01. Let Tr be the rth term of an AP for r = 1, 2, 3, …… If for some distinct positive Integers m and n we have Tm = 1/n and Tn = 1/m, then what is Tmn equal to?

(a) (mn)-1

(b) m-1 + n-1

(c) 1

(d) 0

02. Suppose f(x) is such a quadratic expression that it is positive for all real x. If g(x) = f(x) + f”(x), then for any real x

(a) g(x) < 0

(b) g(x) > 0

(c) g(x) = 0

(d) g(x) ≥ 0

03. Consider the following in respect of matrices A, B and C of same order:

1. (A + B + C)’ = A’ + B’ + C’

2. (AB)’ = A’B’

3. (ABC)’ = C’B’A’

Where A’ is the transpose of the matrix A.

Which of the above are correct?

(a) 1 and 2 only

(b) 2 and 3 only

(c) 1 and 3 only

(d) 1, 2, and 3

04. The sum of the binary numbers (11011)2, (10110110)2 and (10011x0y)2 is the binary number (101101101)2. What are the values of x and y?

(a) x = 1, y = 1

(b) x = 1, y = 0

(c) x = 0, y = 1

(d) x = 0, y = 0

05. Let matrix B be the adjoint of a square matrix A, l be the identity matrix of same order as A. If k (≠0) is the determinant of the matrix A, the what is AB equal to?

(a) l

(b) Kl

(c) k2l

(d) (1/k)l

06. If (0.2)x = 2 and log10 2 = 0.3010, then what is the value of x to the nearest tenth?

(a) – 10.0

(b) – 0.5

(c) – 0.4

(d) – 0.2

07. The total number of 5-digit numbers that can be composed of distinct digits from 0 to 9 is

(a) 45360

(b) 30240

(c) 27216

(d) 15120

08. Consider the following in respect of matrices A and B of same order:

1. A2 – B2 = (A + B) (A – B)

2. (A – I) (I + A) = 0 ↔ A2 = I

Where I is the identity matrix and O is the null matrix.

Which of the above is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

09. What is (2 tan θ/1 + tan2 θ) equal to?

(a) cos 2θ

(b) tan 2θ

(c) sin 2θ

(d) cosec 2θ

10. If sec (θ – α), sec θ and sec (θ + α) are in AP, where cos α ≠ 1, then what is the value of sin2θ + cos α?

(a) 0

(b) 1

(c) – 1

(d) 1/2

11. If A + B + C = 180°, then what is sin 2A – sin 2B – sin 2C equal to?

(a) – 4 sin A sin B sin C

(b) – 4 cos A sin B cos C

(c) – 4 cos A cos B sin C

(d) – 4 sin A cos B cos C

12. A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is 48°. If the height of the balloon above the bridge is 122m, then what is the length of the bridge?

(a) 122 sin 48° m

(b) 122 tan 42° m

(c) 122 cos 48° m

(d) 122 tan 48° m

13. A is an angle in the fourth quadrant. It satisfies the trigonometric equation 3(3 – tan2 A – cot A)2 = 1. Which one of the following is a value of A?

(a) 300°

(b) 315°

(c) 330°

(d) 345°

14. If x, x – y and x + y are the angles of a triangle (not an equilateral triangle) such that tan (x – y), tan x and tan (x + y) are in GP, then what is x equal to?

(a) π/4

(b) π/3

(c) π/6

(d) π/2

15. ABC is a triangle inscribed in a circle with centre O. Let a = ∠BAC, where 45° < 𝛂 <90°. Let 𝛃 = ∠BOC. Which one of the following is correct?

(a) cos β = 1 – tan2 α/1 + tan2 α

(b) cos β = 1 + tan2 α/1 – tan2 α

(c) cos β = 2tan α/1 + tan2 α

(d) sin β = 2 sin2 α

16. If a flag-staff of 6 m height placed on the top of a tower throws a shadow of 2√3 m along the ground, then what is the angle that the sun makes with the ground?

(a) 60°

(b) 45°

(c) 30°

(d) 15°

17. What is tan-1 (1/4) + tan-1 (3/5) equal to?

(a) 0

(b) π/4

(c) π/3

(d) π/2

18. A spherical balloon of radius r subtends an angle α at the eye of an observer, while the angle of elevation of its centre is β. What is the height of the centre of the balloon (neglecting the height of the observer)?

(a) r sin β/sin(α/2)

(b) r sin β/sin(α/4)

(c) r sin (β/2)/sin α

(d) r sin α/sin(β/2)

19. If sin (x + y)/sin (x – y) = a + b/a – b, then what is tan x/tan y is equal to?

(a) a/b

(b) b/a

(c) a + b/a – b

(d) a – b/a + b

20. If sin α + sin β = 0 = cos α + cos β, where 0 < β < α < 2π, then which one of the following is correct?

(a) α = π – β

(b) α = π + β

(c) α = 2π – β

(d) 2α = π + 2β