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If the tangent to the curve y = x+sin y at a point (a,b) is parallel to the line joining (0, 3/2) and (1/2, 2), then:

a) b = (π/2)+a b) |a+b| = 1 c) |b-a| = 1 d) b = a Solution: (dy/dx)p(a,b) = (2-3/2)/(1/2... View Article

Let α and β be the roots of the equation, 5x²+6x-2 = 0. If S_n = αⁿ+βⁿ, n = 1,2,3,.. then

a) 5S6+6S5+2S4 = 0 b) 6S6+5S5 = 2S4 c) 6S6+5S5+2S4 = 0 d) 5S6+6S5= 2S4 Solution: 5x2+6x-2 = 0 Put... View Article

The contrapositive of the statement “If I reach the station in time, then I will catch the train” is

a) If I will catch the train, then I reach the station in time. b) If I do not reach... View Article

If the number of integral terms in the expansion of (3^1/2+5^1/8)ⁿ is exactly 33, then the least value of n is

a) 128 b) 248 c) 256 d) 264 Solution: Tr+1 = nCr(31/2)n-r(51/8)r (n-r)/2 -> n-r = 0, 2, 4, 6,... View Article

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) 1/3 b) 1/4 c) 1/8 d) 1/9 Solution: Total Possibilities = (1, 3), (3, 1), (2, 2), (2, 6),... 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) = 1 b12 = a12(1-e12) 3 =... View Article

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

a) 5π/4 b) 3π/2 c) 7π/4 d) π/2 Solution: 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)) = 2π-tan-1 [(4/3)+(5/12)/(1-(4/3)(5/12) ]- tan-1(16/63) = 2π-tan-1(48+15)/(36-20) –... View Article

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

a) |y’(0)|+|y’’(0)| = 1 b) |y’’(0)| = 0 c) |y’(0)|+|y’’(0)| = 3 d) |y’’(0)| = 2 Solution: yy’ + 2... 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 = 1 b) x-2y+z = 1 c) 2x+y-z = 1 d) x+2y-z = 1 Solution: \(\begin{array}{l}\vec{r} = (1,-2,3)+\lambda... 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) 1 b) -1 c) -3 d) 9 Solution: \(\begin{array}{l}\begin{vmatrix} x-2 & 2x-3 & 3x-4\\ 2x-3&3x-4 & 4x-5\\ 3x-5 &... 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) 1 d) 1+ (51)! Solution: Answer: b

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

1) equivalent to ~p → q 2) equivalent to p → ~q 3) a fallacy 4) a tautology Solution: Answer:... 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) 1/4 2) 2/9 3) 4/9 4) 6/7 Solution: Answer: (2) tan θ = 1/4 tan (θ + β) =... View Article

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

1) 1 / 3 2) 2 / 9 3) −7 / 9 4) −3 / 5 Solution: Answer: (3) 5(tan2... 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 / 55 2) 14 / 45 3) 7 / 55 4) 6 / 55 Solution: Answer: (4) Total... 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 / 16 2) 7 / 64 3) 3 / 16 4) 7 / 32 Solution: Answer: (1) P... 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) 6 2) 4 3) 6 / 25 4) 12 / 5 Solution: Answer: (4) n = 10 p (Probability... 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) 2 2) 5 3) 1 / 8 4) 25 / 8 Solution: Answer: (1)

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√42 2) √42 3) 6√5 4) 3√5 Solution: Answer: (1) Equation of PQ, \(\begin{array}{l}\frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}\end{array} \) Let M be (λ... 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 / √83 2) 5 / √83 3) 10 / √74 4) 20 / √74 Solution: Answer: (1) Let... View Article

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