Q: If 3 tan θ = cot θ where 0 < θ < π/2 , then what is the value of θ ?

3tan θ = cot θ 3tan θ = 1/tan θ tan2θ = 1/3 tan θ = 1/√3 θ = π/6

Q: Consider the following for real numbers α, β, γ and δ ? 1) sec α = 1/4 2) tan β = 20 3) cosec γ = 1/2 4) cos δ = 2 How many of the above statements are not possible?

Sec of any number can never be less than 1 tan can take any value from -∞ to + ∞ cosec of any number can never be less than 1 cos of any number can never be greater than 1 so option 1,3,4 are not possible

Question 1

If 3x= 4y= 12 , then z is equal to

Accepted Answer

Taking log of all 3 we get xln3 = yln4 = zln12 = k z = k/ln12 = k / ln( 3*4) = k/ln3 + ln4 = k / (k/x +k/y) = xy / (x+y)

Question 2

If (4a +7b) (4c-7d) = (4a -7b) (4c +7d, then which one of the following is correct ?

Accepted Answer

(4a+7b)(4c-7d) = (4a-7b)(4c+7d) (4a+7b)/(4a-7b) = (4c+7d)/(4c-7d) Using componendo and dividendo (4a+7b)+(4a-7b) / (4a+7b)-(4a-7b) = (4c+7d)+(4c-7d) / (4c+7d)-(4c-7d) Or 8a/14b = 8c/14d Or a/b = c/d

Question 3

What is the value of 2+2+2+2.......

Accepted Answer

Let 2+√2+√2..... = t 2+√t = t t - 2 = √t t = 4

Question 4

A 12 m long wire is cut into two pieces, one of which is bent into a circle and the other into a square enclosing the circle. What is the radius of the circle?

Accepted Answer

As it is clear that 2r = a where a is the side of the square and R is the radius of circle It is given that 2πr + 4a = 12 a = 12/(π+4)

Question 5

Two equal circles intersect such that each passes through the centre of the other. If the length of the common chord of the circles is 10√3 cm, then what is the diameter of the circle?

Accepted Answer

Let there be 2 circles with centre O1 and OAB is the common chord Since both passes through the center of each other as shown in figure So O1O is the radius of both Let O1O = r = AO1= AO AX = AB / 2 = 5√3 cm (since OX perpendicular to chord bisects it) AOO1 forms an equilateral triangle with on side = radius = r Sin 60 = √3/2 = AX / AO = 5√3/r So r = 10 cm So diameter = 20 cm

Question 6

If the mean of m observations out of n observations is n and the mean of remaining observations is m, then what is the mean of all n observations?

Accepted Answer

Mean of m observations is n Mean of n-m observations is m So total = nm+ (n-m)m Total observations = n Mean = Total / Total observations = (2mn-m^2)/n = 2m –m^2/n

Question 7

If p and q are the roots of the equation x2 –15x + r = 0 and p –q = 1, then what is the value of r ?

Accepted Answer

In the below equation, x2–15x + r = 0 sum of roots = p + q = -(-15)/1 = 15 (sum of roots for equation ax2+ bx + c is -b/a) product of roots = pq = r/1 = r (product of roots for equation ax2+ bx + c is c/a) given p –q = 1 also we know that p+q = 15 subtracting the squares of both (p+q)2 + (p-q)2 = 15^2–1 p2 + q2 + 2pq –p2 –q2 +2pq = 225 -1 4pq = 224 4r = 224 r = 56

Question 8

If 3 tan θ = cot θ where 0 < θ < π/2 , then what is the value of θ ?

Accepted Answer

3tan θ = cot θ 3tan θ = 1/tan θ tan2θ = 1/3 tan θ = 1/√3 θ = π/6

Question 9

Consider the following for real numbers α, β, γ and δ ? 1) sec α = 1/4 2) tan β = 20 3) cosec γ = 1/2 4) cos δ = 2 How many of the above statements are not possible?

Accepted Answer

Sec of any number can never be less than 1 tan can take any value from -∞ to + ∞ cosec of any number can never be less than 1 cos of any number can never be greater than 1 so option 1,3,4 are not possible

Simplification Questions