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If function is differentiable at every point of the domain, then the values of a and b are respectively:

a. 5/2, - 3/2 b. - 1/2, 3/2 c. 1/2, 1/2 d. 1/2, - 3/2 Solution: Answer: (b) f (x) is continuous at x = 1 ⇒ 1 = a + b f (x) is... View Article

The solutions of the equation [1+sin2x sin2x sin2x cos2x 1+cos2x cos2x 4sin2x4 sin2x 1+4sin2x ]=0, (0 lt x lt pi), are:

a. π/6, 5π/6 b. 7π/12, 11π/12 c. 5π/12, 7π / 12 d. π/12, π/6 Solution: Answer: (b) ∴ 2 + 8 sin 2x – 4 sin 2x = 0 ⇒ sin 2x = - 1/2 ⇒ x... View Article

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is:

a. 3b. 2c. 1d. 0Solution:Answer: (b)3x + 4(mx + 1) = 9x(3 + 4m) = 5x = 5/(3 + 4m)(3 + 4m) = ± 1, ± 54m = – 3 ± 1,... View Article

1 / (3² – 1) + 1 / (5² – 1) + 1 / (7² – 1) + ……. + 1 / (201² – 1) is equal to

a. 101/404b. 101/408c. 99/400d. 25/101Solution:Answer: (d)S = ∑r=1100 [1/[(2r + 1)2 - 1]] = ∑r=1100 1/[(2r + 2) (2r)]S = (1/4)... View Article

Let α, β, γ be the roots of the equations, x³ + ax² + bx + c = 0

Let α, β, γ be the roots of the equations, x3 + ax2 + bx + c = 0, (a, b, c ∈ R and a, b and a, b ≠ 0). The system of the... View Article

If the functions are defined as f (x) = √x and g(x) = √(1 – x), then what is the common domain of the following functions: f + g, f – g, f / g, g / f, g – f where (f ± g) (x) = f (x) ± g (x), (f / g) (x) =

a. 0 < x ≤ 1b. 0 ≤ x < 1c. 0 ≤ x ≤ 1d. 0 < x < 1Solution:Answer: (d)f + g = √x + √(1 - x)x ≥ 0 and 1 - x... View Article

Let y = y (x) be the solution of the differential equation xdy – ydx = √(x² – y²) dx

Let y = y (x) be the solution of the differential equation xdy - ydx = √(x2 - y2) dx, x ≥ 1, with y (1) = 0. If the area bounded by... View Article

Let f : R → R satisfy the equation f (x + y) = f (x) . f (y) for all x, y ∈ R and f (x) ≠ 0 for any x ∈ R. If the function f is differentiable at x = 0 and f’ (0) = 3, then lim_h→0 (1 / h) [f (h) – 1] is equal to

Solution:Answer: (3)f (x + y) = f (x) . f (y) thenf (x) = akxf’(x) = (akx ) k ln af’(0) = k ln a = 3 (given f’(0) = 3)a = e3/k... View Article

Let I be an identity matrix of order 2 times 2 and P is a matrix. Then the value of n belongs to N for which Pn = 5I – 8P is equal to

Solution: Answer: (6) P6 = 5I - 8P Thus, n = 6... View Article

If f (x) and g (x) are two polynomials such that the polynomial P (x) = f (x³) + x g (x³) is divisible by x² + x + 1, then P (1) is equal to

Solution:Answer: 0Roots of x2 + x + 1 are ω and ω2 nowQ (ω) = f (1) + ω g (1) = 0 …(1)Q (ω2) = f (1) +... View Article

Let the mirror image of the point (1, 3, a) with respect to the plane r . (2i – j + k) – b = 0 be (-3, 5, 2). Then, the value of |a + b| is equal to

Solution:Answer: (1)Plane : 2x -y + z = bR ≡ [- 1, 4, (a + 2) / 2] on plane- 2 - 4 + (a + 2) / 2 = b⇒ a + 2 = 2b + 12 ⇒ a = 2b +... View Article

Let P (x) be a real polynomial of degree 3 which vanishes at x = – 3. Let P(x) have local minima at x = 1, local maxima at x = -1 and ∫_-1¹ P (x) dx = 18, then the sum of all the coefficients of the polynomial P (x) is equal to

Solution:Answer: (8)P’ (x) = a (x + 1) (x - 1)P (x) = (ax3 / 3) - ax + cP (-3) = 0 (given)⇒ a(- 9 + 3) + C = 0⇒ 6a = C... View Article

Let ⁿC_r denote the binomial coefficient of x^r in the expansion of (1 + x)ⁿ. If ∑_k=0¹⁰ [2² + 3k] ⁿC_k = ɑ . 3¹⁰ + β . 2¹⁰ then ɑ + β is equal to

Solution:n must be equal to 10∑k=010 [22 + 3k] nCk= ∑k=010 [4 + 3k] nCk= 4 ∑k=010nCk + 3 ∑k=010 k nCk= 4 (210) + 3 * 10 * 29= 19... View Article

The term independent of x in the expansion of [(x + 1) / (x^2/3 – x^1/3 + 1) – (x – 1) / (x – x^1/2)]¹⁰, x ≠ 1, is equal to

Solution:Answer: (210)[(x1/3 + 1) - [√x + 1] / [√x]]10 = (x1/3 - x-½)10General term, Tr+1 = 10Cr (x1/3)10-r (- x-½)rFor... View Article

∑_r=1¹⁰ r! (r³ + 6r² + 2r + 5) = α (11!). Then the value of α is equal to

Solution:Answer: (160)Tr = r! ((r + 1) (r + 2) (r + 3) - 9r - 1)= (r + 3) ! - 9r . r! - r!= (r + 3) ! - 9 (r - 1 + 1)) r! - r!= (r + 3) ! - 9 (r... View Article

Let P be a plane containing the line [x – 1] / 3 = [y + 6] / 4 = [z + 5] / 2 and parallel to the line [x – 3] / 4 = [y – 2] / – 3 = [z + 5] / 7. If the point (1, -1, α) lies on the plane P, then the value of |5α| is equal to

Solution:Answer: (38)DR’s of normal (34, -13, -25)P ≡ 34 (x - 1) - 13 (y + 6) - 25 (z + 5)Q (1, -1, α) lies on P.⇒ 3 (1 -... View Article

Let a tangent be drawn to the ellipse (x² / 27) + y² = 1 at (3√3 cos θ, sin θ) where θ ∈ (0, π / 2). Then the value of θ such that the sum of intercepts on axes made by a tangent is minimum is equal to:

a. π/8b. π/6c. π/3d. π/4Solution:Answer: (b)Equation of tangent [x/3√3] cos θ + y sin θ = 1A [(3√3/cos... View Article

Let y = y (x) be the solution of the differential equation dy / dx = (y + 1) [(y + 1) e^{x^2/2} – x], 0 < x < 2.1, with y (2) = 0. Then the value of dy / dx at x = 1 is equal to:

a. [e5/2]/[1 + e2]2b. [5e1/2]/[e2 + 1]2c. - 2e2/(1 + e2)2d. [-e3/2]/[e2 + 1]2Solution:Answer: (d)dy/dx = (y + 1) [(y + 1) ex^2/2 - x]⇒ [-... View Article

Let f : R → R be a function defined as

Let f : R → R be a function defined as . If f is continuous at x = 0, then the value of a + b is equal toa. - 2b. - 2/5c. - 3/2d. -... View Article

Consider a hyperbola H : x² – 2y² = 4……

Consider a hyperbola H : x2 - 2y2 = 4. Let the tangent at a point P (4, √6) meet the x-axis at Q and latus rectum at R (x1, y1), x1 > 0.... View Article