Trigonometric Ratios & Identities PYQs Set 15 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 15 (Q421-450)
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421. $All possible values of θ \in [0, 2 π] for which sin 2 θ + tan 2 θ > 0 lie in:$ (1) $(0, \frac{π}{2}) \cup (π, \frac{3 π}{2})$ (2) $(0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (π, \frac{5 π}{4}) \cup (\frac{3 π}{2}, \frac{7 π}{4})$ (3) $(0, \frac{π}{2}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (π, \frac{7 π}{6})$ (4) $(0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (\frac{3 π}{2}, \frac{11 π}{6})$ 422. $If the points P and Q are respectively the circumcenter and the orthocentre of a ∆ ABC, then P A + P B + P C is equal to_______$ (1) $2 QP$ (2) $2 P Q$ (3) $P Q$ (4) $QP$ 423. $The angle of elevation of the top of a TV tower from three points A, B and C in a straight line through the foot of the tower are α, 2 α and 3 α respectively. If AB = a, the height of the tower is$ (1) $a sin 2 α$ (2) $a sin 3 α$ (3) $a tan α$ (4) $a sin α$ 424. $Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If ∠BPC = β, then tanβ is equal to$ (1) $\frac{6}{7}$ (2) $\frac{1}{4}$ (3) $\frac{2}{9}$ (4) $\frac{4}{9}$ 425. $The sides of a triangle are sinα, cosα and 1 + sin α cos α for some 0 < α < \frac{π}{2} . Then the greatest angle of the triangle is :$ (1) $60º$ (2) $90º$ (3) $120º$ (4) $150º$ 426. $If the angles A, B and C of a triangle are in an Arithmetic Progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression \frac{a}{c} sin 2 C + \frac{c}{a} sin 2 A is$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $1$ (4) $3$ 427. $ABC is a triangle with ∠ A = 30^{\circ}, BC = 10 cms . The area of the circum-circle of the triangle is$ (1) $100 π sq. cms$ (2) $5 sq. cms$ (3) $25 sq . cms$ (4) $100 π 3 sq. cms$ 428. $Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD then the height of P (in m) above the line AC is:$ (1) $6$ (2) $20/3$ (3) $10/3$ (4) $5$ 429. $Given 0 \leq x \leq \frac{1}{2}, then the value of tan (sin^{- 1} (\frac{x}{2} + \frac{1 - x ^{2}}{2}) - sin^{- 1} x) is$ (1) $1$ (2) $3$ (3) $-1$ (4) $\frac{1}{3}$ 430. $PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45º, 30º and 30º, then the height of the tower (in m) is :$ (1) $1003$ (2) $502$ (3) $100$ (4) $50$ 431. $In any triangle ABC, if a : b : c = 2 : 3 : 4, then R : r =$ (1) $8 : 3$ (2) $16 : 9$ (3) $5 : 16$ (4) $16 : 5$ 432. $If in a triangle ABC,AB = 5 units, and radius of circumcircle of ΔABC is 5 units, then the area (in sq. units) of ΔABC is:$ (1) $10 + 62$ (2) $8 + 22$ (3) $6 + 83$ (4) $4 + 23$ 433. $If r = in radius, Δ = area of △ ABC, s = semi-perimeter then which of the following is true?$ (1) $Δ = r + s$ (2) $Δ = \frac{r}{s}$ (3) $Δ = (rs)^{2}$ (4) $Δ = rs$ 434. $Let ABC be an acute-angled triangle with area R Then a^{2} b^{2} - 4 R^{2} + b^{2} c^{2} - 4 R^{2} + c^{2} a^{2} - 4 R^{2} =$ (1) $a + b + c$ (2) $a^{2} + b^{2} + c^{2}$ (3) $\frac{a ^{2} + b ^{2} + c ^{2}}{2}$ (4) $2 a^{2} + b^{2} + c^{2}$ 435. $If a and b respectively represent the lengths of a side and a diagonal of a regular pentagon that is inscribed in a circle, then \frac{b}{a} =$ (1) $2 sin \frac{π}{5}$ (2) $2 cos \frac{π}{5}$ (3) $cos \frac{π}{5}$ (4) $sin \frac{π}{5}$ 436. $The sides of a triangle are in the ratio 1 : 3 : 2 . Then the angles are in the ratio$ (1) $1 : 2 : 3$ (2) $1 : 2 : 4$ (3) $1 : 4 : 5$ (4) $1 : 3 : 5$ 437. $A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18m away from the base of the pole, then the height of the pole (in meters) is :$ (1) $12\sqrt15$ (2) $12\sqrt10$ (3) $8\sqrt10$ (4) $6\sqrt10$ 438. $In a triangle A B C, A D and B E are medians. If A D = 4, ∠ D A B = \frac{π}{6} and ∠ A B E = \frac{π}{3} then the area of △ A B C is$ (1) $\frac{14}{3 3}$ (2) $\frac{28}{3 3}$ (3) $\frac{11}{3 3}$ (4) $\frac{32}{3 3}$ 439. $In a triangle PQR, P is the largest angle and cosP = \frac{1}{3} . Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)$ (1) $16$ (2) $18$ (3) $24$ (4) $22$ 440. $From an aeroplane flying, vertically above a horizontal road, the angles of depression of two consecutive stones on the same side of the aeroplane are observed to be 30^{\circ} and 60^{\circ} respectively. The height at which the aeroplane is flying in km is$ (1) $\frac{4}{3}$ (2) $\frac{3}{2}$ (3) $\frac{2}{3}$ (4) $2$ 441. $Angles of elevation of the top of a tower from three points (collinear) A, B and C on a road leading to the foot of the tower are 30^{\circ}, 45^{\circ} and 60^{\circ} respectively. The ratio of AB to BC is$ (1) $3 : 1$ (2) $3 : 2$ (3) $1 : 2$ (4) $2 : 3$ 442. $In an isosceles right-angled triangle, a straight line is drawn from the mid point of one of the equal sides to the opposite vertex. Then a pair of possible values of the cotangents of the two angles so formed at that vertex are$ (1) $1 and 2$ (2) $2 and 3$ (3) $3 and 4$ (4) $4 and 5$ 443. $In △ ABC, if a = 1, b = 2, ∠ C = 60^{\circ} then 4 Δ^{2} + c^{2} =$ (1) $6$ (2) $3$ (3) $\frac{3}{2}$ (4) $9$ 444. $A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30°, After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60°. Then the time taken (in minutes) by him, from B to reach the pillar, is$ (1) $10$ (2) $20$ (3) $5$ (4) $6$ 445. $In a triangle the sum of two sides is x and the product of the same two sides is y. If x2 - c2 = y, where c is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is$ (1) $\frac{3 y}{2 x ( x + c )}$ (2) $\frac{3 y}{2 c ( x + c )}$ (3) $\frac{3 y}{4 x ( x + c )}$ (4) $\frac{3 y}{4 c ( x + c )}$ 446. $If sin A = \frac{- 7}{25}, cos B = \frac{8}{17}, A does not lie in the 3^{rd} quadrant and B does not lie in the 1^{s t} quadrant, then 8 tan A - 5 cot B =$ (1) $0$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $1$ 447. $Let in a right angled triangle, the smallest angle be θ . If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then s in θ is equal to:$ (1) $\frac{5 + 1}{4}$ (2) $\frac{5 - 1}{2}$ (3) $\frac{2 - 1}{2}$ (4) $\frac{5 - 1}{4}$ 448. $For what values of x, the following identity is valid and holds? tan h^{- 1} (x) = \frac{1}{2} log_{e} (\frac{1 + x}{1 - x}) .$ (1) $(- \infty, \infty)$ (2) $(1, \infty)$ (3) $(- \infty, 1)$ (4) $(- 1, 1)$ 449. $Two parallel chords of a circle of radius 2 are at a distance 3 + 1 apart. If the chords subtend at the center, angles \frac{π}{k} and \frac{2 π}{k}, where k > 0, then the value of [k], is [JEE (Main) 2010]$ 450. $The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be 45^{\circ} . After walking a distance of 80 meters towards the top, up a slope inclined at an angle of 30^{\circ} to the horizontal plane, the angle of elevation of the top of the hill becomes 75^{\circ} . Then the height of the hill (in meters) is$