Trigonometric Equations PYQs Set 2 (PDF)

Set 2 of 2 · Questions 31-47 of 47

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Equations · Set 2 (Q31-47)
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31. $A horizontal park is in the shape of a triangle OAB with AB = 16. A vertical lamp post OP is erected at the point O such that ∠ PAO = ∠ PBO = 15° and ∠ PCO = 45°, where C is the midpoint of AB. Then (O P)^{2} is equal to$ (1) $\frac{32}{3} (3 - 1)$ (2) $\frac{32}{3} (2 - 3)$ (3) $\frac{16}{3} (3 - 1)$ (4) $\frac{16}{3} (2 - 3)$ 32. $The number of solutions of the equation x + 2 tan x = \frac{π}{2} in the interval [0, 2 π] is:$ (1) $3$ (2) $4$ (3) $2$ (4) $5$ 33. $The sum of all values of x in [0, 2π], for which, sin x + sin 2x + sin 3x + sin 4x = 0 is equal to :$ (1) $8π$ (2) $11π$ (3) $12π$ (4) $9π$ 34. $If n is the number of solutions of the equation 2 cos x (4 sin (\frac{π}{4} + x) sin (\frac{π}{4} - x) - 1) = 1, x \in [0, π] and S is the sum of all these solutions, then the ordered pair (n,S) is :$ (1) $(3, 13π/9)$ (2) $(2, 2π/3)$ (3) $(2, 8π/9)$ (4) $(3, 5π/3)$ 35. $The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is cos^{- 1} (\frac{3}{13}) . If the distance of the point B from the tower is 15 units, then cot a is equal to :$ (1) $\frac{6}{5}$ (2) $\frac{9}{5}$ (3) $\frac{4}{3}$ (4) $\frac{7}{3}$ 36. $If for x \in (0, \frac{π}{2}), lo g_{10} sin x + lo g_{10} cos x = - 1 and lo g_{10} (sin x + cos x) = \frac{1}{2} (lo g_{10} n - 1), n > 0, then the value of n is equal to :$ (1) $20$ (2) $12$ (3) $9$ (4) $16$ 37. $The sum of solutions of the equation \frac{c o s x}{1 + s i n x} = ∣ tan 2 x ∣, x \in (- \frac{π}{2}, \frac{π}{2}) - {\frac{π}{4}, - \frac{π}{4}} is$ (1) $- \frac{11 π}{30}$ (2) $\frac{π}{10}$ (3) $- \frac{7 π}{30}$ (4) $- \frac{π}{15}$ 38. $The number of solutions of the equation 3 2^{t a n^{2} x} + 3 2^{s e c^{2} x} = 81, 0 \leq x \leq \frac{π}{4} is:$ (1) $3$ (2) $1$ (3) $0$ (4) $2$ 39. $The number of solutions of \|cos x\| = sin x, such that - 4 π \leq x \leq 4 π is :$ (1) $4$ (2) $6$ (3) $8$ (4) $12$ 40. $The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is 45°. Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is 60°. If ∠ BAQ = 30°, AB = d and the area of the trapezium PQRB is α, then the ordered pair (d, α) is :$ (1) $(10 (3 - 1), 25)$ (2) $(10 (3 - 1), \frac{25}{2})$ (3) $(10 (3 + 1), 25)$ (4) $(10 (3 + 1), \frac{25}{2})$ 41. $The number of solutions of sin^{2} x + (2 + 2 x - x^{2}) sin x - 3 (x - 1)^{2} = 0, where - π \leq x \leq π, is_______$ 42. $Let S = {θ \in (0, 2 π) : 7 cos^{2} θ - 3 sin^{2} θ - 2 cos^{2} 2 θ = 2} . Then, the sum of roots of all the equations x^{2} - 2 (tan^{2} θ + cot^{2} θ) x + 6 sin^{2} θ = 0 θ \in S, is________.$ 43. $The number of values of x in the interval (\frac{π}{4}, \frac{7 π}{4}) for which 14 cose c^{2} x - 2 s i n^{2} x = 21 - 4 co s^{2} x holds, is________$ 44. $The number of elements in the set S = {θ \in [- 4 π, 4 π] : 3 cos^{2} 2 θ + 6 cos 2 θ - 10 cos^{2} θ + 5 = 0} is________.$ 45. $If m and n respectively are the numbers of positive and negative value of θ in the interval [- π, π] that satisfy the equation cos 2 θ cos \frac{θ}{2} = cos 3 θ cos \frac{9 θ}{2} then mn is equal to ________$ 46. $The number of solutions of the equation 2 θ - cos^{2} θ + 2 = 0 is R is equal to______.$ 47. $The number of solutions of the equation sin x = co s^{2} x in the interval (0,10) is .$