1 . Solve the inequality 6 + 5x - x² ≥ 0.

2 . Find the domain and range of the function f(x) = √(6 - x - x²).

3 . Prove that the function f: R → R defined by f(x) = 3x - 1 is bijective.

4 . Expand e^x about x = 0 by using the Maclaurin series.

5 . Find the inverse matrix of the matrix (1 4 1 3 3 -2 0 4 1).

6 . There are 7 men and 3 ladies. Find the number of ways in which a committee of 6 persons can be formed if the committee should have at least one lady.

7 . Find the equation of a parabola having vertex (-1, 2) and directrix x = 4.

8 . a) If A and B be two subsets of universal set U such that n(U) = 350, n(A) = 100, n(B) = 150 and n(A ∩ B) = 50 then find n(A' ∩ B'). b) If a, b, c are in A.P., b, c, d are in G.P. and c, d, e, are in H.P. Then prove that a, c, e are in G.P.

9 . a) Prove that (1 a bc 1 b ca 1 c ab) = (a - b)(b - c)(c - a). b) By using the vector method prove that cos(A - B) = cos A cos B + sin A sin B.

10 . a) Define a parabola with different parts using the figure and Derive the standard equation of parabola y² = 4ax. b) In how many ways letters of the word “ARRANGE” be arranged so that all the vowels are always together?

11 . Solve the inequality 3 + 2x - x² ≥ 0.

12 . Find the domain and range of the function f(x) = √(6 - x - x²).

13 . If a, b, c, and d are in G.P. prove that a², b², c², d² are also in G.P.

14 . Prove that | 1 + x 1 1 | | 1 1 + y 1 | = xyz (1/x + 1/y + 1/z + 1) | 1 1 1 + z |

15 . Find the equation of the ellipse whose latus rectum is 5 and the eccentricity is 1/√2.

16 . If a⃗ · b⃗ = √3 and a⃗ × b⃗ = (1, 2, 2) find the angle between a⃗ and b⃗.

17 . How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, and 6?

18 . a) Prove that (3 + 4i)/(1 - i) + (3 - 4i)/(1 + i) is a real number. b) If x² + y² = 11xy, prove that log((x - y)/3) = 1/2 (log x + log y).

19 . a) Find the Maclourin series of the function f(x) = cosx. b) Take any matrix of order 3 × 3 and express it as a sum of symmetric and skew-symmetric matrix.

20 . a) Find the equation of a hyperbola in standard form having focus (-2,0) and Diretrix x = -1/2. b) In an examination paper on mathematics, 20 questions are set. In how many different ways you can choose 18 questions to answer?

31 . Attempt any SIX questions. [6×5=30] Solve the inequality 3 + 2x − x² ≥ 0.

32 . Find the domain and range of the function f(x) = √6 − x − x².

33 . If a, b, c, and d are in G.P. prove that a², b², c², d² are also in G.P.

34 . Prove that

35 . Find the equation of the ellipse whose latus rectum is 5 and the eccentricity is 1/√2.

36 . If a→. b→ = √3 and a→ × b→ = (1, 2, 2) find the angle between a→ and b→.

37 . How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, and 6?

38 . Attempt any TWO questions. [2×10=20] a) Prove that 3+4i / 1−i + 3−4i / 1+i is a real number. b) If x² + y² = 11xy, prove that log(x−y/3) = 1/2 (log x + log y).

39 . a) Find the Maclourin series of the function f(x) = cosx. b) Take any matrix of order 3 × 3 and express it as a sum of symmetric and skew-symmetric matrix.

40 . a) Find the equation of a hyperbola in standard form having focus (-2,0) and Diretrix x = −1/2. b) In an examination paper on mathematics, 20 questions are set. In how many different ways you can choose 18 questions to answer?