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 $f(x)=\frac{2x+1}{3-x}$

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