HSC +2 Maths - Sample 5 Mark Questions - Part3

For Plus2 Maths Part1 and Part2 Important questions , watch these pages.

1) Examine the consistency of the system of equations:

       2x + 3y + 7z = 5,

       3x + y – 3z = 13,

       2x + 19y – 47z = 32.

2) Find the adjoint of the matrix $ \begin{bmatrix} 1\;\;\;\;\;\;\; 2 \\ 3\;\;\; -5 \end{bmatrix} $ and verify the result A(adj A) = (adj A) A = |A|.I

3)       (1) For any vector $ \overrightarrow{r} $ prove that $ \overrightarrow{r}=(\overrightarrow{r}.\overrightarrow{i})\overrightarrow{i}+(\overrightarrow{r}.\overrightarrow{j})\overrightarrow{j}+(\overrightarrow{r}.\overrightarrow{k})\overrightarrow{k} $

          (2) Find the projection of the vector $ 7\overrightarrow{i}+\overrightarrow{j}-4\overrightarrow{k} $ on $ 2\overrightarrow{i}+6\overrightarrow{j}+3\overrightarrow{k} $.

4) Find the vector and Cartesian equation of the sphere on the join of the points A and B having position vectors $ 2\overrightarrow{i}+6\overrightarrow{j}-7\overrightarrow{k} $ and $-2\overrightarrow{i}+4\overrightarrow{j}-3\overrightarrow{k} $ respectively as a diameter. Find also the centre and radius of the sphere.

5) If P represents the variable complex number z, find the locus of P, if |2z-1| = |z-2|.

6) The headlight of a motor vehicle is a parabola reflector of diameter 12 cm and depth 4 cm. Find the position of bulb on the axis of the reflector for effective functioning of the head.

7) Obtain the Maclaurin’s series for $ log_{e}(1+x) $.

8) Verify Lagrange’s law of mean for the function $ f(x)=x^{3}-5x^{2}-3x $ on [1, 3].

9) Given $ w=\frac{x}{x^{2}+y^{2}} $, where x=cos t; y=sin t; Find $ \frac{\mathrm{dw} }{\mathrm{d} t} $

10) Evaluate: $ \int_{0}^{2\pi}sin^{9}\frac{x}{4} \; dx $

11) Solve: $ x\;dy=(y+4x^{5}\;e^{x^{4}}) \;dx $

12) G is a group, $ a, b \;\varepsilon \;G $. Prove that $ (a*b)^{-1}=b^{-1}*a^{-1} $

13) Construct the truth table for $ (p \; \wedge \;q) \;\vee \;(\sim r) $.

14) The overall percentage of passes in a certain examination is 80. If 6 candidates appear in the examination, what is the probability that at least 5 pass in examination?

15)      (1) State and prove cancellation laws on groups.
                                       (or)
           (2) Four coins are tossed simultaneously. What is the probability of getting (i) exactly 2 heads; (ii) at least 2 heads; (iii) almost 2 heads?

16) Find the shortest distance between the parallel lines $ \overrightarrow{r}=(i-j)+t(2i-j+k) $ and $ \overrightarrow{r}=(2i+j+k)+s(2i-j+k) $.

17) If n is a positive integer prove that $ (\sqrt3+i)^{n}+(\sqrt3-i)^{n}=2^{n+1}\;cos(\frac{n\pi}{6}) $

18) Find the equation to the two tangents that can be drawn from the point (1, 2) to the hyperbola $ 2x^{2}-3y^{2}=6 $.

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

20) Prove that $ sin\;x<x<tan\;x, x\epsilon (0,\frac{\pi}{2}) $