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Group A
Rewrite the correct options for each question in your answer sheet. 11x1=11
1. The property of cube roots of unity where ⩊ the cube root of unity is.
A) 1 + ⩊ + ⩊ˆ2 = 0
B) ⩊ˆ3 + 1 = 0
C) ⩊ = -⩊ˆ2
D) ⩊ + ⩊ˆ2 = 1
2. Which of the following is not true?
A) The set of integers under addition is an abelian group.
B) The set of two order matrices is an abelian group under multiplication.
C) The set of rational numbers under multiplication is an abelian group.
D) The set of natural numbers is an abelian group under addition.
3. Sin-1 x is equal to
A) cos-1 √xˆ2 - 1
B) tan-1 (√1 - xˆ2)
C) cosec-1 √1 - xˆ2
D) cos-1 √1 - xˆ2
A) nπ +- π/6
B) 2nπ + π/6
C) 2nπ + 5π/6
D) nπ + 5π/6
B) a ⃗ˆ2 - b ⃗ˆ2
C) 2 (a ⃗ x b ⃗ )
D) 2 (a ⃗ˆ2 - b ⃗ˆ2)
6. The eccentricity of the ellipse xˆ2/25 + yˆ2/16 = 1 is
A) 3/5
B) 5/3
C) 3/4
D) 4/3
7. The range of probability of any event A is
A) 0 ≤ P(A) ≤ 1
B) 0 ≤ P(A) < 1
C) -1 ≤ P(A) ≤ 1
D) 0 < P(A) ≤ 1
8. Tangent to the curve y=xˆ3 at x = 2 and x = -2 are
A) parallel
B) perpendicular
C) intersecting
D) Does not exist
9. The integrating factor of different equation sinx dy/dx + ycosx = x sinx is
A) sin x
B) cos x
C) tan x
D) cot x
10. During the process of solving a system of linear equations in x, y, and z if we get 0 x z = 0, then the system is
A) inconsistent and has no solution
B) consistent and has a unique solution
C) inconsistent but has no solution
D) consistent and has infinitely solution
11. A force acts on a body of mass m for a time 't' and changes its velocity from 'u' to 'v'. The change in momentum of the body in time t is
A) mu -mv / t
B) mv - mu / t
C) tm - tv / uv
D) u - v / mt
Or
In an input-output model if A = input coefficients of the matrix. 1 be unit matrix then each diagonal element of the (1-A) is positive and [1-A] is positive then the system is called
A) viable
B) unviable
C) unit elastic
D) elastic
Group B
8x5=40
12. a) Define binomial expansion. (1)
b) Find the middle term of (2 + 3x)ˆ30. (2)
c) Find the sum of n binomial coefficients in the expansion of (l + x)ˆn when x=1. (1)
d) What do you mean by the argument of a complex number? (1)
13. a) Prove that 1ˆ2 + 2ˆ2 + 3ˆ2 +...+ nˆ2 = n (n+1) (2n+1) / 6 by mathematical induction. (4)
b) Find the sum of the square of the first 20 natural numbers. (1)
14. a) Find the equation of the plane through (-3, 2, -1) and parallel to the plane 2x+5y-2 = 0. (2)
b) Using the vector method, prove that SinA / a = SinB / b = SinC / c in any ∆ABC. (3)
15. a) Find the derivative of (tanx)ˆlogx. (2)
b) Solve: (1+x)dy=dx(1+xy-x). (3)
16. The regression equations related to age (x) and blood pressure (y) are given below:
0.22x - y + 81.1175 = 0 and x - 1.623y + 105.695 = 0
a) Find the regression coefficient. (2)
b) Find the correlation coefficient. (1)
c) Find the mean of x and mean of y. (2)
17. a) Define proper rational fraction with an example integrate ∫ 2xˆ2 + 3 / xˆ3 + 3xˆ2 + 2x dx using concept of partial fraction. (1+4)
18. Using the simplex method, maximize Z (x,y)=7x+y subject to constraints 3x + 4y ≤ 12, y - x ≤ 2, x, y ≥ 0. (5)
19. A particle is thrown with an initial velocity of 120msˆ-1 at an angle of 30° above the horizon. if g= 10msˆ-2 then find
a) the time to attain the highest point. (1)
b) the time of flight (1)
c) the horizontal range (1)
d) the greatest height reached. (1)
e) the maximum horizontal range. (1)
Or
Given is the transition matrix. (5)
| Producing sector | Purchasing sector | Final demand | |
|---|---|---|---|
| Agriculture | Industry | ||
| Agriculture Industry | 300 400 | 600 1200 | 100 400 |
Group C
3x8=24
20. A mixture is to be prepared of three foods A, B, and C which contains nutrients P, Q, and R as shown in the table below. The total quantity of 30 units of P, 36 units of Q, and 30 units of R respectively be required.
| Foods | P | Q | R |
|---|---|---|---|
| A | 2 | 2 | 4 |
| B | 3 | 4 | 0 |
| C | 4 | 3 | 5 |
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