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Mere Teacher · Previous Year Questions
JEE Main Mathematics · Continuity & Differentiability · Set 1 (Q1-30)
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1.
If f(x)=x2ex2−cosx, for x=0 is continuous at x=0, then value of f(0) is
(1)32
(2)25
(3)1
(4)23
2.
f(x)=xx has stationary point at
(1)x=e
(2)x=e1
(3)x=1
(4)x=e
3.
If sin−1a is the acute angle between the curves x2+y2=4x and x2+y2=8 at (2,2), then a=
(1)1
(2)0
(3)21
(4)23
4.
The derivative of tan−1(sinx+cosxsinx−cosx), with respect to 2x, where (x∈(0,2π)) is :
(1)1
(2)32
(3)21
(4)2
5.
If f(x)=x2−2x+4, then the set of values of x satisfying f(x−1)=f(x+1) is
(1){−1}
(2){−1,1}
(3){1}
(4){1,2}
6.
The function f(x)=x2+x54
(1)is increasing and has minimum value 27 in the interval (0,∞)
(2)is decreasing and has neither maximum nor minimum in the interval (−∞,0)
(3)has maximum value 27 in the interval (−∞,∞)
(4)is increasing and has neither maximum nor minimum values in the interval (−∞,∞)
7.
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :
(1)12.5
(2)10
(3)25
(4)30
8.
If the function f(x)=x(x+3)e2−x satisfies all the conditions of Rolle's theorem in [−3,0], then a root of f’(x)=0 is
(1)3
(2)-1
(3)-2
(4)-3
9.
If y=(sin−1x)2+(cos−1x)2, then (1−x2)y2−xy1=
(1)1
(2)4
(3)-4
(4)-1
10.
Let a and b be real constants such that the function f defined by f(x)=(x2+3x+a,x≤1bx+2,x>1) be differentiable on R. Then, the value of ∫−22f(x)dx equals
(1)615
(2)619
(3)21
(4)17
11.
If y=(x−sinx)+(x−sinx)+(x−sinx)……….then dxdy=
(1)2y−11−cosx
(2)2y−11+cosx
(3)2y+11−cosx
(4)2y−11−sinx
12.
If f(x)=(x−1logx,ifx=1k,ifx=1) is continuous at x=1, then the value of k is
(1)0
(2)-1
(3)1
(4)e
13.
The value of c of Lagrange's mean value theorem for f(x)=25−x2 on [1,5] is
(1)15
(2)5
(3)10
(4)1
14.
If log10(x2+y2x2−y2)=2, then dxdy=
(1)−101y99x
(2)101y99x
(3)−101x99y
(4)101x99y
15.
If y(θ)=cos3θ+4cos2θ+5cosθ+22cosθ+cos2θ, then at θ=2π,y’’+y’+y is equal to :
(1)21
(2)1
(3)2
(4)23
16.
If f(a)=log(1+a1−a) for a=-1,1, then the set of values of all ' a ', for which f(1+a22a)>0 is
(1)(0,∞)−{1}
(2)(−∞,0)−{−1}
(3)(−∞,∞)−{−1,1}
(4)(−1,1)
17.
If xy=ex−y, then dy/dx at x=1 is
(1)e
(2)1
(3)0
(4)-1
18.
If f(x)=(xa|x|+x2−2(sin|x|)(cos|x|),x=0b,x=0) is continuous at x=0, then a+b is equal to
(1)4
(2)1
(3)2
(4)0
19.
If x+y=k,x>0,y>0 then x2+y2 is minimum, if
(1)x>y
(2)x<y
(3)x=y
(4)x=2y
20.
Let f be any continuous function on [0,2] and twice differentiable on (0,2). If f(0)=0,f(a)=1 and f(b)=2, then
(1)f"(x)=0for all x∈(0,2)
(2)f"(x)=0 for some x∈(0,2)
(3)for some x∈[0,2]
(4)f"(x)>0for all x∈(0,2)
21.
If f(x)=(x−2x2−(a+2)x+a,x=22,x=2)is continuous at x=2, then the value of a is
(1)-1
(2)-6
(3)0
(4)1
22.
If f:N→R is defined by f(1)=−1 and f(n+1)=3f(n)+2 for ≥1, then f is
(1)one-one
(2)onto
(3)a constant function
(4)f(n)>0 for n>1
23.
If x=e(yx), then dxdy=
(1)xlogyx−y
(2)xlogxx−y
(3)ylogxx−y
(4)xlogxx+y
24.
If ey+xy=e the ordered pair (dxdy,dx2d2y) atx=0 is equal to
(1)(e1,e21)
(2)(e1,e2−1)
(3)(e−1,e2−1)
(4)(e−1,e21)
25.
The maximum slope of the curve y=21x4−5x3+18x2−19x occurs at the point:
(1)(2,9)
(2)(2,2)
(3)(3,221)
(4)(0,0)
26.
The approximate value of (8.01)4/3+(8.01)2 upto 3 decimal places is
(1)80.116
(2)80.216
(3)80
(4)80.186
27.
If y=emsin−1x and (1−x2)(dxdy)2=Ay2, then A=
(1)m
(2)−m
(3)m2
(4)−m2
28.
The number of points, where the function f : R → R, f (x) = |x – 1| cos |x – 2| sin |x – 1| + (x - 3) |x2 - 5x + 4|, is NOT differentiable, is :
(1)1
(2)2
(3)3
(4)4
29.
If f(x)=sin(cos−1(1+22x1−22x)and its first derivative with respect to x is−abloge2 when x = 1, where a and b are integers, then the minimum value of (a2−b2)is ____.
30.
Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6is unity and it has extrema at x=−1 and x=1 . If limx→0x3f(x)=1, then 5.f(2) is equal to
