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If e₁ and e₂ are the eccentricities of the ellipse

a) 14 b) 15 c) 17 d) 16 Solution: e1 = √(1-(4/18)) = √7/3 e2 = √(1+(4/9)) = √13/3 Since (e1, e2) lies on the ellipse 15x2 + 3y2 = k... View Article

If for some α and β in R, the intersection of the following three planes x + 4y − 2z = 1, x + 7y − 5z =β and x + 5y + αz = 5 is a line in ³, then α+β is equal to:

a) 0b) 10c) -10d) 2Solution:The given planes intersect in a lineD = Dx = Dy = Dz = 0D = 0 35 - 5β -20 + 4β -2 = 0β= 13α+ β= 10Answer: (b)... View Article

Solve integral problem – Step by Step Solution

a) 2 b) 4 c) 22 d) π2 Solution: Let I = …(i) I = = ..(ii) Adding (i) and (ii) , , …(iii) = ..(iv) Adding (iii) and... View Article

The number of real roots of the equation, e^4x + e^3x – 4e^2x + 1 = 0 is

a) 3b) 4c) 1d) 2Solution:e4x + e3x - 4e2x + ex + 1 = 0e2x + ex - 4 + (1/ex) + (1/e2x) = 0(e2x + (1/e2x)) + (ex + 1/ex) - 4 = 0(ex + 1/ex)2 - 2 +... View Article

The value of cos³ (π/8) cos (3π/8) + sin³ (π/8) sin (3π/8) is:

a) 1/4b) 1/2√2c) 1/2d) 1/√2Solution:cos3 (π/8) cos (3π/8) + sin3 (π/8) sin (3π/8) = cos3 (π/8) [ 4 cos3 (π/8) - 3 cos (π/8)] + sin3 (π/8)[ 3 sin... View Article

Let f be any function continuous on [ , ] and twice differentiable on ( , ). If for all ∈( , ),f’(x)> 0 and f’’(x) < 0, then for any c ∈ (a,b), (f(c) – f(a)])- (f(b) – f(c)]) is greater than

a) (b-c)/(c-a)b) 1c) (c-a)/(b-c)d) (b+a)/(b-a)Solution:c∈(a,b) and f is twice differentiable and continuous function on ( , )LMVT is... View Article

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at the rate of 50 cm³/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which the thickness of ice decreases, is:

a) 5/6πb) 1/54πc) 1/36πd) 1/18πSolution:Let thickness of ice be x cm.Therefore, net radius of sphere = (10+x) cmVolume of sphere V =... View Article

The product 2^1/4×4^1/16×8^1/48×16^1/128… ..∞ to is equal to

a) 21/4b) 2c) 21/2d) 1Solution:21/4×41/16×81/48×161/128… ∞= 21/4×22/16×23/48×….∞21/4×21/8×21/16×… ∞= 21/4 + 1/8 + 1/16 …∞ = √2Answer: (c)... View Article

If C be the centroid of the triangle having vertices (3, -1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0, then the line passing through the points C and P also passes through the point:

a) (-9, -7)b) (-9, -6)c) (7, 6)d) (9, 7)Solution:Coordinates of C are ((3+1+2)/3 , (-1+3+4)/3) = (2,2)Point of intersection of two linesx + 3y -... View Article

Let a be an integer such that all the real roots of the polynomial 2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to

Solution:Answer: 2Let f (x) = 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10f’ (x) = 10 (x4 + 2x3 + 3x2 + 2x + 1)= 10x2 [x2 + (1 / x2) + 2 [x + (1 /... View Article

If f (x) = ∫ [5x⁸ + 7x⁶] / [x² + 1 + 2x⁷]² dx, (x ≥ 0), f (0) = 0 and f (1) = 1 / k, then the value of K is

Solution:Answer: (28)The normal of plane = PQ = - 2i + 6j - 6ka = –2, b = 6, c = –6 &The equation of plane is – 2x + 6y... View Article

Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, –3, 1) and (2, 3, –5) at the right angles. If a, b, c, d are integers, then the minimum value of (a² + b² + c² +d²) is

Solution:Answer: (6)In equilateral Δ,z12 + z22 + z32 = z1z2 + z2z3 + z1z3z12 + z22 = z1z2 [because z3 = 0](z1 + z2)2 = 3z1z2a2 = 36|a| = 6... View Article

Let z₁, z₂ be the roots of the equations z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is

Solution:Answer: (1)Case I : x ∈ [0, π / 2] ⋃ [π, 3π / 2]cot x = cot x + (1 / sin x) ⇒ not possibleCase II : x ∈ [π /... View Article

The number of solutions of the equation |cot x| = cot x + (1 / sin x) in the interval [0, 2π] is

The missing value in the following figure is ______. Solution:Answer: (4)424 has base 4 (= 12 – 8)36 has base 3 (= 7 – 4)(?) will... View Article

The missing value in the following figure is

Solution:Answer: (80)OA ⟂ OB(1 / p2) (- 1 / q2) = - 1p2q2 = 1p [(p + q) / 2, [(1 / p) - (1 / q)] / 2] lies on x2 y2 = 1(p + q)2 [(1 / p) - (1 /... View Article

A square ABCD has all its vertices on the curve x²y² = 1. The midpoints of its sides also lie on the same curve. Then, the square of the area of ABCD is

Solution:Answer: (35)x1 + x2 + …. + x25 = 25 × 40 = 1000[x1 + x2 + …. + x25 - 60 + a] / 25 = 391000 – 60 + a = 25... View Article

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is

Solution:Answer: (512)I = 2 ∫04 f (x2) dx ……….(1)I = 2 ∫04 f ((4 - x)2) dx ……….(2)Adding... View Article

Let f (x) and g (x) be two functions satisfying f (x²) + g (4 – x) = 4x³ and g (4 –x) + g(x) = 0, then the value of ∫_-4⁴ f (x²) dx is

Solution:Answer: (4)x – 2y + 2z + λ = 0Now givend = |1 - 4 + 6 + λ| / √9 = 1|λ + 3| = 3λ + 3 = ±... View Article

The equation of the planes parallel to the plane x – 2y + 2z – 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d = 0. If (b – d) = K (c – a), then the positive value of K is

Solution:Answer: (300)... View Article

The number of times digit 3 will be written when listing the integers from 1 to 1000 is

a. 5 √2x + 4y - 15 + 4√2 = 0b. 4 √2x - 5y - 5 + 4√2 = 0c. 4 √2x + 5y - 4√2 = 0d. 4 √2x + 5y - (15 +... View Article