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Let [x] denote the greatest integer less than or equal to x.

Let [x] denote the greatest integer less than or equal to x. If for n ∈ N, (1 - x + x3)n = ∑j=03n ajxj, then ∑j=03n/2 a2j + 4... View Article

The locus of the midpoints of the chord of the circle, x²+ y²= 25 which is tangent to the hyperbola, x² / 9 – y² / 16 = 1 is

a. (x2 + y2)2 - 16x2 + 9y2 = 0 b. (x2 + y2)2 - 9x2 + 144y2 = 0 c. (x2 + y2)2 - 9x2 - 16y2 = 0 d. (x2 + y2)2 - 9x2 + 16y2 = 0 Answer: (d) The... View Article

If for x ∈ (0, π / 2), log₁₀sin x + log₁₀cos x = –1 and log₁₀(sin x+cos x) = 1 / 2 (log₁₀ n – 1), n>0, then the value of n is equal to:

a. 16 b. 20 c. 12 d. 9 Answer: (c) log10sin x + log10cos x = –1 sinx * cosx = 1 / 10 --- (1) log10 (sinx + cosx) = (1 / 2) (log10 n... View Article

Let [latex]A=\begin{bmatrix} i&-i \\ -i&i \end{bmatrix}[/latex].

Let has : a. No solution b. Exactly two solutions c. A unique solution d. Infinitely many solutions Answer: (a)... View Article

Which of the following Boolean expression is a tautology?

a. (p ∧ q) ∧ (p → q) b. (p ∧ q) ∨ (p ∨ q) c. (p ∧ q) ∨ (p → q) d. (p ∧ q) → (p → q) Answer:... View Article

If y = y(x) is the solution of the differential equation, dy / dx + 2y tanx = sinx, y (π / 3) = 0, then the maximum value of the function y(x) over R is equal to

a. 8 b. 1 / 2 c. - 15 / 4 d. 1 / 8 Answer: (d) dy / dx + 2y tanx = sinx IF = eln (sec2x) = sec2 x y sec2x = ∫tanx secx dx = secx + c... View Article

The number of roots of the equation, (81)^sin²x + (81)^cos²x = 30 in the interval [0, π] is equal to :

a. 3 b. 2 c. 4 d. 8 Answer: (c) (81)sin2x + (81)cos2x = 30 (81)sin2x + [81 / (81)sin2x] = 30 Let (81)sin2x = t t + (81 / t) = 30 ⇒... View Article

Let S_k = ∑_r=1^k tan^-1[(6^r) / (2^r+1 + 3^2r+1)].

Let Sk = ∑r=1k tan-1[(6r) / (2r+1 + 32r+1)]. Then limk→∞ Sk = a. tan-1 (3 / 2) b. cot-1 (3 / 2) c. π / 2 d. tan–1(3)... View Article

If the three normals drawn to the parabola, y²= 2x pass through the point (a, 0) a ≠ 0, then ‘a’ must be greater than

a. 1 b.1 / 2 c. - 1 / 2 d. –1 Answer: (a) Let the equation of the normal be y = mx – 2am – am3. Here 4a = 2 a = 1 / 2 y... View Article

Let the position vectors of two points P and Q be 3i – j + 2k and i + 2j – 4k, respectively.

Let the position vectors of two points P and Q be 3i - j + 2k and i + 2j - 4k, respectively. Let R and S be two points such that the direction... View Article

If n is the number of irrational terms in the expansion of [3^1/4 + 5^1/8]⁶⁰, then (n – 1) is divisible by

a. 8 b. 26 c. 7 d. 30 Answer: (b) Tr+1 = 60Cr (31/4)60-r (51/8)r It is rational if [60 - r] / 4, (r / 8), both are whole numbers, r ∈... View Article

Area included between curves y = x² – 3x + 2 and y = – x² + 3x – 2 is

1) 1/6 sq unit 2) 1/2 sq unit 3) 1 sq unit 4) 1/3 sq unit 5) None of these Solution: (5) None of these No area is... View Article

The area of the region bounded by the curves y² = 2x + 1 and x – y = 1 is

1) 2/3 2) 4/3 3) 8/3 4) 11/3 5) None of these Solution: (5) None of these... View Article

Find the volume of the solid obtained by revolving the loop of the curve 2ay² = x (x − a)² about the x-axis, a > 0.

1) (1 / 2) π3a2 cu units 2) πa3 / 24 cu units 3) (1 / 2) π2a3 cu units 4) π2 a3 cu units Solution: (2) πa3 / 24 cu units... View Article

The volume of the solid generated by revolving the region bounded by y = 2x² + 1 and y = 2x + 11 about x-axis is

1) 104π/15 cu units 2) 42π/15 cu units 3) 52π/15 cu units 4) None of these Solution: (4) None of these... View Article

The volume of the solid formed by rotating the area enclosed between the curve y² = 4x, x = 4 and x = 5 about the x-axis is (in cubic units)

1) 18π 2) 36π 3) 9π 4) 24π Solution: (1) 18π Volume of the solid = ∫45 πy2 dx = π ∫45 4x dx = 4π (x2 / 2)45 = 18π cu units... View Article

The length of the parabola y² = 12x cut off by the latus rectum is

1) 6 [√2 + log (1 + √2)] 2) 3 [√2 + log (1 + √2)] 3) 6 [√2 - log (1 + √2)] 4) 3 [√2 - log (1 + √2)] Solution: (1) 6 [√2 + log (1 +... View Article

The volume of the solid generated by revolving about the y-axis bounded by the parabola y = x² and x = y² is

1) (2/15) π 2) (24/5) π 3) 3π/10 4) (5/24) π Solution: (1) (2/15) π... View Article

If A is the area of the region bounded by the curve y = √3x + 4, x-axis and the lines x = – 1 and x = 4 and B is the area bounded by curve y² = 3x + 4, x-axis and the lines x = – 1 and x = 4, then A : B =

1) 1 : 1 2) 2 : 1 3) 1 : 2 4) None of these Solution: (1) 1 : 1 y = √3x + 4 y2 = 3 [x + (4 / 3)] A = ∫-14 √3x + 4 dx Let 3x + 4... View Article

The area cut off by the latus rectum from the parabola y² = 4ax is

1) (8/3) a sq units 2) (8/3) √a sq units 3) (3/8) a2 sq unit 4) (8/3) a2 sq unit Solution: (4) (8/3) a2 sq unit... View Article