Q.1. Suppose sin3x.sin 3x =cos mx is an identity in x, where C_{0}, C_{1}, ___C_{n} are constants, and C_{n} ≠ 0. Then the value of n is _________. (1981  2 Marks)
Ans. 6
Sol. sin^{3}x . sin 3x =
cos mx
sin^{3}x.sin 3x =[3 sinx – sin3x] sin 3x
=
=
= [ cos 6x + 3 cos 2x  3 cosx  1]
We observe that on LHS 6 is the max value of m.
∴n = 6
Q.2. The solution set of the system of equations x + y =, cosx + cosy = where x and y are real, is _________. (1987  2 Marks)
Ans. φ
Sol. The equations are x + y = 2π /3 ... (i)
cos x + cos y = 3/2 ...(ii)
From eq.
(ii)
⇒ [Using eq. (i)]
⇒
⇒
Which has no solution.
∴ The solution of given equations is φ.
Q.3. The set of all x in the interval [0,π] for which 2sin^{2} x – 3 sin x + 1 ≥ 0, is _________. (1987  2 Marks)
Ans.
Sol. We have 2 sin^{2} x – 3 sin x + 1 ≥ 0
⇒ (2 sin x – 1) (sin x – 1) ≥ 0
⇒
But we know that sin x ≤ 1 and sin x ≥ 0 for x ∈ [0, π]
⇒ either sin x = 1 or
⇒ either x = π/2 or
Combining, we get
Q.4. The sides of a triangle inscribed in a given circle subtend angles α, β and γ at the centre. The minimum value of the arithmetic mean of andis equal to _________. (1987  2 Marks)
Ans.
Sol. We know that A.M. ≥ G.M.
⇒ Min value of AM. is obtained when AM = GM
⇒ The quantities whose AM is being taken are equal.
i.e., Cos
⇒ sin α = sin β = sinγ
Also α +β + γ= 360° ⇒ α = β = γ=120° = 2π/3
∴ Min value of A.M.
Q.5. The value of is equal to _________. (1991  2 Marks)
Ans.
Sol.
=
Q.6. If K = sin(π/18) sin(5π/18) sin(7π/18), then the numerical value of K is _________ . (1993  2 Marks)
Ans.
Sol.
[Using
Q.7. If A > 0,B>0 and A + B = π/3, then the maximum value of tan A tan B is_________ . (1993  2 Marks)
Ans.
Sol. A + B = π/3 ⇒ tan (A +B ) =
[where y = tan A tan B]
⇒ tan^{2} A + (y – 1) tan A + y = 0
For real value of tan A, 3(y – 1)^{2} – 4y ≥ 0
⇒
But A, B > 0 and A + B = π/3
⇒ A, B < π/3
⇒ tan A tan B < 3
i.e., max. value of y is 1/3.
Q.8. General value of θ satisfying the equation tan^{2}θ + sec 2θ = 1 is_________ . (1996  1 Mark)
Ans.
Sol. tan^{2}θ + sec 2θ= 1
where t = tanθ
which means θ = nπ and θ =nπ±π/3
Q.9. The real roots of the equation cos^{7 }x + sin^{4} x=1 in the interval (–π, π) are ______, ______, and _________ . (1997  2 Marks)
Ans.
Sol. cos^{7} x = 1– sin^{4} x = (1– sin^{2} x) (1 + sin^{2} x) = cos^{2} x (1 + sin^{2} x)
∴ cos x = 0
or x = π/2, – π/2
or cos^{5} x = 1 + sin^{2} x or cos^{5} x – sin^{2} x = 1
Now maximum value of each cos x or sin x is 1.
Hence the above equation will hold when cos x = 1 and sin x = 0. Both these imply x = 0
Hence
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