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# Mathematics - I

### Examination 2019

#### Group B

Attempt any Six question

2. In a class of 100 students 40 students failed in Mathematics, 70 failed in English and 20 failed in both subject. Find
a. How many students passed in both subject ?
b. How many students passed in Mathematics only?
c. How many student failed in mathematics only?

3. Find the domain and range of the function $f\left(x\right)=\frac{2x+1}{3-x}$

4. Find the Maclurin series of the function $f\left(x\right)=\mathrm{sinx}$

5. Prove that $\left[\begin{array}{ccc}1& x& {x}^{2}\\ 1& y& {y}^{2}\\ 1& z& {z}^{2}\end{array}\right]$ = (x - y)(y - z)(z - x)

6. Find a unit vector perpendicular to the plane containing points p(1,-1,0),Q(2,1,-1) and R(-1,1,2)

7. In how many ways can be letter of words "Sunday" be arraged? How many of these arrangement begin with S? How many begin with S and don't end with y?

8. If x + iy = $\sqrt{\frac{1+i}{1-i}}$ then show that x2 + y2 = 1

#### Group C

Attempt any Two question

9.a) Define conic section. Find the coordinates of vertices, eccentricity and foci of the ellipse
9x2 + 4y2 - 18x - 16y - 11 = 0

b) If T : R2 ⟶ R3 defined by T(x1, x2) = (x1 + x2, x2,x1) be the linear transformation. then find matrix associated with linear map T.

10. Define irrational number. Prove that √2 is an irrational number.

If function f: R ⟶ R defined by f(x) = 2x + 1 and g:R ⟶ R defined by g(x) = x2 - 2. Find the formulae for composite function f * g and g * f and also verify that f * g ≠ g* g.

11. a) If arithmetic mean, geometic mean

I there is