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The sum of the series ∑_n=1^∞∞[n²+ 6n + 10] / (2n + 1)! is equal to:

a. (41/ 8) e + (19 / 8) e-1 - 10b. (- 41/ 8) e + (19 / 8) e-1 - 10c. (41/ 8) e - (19 / 8) e-1 - 10d. (41/ 8) e + (19 / 8) e-1 +... View Article

Let L be a line obtained from the intersection of two planes

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P (ɑ, β, γ) is the foot of the... View Article

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is

a) 3√(5/2)b) 3√5/2c) √10d) 2√10Solution:As we know centroid divides line joining circumcentre and orthocentre internally 1:2.So, C(6,2)AC =... View Article

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series.

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series. 12 +2.22+32+2.42+52 + 2.62+..If B-2A = 100λ,... View Article

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is

a) at least 500 but less than 750b) at least 750 but less than 1000c) at least 1000d) less than 500Solution:We have 6 novels and 3 dictionaries.... View Article

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR…

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... View Article

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x+y+z = 7 is

a) 1/3b) √(2/3)c) 2/√3d) 2/3Solution:Direction ratios of AB = (1,0,1)Let be angle between line AB and normal of plane.So cos = 1×1+0×1+1×1)/√2√3=... View Article

Let u be a vector coplanar with the vectors ….

Let u be a vector coplanar with the vectors Answer: (c)... View Article

Let y = y(x) be the solution of the differential equation sin x(dy/dx)+y cosx = 4x, x∈(0,π). If y(π/2) = 0, then y(π/6) is equal to

a) (-8/9)π2b) (-4/9)π2c) (4/9√3)π2d) (-8/9√3)π2Solution:(dy/dx)+y cotx = 4x/sin xIF = e∫cot xdx = elog|sin x| = sin x as x∈(0,π)y... View Article

Let S = {t∈R: f(x) = |x-π|(e^|x|-1)sin x is not differentiable at t}. Then the set S is equal to:

a) {π}b) {0,π}c) (an empty set)d) {0}Solution:f(x) = |x-π|(e|x|-1) sinxObviously differentiable at x = 0Check at x = π So,... View Article

Solve the integral of fraction – Step by Step Solution

a) 1/(1+cot3x) + C b) -1/(1+cot3x) + C c) 1/3(1+tan3x) + C d) -1/3(1+tan3x) + C where C is the constant of integration. Solution: Answer: (d)... View Article

If ∑⁹_i=1 (x_i-5) = 9 and ∑⁹_i=1 (x_i-5)² = 45, then the standard deviation of the 9 items x₁, x₂,…x₉ is:

a) 2b) 3c) 9d) 4Solution:Variance = (1/n) Ʃi=1n xi2-[(1/n) Ʃi=1n xi]2= (1/9)×45-((1/9)×9)2= 5-1= 4S.D = √4 = 2Answer: (a)... View Article

For each t ∈ R, let [t] be the greatest integer less than or equal to t.

For each t ∈ R, let [t] be the greatest integer less than or equal to t. Then a) is equal to 120b) does not exist (in R)c) is equal to 0d) is... View Article

Let g(x) = cos x², f(x) = √x and α, β(α< β) be the roots

Let g(x) = cos x2, f(x) = √x and α, β(α< β) be the roots of the quadratic equation 18x2-9πx+π2 = 0. Then the area... View Article

The value of [latex]\int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}\frac{\sin ^{2}x}{1+2^{x}}dx[/latex] is:

a) 4π b) π/4 c) π/8 d) π/2 Solution: Answer: (b)... View Article

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is

a) 3x+2y = xyb) 3x+2y = 6xyc) 3x+2y = 6d) 2x+3y = xySolution: line y-3 = m(x-2)y-3 = mx-2mP = ((2m-3)/m, 0)Q = (0, 3-2m)Let R(α, β)So... View Article

Let a₁, a₂, a₃,..a₄₉ be in A.P

Let a1, a2, a3,..a49 be in A.P such that Ʃk=012 a4k+1 = 416 and a9+a43 = 66. If a12+a22+…+a172 = 140m, then m is equal to:a) 34b) 33c) 66d)... View Article

The sum of the coefficients of all odd degree terms in the expansion of (x+√(x³-1))⁵+(x-√(x³-1))⁵, (x>) is

a) 1b) 2c) -1d) 0Solution:(a+b)n+(a-b)n = 2(nC0an + nC2an-2b2+ nC4an-4b4…)(x+√(x3-1))5+(x-√(x3-1))5, (x>)=... View Article

If [latex]\begin{vmatrix} x-4 & 2x &2x \\ 2x& x-4 & 2x\\ 2x& 2x & x-4 \end{vmatrix}[/latex] = (A+Bx)(x-A)² then the ordered pair (A, B) is equal to

a) (-4, 5)b) (4, 5)c) (-4, -5)d) (-4, 3)Solution: = (5x-4)(-1)(x+4)(x-4-2x)= -1(5x-4)(x+4)(-x-4)= (5x-4)(x+4)(x+4)2 = (-4+5x)(x-(-4))2 Hence A =... View Article

Tangent and normal are drawn at P(16, 16) on the parabola y² = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tan θ is

a) 3b) 4/3c) 1/2d) 2Solution: Tangent and normal are drawn at P (16, 16) on y2 = 16xy2 = 16xSlope of tangent dy/dx = 16/2y= 8/y(dy/dx)(16,16) =... View Article

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