xe^{-2x}\: dx $

11) Evaluate the problem using properties of integration:

$ \int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}x^3 \:cos^{3}\: x \:dx $

12) Solve: $ (x^2-yx^2)dy+(y^2+xy^2)dx=0 $

13) Solve: $ \frac{\mathrm{d} y}{\mathrm{d} x}+\frac{y}{x}=sin(x^2) $

14) State and prove reversal law of the group (G ; *)

15) Show that the set G of all positive rationals forms a group under the composition * defined by

a*b = ab/3, for all a,b E G

                                           (OR)

The probability density function of a random variable x is:

$ f(x)=\begin{cases} kx^{\alpha-1} \: e^{-\beta x^{\alpha}} \ x, \alpha, \beta>0 \\  0 \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \text{ elsewhere }\end{cases} $

Find (i) k                          (ii) P(X>10)