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If the sides AB, BC, and CA of a triangle ABC have, 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices is equal to:

a. 360 b. 240 c. 333 d. 364 Answer: (c) Total number of triangles = 3C1 × 5C1 × 6C1 + 3C1 × 5C2 + 5C1 × 3C2 +... View Article

Let S₁, S₂ and S₃ be three sets defined

Let S1, S2 and S3 be three sets defined as S1 = {z ∈ C : |z - 1| ≤ √2}, S2 = {z ∈ C : Re ((1 - i) z) ≥ 1}, S3 =... View Article

If the curve y = y(x) is the solution of the differential equation

If the curve y = y(x) is the solution of the differential equation 2 (x2 + x5/4) dy - y (x + x1/4) dx = 2x9/4 dx, x > 0 which passes through... View Article

The value of the limit lim_θ→0 [tan (π cos² θ)] / [sin (2π sin² θ)] is equal to

a. - 1 / 2 b. - 1 / 4 c. 0 d. 1 / 4 Answer: (a) Given, limθ→0 [tan (π cos2 θ)] / [sin (2π sin2 θ)] =... View Article

Let y = y (x) be the solution of the differential equation cosx (3 sinx + cosx + 3) dy = (1 + y sinx (3 sinx + cosx + 3)) dx, 0 ≤ x ≤ π / 2, y(0) = 0. Then, y (π / 3) is equal to

a. 2 loge [(2√3 + 10) / 11] b. 2 loge [(√3 + 7) / 2] c. 2 loge [(3√3 - 8)] / 4 d. 2 loge [(2√3 + 9) / 6] Answer: (a)... View Article

If the integral ∫₀¹⁰ [sin 2πx] / e^x-[x]dx = ɑe^-1 + βe^-½ + ,

If the integral ∫010 [sin 2πx] / ex-[x] dx = ɑe-1 + βe-½ + , where ɑ, β, are integers and [x] denotes the greatest... View Article

If x, y, z are in arithmetic progression with common difference d

If x, y, z are in arithmetic progression with common difference d, x ≠ 3d, and the determinant of the matrix is zero, then the value of k2... View Article

Let f : R → R be defined as f (x) = e^–xsinx.

Let f : R → R be defined as f (x) = e–x sinx. If F : [0,1] R→ is a differentiable function such that F (x) = ∫0x f (t) dt,... View Article

Let L be a tangent line to the parabola y² = 4x – 20 at (6, 2). If L is also a tangent to the ellipse (x² / 2) + (y² / b) = 1, then the value of b is equal to:

a. 20 b. 14 c. 16 d. 11 Answer: (b) Parabola y2 = 4x – 20 Tangent at P (6, 2) will be 2y = 4 [(x + 6) / 2] – 20 2y = 2x + 12... View Article

Let O be the origin. Let OP = xi + yj – k and OQ = – i + 2j + 3xk

Let O be the origin. Let OP = xi + yj - k and OQ = - i + 2j + 3xk, x, y ∈ R, x > 0, be such that |PQ| = √20 and the vector OP is... View Article

Consider the function f : R → R defined by f(x) = [latex]left{begin{matrix} (2-sin(1/x)|x|), &xneq 0 \ 0&x=0 end{matrix}right.[/latex]. Then f is

a. monotonic on (0, ∞) only b. Not monotonic on (–∞, 0) and (0, ∞) c. monotonic on (–∞, 0) only d. monotonic... View Article

If the equation of the plane passing through the mirror image of a point

If the equation of the plane passing through the mirror image of a point (2, 3, 1) with respect to line (x + 1) / 2 = (y - 3) / 1 = (z + 2) / -1... View Article

The number of solutions of the equation x + 2 tanx = π / 2 in the interval [0, 2π] is

a. 5 b. 2 c. 4 d. 3 Answer: (d) x + 2 tan x = π / 2 in [0, 2π] 2 tan x = π / 2 – x 2 tan x = π / 2 – x tan x = π / 4... View Article

Let a computer program generate only the digits 0 and 1

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with a probability of occurrence of 0 at even places... View Article

Let the tangent to the circle x² + y² = 25 at the point R (3, 4)

Let the tangent to the circle x2 + y2 = 25 at the point R (3, 4) meet the x-axis and y-axis at points P and Q, respectively. If r is the radius... View Article

A transmitting station releases waves of wavelength 960 m. A capacitor of 256 μF is used in the resonant circuit. The self inductance of coil necessary for resonance is _____ × 10^–8H.

Answer: 10 At resonance ωr = 1/√(Lc) ∴ 2πf = 1/√(Lc) ∴ 4π2c2/λ2 = 1/Lc... View Article

A coil of inductance 2 H having negligible resistance is connected to a source of supply whose voltage is given by V = 3t volt. (where t is in second). If the voltage is applied when t = 0, then the energy stored in the coil after 4 s is _______ J.

Answer: 144 L di/dt = ε = 3t L ∫di = 3 ∫t dt Li = 3t2/2 i = 3t2/2L Energy stored in the coil, E = ½ Li2 = ½ L (3t2/2L)2 = ½ × 9t4/4L =... View Article

A monoatomic gas of mass 4.0 u is kept in an insulated container. Container is moving with a velocity 30 m/s. If container is suddenly stopped then change in temperature of the gas (R=gas constant) is x/3R. Value of x is ______.

Answer: 3600 ΔKE = ΔU ΔU = nCVΔT ½ mv2 = (3/2) nRΔT mv2/3nR = ΔT 4×(30)2/3×1×R... View Article

Circuit of potentiometer

In the given circuit of potentiometer, the potential difference E across AB (10 m length) is larger than E1 and E2 as well. For key K1 (closed),... View Article

In a certain thermodynamic process, the pressure of a gas depends on its volume as kV³. The work done when the temperature changes from 100⁰C to 300⁰C will be ____ nR, where n denotes number of moles of a gas.

Answer: 50 P = kv3 pv–3 = k x = –3 w = nR(T1 - T2)/(x - 1) = nR(100-300)/(-3 -1) = nR(-200)/-4 = 50nR... View Article

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