Math

What is an arrangement of the natural numbers called?
a. Induction
b. Permutation
c. Combination
d. Expectation

Let ω be the cube roots of unity. Under which operation is the set closed?
a. Addition
b. Subtraction
c. Multiplication
d. Division

What is the domain of the given function?
a. x ≥ 1 or x ≤ −1
b. (−∞, ∞)
c. −1 < x < 1
d. −1 ≤ x ≤ 1

ABCD is a parallelogram. Which one of the following represents area of the parallelogram?
a. Magnitude of vector product of two vectors along AB and BD
b. Magnitude of vector product of two vectors along AB and DC
c. Magnitude of vector product of two vectors along AC and BC
d. Magnitude of vector product of two vectors along AB and AD

If a conic section has eccentricity e, what is the equation of that conic section?
a. x²/a² − y²/b² = 1
b. x²/a² + y²/b² = 1
c. x²/a² + y²/a² = 1d. x²/b² + y²/b² = 1

If the given trigonometric equation holds for integer n, what is the general value of the angle?
a. 2nπ ± 2π/3
b. nπ + (−1)ⁿπ/3
c. nπ − π/3
d. nπ + π/3

Let A and B be two dependent events. If the given probabilities are known, what is the value of the required probability?
a. Equal to P(B/A)
b. Equal to P(A)
c. Less than P(A ∩ B)
d. Less than P(B/A)

The edge of a cube increases from the given value to the given value. What would be the approximate increment in volume?
a. 10³ cm³
b. 10.025³ cm³
c. 7.5187 cm³
d. 7.5 cm³

What is the integrating factor of the differential equation given below?

cos²x(dy/dx) + y = 1

a. tan x

b. e^{tan x}
c. sec² x
d. e^(sec x)

What is the number of solutions of the system of linear equations given below?

x + y = 5 and x + y = 7

Forces F₁ and F₂ are acting along ceiling and floor of a rectangular room. What is the nature of the forces?
a. Like
b. Unlike
c. Collinear
d. Parallel

For any positive integer n,

Given expression

a. How many terms are there in the expressions?

b. Write the binomial coefficients in the expansion.

c. Write the general term of the expansion.

d. Write the relation among C(n,r−1), C(n+1,r) and C(n,r).

e. What is the value of C0 + C1 + C2 + ⋯ + Cn?

a) Using the principle of mathematical induction, show that: 1+2+3+…. +n < (1/8)(2n+1)²
b) Find the quadratic equation whose one of the roots is 2+√3.

a) Given y = sin^-1x and y>0, express cosy and tany in terms of x.
b) If a, b and c are any three vectors such that a × b = a × c for a ≠ (0,0)., show that b = c.

The price in Rupees and demand in unit of 6 days of a week is given as:

Price (X)	10	12	13	12	16	15
Demand (Y)	40	38	43	45	37	43

Calculate the Pearson’s coefficient of correlation and the regression coefficients of Y on X.

a) Define L.Hospital's rule. 
b) Write the slope of the tangent and normal to the curve y = f(x) at (x₁,y₁). 
c) Write the integral of ∫ 1/(x²+a²) dx 
d) What is the integral of ∫ Sinhx dx 

Using simplex method, maximize P(x,y)=15x+10y subject to 2x+y≤10, x+3y≤10, x, y≥ 0

A particle is projected with a velocity 'v' and greatest height is 'H', prove the horizontal range R is:

R = √(H(v²/2g − H))

OR,

The cost function C(x) in thousands of rupees for producing x units of maths textbooks is given by

C(x)=30+20x-0.5x², 0≤x≤15

a) Find the marginal cost function. 
b) Find the marginal cost for producing 12,000 maths textbooks.

a) Using matrix methods, solve the following system of linear equations:
x+y+z = 4, 2x+y-3z = -9, 2x-y+z = -1

b) Apply De-moivre's theorem to find the value of [2(cos15° + isin15°)]⁶

c) Prove that (1 + 1/1! + 1/2! + 1/3! + ⋯)(1 − 1/1! + 1/2! − 1/3! + ⋯) = 1 

a) Find the direction cosines of the line joining the points (4,4,-10) and (-2,2,4). 
b) Find the angle between the two diagonals of a cube. 
c) Find the vertices of the conic section:
16(y-1)² -9(x-5)² = 144. 

a) If the limiting value of (f(x)-5)/(x-3) at x=3 is 2 by using L'Hospital's rule, find the appropriate value of f(x).

b) Write any one homogeneous differential equation in (x,y) and solve it

c) The concept of anti-derivative is necessary for solving a differential equation. Justify this statement with example.