The smallest positive x satisfying the equation
 is

a)

b)

c)

d)

The question contains two statements: Statement 1(assertion) and Statement 2(reason). The question has four alternative choices, only one of which is correct. You have to select one of the codes (a), (b), (c), (d)
Statement 1: The equation sin (cosx) = cos (sinx) does not possess real roots

Statement 2: If sinx > 0 then 2nπ < x < (2n + 1)π, n ε I

a) Statement 1 is false. Statement 2 is true.

b) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1.

c) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1.

d) Statement 1 is true. Statement 2 is false.

The base of a cliff is circular. From the extremities of a diameter of the base, the angle of elevation of the top of the cliff are 30 and 60. If the height of the cliff is 500m, then the diameter of the base of the cliff is

a)  m

b)  m

c)

d)  m

The maximum value of  in the interval  is attained at

a)

b)

c)

d)

Prove that
 

Prove that

Prove that
 

Sum the series
 

Obtain explicit value of the series
 

Obtain explicit value of the series
 

Prove that
 

Prove that
 

Prove that
 

Prove that
 

Compute the product
 .  .  .  

Compute the product
 .  .  .

Show that if angles a and b satisfy
 
Then prove that sina sinb = ± sin45°

Find the range of the function
f : ℝ → ℝ, f (x) = (sinx + 1)(cosx + 1)

Let a, b, c, d ε [0, π] be such that
2cosa + 6cosb + 7cosc + 9cosd = 0 and
2sina – 6sinb + 7sinc – 9sind = 0
Prove that 3cos (a + d) = 7cos (b + c)

Find the value of the parameter u for which the expression
 
does not depend on x

a) x = (k ± 1) π

b)

c)

d)

Consider a triangle whose angles satisfy the equality
 
 

a) ABC is a right angled triangle

b) ABC is an isosceles triangle

c) ABC is an equilateral triangle

d) None of the above

Prove that
 

Solve the equation
 

a) (2k + 1) π

b) (4k + 1) π

c)

d)