# +2 Maths Important Questions AA5- Part 2

In this post, we present some sample 10 mark questions in +2 Mathematics subject. There are a total of 15 ten mark questions to be solved and practice these questions well from an examination angle. Post your answers or solution approach to these questions and share it with rest of the readers
1) Solve the non-homogenous equations of three unknowns:

x+y+2z=6;                 3x+y-z=2;                 4x+2y+z=8

2) Prove that cos (A-B) = cos A cos B + sin A sin B

3) Find the vector and cartesian equation of the plane through the points (1, 2, 3) and (2, 3, 1)

perpendicular to the plane 3x – 2y +4z – 5 =0

4) If $\alpha$ and $\beta$ are the roots of the equation

$x^2-2px+(p^2+q^2)=0$ and $tan \Theta=\frac{q}{y+p}$,

show that $\frac{(y+\alpha)^n-(y+\beta)^n}{\alpha - \beta}=q^{n-1}\frac{sin \: n\Theta}{sin^n \: \Theta}$

5) P represents the variable complex number z. Find the locus of P, if $arg(\frac{z-1}{z+3})=\frac{\pi}{2}$

6) Find the equations of directrices, latus rectum and lengths of latus rectums of the following ellipses

$3x^2+2y^2-30x-4y+23=0$

7) Find the equation of the hyperbola whose foci are $(0, \pm \sqrt10)$ and passing through (2, 3).

8) Find the equation of the asymptotes to the hyperbola:

i) $36x^2-25y^2=900$

ii) $8x^2+10xy-3y^2-2x+4y-2=0$

9) If the curve $y^2=x$ and $xy=k$ are orthogonal then prove that $8k^2=1$

10) Show that the volume of the largest right circular cone that can be inscribed in a sphere of

radius a is 8/27 (volume of the sphere).

11) Find the length of the curve $x=a(t-sint), \: y=a(1-cost)$

between $t=0 \: and \: \pi$

12) Find the area of the loop of the curve $3ay^2=x(x-a)^2$

13) Solve: $\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{y(x-2y)}{x(x-3y)}$

14) Solve the differential equation: $(D^2-6D+9)y=x+e^{2x}$

15) Show that the set G of all matrices of the form $\begin{bmatrix} x\:\:\:\:\: x \\ x\:\:\:\:\: x \\ \end{bmatrix}$,

where x E R-{0} is a group under matrix multiplication.

(OR)

In a Binomial distribution, if n=5 and P(X=3)=2P(X=2) find P. Also find P(X=2)

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)