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Let [t] denote the greatest integer ≤ t. Then the equation in x, [x]²+2[x+2]-7 = 0 has:

a) exactly four integral solutions. b) infinitely many solutions. c) no integral solution. d) exactly two solutions. Solution: [x]2+2[x+2]-7 =... View Article

The value of [latex]\sum_{r=0}^{20} \: ^{50-r}C_{6}[/latex] is equal to

a) 51C7–30C7 b) 51C7+30C7 c) 50C7–30C7 d) 50C6–30C6 Solution: \(\begin{array}{l}\sum_{r=0}^{20} \: ^{50-r}C_{6}\end{array} \) = 50C6+49C6+48C6+…+31C6+30C6 Add and subtract 30C7 Using... View Article

Let y = y(x) be the solution of the differential equation, xy’-y = x²(xcosx+sinx),x > 0. If y(π) = π, then y’’(π/2)+y(π/2) is equal to:

a) 2+(π/2)+(π2/4) b) 2+(π/2) c) 1+(π/2) d) 1+(π/2)+ (π2/4) Solution: xy’-y = x2(x cosx + sinx) x > 0, y(π)... View Article

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^thA.M. is equal to 2^nd G.M., then m is equal to

Solution: 3, …., 243 (m A.M) d = (b-a)/(n+1) = (243-3)/(m+1) = 240/(m+1) 4th AM = 3+4d = 3+4(240/m+1) 3,... View Article

Let R₁ and R₂ be two relations defined as follows

Let R1 and R2 be two relations defined as follows: R1 = {(a,b) ∈ R2: a2+b2 ∈ Q} and R2... View Article

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec), when the length of a side of the cube is 10cm, is

a) 9 b) 10 c) 18 d) 20 Solution: A = 6a2 A is the side of cube. dA/dt =... View Article

Let p, q, r be three statements such that the truth value of (p˄q) -> (∼q˅r) is F. Then the truth values of p, q, r are respectively:

a) F, T, F b) T, F, T c) T, T, F d) T, T, T Solution: (p˄q) -> (∼q˅r)... View Article

If the sum of the series [latex]20+19frac{3}{5}+19frac{1}{5}+18frac{4}{5}+…[/latex] upto n^thterm is 488 and the n^th term is negative, then:

a) n = 60 b) n = 41 c) nth term is -4 d) nth term is \(\begin{array}{l}-4\frac{2}{5}\end{array} \) Solution:... View Article

If [latex]\int \sin ^{-1}\left ( \sqrt{\frac{x}{1+x}} \right )dx= A(x)\tan ^{-1}(\sqrt{x})+B(x)+C[/latex] , where C is a constant of integration, then the ordered pair (A(x), B(x)) can be:

a) (x+1, -√x) b) (x-1, -√x) c) (x+1, √x) d) (x-1, √x) Solution: Given \(\begin{array}{l}\int \sin ^{-1}\left ( \sqrt{\frac{x}{1+x}} \right... View Article

The probability that a randomly chosen 5 digit number is made from exactly two digits is

a) 134/104 b) 121/104 c) 135/104 d) 50/104 Solution: Total case = 9(104) First case : Choose two non-zero digits... View Article

Let x_i be ten observations of a random variable X, then the standard deviation of these observations is

Let xi (1≤ i ≤10)be ten observations of a random variable X. If \(\begin{array}{l}\sum_{i=1}^{10}(x_{i}-p)=3\end{array} \) and \(\begin{array}{l}\sum_{i=1}^{10}(x_{i}-p)^{2}=9\end{array} \) where 0... View Article

[latex]\lim_{x\to a}\frac{(a+2x)^{\frac{1}{3}}-(3x)^{\frac{1}{3}}}{(3a+x)^{\frac{1}{3}}-(4x)^{\frac{1}{3}}}[/latex] is equal to

a) (2/9)4/3 b) (2/3)4/3 c) (2/3)(2/9)1/3 d) (2/9)(2/3)1/3 Solution: Apply L hospital rule ⇒ [(2/3)(3a)-2/3-1/32/3(a-2/3)]/[(1/3)(4a)-2/3-(1/3)41/3a-2/3] = (2/3)×(2/9)1/3 Answer: c

The plane which bisects the line joining the points (4,-2,3) and (2,4,-1) at right angles also passes through the point

a) (0,-1,1) b) (4,0,1) c) (4,0,-1) d) (0,1,-1) Solution: a = 2, b = -6 c = 4 Equation of... View Article

Let the latus rectum of the parabola y² = 4x be the common chord to the circles C₁and C₂ each of them having radius 2√5. Then, the distance between the centres of the circles C₁ and C₂is

a) 8 b) 8√5 c) 4√5 d) 12 Solution: C1C2 = 2C1A (C1A)2 + 4 = ( 2√5)2 C1A =... View Article

The set of all real values of λ for which the quadratic equations, (λ²+1)x²-4λx+2 = 0 always have exactly one root in the interval (0,1) is

a) (-3,-1) b) (2,4) c) (1,3) d) (0,2) Solution: f(0) f(1) ≤ 0 ⇒ (2) [λ2-4λ +3] ≤0 (λ-1)(λ- 3)... View Article

If z₁, z₂ are complex numbers such that…

If z1, z2 are complex numbers such that Re(z1) = |z1-1|, Re(z2) = |z2-1| and arg (z1-z2) = π/6, then... View Article

Let e₁ and e₂ be the eccentricities of the ellipse, (x²/25)+(y²/b²) = 1…

Let e1 and e2 be the eccentricities of the ellipse, (x2/25)+(y2/b2) = 1 (b<5) and the hyperbola, (x2/16)-(y2/b2) = 1... View Article

If a triangle ABC has vertices A(-1,7), B(-7,1) and C(5,-5), then its orthocentre has coordinates

a) (-3,3) b) (-3/5, 3/5) c) (3/5, -3/5) d) (3,-3) Solution: mAH. mBC = -1 ⇒ ((y-7)/(x+1))(1+5)/(-7-5) = -1 ⇒... View Article

If the term independent of x in the expansion of ((3/2)x² -(1/3x))⁹ is k, then 18 k is equal to

a) 5 b) 9 c) 7 d) 11 Solution: Tr+1 = \(\begin{array}{l}^{^{9}}C_{r}\left ( \frac{3}{2}x^{2} \right )^{9-r}\left ( \frac{-1}{3x} \right )^{r}\end{array}... View Article

If the value of the integral [latex]int_{0}^{frac{1}{2}}frac{x^{2}}{(1-x^{2})^{frac{3}{2}}}dx[/latex] is k/6, then k is equal to

a) 2√3+π b) 3√2+π c) 3√2-π d) 2√3-π Solution: \(\begin{array}{l}\int_{0}^{\frac{1}{2}}\frac{x^{2}}{(1-x^{2})^{\frac{3}{2}}}dx\end{array} \) Put x = sinθ \(\begin{array}{l}\int_{0}^{\frac{\pi }{6}}\frac{\sin ^{2}\theta }{\cos ^{3}\theta... View Article

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