# HSC +2 Maths - Practice 10 Mark Questions - Part 1

This page has all useful and important sample Plus2 Maths 10 Mark Questions.

1) If $A=\frac{1}{3}\begin{bmatrix} 2\;\;\;\;\;\; 2\;\;\;\;\;\; 1 \\ -2\;\;\;\;\; 1\;\;\;\;\; 2 \\ 1\;\;\;\; -2\;\;\;\;\; 2 \\ \end{bmatrix}$, prove that $A^{-1} = A^{T}$

2) A small seminar hall can hold 100 chairs. Three different colors (red, blue and green) of chairs are available. The cost of a red chair is Rs. 240, cost of a blue chair is Rs. 260 and the cost of the green chair is Rs. 300. The total cost of chair is Rs. 25,000. Find at least 3 different solution of the number of chairs in each color to be purchased.

3) For what value of $\mu$ the equations.

$x + y + 3z = 0; \;4x + 3y + \mu z = 0; \;2x + y + 2z = 0$ have a (i) trivial solution. (ii) non-trivial solution.

4) Prove by vector method: cos(A + B) = cos A cos B – sin A sin B

5) Find $(\overrightarrow{a}\times\overrightarrow{b}).(\overrightarrow{c}\times\overrightarrow{d})$ if $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}, \;\overrightarrow{b}=2\overrightarrow{i}+\overrightarrow{k}, \; \overrightarrow{c}=2\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}, \; \overrightarrow{d}=\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}$

6) If the vector and cartesian equation of the plane passing through the points A(1, –2, 3) and B(-1, 2, –1) and is parallel to the line $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z-1}{4}$

7) Find the coordinates of the centre and the radius of the sphere whose vector equation is: $\overrightarrow{r}^2-\overrightarrow{r}.(8\overrightarrow{i}-6\overrightarrow{j}+10\overrightarrow{k})-50=0.$

8) If $(a_1+ib_1)(a_2+ib_2)...........(a_n+ib_n)=A+iB$, prove that

(i) $(a_1^2+b_1^2)(a_2^2+b_2^2)...........(a_n^2+b_n^2)=A^2+B^2$

(ii) $tan^{-1}(\frac{b_1}{a_1})+tan^{-1}(\frac{b_2}{a_2})+..............+tan^{-1}(\frac{b_n}{a_n})=k\pi+tan^{-1}(\frac{B}{A}), \;k\;\varepsilon \;Z$

9) Solve the equation $x^4-4x^3+11x^2-14x+10=0$ if one root is 1+2i.

10) If $\alpha$ and $\beta$ are the roots of $x^2-2x+2=0$ and $cot \; \Theta=y+1$, show that $\frac{(y+\alpha)^n-(y+\beta)^n}{\alpha - \beta}=\frac{sin \; n\Theta}{sin^n \; \Theta}$

11) Find the values of $(\sqrt3+i)^{\frac{2}{3}}$

12) A cable of suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally, The distance between two towers is 1500 ft, the points of support of the cable on the towers are 200 ft above the road way and the lowest point on the cable is 70 ft above the roadway. Find the vertical distance to the cable (parallel to the roadway) from a pole whose height is 122 ft.

13) Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram: $16x^2+9y^2+32x-36y=92$

14) Prove that the line $5x+12y=9$ touches the hyperbola $x^2-9y^2=9$ and find its point of contact.

15) 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).

OR

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.