Practice Questions

Q61.Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m ) above the line AC is : (1) 20/3 (2) 5 (3) 10/3 (4) 6

Q62.Let [t] denote the greatest integer ≤t and x→0x[lim discontinuous, when x is equal to: (1) √A + 1 (2) √A + 5 (3) √A + 21 (4) √A

Q62.The values of λ and μ for which the system of linear equations x + y + z = 2, x + 2 y + 3 z = 5, x + 3y + λz = μ has infinitely many solutions, are respectively (1) 6 and 8 (2) 5 and 7 (3) 5 and 8 (4) 4 and 9 ∞ Σ = 2, x, y ∈N , where N is the set of all natural numbers, then the value

Q62.If the minimum and the maximum values of the function f : [ π4 , π2 ] →R, defined by −sin2 θ −1 −sin2 θ 1 f(θ) = −cos2 θ −1 −cos2 θ 1 are m and M respectively, then the ordered pair (m, M) is equal to : 12 10 −2 (1) (0, 2√2) (2) (−4, 0) (3) (−4, 4) (4) (0, 4)

Q62.Let S , be the set of all functions f : [0, 1] →R, which are continuous on [0, 1], and differentiable on (0, 1). Then for every f in S , there exists c ∈(0, 1), depending on f , such that. f '(c) (1) |f(c) −f(1)| < (1 −c) f '(c) (2) f(1)−f(c)1−c = (3) |f(c) + f(1)| < (1 + c) f '(c) (4) |f(c) −f(1)| < f '(c)

Q62.The inverse function of f(x) = 82x−8−2x , x ∈(−1, 1), is __________. 82x+8−2x (1) 4 1 loge( 1+x1−x ) (2) 14 loge( 1−x1+x ) (3) 1 4 (loge) loge( 1−x1+x ) (4) 14 log8( 1+x1−x ) = π6 , then

Q62.The domain of the function f(x) = sin−1( |x|+5x2+1 ) is (−∞, −a] ∪[a, ∞), then a is equal to (1) √17 (2) √17−1 2 2 (3) 1+√17 (4) √17 2 2 + 1 Q63. ⎧ aex + be−x, −1 ≤x < 1 If a function f(x) defined by f(x) = 1 ≤x ≤3 be continuous for some a, b, c ∈R and ⎨ cx2, ⎩ ax2 + 2cx, 3 < x ≤4 f ′(0) + f ′(2) = e, then the value of a is (1) 1 (2) e e2−3e+13 e2−3e−13 (3) e (4) e e2+3e+13 e2−3e+13

Q62.Let y = y(x) be a function of x satisfying y√1 −x2 = k −x√1 −y2 where k is a constant and y( 21 ) = −14 .Then dx dy at x = 12 , is equal to (1) −√54 (2) −√52 (3) 2 (4) √5 √5 2

Q62.Let f : R →R be a function defined by f(x) = max {x, x2}.Let S denote the set of all points in R,where f is not differentiable.Then : (1) {0, 1} (2) {0} (3) ϕ (an empty set) (4) {1} π ,

Q62.If y(α) = √2( (1) 4 (2) 43 (3) −4 (4) −14

Q62.A survey shows that 63% of the people in a city read newspaper A whereas 76% read news paper B. If x% of the people read both the newspapers, then a possible value of x can be: (1) 29 (2) 37 (3) 65 (4) 55 where i = √−1, then which one of the following is not (θ = 24π ) and A5 = [ ac bd ],

Q62.Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector 1 0 0 b on the right side is equal to b1, b2 and b3 respectively. If x1 = ⎡ 1 ⎤, x2 = ⎡2 ⎤, x3 = ⎡0⎤ ; 1 1 1 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 1 0 0 b1 = ⎡ 0 ⎤, b2 = ⎡ 2 ⎤, b3 = ⎡0 ⎤, then the determinant of A is equal to 0 0 2 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (1) 4 (2) 2 (3) 1 (4) 3 2 2

Q62.If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for some k ∈R, then x + ( yz ) is equal to : (1) −3 (2) 9 (3) 3 (4) −9

Q63.The value of c, in the Lagrange’s mean value theorem for the function f(x) = x3 −4x2 + 8x + 11, when x ∈[0,1], is (1) 4−√5 (2) 4−√7 3 3 (3) 2 (4) √7−2 3 3

Q63.The set of all real values λ for which the function f(x) = (1 −cos2 x). (λ + sin x), xε (−π2 2 ), has exactly one maxima and exactly one minima, is : (1) (−12 , 12 ) −{0} (2) (−32 , 32 ) (3) (−12 , 12 ) (4) (−32 , 32 ) −{0}

Q63.Let xk + yk = ak, (a, k > 0) and dx 1 dy + ( xy ) 3 = 0, then k is (1) 3 (2) 4 2 3 (3) 32 (4) 13

Q63.Let f be any function continuous on [a, b] and twice differentiable on (a, b) . If all x ∈(a, b), f '(x) > 0 and f ''(x) < 0 , then for any c ∈(a, b), f(c)−f(a)f(b)−f(c) (1) b+a (2) 1 b−a (3) b−c (4) c−a c−a b−c

Q63.If y2 + loge(cos2 x) = y, x ∈(−π2 , π2 ) then : (1) y′′(0) = 0 (2) |y′(0)| + |y′′(0)| = 1 (3) |y′′(0)| = 2 (4) |y′(0)| + |y′′(0)| = 3

Q63.Let λ ∈R. The system of linear equations 2x1 −4x2 + λx3 = 1 x1 −6x2 + x3 = 2 λx1 −10x2 + 4x3 = 3 is inconsistent for : (1) exactly one positive value of λ (2) exactly one negative value of λ (3) every value of λ (4) exactly two values of λ

Q63.If f(x + y) = f(x) f(y) and x=1f(x) of f(4) is f(2) (1) 2 (2) 1 3 9 (3) 1 (4) 4 3 9

Q63.If x = 2 sin θ −sin 2θ and y = 2 cos θ −cos 2θ , θ ∈[0, 2π], then d2y at θ = π is: dx2 (1) 4 3 (2) −38 (3) 2 3 (4) −34

Q63.Suppose f(x) is a polynomial of degree four having critical points at −1, 0, 1. If T = {x ∈R |f(x) = f(0)}, then the sum of squares of all the elements of T is : (1) 4 (2) 6 (3) 2 (4) 8

Q63.The minimum value of 2sin x + 2cos x is : −1+ (1) √2 1 (2) 2−1+√2 2 (3) 21−√2 (4) 2 1−1√2 4 + tan−1 x, |x| ≤1 is :

Q63.If A = [ cosisinθθ cosisinθθ ], true? (1) 0 ≤a2 + b2 ≤1 (2) a2 −d2 = 0 (3) a2 −c2 = 1 (4) a2 −b2 = 12 cos = a2 −b2 , where a > b > 0, then dxdy at ( π4 , π4 ) is:

Q63.The length of the perpendicular from the origin, on normal to the curve, x2 + 2xy −3y2 = 0, at the point (2, 2), is. (1) √2 (2) 4√2 (3) 2 (4) 2√2 ∫x0 tsin(10t)dt , is equal to