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0% found this document useful (0 votes)
48 views3 pagesMaths
Uploaded by
Archit GuptaCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 3
A causa uci Ld lll
1. The number of tangents to the curvex*?+y"?=2a°?,a>0,
which are equally inclined to the axes
(2 Q1 Bo a4
ye angle made by the tangent of the curve.x=a (t+ sint cost),
a (1 + sint)? with the x-axis at any point on it is
1 I-sine
Linea
(1) e424) Qie,
1 L+sine
dey. 4
@ Gere. oS
if m is the slope of a tangent to the curve e” = 1 +.x°, then
(1) [ml > 1 Q) m>1
(3) m>-1 (4) mis
4. Ifateach point ofthe curve y=.° —ax? ++ 1, the tangent
is inclined at an acute angle with the positive direction of
the x-axis, then
(ly a>0
B 3s0s8
5. The slope of the tangent to the curve y = ¥4—x7 at the
pojnt where the ordinate and the abscissa are equal is
a ot
@) 0 (4) none of these
6. ‘The curve given by x + y=” has a tangent parallel to the
_y-axis atthe point
() @1) 11,0)
QB) (1) (4) none of these
7. Ifthe line joining the points (0, 3) and (5, ~2) isa tangent
@ asB
(4) none of these
to the curve y = ©, then the value of eis
vs
at @-2
Ws (4) none of these
8. A differentiable function y =/{x) satisfies f"(x) = fo) + 5
and (0) = 1. Then the equation of tangent at the point
where the curve erosses y-axis is
() x-y+1=0 Q) x-2y+1=0
VA) 6r-y+1=0 (4) x-2y-1=0
9 The distance between the origin and the tangent to the
curve y = e* +x? drawn at the point x =0 is
1 z
VE Os
-I 2
OF OF
10. The point on the curve 3y = 6x ~ 5x° the normal at which
passes through the origin is
() 413) WZ 1-18)
(3) (2,- 28/3) (4) none of these
A. The normal to the curve 2x7 + y/
cuts the curve again at
(3-5) (35)
3) €2,-2) (4) none of these
12 at the point (2,25
1
12, Atwhat points ofeurvey= 23° +5
equal angle with the axes?
vale (9)
Q (
the tangent makey
La
10
7 3) ana »
13. The equation of the tangent to the curve y = be “* at
point where it crosses the y-axis is
Q) ax+by=1
@) a-ty=1 =
b
14, The ae of intersection of the normals at the poi
Fy otto» and 9x2 + 25y°=225
(sz V2 2.
x x x
Mo yt OF oF
15. A function y = f(x) has a second-order derivati
£78) = OCs ~ 1). IF ts graph passes through the poi
(2, 1) and at that point tangent to the graph is 5
then the value of (0) is
ai go @)2 ao
16, The curvex+y—log(x+y) =2x+ 5 has a verti
at the point (a, 8). Then a+ Bis equal to
Or gt @) 2 o2
ange
17. A curve is defined parametrically by
log.(#*), where t is a parameter. Then the equation of th
tangent line drawn to the curve at r= 1 is
Mya 2ev @ y-2e
@y>Sxtt @) y= Sx-1
18, Ifx+4y= 14 isa normal to the curve y?= cee’ — Bat (2.3)
then the value of c+ Bis
Ge -@-5 @7 a 19, In the curve represented parametrically by the equations
x=2Incot +1 and y= tan 1+ cot t,
(1) tangent and normal intersect at the point 2, 1)
(2) normal at = 72/4 is parallel to the y-axis
) tangent at 1= 7/4 is parallel tothe line y = x
Of tangent at = 7/4 is parallel tthe x-axis
20, The abscissas of points P and Q on the curve y= e' + e*
such that tangents at P and Q make 60° with the x-axis are
a) in (4 and oS *)
2
4) »(44)
2
a4)
@) nf ;
ff)
4) In (
21, Ifa variable tangent to the curve xy = c? makes intercepts a,
‘bon.x-and y-axes, respectively, then the value of a°b is
4
yeas
Qs
@ 27e @ oe
22, Let C be the curve y = x° (where x takes all real values).
‘The tangent at A meets the curve again at B. If the gradient
at Bis K times the gradient at 4, then K is equal to
1
Ws a2 @) -2 a5
23. ‘The equation of the line tangent to the curve
xsiny +y sin x= rat the point ( 2) is
272.
()) x+y=28 Qx-y=0
@) 2-y= a2 Wortyee
24, The xintercept ofthe tangent at any arbitrary point of the
curve © + 4 = 1 is proportional to
Be oY,
(1) square of the abscissa of the point of tangeney
2) square root of the abscissa of the point of tangency
WZ) cube of the abscissa of the point of tangency
(4) cube root of the abscissa of the point of tangency
=e, the mean
28, At any point on the curve 2x
Proportional between the abscissa and the difference
between the abscissa and the sub-normal drawn to the
Curve at the same point is equal to
VS) ordinate (2) radius vector
(3) a-intercept of tangent (4) sub-tangent
26. Given g(x) = £+2 and the line 3x + y~ 10 = 0. Then the
line is x1
2) normal to 2(0)
(4) none of these
WG; tangent 10 g(x)
G3) chord of gx)
27.
28.
29.
30,
31.
32.
33.
MM,
35,
37.
Application of Derivatives 5.25
If the length of sub-normal is equal to the length of sub-
tangent at any point (3, 4) on the curve y =f (x) and the
tangent at (3, 4) to y = f(x) meets the coordinate axes at A
and B, then the maximum area of the triangle OAB, where
is origin, is
() 452 YZ) 492 @) 252
The number of points in the rectangle {(x,y)|- 12x 12 and
-3$y'$3} which lie on the curve y=x+ sin x and at which
the tangent to the curve is parallel tothe x-axis is
yo @2 @ 4 (8
Tangent of acute angle between the curves y = |x? — 1| and
y= 7—2 at their points of intersection is
sv3 wy oe
je (2)
oF @
The lines tangent to the curves y° —x°y + Sy 2x=0 and
xt— ay? + Sx + 2y = 0 atthe origin intersect at an angle @
(4) 812
33
>
35
equal to
DE @F wt ay =
ONE ord OQ) Vo
The two curves.
‘Then a? is equal to
1 1
Or Ys
The tangent to the curve y= e at a point (0, 1) meets the
x-axis at (a, 0), where a € (~2,~ 1]. Then ke
(), E120) @ 1-12)
8) 0,1] wun
The curves 4x + 99? = 72 and xy? = 5 at (3, 2)
(J) touch each other Pf cut orthogonally
(3) intersect at 45° (4) intersect at 60°
y= a’ cut orthogonally at a point.
@3 @)2
‘The coordinates of a point on the parabola y* = &x whose
distance from the circle x° + (y+ 6)? = 1 is minimum is
() 24) F 2-4)
(3) (18,- 12) (4) (8,8)
At the point P(a, a") on the graph of y = x", (n € N), in the
first quadrant, a normal is drawn. The normal intersects the
1
y-axis at the point (0, b). If lim,
ow 2
Ol @3 2 @4
Let/ be a continuous and differentiable function. If the
tangent to y = f(x) atx =a is also the normal to y = f(x) at
x= b, then there exists at least one c € (a, b) such that
(s@=0 @) fe>0
GB) O<0 ) none of these
A point on the parabola 18x at which the ordinate
increases at twice the rate of the abscissa is
then m equals,
() 2,6) 2) Q,-6)
9 99)
3) [=
ol 4 (3.3) 5.26 Calculus
38. The rate of change of the volume of a sphere watt. its
surface area, when the radius is 2 em, is
wi 2
@) 3 wa
39. If there is an error of k% in measuring the edge of a cube,
then the percent error in estimating its volume is
ak Pk
k
3) £
On
(4) none of these
40. A lamp of negligible height is placed on the ground ¢, away
ee
froma wall, A man fm tis walking ata speed of [Fis
from the lamp to the nearest point on the wall. When he is,
midway between the lamp and the wall, the rate of change
in the length of this shadow on the wall is
o lms wv
hy
=2 mis
2 m/s
5
41, The function f(x) =.x(x + 3)e satisfies all the conditions
‘of Rolle’s theorem on [-3, 0]. The value of ¢ which verifies
Rolle’s theorem is
v) 2 @) -1
@) 0 4) 3
42. The radius of a right circular cylinder increases at the
rate of 0.1 cm/min, and the height decreases at the rate of
0.2 cm/min, The rate of change of the volume of the
cylinder, in cm*/min, when the radius is 2 em and the
height is 3 cm is
@) -Sms ®
ar
1) 523 2) - =
(1) 2 Oss
3x Qn
3) - == a
Oars «4
43. A cube of ice melts without changing its shape at the
uniform rate of 4 m/min, The rate of change of the
surface area of the cube, in em’/min, when the volume of
the cube is 125 cm’, is
() -4 Bs - 16/5
@) - 166 (4) sis
YF The radius ofthe base of a cone is increasing atthe rate of|
3 cm/min and the altitude is decreasing at the rate of 4 em!
in, The rate of change of lateral surface when the radius
is 7 cm and altitude is 24 em is
A) 108m em*imin (2) Txem*/min
(3) 27m em*/min (4) none of these
45. ILf(x) =x + 2x~ 1, then f(x) has a zero between x= 0 and
- The theorem that best describes this is
(1) mean value theorem
(2) maximum-minimum value theorem
7) intermediate value theorem
(4) none of these
xsin™ for x>0
46, Consider the function f(x) = x
0. for r=0
Then, the number of points in (0, 1) where the derivative
f'(@) vanishes is,
(yo Qt @) 2 WA infinite
47. Let f(x) and g(x) be differentiable for 0