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A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is

a) 1/3b) 1/4c) 1/8d) 1/9Solution:Total Possibilities = (1, 3), (3, 1), (2, 2), (2, 6), (6, 2) (4, 4), (3, 5), (5, 3) (6, 6)(A ∩ B) = (4,4)n(A ∩... View Article

A hyperbola having the transverse axis of length √2 has the same foci as that of the ellipse 3x²+4y²= 12, then this hyperbola does not pass through which of the following points

a) (√(3/2), 1/√2)b) (1, -1/√2)c) (1/√2, 0)d) (-√(3/2),1)Solution:(x2/4)+(y2/3) = 1b12 = a12(1-e12)3 = 4(1-e12)e1 = 1/2focus = (± a1e1, 0)=... View Article

2π-(sin^-1(4/5)+sin^-1(5/13)+sin^-1(16/65)) is equal to

a) 5π/4b) 3π/2c) 7π/4d) π/2Solution:2π-(sin-1(4/5)+sin-1(5/13)+sin-1(16/65)) = 2π-(tan-1(4/3)+tan-1(5/12)+tan-1(16/63))=... View Article

If y²+log_e(cos²x) = y, x∈(-π/2, π/2), then

a) |y’(0)|+|y’’(0)| = 1b) |y’’(0)| = 0c) |y’(0)|+|y’’(0)| = 3d) |y’’(0)| = 2Solution:yy’ + 2 (-tanx) = y’ ....(1)diff. w.r.t.x2y’’+2(y’)2 -2 sec2... View Article

The foot of the perpendicular drawn from the point (4,2,3) to the line joining the points (1,2,3) and (1,1,0) lies on the plane:

a) x-y-2z = 1b) x-2y+z = 1c) 2x+y-z = 1d) x+2y-z = 1Solution: (-3, 3λ-4, -3λ).(0,3, -3) = 0⇒ 0+9λ-12+9λ = 0⇒ λ= 12/18 = 2/3m =... View Article

If Δ = [latex]\begin{vmatrix} x-2 & 2x-3 & 3x-4\\ 2x-3&3x-4 & 4x-5\\ 3x-5 & 5x-8 &10x-17 \end{vmatrix}[/latex] = Ax³+Bx²+Cx+D, then B+C is equal to:

a) 1b) -1c) -3d) 9Solution: = Ax3+Bx2+Cx+DR2 ⇒ R2 -R1 R3 ⇒ R3 -R2 ⇒ = Ax3+Bx2+Cx+D⇒ (x-1)(x-2){(x-2)(6- 2)-(2x-3)(6-1)+(3x-... View Article

The value of (2.¹P₀-3.²P₁+4.³P₂-….. up to 51^th term) +(1!-2!+3!-…… up to 51^th term) is equal to

a) 1-51(51)!b) 1+(52)!c) 1d) 1+ (51)!Solution:Answer: b... View Article

The following statement (p → q) → [(~p → q) → q] is

1) equivalent to ~p → q2) equivalent to p → ~q3) a fallacy4) a tautologySolution:Answer: (4)... View Article

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If ∠BPC = β, then tan β is equal to

1) 1/42) 2/93) 4/94) 6/7Solution:Answer: (2) tan θ = 1/4tan (θ + β) = 1/2... View Article

If 5(tan² x − cos² x) = 2cos²x + 9, then the value of cos 4x is

1) 1 / 32) 2 / 93) −7 / 94) −3 / 5Solution:Answer: (3)5(tan2 x − cos2 x) = 2cos2x + 95 sec2 x - 5 = 9 cos2 x + 7Let cos2 x = t(5 / t) = 9t +... View Article

If two different numbers are taken from the set {0, 1, 2, 3, ……, 10}; then the probability that their sum, as well as absolute difference, are both multiples of 4, is

1) 12 / 552) 14 / 453) 7 / 554) 6 / 55Solution:Answer: (4) Total number of ways = 11C2 = 55Favourable ways are (0, 4), (0, 8), (4, 8), (2, 6),... View Article

For three events A, B and C, P (Exactly one of A or B occurs) = P (Exactly one of B or C occurs) = P (Exactly one of C or A occurs) = 1 / 4 and P (All the three events occur simultaneously) = 1 / 16. Then the probability that at least one of the events occurs is :

1) 7 / 162) 7 / 643) 3 / 164) 7 / 32Solution:Answer: (1) P (A) + P (B) – P (A ⋂ B) = 1 / 4P (B) + P (C) – P (B ⋂ C) = 1 / 4P (C) + P (A) – P (A... View Article

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is

1) 62) 43) 6 / 254) 12 / 5Solution:Answer: (4)n = 10p (Probability of drawing a green ball) = 15 / 25p = 3 / 5, q = 2 / 5var (X) = n.p.q= 10 * (3... View Article

Let a = 2i + j – 2k and b = i + j. Let c be a vector such that |c – a| = 3, |(a x b) x c| = 3 and the angle between c and a × b be 30^o. Then a ⋅ c is equal to

1) 22) 53) 1 / 84) 25 / 8Solution:Answer: (1)... View Article

If the image of the point P (1, −2, 3) in the plane, 2x + 3y − 4z + 22 = 0 measured parallel to the line, (x / 1) = (y / 4) = (z / 5) is Q, then PQ is equal to

1) 2√422) √423) 6√54) 3√5Solution:Answer: (1) Equation of PQ, Let M be (λ + 1, 4λ - 2, 5λ + 3) As it lies on 2x + 3y – 4z + 22 = 0λ = 1For... View Article

The distance of the point (1, 3, −7) from the plane passing through the point (1, −1, −1), having normal perpendicular to both the lines [latex]frac{x-1}{1}=frac{y+2}{-2}=frac{z-4}{3}[/latex] and [latex]frac{x-2}{2}=frac{y+1}{-1}=frac{z+7}{-1}[/latex] is:

1) 10 / √832) 5 / √833) 10 / √744) 20 / √74Solution:Answer: (1) Let the plane be a (x - 1) + b (y + 1) + c (z + 1) = 0.It is perpendicular to... View Article

A hyperbola passes through the point P (√2, √3) and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point

1) (2√2 , 3√3)2) (√3, √2)3) (− √2 , –√3)4) (3√2 , 2√3)Solution:Answer: (1) [x2 / a2] - [y2 / b2] = 1a2 + b2 = 4 and [2 / a2] - [3 / b2] = 1(2 /... View Article

The eccentricity of an ellipse whose centre is at the origin is 1 / 2. If one of its directrices is x = −4, then the equation of the normal to it at (1, (3 / 2)) is

1) 4x − 2y = 12) 4x + 2y = 73) x + 2y = 44) 2y − x = 2Solution:Answer: (1) x = –4e = 1 / 2(-a / e) = -4-a = -4 * ea = 2Now, b2 = a2 (1 - e2) =... View Article

The radius of a circle, having minimum area, which touches the curve y = 4 − x² and the lines, y = |x| is

1) 2 (√2 - 1)2) 4 (√2 - 1)3) 4 (√2 + 1)4) 2 (√2 + 1)Solution:Answer: (2) x2 = - (y - 4)Let a point on the parabola P [(t/2) , (4 -... View Article

Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

1) (1, 3/4)2) (1, -3/4)3) (2, 1/2)4) (2, -1/2)Solution:Answer: (3) (k2 - 7k + 10) + 4k2 + 20k = ± 565k2 + 13k - 46 = 05k2 + 13k + 66 = 05k2... View Article