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LEVEL-1
LEVEL-1
(Practice
(Practice Question)
Question) Tangent
Tangent &
& Normal
Normal
Question
Question
based on
based on Equation of tangent & related factsEquation of tangent & related facts
Q.1
Q.1 If tangent to the curve y = f(x) at any point isIf tangent to the curve y = f(x) at any point is par
parallallel el to to y-ay-axisxis, , then then at at thathat t popoint int dy/ddy/dxx equals-(A) (A) 0 0 (B) (B) 1 1 (C)(C)
(D) – 1 (D) – 1 Q.2Q.2 If normal to the curve y = f(x) at a pointIf normal to the curve y = f(x) at a point makes 135º angle with x- axis, then at that makes 135º angle with x- axis, then at that point dy/dx
point dy/dx equals-(A)
(A) 1 1 (B) (B) –1 –1 (C) (C) 00 (D)
(D)
Q.3Q.3 The slope of the curve y = sin x + cosThe slope of the curve y = sin x + cos22 x is x is zero at the point,
zero at the point, where-(A) x = (A) x = 4 4
(B) (B) x x == 2 2
(C) x = (C) x =
(D) (D) No No wherewhere Q.4Q.4 The The slope slope of of the the tangent tangent to to the the curvecurve x
x22 + 2y = 8x + 2y = 8x – 7 at the point x = 5 is -– 7 at the point x = 5 is -(A)
(A)
/4 /4 (B)(B)
/3 /3 (C) (C) 33
/4/4 (D)(D)
/2 /2 Q.5Q.5 The equation of the tangent to the curveThe equation of the tangent to the curve y = cos x at x =
y = cos x at x =
/3 is-/3 is-(A) 3x – 2 (A) 3x – 2 33y =y =
+ + 33 (B) 3x + 2 (B) 3x + 2 33y =y =
+ + 33 (C) 3x + 2 (C) 3x + 2 33y =y =
– – 33 (D) None of these (D) None of these Q.6Q.6 The equation of tangent to the curveThe equation of tangent to the curve y = 2 sin x + sin 2x at the point x =
y = 2 sin x + sin 2x at the point x =
/3 is-/3 is-(A)(A) 2y 2y = = 3 3 (B) (B) 3y 3y == 22
(C) 2y = 3
(C) 2y = 3 33 (D) (D) 2y 2y = = 33 Q.7
Q.7 The equation of the tangent to the curve 6y = 7 –The equation of the tangent to the curve 6y = 7 – x
x33 at point (1, 1) is- at point (1, 1) is-(A)
(A) 2x 2x + + y y = = 3 3 (B) (B) x x + + 2y 2y = = 33 (C)
(C) x x + + y y = = 1 1 (D) (D) x x + + y y + + 2 2 = = 00
Q.8
Q.8 The equation of the tangent to the curveThe equation of the tangent to the curve 1/
1/ xx + 1/+ 1/ yy= 2/= 2/ aa at point (a, a) is- at point (a, a) is-(A) a/ (A) a/ xx + a/+ a/ yy= 2= 2 aa (B) x + y = (B) x + y = 2a2a (C) (C) xx ++ yy = 2= 2 aa (D) None of these (D) None of these Q.9
Q.9 y = x –11 is a tangent to the curve y = xy = x –11 is a tangent to the curve y = x33 – 11x + 5 – 11x + 5 at the
at the point-(A)
(A) (2,–9) (2,–9) (B) (B) (3, (3, –8)–8) (C)
(C) (11, (11, 0) 0) (D) (D) None None of of thesethese Q.10
Q.10 If tangent at a point of the curve y = f(x) isIf tangent at a point of the curve y = f(x) is perpendicu
perpendicular lar to to 2x 2x – – 3y 3y = = 5, 5, then then at at thatthat point dy/dx
point dy/dx equals-(A)
(A) 2/3 2/3 (B) (B) –2/3–2/3 (C)
(C) 3/2 3/2 (D) (D) –3/2–3/2 Q.11
Q.11 At what point the tangent to the curveAt what point the tangent to the curve xx ++
y
y== aa is is perpendicular perpendicular to to thethe x-
x- axis-(A)
(A) (0, (0, 0) 0) (B) (B) (a, (a, a)a) (C)
(C) (a, (a, 0) 0) (D) (D) (0, (0, a)a) Q.12
Q.12 At what point of the curve y = 2xAt what point of the curve y = 2x22 – x + 1 – x + 1 tangent is parallel to y = 3x + 4 tangent is parallel to y = 3x + 4 (A) (A) (0, (0, 1) 1) (B) (B) (1, (1, 2)2) (C) (C) (–1, (–1, 4) 4) (D) (D) (2, (2, 7)7) Q.13
Q.13 If tangent of the curve x = tIf tangent of the curve x = t22 – 1, – 1, y = y = tt22 – t is – t is perpendicu
perpendicular to x- axis, then-lar to x- axis, then-(A)
(A) t t = = 0 0 (B) (B) t t = = 1/1/ 22
(C) t =
(C) t =
(D) (D) t t = = –1/–1/ 33Q.14
Q.14 The equation The equation of of tangent tangent to to the the curve curve y y == 1 – e
1 – ex/2x/2 at the point where it meets y- axis is- at the point where it meets y- axis is-(A)
(A) x x + + 2y 2y = = 0 0 (B) (B) 2x 2x + + y y = = 00 (C)
Q.15
Q.15 The point where the tangent line to the curveThe point where the tangent line to the curve y = e
y = e2x2x at (0, 1) at (0, 1) meets x- axis is-meets x- axis is-(A)
(A) (1, (1, 0) 0) (B) (B) (–1, (–1, 0)0) (C)
(C) (–1/2, (–1/2, 0) 0) (D) (D) None None of of thesethese Q.16
Q.16 At what point the slope of the tangent to theAt what point the slope of the tangent to the curve x
curve x22 + y + y22 – 2x –3 = – 2x –3 = 0 is zero-0 is zero-(A)
(A) (3, (3, 0); 0); (–1, (–1, 0) 0) (B) (B) (3, (3, 0) 0) ; ; (1, (1, 2)2) (C)
(C) (–1, (–1, 0); 0); (1, (1, 2) 2) (D) (D) (1, (1, 2) 2) ; ; (1, (1, –2)–2) Q.17
Q.17 The The equation equation of of the the tangent tangent to to the the curve curve y y == x
x22 + 1 at point (1, 2) is- + 1 at point (1, 2) is-(A)
(A) y y = = 2x 2x (B) (B) x x + + 2y 2y = = 55 (C)
(C) 2x 2x + + y y = = 4 4 (D) (D) None None of of thesethese Q.18
Q.18 The equation of tangent at the point (atThe equation of tangent at the point (at22, , atat33)) on the curve ay
on the curve ay22 = x = x33 is- is-(A) 3tx– 2y = at
(A) 3tx– 2y = at33 (B) (B) tx tx – – 3y 3y = = atat33 (C) 3 tx + 2y = at
(C) 3 tx + 2y = at33 (D) (D) None None of of thesethese Q.19
Q.19 The slopes of the tangents to the curveThe slopes of the tangents to the curve y = (x +1) (x – 3) at the points where it y = (x +1) (x – 3) at the points where it crosses x- axis
crosses x- axis are-(A)
(A) ±2 ±2 (B) (B) ±3±3 (C)
(C) ±4 ±4 (D) (D) None None of of thesethese Q.20
Q.20 The coordinates of the point on the curveThe coordinates of the point on the curve y
y = = xx22 + 3x + 4, the tangent at which passes + 3x + 4, the tangent at which passes through the origin
through the origin are-(A)
(A) (–2, (–2, 2), 2), (2, (2, 14) 14) (B) (1, (B) (1, –1), –1), (3, (3, 4)4) (C)
(C) (2, (2, 14), 14), (2, (2, 2) 2) (D) (D) (1, (1, 2), 2), (14, (14, 3)3) Q.21
Q.21 The angle made by tangent at the point (2, 0)The angle made by tangent at the point (2, 0) of the curve y = (x – 2) (x
of the curve y = (x – 2) (x – 3) with x- axis is-– 3) with x- axis is-(A) (A)
/4 /4 (B)(B)
/2 /2 (C)(C) 4 4 3 3
(D) (D)
Q.22Q.22 If the curve y = xIf the curve y = x22 + bx + c, touches the line + bx + c, touches the line y = x at the point (1, 1), then values of b and c y = x at the point (1, 1), then values of b and c are-(A) (A) –1, –1, 2 2 (B) (B) –1, –1, 11 (C) (C) 2, 2, 1 1 (D) (D) –2, –2, 11 Q.23
Q.23 The line x/a + y/b = 1 touches the curve y = beThe line x/a + y/b = 1 touches the curve y = be –x/a –x/a at the at the point-(A) (A) (0, (0, a) a) (B) (B) (0, (0, 0)0) (C) (C) (0, (0, b) b) (D) (D) (b, (b, 0)0) Q.24
Q.24 The straight line x + y = a will be tangent toThe straight line x + y = a will be tangent to
9 9 x x22 + + 16 16 y y22 = 1, if a equals = 1, if a equals to-(A) (A) 8 8 (B) (B) ± ± 55 (C) (C) ±1 ±1 (D) (D) ± ± 66 Q.25
Q.25 A tangent to the curve y = xA tangent to the curve y = x22 + 3x passes + 3x passes through a point (0, –9) if it is drawn at the through a point (0, –9) if it is drawn at the point-(A) (A) (–3, (–3, 0) 0) (B) (B) (1, (1, 4)4) (C) (C) (0, (0, 0) 0) (D) (D) (–4, (–4, 4)4) Q.26
Q.26 The coordinates of the points on the curveThe coordinates of the points on the curve x = a (
x = a (
+ sin + sin
), y = a (1– cos), y = a (1– cos
), where), where tangent is inclined an angletangent is inclined an angle
/4 to the x-axis/4 to the x-axis areare -(A)
(A) (a, (a, a) a) (B)(B)
aa ,, 1 1 2 2 aa (C) (C)
aa ,, 1 1 2 2 aa (D)(D)
1 1 2 2 aa ,, aa Q.27Q.27 If the area of the triangle included betweenIf the area of the triangle included between the axes and any tangent to the curve the axes and any tangent to the curve xy
xynn = a = an+1n+1 is constant, then value of n is- is constant, then value of n is-(A)
(A) –1 –1 (B) (B) 1 1 (C) (C) .2 .2 (D) (D) –2–2 Q.28
Q.28 The angle made by the tangent to the curveThe angle made by the tangent to the curve x
x = = eett cos t, y = e cos t, y = ettsin t at point t =sin t at point t =
/4 with/4 with x– axis isx– axis is -(A)
(A) 0 0 (B)(B)
/4 /4 (C)(C)
/3 /3 (D)(D)
/2/2 Q.29Q.29 The points at which the tangent to the curveThe points at which the tangent to the curve y = x
y = x33 + 5 is perpendicular to the line x + 3y = 2 + 5 is perpendicular to the line x + 3y = 2 are are -(A) (A) (6, (6, 1), 1), (–1, (–1, 4) 4) (B) (B) (6, (6, 1), 1), (4, (4, –1)–1) (C) (C) (1, (1, 6), 6), (1, (1, 4) 4) (D) (D) (1, (1, 6), 6), (–1, (–1, 4)4) Q.30
Q.30 The coordinates of any point P on a curve areThe coordinates of any point P on a curve are represented by x = represented by x = 2 2 1 1 tt22, y =, y = 3 3 1 1 tt33, where t is a, where t is a parameter,
parameter, then then equation equation of of tangent tangent to to thethe curve at P curve at P is-(A) 6tx – 6y = t (A) 6tx – 6y = t33 (B) (B) 4tx 4tx + + 3y 3y = = tt33 (C) 3tx + 2y = t (C) 3tx + 2y = t33 (D) (D) 3tx 3tx + + y y = = tt33
Q.15
Q.15 The point where the tangent line to the curveThe point where the tangent line to the curve y = e
y = e2x2x at (0, 1) at (0, 1) meets x- axis is-meets x- axis is-(A)
(A) (1, (1, 0) 0) (B) (B) (–1, (–1, 0)0) (C)
(C) (–1/2, (–1/2, 0) 0) (D) (D) None None of of thesethese Q.16
Q.16 At what point the slope of the tangent to theAt what point the slope of the tangent to the curve x
curve x22 + y + y22 – 2x –3 = – 2x –3 = 0 is zero-0 is zero-(A)
(A) (3, (3, 0); 0); (–1, (–1, 0) 0) (B) (B) (3, (3, 0) 0) ; ; (1, (1, 2)2) (C)
(C) (–1, (–1, 0); 0); (1, (1, 2) 2) (D) (D) (1, (1, 2) 2) ; ; (1, (1, –2)–2) Q.17
Q.17 The The equation equation of of the the tangent tangent to to the the curve curve y y == x
x22 + 1 at point (1, 2) is- + 1 at point (1, 2) is-(A)
(A) y y = = 2x 2x (B) (B) x x + + 2y 2y = = 55 (C)
(C) 2x 2x + + y y = = 4 4 (D) (D) None None of of thesethese Q.18
Q.18 The equation of tangent at the point (atThe equation of tangent at the point (at22, , atat33)) on the curve ay
on the curve ay22 = x = x33 is- is-(A) 3tx– 2y = at
(A) 3tx– 2y = at33 (B) (B) tx tx – – 3y 3y = = atat33 (C) 3 tx + 2y = at
(C) 3 tx + 2y = at33 (D) (D) None None of of thesethese Q.19
Q.19 The slopes of the tangents to the curveThe slopes of the tangents to the curve y = (x +1) (x – 3) at the points where it y = (x +1) (x – 3) at the points where it crosses x- axis
crosses x- axis are-(A)
(A) ±2 ±2 (B) (B) ±3±3 (C)
(C) ±4 ±4 (D) (D) None None of of thesethese Q.20
Q.20 The coordinates of the point on the curveThe coordinates of the point on the curve y
y = = xx22 + 3x + 4, the tangent at which passes + 3x + 4, the tangent at which passes through the origin
through the origin are-(A)
(A) (–2, (–2, 2), 2), (2, (2, 14) 14) (B) (1, (B) (1, –1), –1), (3, (3, 4)4) (C)
(C) (2, (2, 14), 14), (2, (2, 2) 2) (D) (D) (1, (1, 2), 2), (14, (14, 3)3) Q.21
Q.21 The angle made by tangent at the point (2, 0)The angle made by tangent at the point (2, 0) of the curve y = (x – 2) (x
of the curve y = (x – 2) (x – 3) with x- axis is-– 3) with x- axis is-(A) (A)
/4 /4 (B)(B)
/2 /2 (C)(C) 4 4 3 3
(D) (D)
Q.22Q.22 If the curve y = xIf the curve y = x22 + bx + c, touches the line + bx + c, touches the line y = x at the point (1, 1), then values of b and c y = x at the point (1, 1), then values of b and c are-(A) (A) –1, –1, 2 2 (B) (B) –1, –1, 11 (C) (C) 2, 2, 1 1 (D) (D) –2, –2, 11 Q.23
Q.23 The line x/a + y/b = 1 touches the curve y = beThe line x/a + y/b = 1 touches the curve y = be –x/a –x/a at the at the point-(A) (A) (0, (0, a) a) (B) (B) (0, (0, 0)0) (C) (C) (0, (0, b) b) (D) (D) (b, (b, 0)0) Q.24
Q.24 The straight line x + y = a will be tangent toThe straight line x + y = a will be tangent to
9 9 x x22 + + 16 16 y y22 = 1, if a equals = 1, if a equals to-(A) (A) 8 8 (B) (B) ± ± 55 (C) (C) ±1 ±1 (D) (D) ± ± 66 Q.25
Q.25 A tangent to the curve y = xA tangent to the curve y = x22 + 3x passes + 3x passes through a point (0, –9) if it is drawn at the through a point (0, –9) if it is drawn at the point-(A) (A) (–3, (–3, 0) 0) (B) (B) (1, (1, 4)4) (C) (C) (0, (0, 0) 0) (D) (D) (–4, (–4, 4)4) Q.26
Q.26 The coordinates of the points on the curveThe coordinates of the points on the curve x = a (
x = a (
+ sin + sin
), y = a (1– cos), y = a (1– cos
), where), where tangent is inclined an angletangent is inclined an angle
/4 to the x-axis/4 to the x-axis areare -(A)
(A) (a, (a, a) a) (B)(B)
aa ,, 1 1 2 2 aa (C) (C)
aa ,, 1 1 2 2 aa (D)(D)
1 1 2 2 aa ,, aa Q.27Q.27 If the area of the triangle included betweenIf the area of the triangle included between the axes and any tangent to the curve the axes and any tangent to the curve xy
xynn = a = an+1n+1 is constant, then value of n is- is constant, then value of n is-(A)
(A) –1 –1 (B) (B) 1 1 (C) (C) .2 .2 (D) (D) –2–2 Q.28
Q.28 The angle made by the tangent to the curveThe angle made by the tangent to the curve x
x = = eett cos t, y = e cos t, y = ettsin t at point t =sin t at point t =
/4 with/4 with x– axis isx– axis is -(A)
(A) 0 0 (B)(B)
/4 /4 (C)(C)
/3 /3 (D)(D)
/2/2 Q.29Q.29 The points at which the tangent to the curveThe points at which the tangent to the curve y = x
y = x33 + 5 is perpendicular to the line x + 3y = 2 + 5 is perpendicular to the line x + 3y = 2 are are -(A) (A) (6, (6, 1), 1), (–1, (–1, 4) 4) (B) (B) (6, (6, 1), 1), (4, (4, –1)–1) (C) (C) (1, (1, 6), 6), (1, (1, 4) 4) (D) (D) (1, (1, 6), 6), (–1, (–1, 4)4) Q.30
Q.30 The coordinates of any point P on a curve areThe coordinates of any point P on a curve are represented by x = represented by x = 2 2 1 1 tt22, y =, y = 3 3 1 1 tt33, where t is a, where t is a parameter,
parameter, then then equation equation of of tangent tangent to to thethe curve at P curve at P is-(A) 6tx – 6y = t (A) 6tx – 6y = t33 (B) (B) 4tx 4tx + + 3y 3y = = tt33 (C) 3tx + 2y = t (C) 3tx + 2y = t33 (D) (D) 3tx 3tx + + y y = = tt33
Q.31
Q.31 If at a point to a curve, tangent isIf at a point to a curve, tangent is perpendicu
perpendicular to y- axis then at that plar to y- axis then at that point- oint-(A)
(A) dy/dx dy/dx = = 0 0 (B) (B) dx/dy dx/dy = = 00 (C)
(C) dy/dx dy/dx = = 1 1 (D) (D) dy/ddy/dx x = = –1–1 Q.32
Q.32 The tangent to the curve (x – 2)The tangent to the curve (x – 2)44 + (y – 1) + (y – 1)44 = 81 = 81 at the point (5, 1)
at the point (5, 1) is-(A)
(A) 2x 2x + + y y = = 1 1 (B) (B) x x + + 5y 5y = = 1010 (C)
(C) y y = = 1 1 (D) (D) x x = = 55 Q.33
Q.33 If the slope of the tangent to tIf the slope of the tangent to t he curvehe curve
xy+ax–2y =0 at point (1, 1) is 2, then a xy+ax–2y =0 at point (1, 1) is 2, then a
equals-(A)
(A) 0 0 (B) (B) 1 1 (C) (C) 2 2 (D) (D) 33 Q.34
Q.34 The equation of the tangents to the curveThe equation of the tangents to the curve y = (x
y = (x33 –1) (x–2) at the points where it meets –1) (x–2) at the points where it meets x–axis x–axis are-(A) y + 3x = 3, (A) y + 3x = 3, y – 7x – 14 = 0y – 7x – 14 = 0 (B) y – 3x = (B) y – 3x = 3, y – 7x + 14 = 3, y – 7x + 14 = 00 (C) y + 3x = (C) y + 3x = 3, y – 7x + 14 = 3, y – 7x + 14 = 00 (D) None of these (D) None of these Q.35
Q.35 The point where the tangent to the curveThe point where the tangent to the curve xy + 4 = 0 is equally inclined with both axes xy + 4 = 0 is equally inclined with both axes are
are -(A) (±1,
(A) (±1,4) 4) (B) (B) (±4,(±4, 1)1) (C) (±2,
(C) (±2, 2) 2) (D) (D) None None of of thesethese Q.36
Q.36 The abscissa of the point, where the tangentThe abscissa of the point, where the tangent to the curve y
to the curve y22 = 4a{x + = 4a{x + a sin (x/a)} is parallela sin (x/a)} is parallel to x- axis
to x- axis is-(A)
(A) 1 1 (B) (B) –1–1 (C) a
(C) a
(D) (D) None None of of thesethese QuestionQuestion
based on
based on
Length of intercepts made on axes by Length of intercepts made on axes by the Tangent and Normal
the Tangent and Normal
Q.37
Q.37 The sum of the intercepts made by a tangentThe sum of the intercepts made by a tangent to the curve
to the curve xx ++ yy= 4 at point (4, 4) on= 4 at point (4, 4) on coordinate axes
coordinate axes is-(A) 4
(A) 4 22 (B) (B) 66 33
(C) 8
(C) 8 22 (D)(D) 256256
Q.38
Q.38 The abscissa of the point on the curve ayThe abscissa of the point on the curve ay22 == x
x33, the normal at which cuts off equal, the normal at which cuts off equal intercept
intercepts from ts from the axes is-he axes is-(A)
(A) 1 1 (B) (B) 4a/34a/3 (C)
(C) 3 3 (D) (D) 4a/94a/9 Q.39
Q.39 If the tangents at any point on the curve xIf the tangents at any point on the curve x44 + y + y44 = = aa44 cuts off intercept p and q on the axes, the cuts off intercept p and q on the axes, the value of p
value of p –4/3 –4/3 + q + q –4/3 –4/3 is- is-(A) a
(A) a –4/3 –4/3 (B) (B) aa –1/2 –1/2 (C) a
(C) a1/21/2 (D) None of these(D) None of these Q.40
Q.40 If tangent at any point of the curve y = f(x)If tangent at any point of the curve y = f(x) makes equal intercepts with positive
makes equal intercepts with positive directiondirection of coordinate axes, then at that point (dy/dx) of coordinate axes, then at that point (dy/dx) equals-(A) (A) 0 0 (B) (B) 11 (C) (C) – – 1 1 (D)(D)
Q.41Q.41 At what point on the curve y = eAt what point on the curve y = e –x –x, the, the tangent cuts intercept equal in length on tangent cuts intercept equal in length on coordinate
coordinate axes-(A)
(A) (0, (0, 1) 1) (B) (B) (–1, (–1, e)e) (C)
(C) (1, (1, 1/e) 1/e) (D) (D) (–1, (–1, 1/e)1/e) Question
Question
based on
based on Equation of Normal & related factsEquation of Normal & related facts
Q.42
Q.42 The equation of the normal to the curve yThe equation of the normal to the curve y22 = 4ax = 4ax at point (a, 2a)
at point (a, 2a) is-(A)
(A) x x – – y y + + a a = = 0 0 (B) (B) x x + + y y – – 3a 3a = = 00 (C) x
(C) x + 2y + 2y + 4a + 4a = 0 = 0 (D) x (D) x + y + y + 4a + 4a = 0= 0 Q.43
Q.43 The equation of the normal to the curveThe equation of the normal to the curve x = at
x = at22, y = 2at at 't' point is-, y = 2at at 't' point is-(A) ty = x + at
(A) ty = x + at22 (B) (B) y y + + tx tx – – 2at 2at – – atat33 = 0 = 0 (C) y = tx –2at – at
(C) y = tx –2at – at33 (D) (D) None None of of thesethese Q.44
Q.44 The equation of normal to the curve xThe equation of normal to the curve x2/2/33 + + yy2/2/33 = = aa2/2/33 at the point (a, 0)
at the point (a, 0) is-(A)
(A) x x = = a a (B) (B) x x = = –a–a (C)
(C) y y = = a a (D) (D) y y = = –a–a Q.45
Q.45 The equation of normal to the curve y = eThe equation of normal to the curve y = exx at at the point (0, 1)
the point (0, 1) is-(A)
(A) x x + + y y = = 1 1 (B) (B) x x – – y y = = 11 (C)
Q.46 The equation of normal to the curve y2 = 16x at the point (1, 4)
is-(A) 2x + y = 6 (B) 2x – y + 2 = 0 (C) x + 2y = 9 (D) None of these Q.47 The equation of normal to the curve y = tan x
at the point (0, 0)
is-(A) x + y = 0 (B) x – y = 0 (C) x + 2y = 0 (D) None of these Q.48 The equation of the normal to the curve 2y = 3 –
x2 at (1, 1)
is-(A) x + y = 0 (B) x + y + 1 = 0 (C) x – y + 1 = 0 (D) x – y = 0
Q.49 The equation of normal to the curve y = x3 – 2x2 + 4 at the point x = 2
is-(A) x + 4y = 0 (B) 4x – y = 0 (C) x + 4y = 18 (D) 4x – y = 18
Q.50 If x = t2 and y = 2t, then equation of normal at t = 1
is-(A) x + y + 3 = 0 (B) x + y + 1 = 0 (C) x + y – 1 = 0 (D) x + y – 3 = 0
Q.51 If equation of normal at a point (m2, –m3) on the curve x3 – y2 = 0 is y = 3mx – 4m3, then m2
equals-(A) 0 (B) 1 (C) –2/9(D) 2/9
Q.52 The normal to the curve x + y= a is perpendicular to x-axis at the
point-(A) (0, a) (B) (a, 0) (C) (a/4, a/4) (D) No where
Q.53 The slope of the normal to the curve x = a(
– sin
), y = a(1– cos
) at point
=
/2is-(A) 0 (B) 1
(C) –1 (D) 1/ 2
Q.54 The normal to the curve x = a (1 + cos
), y = a sin
at the point
always passes through a fixed point whichis-(A) (a, a) (B) (a, 0)
(C) (0, a) (D) None of these
Q.55 The equation of the normal to the curve y2 = x3 at the point whose abscissa is 8,
is-(A) x ± 2 y = 104 (B) x ±3 2 y = 104 (C) 3 2 x ± y = 104 (D) None of these Q.56 The equation of normal to the curve x2 + y2 =
a2 at the point
2 a 3 , 2 a is-(A) 3x – y = 0 (B) x + y = 0 (C) x + 3y = 2a (D) 3x + y = 2a Q.57 The equation of the normal to the curvex = a cos3 t, y = a sin3 t at 't' point is-(A) x cos t + y cos t = a cos 2t (B) x cos t – y sin t = a sin 2t (C) x cos t – y sin t = a cos 2t (D) x cos t + y sin t = a sin 2t
Q.58 The length of perpendicular drawn from the origin to the normal at any point
of the curve x = a cos3
, y = a sin3
is-(A) a sin 2
(B) a cos 2
(C) a/2 sin 2
(D) a/2 cos 2
Q.59 The value of dy/dx at the point where normal to the curve y = f(x) make equal intercepts with positive direction of coordinates axes is-(A) 0 (B) 1 (C) –1 (D)
Q.60 The points on the curve 9y2 = x3 where thenormal to the curve cuts equal intercepts from the axes
are-(A) (4, 8/3), (4, –8/3) (B) (1, 1/3), (1, –1/3) (C) (0, 0)
(D) None of these
Q.61 The distance of normal from origin at any point
to the curve x = a (cos
+
sin
),y = a (sin
–
cos
)Question
based on Angle of intersection of two curves
Q.62 The angle of intersection between the curve y2 = 16 x and 2x2 + y2 = 4
is-(A) 0º (B) 30º (C) 45º (D) 90º Q.63 The angle of intersection between the curve
y = 4x2 and y = x2 is
-(A) 0º (B) 30º (C) 45º (D) 90º Q.64 The angle of intersection between the curves
y2 = 8x and x2 = 4y – 12 at (2, 4) is -(A) 90º (B) 60º
(C) 45º (D) 0º
Q.65 The angle of intersection between curves y = x3 and 6y = 7 – x2 at point (1, 1) is-(A)
/4 (B)
/3(C)
`/2 (D) None of these Q.66 The angle of intersection of curves 2y = x3and y2 = 32x at the origin is-(A)
/6 (B)
/4 (C)
/2 (D) None of these Q.67 If curves 2 2 a x + 2 2 b y = 1 and 2 2 x – 2 2 m y = 1 intersect orthogonally,then-(A) a2 + b2 = 2 + m2
(B) a2 – b2 =2 – m2
(C) a2 – b2 =2 + m2
(D) a2 + b2 =2 – m2
Q.68 The angle of intersection between the curve x2 = 32 y and y2 = 4x at point (16, 8) is-(A) 60º (B) 90º
(C) tan –1(3/5) (D) tan –1(4/3)
Q.69 The angle of intersection between the curves x3 – 3xy2+ 2 = 0 and 3x2 y – y3 – 2 = 0 is-(A) 45º (B) 90º (C) 60º (D) 30º Q.70 If the curves 2 2 a x + 4 y2 = 1 and y3 = 16x
intersect at right angle, then a2
equals-(A) 1 (B) 3/4
(C) 4/3 (D) any number
Q.71 The angle of intersection between the curves y2 = 2x/
and y = sin xis-(A) tan –1 (–1/
) (B) cot –1 (1/
) (C) tan –1
(D) cot –1
Questionbased on
Length of tangent, normal, subtangent & subnormal
Q.72 For a curve 2 2 tangent) of length ( normal) of length (
is equal to-(A) (subnormal)/ (subtangent)
(B) (subtangent)/ (subnormal) (C) (subtangent/subnormal) (D) constant
Q.73 At any point of a curve (subtangent) × (subnormal) is equal to the square of
the-(A) slope of the tangent at that point (B) slope of the normal at that point (C) abscissa of that point
(D) ordinate of that point
Q.74 At any point of a curve
subtangent subnormal
is equal to
-(A) the abscissa of that point (B) the ordinate of that point
(C) slope of the tangent at that point (D) slope of the normal at that point
Q.75 The length of the subtangent at any point of the curve y = ax3
is-(A) x (B) x/3
(C) x/a (D) ax
Q.76 The length of subtangent to the curve x2 + xy + y2 = 7 at the point (1, –3)
is-(A) 3 (B) 5
Q.77 The length of subnormal at any point to the parabola y2 = 4ax
is-(A) 1 (B) 2
(C) 2a (D) 4a
Q.78 The length of subtangent at any point to the curve y = be –a/x is proportional
to-(A) x3 (B) y (C) x2 (D) xy Q.79 The length of the subtangent to the curve
y = (x – 2) (x + 2) at point (2, 0) is-(A) 4 (B) 2 (C) 0 (D) 1 Q.80 The length of the subtangent to the curve
x + y = 3 at the point (4, 1) is-(A) 2 (B) 1/2 (C) 3 (D) 4 Q.81 If at any point (x1, y1) on the curve the
subtangent and subnormal are equal, then the length of tangent is equal to
-(A) y1 (B) 2 y1 (C) 2y1 (D) None of these Q.82 For a curve tangent of length normal of length equals-(A) subtangent (B) subnormal (C) slope of tangent (D) slope of normal Q.83 The length of subnormal to the curve y2 = 12
ax at any point
is-(A) 2a (B) 4a
(C) 6a (D) 8a
Q.84 The length of subtangent at any point of the curve y = bex/a
is-(A) ab (B) a
(C) b (D) b/a
Q.85 The length of normal to the curve x = a(t + sin t), y = a(1– cos t) at any point t is -(A) a sin t
(B) 2a sin3 (t/2) sec(t/2) (C) 2a sin (t/2) tan (t/2) (D) 2a sin (t/2)
Q.86 The length of the tangent at any point to the curve x2/3 + y2/3 = a2/3 which is intercepted between the axes,
is-(A) a (B) 2a
(C) a (D) a/2
Q.87 At a point to the parabola y2 = 16 (x – 2) (A) length of tangent is constant
(B) length of normal is constant (C) length of s ubtangent is constant (D) length of subnormal is constant Question
based on Point of inflexion
Q.88 Point of inflexion to the curve y = x1/3 is-(A) (1, 1) (B) (0, 0)
(C) (1, 0) (D) (0,1) Question
based on Rolle’s Theorem
Q.89 If the polynomial equation anxn + an–1xn–1 + ...
+ a2x2 + a1x + a0 = 0; n positive integer, has
two different real roots
and
, then between
and
, the equation nanxn–1 + (n – 1) an–1xn–2+ .... + a1= 0 has
(A) exactly one root (B) atmost one root (C) atleast one root (D) No root
Q.90 If f(x) = b tan a tan x tan b cos a cos x cos b sin a sin x sin , where 0 < a < b < 2
, then the equation f ' (x) = 0 has,
in the interval (a,
b)-(A) atleast one root (B) atmost one root (C) no root (D) None of these Q.91 If a + b + c = 0, then the equation 3ax2 + 2bx + c = 0
has, in the interval (0, 1)
(A) atleast one root (B) atmost one root (C) no root (D) None of these
Q.92 If 1 n a0
+ n a1 + 1 n a2
+...+ 2 an1 + an = 0, then the equation a0xn + a1xn–1 + ...+ an–1x + an = 0has, in the interval (0, 1), (A) exactly one root (B) atleast one root (C) atmost one root (D) No root.
Q.93 If 27a + 9b + 3c + d = 0, then the equation 4ax3 + 3bx2 + 2cx + d = 0, has atleast one real root lying between –
(A) 0 and 1 (B) 1 and 3
(C) 0 and 3 (D) None of these Q.94 The value of c for the function f(x) = log sin x
in the interval
6 5 , 6 is-(A) 4
(B) 2
(C) 3 2
(D) None of these Q.95 If the function f(x) = x3 – 6x2 + ax + bdefined on [1, 3], satisfies the Rolle's theorem for c = 3 1 3 2
, then-(A) a = 11, b = 6 (B) a = –11, b = 6 (C) a = 11, b
R (D) None of these Q.96 If 1 C0 + 2 C1 + 3 C2= 0, where C0C1C2 are all real,
then the quadratic equation C2x2 + C1x + C0 = 0
has
(A) at least one root in (0, 1)
(B) one root in (1, 2) and the other in (3, 4) (C) one root in (–1, 1)and the other in (–5, –2) (D) both roots imaginary
Question
based on Langrange’s mean value theorem Q.97 If f(x) is differentiable in the interval [2, 5],
where f(2) = 5 1 and f(5) = 2 1
, then there exists a number c, 2 < c < 5 for which f' (c) =
(A) 2 1 (B) 5 1 (C) 10 1 (D) None of these
Q.98 Let f(x) and g(x) be differentiable for 0
x
1, such that f(0) = 2, g(0) = 0, f(1) = 6. Let there exist a real number c in [0,1] such that f ' (c) = 2g'(c), then the value of g(1) mustbe-(A) 1 (B) 2 (C) – 2 (D) – 1
Question
based on Rate Measure
Q.99 A balloon, which always remains spherical, has a variable diameter
2 3
(2x + 3). The rate of change of volume with respect to x will be-(A) 8 27
(2x – 3)2 (B) 8 27
(2x + 3)2 (C) 8 27
(3x + 2)2 (D)
27 8 (2x + 3)2Q.100 The rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm,
is-(A) 1 cm (B) 4 cm (C) 3 cm (D) 2 cm
Q.101 The rate of change of the volume of a cone with respect to the radius of its base
is-(A)
r 2h (B) 3 4
rh (C) 3 4
r 2h (D) 3 2
rhQ.102 A stone is dropped into a quiet lake and waves move in a circle at a speed of 3.5 cm/sec. At the instant when the radius of the circular wave is 7.5 cm, The enclosed area increases as fastly as
(A) 52.5
cm2/sec (B) 50.5
cm2/sec (C) 57.5
cm2/sec (D) 62.5 cm2/secQ.103 The side of a square sheet is increasing at the rate of 4 cm per minute. The rate by which the area increasing when the side is 8 cm l ong
is-(A) 60 cm2/minute (B) 66cm2/minute (C) 62 cm2/minute (D) 64cm2/minute
Q.104 The side of a square is increasing at the rate of 0.2 cm/sec, then the rate of increase of the perimeter of the square is–
(A) 0.7 cm/sec (B) 0.8 cm/sec (C) 0.5 cm/sec (D) 0.6 cm/sec
Q.105 The radius of a circle is increasing at the rate of 0.7 cm/sec. The rate of increase of its circumference is –
(A) 0.7 cm/sec (B) 2.1 cm/sec (C) 1.4
cm/sec (D) 2.8 cm/secQ.106 The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec, then the rate of increase of its surface area, when the radius is 7 cm.
(A) 12.2
cm2/sec (B) 11.2
cm2/sec (C) 10.2
cm2/sec (D) 9.2
cm2/secQ.107 A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimeters of gas per second. The rate at which the radius of the balloon is increasing when the radius is 15 cm is
-(A)
1 cm/sec (B)
2 cm/sec (C)
cm/sec (D) 2
cm/secQ.108 The radius of an air bubble is increasing at the rate of 0.5 cm/sec. The rate by which the volume of the bubble is increasing when the radius is 1 cm, is –
(A)
cm3/sec (B) 3
cm3/sec (C) 2
cm3/sec (D)
cm3/secQ.109 A particle moves along the curve y = x2 + 2x. Then the points on the curve are the x and y coordinates of the particle changing at the same rate,
are-(A)
2 1 , 4 3 (B)
4 3 , 2 1 (C)
2 1 , 4 3 (D)
4 3 , 2 1Q.110 The point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate is
(A) (4, 2) (B) (– 4, – 2) (C) (2, 4) (D) (– 2, – 4)
LEVEL- 2
Q.1 The equation of normal to the curve
2 2 a x – 2 2 b y
= 1 at the point (a sec
, b tan
)is-(A)
sec ax +
tan by = a2 + b2 (B)
sec ax –
tan by = a2 – b2 (C)
sec ax +
tan by = a2 – b2 (D)
sec ax +
tan by = a – bQ.2 The equation of tangent to the curve
n a x
+ n b y
= 2 at the point (a, b) for all values of n is-(A) a x + b y = 1 (B) a x + b y = 2 (C) a x + b y = 2 1 (D) x a + y b = 2
Q.3 If length of subnormal at any point of the curve yn = an–1 x is constant, then n
equals-(A) 1 (B) 3 (C) 2 (D) 0
Q.4 The angle of intersection between the curves r = a sin(
–
) and r = b cos (
–
)is-(A)
–
(B)
+
(C) 2
+
+
(D) 2
+
–
Q.5 The length of subtangent at the point x = a of the curve ay2 = (a + x)2 (3a – x)
is-(A) a (B) 2a
(C) 4a (D) 6a
Q.6 If the line ax + by + c = 0 is a normal to the curve xy = 1,
then-(A) a, b
R (B) a > 0, b > 0(C) a < 0, b > 0 or a > 0, b < 0 (D) a < 0, b < 0
Q.7 If the tangent to the curve f(x) = x2 at any point (c, f(c)) is parallel to line joining the points (a, f(a)) and (b, f(b)) on the curve, then a, c, b are in-(A) H.P. (B) G.P.
(C) A.P. (D) A.P. and G.P. both Q.8 The set of points where the tangent to the curve
y3 – 3xy + 2 = 0 is horizontal is-(A) {(1, 1)} (B) {(0, 0)} (C) {(0, 1)} (D)
Q.9 At what points the tangent line to the curve y = cos (x + y), (–2
x
2
) is parallel to x + 2y =0-(A) (
/2, 0) (B) (–
/2, 0) (C) (3
/2, 0) (D) (–3
/2,
/2)Q.10 The equation of one of the tangents to the curve y = cos (x + y), –2
x
2
, that is parallel to the line x + 2y = 0is-(A) x + 2y – 1 = 0 (B) 2x + 4y =
(C) x –2y + 1 = 0 (D) 4x – 8y +
= 0 Q.11 The area of triangle formed by tangent to thehyperbola 2xy = a2 and coordinates axes is-(A) a2 (B) 2a2
(C) a2/2 (D) 3a2/2
Q.12 If the tangent to the curve 2y3 = ax2 + x3 at a point (a, a) cuts off intercepts p and q on the coordinates axes, where p2 + q2 = 61, then a
equals-(A) 30 (B) –30
(C) 0 (D) ±30
Q.13 If
be the angle of intersection between the curves y = ax and y = bx, then tan
is equal to-(A) b log a log 1 b log a log
(B) b log a log 1 b log a log
(C) b log a log 1 b log a log
(D) None of theseQ.14 The point on the curve y = x2 – 3x + 2 at which the tangent is perpendicular to the line y = x is-(A) (0, 2) (B) (1, 0)
(C) (–1, 6) (D) (2, –2)
Q.15 The distance between the origin and the normal to the curve y = e2x + x2 at the point x = 0 is-(A) 2 5 (B) 2/ 5
(C) 5 (D) None of these
Q.16 If the curve y = ax2 – 6x + b passes through (0, 2) and has its tangent parallel to x-axis at x = 3/2, then the value of a and b
are-(A) 2, 2 (B) –2, –2 (C) –2, 2 (D) 2, –2
Q.17 Tangents are drawn from origin to the curve y = sin x, then point of contact lies
on-(A) x2 = y2 (B) x2y2 = x2 – y2 (C) x2y2= 0 (D) None of these
Q.18 If at any point S of the curve by2 = (x + a)3, the relation between subnormal SN and subtangent ST be p(SN) = q(ST)2 then p/q is equal to -(A) 27 b 8 (B) 27 a 8 (C) a b (D) None of these
Q.19 If a, b, c be non-zero real numbers such that
1 0 8x) cos 1 ( (ax2+ bx + c) dx
2 0 8 ) x cos 1 ( (ax2+ bx + c) dx = 0,then the equation ax2 + bx + c = 0 will have-(A) one root between 0 and 1 and other root
between 1 and 2
(B) both the roots between 0 and 1 (C) both the roots between 1 and 2 (D) None of these
Q.20 The value of c in Lagrange's theorem for the function f(x) =
0 x , 0 0 x , x 1 cos xin the interval [– 1, 1] is (A) 0 (B) 2 1 (C) – 2 1
(D) Non existent in the interval Q.21 Let f be a function which is continuous and
differentiable for all real x. If f(2) = – 4 and f ' (x)
6 for all x
[2, 4],then-(A) f (4) < 8 (B) f (4)
8 (C) f (4)
12 (D) None of theseQ.22 If f(x) and g(x) are differentiable functions for 0
x
1 such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2, then in the interval (0, 1),(A) f ' (x) = 0 for all x
(B) f ' (x) = 2g'(x) for atleast one x
(C) f ' (x) = 2g'(x) for atmost one x
(D) None of these
Q.23 Let f(x) = (x – 4) (x – 5) (x – 6) (x – 7) then (A) f '(x) = 0 has four roots
(B) Three roots of f'(x) = 0 lie in
(4, 5)
(5, 6)
(6, 7)(C) The equation f '(x) = 0 has only one root
(D) Three roots of f'(x) = 0 lie in
(3, 4)
(4, 5)
(5, 6).Q.24 The number of values of k for which the equation x3 – 3x + k = 0 has two distinct roots lying in the interval (0, 1)
are-(A) three (B) two
(C) infinitely many
(D) no value of k satisfies the requirement. Q.25 A man 2 metres high, walks at a uniform speed
of 6 metre per minute away from a lamp post, 5 metres high. The rate at which the length of his shadow increases is –
(A) 1 metres/minute (B) 2 metres/minute (C) 4 metres/minute (D) 3 metres/minute
Q.26 A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground
away from the wall, at the rate of 2 m/sec. How fast its height on the wall decreasing when the foot of the ladder is 4 m away from the wall (A) 3 4 m/sec (B) 3 8 m/sec (C) 3 10 m/sec (D) 3 6 m/sec
Q.27 Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand-cone
increasing when the height is 4 cm (A) 48
cm/sec (B)
48 1 cm/sec (C) 58
cm/sec (D)
58 1 cm/secQ.28 A man 2 metres high walks at a uniform peed of 5 km/hr away from a lamp-post 6 metres high. The rate at which the length of his shadow increases, is –
(A) 2 km/hr (B) 2.5 km/hr (C) 4 km/hr (D) 3 km/hr
Q.29 Water is dripping out from a conical funnel of semi-vertical angle
4
at the uniform rate of 2 cm3/sec in its surface area through a tiny hole at the vertex in the bottom. When the slant height of the water is 4 cm, then the rate of decrease of the slant height of the water, is – (A)
4 6 cm/sec (B)
4 5 cm/sec (C)
4 2 cm/sec (D)
4 3 cm/secQ.30 An inverted cone has a depth of 10 cm and a base of radius 5 cm. Water is poured into it at the rate of 3/2 c.c. per minute. The rate at which the level of water in the cone is rising when the depth is 4 cm is
-(A)
8 5 cm/min (B)
8 2 cm/min (C)
8 7 cm/min (D)
8 3 cm/minQ.31 The surface area of a spherical bubble is increasing at the rate of 2 cm2/s. When the radius of the bubble is 6 cm, then the rate by which the volume of the bubble increasing is-(A) 6 cm3/sec (B) 9 cm3/sec
(C) 3 cm3/sec (D) 12 cm3/sec
Q.32 The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, then the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively is
-(A) 0.75 cm/sec (B) 0.25 cm/sec (C) 1 cm/sec (D) 0.50 cm/sec
LEVEL- 3
Q.1 If 2a + 3b + 6c = 0, then at least one root of the equation ax2+ bx + c = 0 lies in the interval -(A) (0, 1) (B) (1, 2)
(C) (2, 3) (D) None of these
Q.2 For the parabola y2 = 4ax, the ratio of the subtangent to the abscissa is
-(A) 1 : 1 (B) 2 : 1 (C) x : y (D) x2: y
Q.3 The chord joining the points where x = p and x = q on the curve y = ax2+ bx + c is parallel to the tangent at the point on the curve whose abscissa is -(A) ( p q) 2 1
(B) ( p q) 2 1
(C) 2 pq (D) none of theseQ.4 If the tangent at any point on the curve x4 + y4 = c4 cuts off intercepts a and b on the coordinate axes, the value of a –4/3 + b –4/3
is-(A) c –4/3 (B) c –1/2
(C) c1/2 (D) none of these
Q.5 The tangent to the curve x = a(
– sin
), y = a(1 + cos
) at the points
= (2k + 1)
, k
z are parallel to :(A) y = x (B) y = –x (C) y = 0 (D) x = 0
Q.6 A balloon is pumped at the rate of a cm3/minute. The rate of increase of its surface area when the radius is b cm,
is-(A) 4 2 b a 2 cm2/min (B) b 2 a cm2/min (C) b a 2 cm2/min (D) none of these
Q.7 x and y are the sides of two squares such that y = x – x2. The rate of change of the area of the second square with respect to that of the first square
is-(A) 2(1 –x2) x (B) 2x2 –3x + 1 (C) 2(2x2–3x + 1) (D) None of these
Q.8 Let the equation of a curve be x = a(
+ sin
), y = a (1 – cos
). If
changes at a constant rate k then the rate of change of the slope of the tangent to the curve at
=3
is-(A) 3 k 2 (B) 3 k (C) k (D) none of theseQ.9 On the curve x3 = 12y the abscissa changes at a faster rate than the ordinate. Then x belongs to the interval
-(A) (– 2, 2) (B) (–1, 1)
(C) (0, 2) (D) none of these
Assertion & Reason type Questions
All questions are Assertion & Reason type questions. Each of these questions contains two statements: Statement-I (Assertion) and Statement-II (Reason). Answer these questions from the following four option. (A) Statement- I and Statement- II are true,
and Statement - II is the correct explanation of Statement– I.
(B) Statement - I and Statement - II are true
but Statement - II is not the correct explanation of Statement – I.
(C) Statement - I is true but Statement - II is false. (D) Statement - I is false but Statement - II is true.
Q.10 Statement I : If the curves 4 y a x 2 2 2
= 1 and y3 = 16x intersect at right angles then a2 is equal to 2/3Statement II : If two curves cut each other orthogonally, then product of slopes of tangent at point of intersection is equal to –1.
Q.11 Statement I : Subnormal length to xy = e2 at any point varies directly as cube of ordinate Statement II : Length of subnormal = |y
dx dy | and dx dy
is given curve is proportional to y2
Q.12 Statement I : The slope of normal at the point with abscissa x = –2 of the graph of the
function f(x) = |x2 – |x|| is 1/3.
Statement II : at x = –2, the slope of tangent
dx dy
on the curve is (–3) and normal perpendicular to tangent.
Q.13 Statement I : If a function is continuous and differentiable in [a, b] and f(a)
f(b) then there exist a point c
(a, b) such that f
(c) = 0Statement II : According to Rolle's theorem. If a function is continuous and differentiable in [a, b] and f(a) = f(b) then there exist a point c
(a, b) such that f
(c) = 0Column Matching Questions
Match the entry in Column 1 with the entry in Column 2.
Q.14 Column 1 Column II
(A) The slope of the curve 2y2 = ax2+ b at (P) a – b = –6 (1, 1) is –1, then
(B) If (a, b) be the point on the curve 9y2 = x3 (Q) a – b = 7/2 where normal to the curve makes equal intercepts
with the axes, then
(C) If the tangent at a point (1, 2) on the curve (R) a – b = 4/3
2 7 bx ax
y
2
be parallel to the normal at (– 2, 2) on the curve y = x2 + 6x + 10, then(D) If the tangent to the curve xy + ax + by = 0 at (S) a – b = 3 (1, 1) is inclined at an angle tan –1 2 with x-axis, then
Q.15 Column 1 Column II
(A) The equation x log x = 3 – x has at least (P) (0, 1) one root in
(B) If 27a + 9b + 3c + d = 0, Then the equation (Q) (1, 3) 4ax3 + 3bx2 + 2cx + d = 0 has at least one root in
(C) If c = 3 & x 1 x ) x ( f
then interval in (R) (0, 3) which LMVT is applicable for f(x) is(D) If
2 1
c
& f(x) = 2x –x2, then interval in (S) (–1, 1) which LMVT is applicable for f(x) isANSWER KEY (Tangent & Normal)
LEVEL- 1
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. C A B C B C B B A D D B A A C D A A C A Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Ans. C B C B A C B D D A A D B C C C D D A C Q.No. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Ans. A B B A A C A D C D D B C B B A C B B A Q.No. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 Ans. A D A D C C C C B C D A D C B C C C C A Q.No. 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Ans. B C C B C A D B C A A B C B C A C B B C Q.No. 101 102 103 104 105 106 107 108 109 110 Ans. D A D B C B A C B CLEVEL- 2
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. A B C D C C C D A B A D A B B A B A A D Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 Ans. B B B D C B B B C D A BLEVEL- 3
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 Ans . A B A A C C B D A D A A D 14.(A)
P (B)
R (C)
Q (D)
S 15.(A)
Q (B)
R (C)
Q (D)
PLEVEL- 1
(Monotonicity)
Question
based on Monotonicity in algebraic function Q.1 When x < 0, function f(x) = x2 is
-(A) decreasing (B) increasing (C) constant (D) not monotonic
Q.2 When x > 1, function f(x) = x3 is -(A) increasing (B) decreasing (C) constant (D) not monotonic Q.3 In the interval (0, 1), f(x) = x2 – x + 1 is -(A) monotonic (B) not monotonic (C) decreasing (D) increasing Q.4 f(x) = x + 1/x, x
0 is increasing when -(A) | x | < 1 (B) | x | > 1 (C) | x | < 2 (D) | x | > 2 Q.5 The function f(x) = x | x | (x
0), x > 0 is -(A) decreasing (B) increasing (C) constant function (D) None of these Q.6 When x
(0, 1), function f(x) = x 1 is (A) increasing (B) decreasing(C) neither increasing nor decreasing (D) constant
Q.7 Function f(x) = 3x4 + 7x2 + 3 is (A) monotonically increasing (B) monotonically decreasing (C) not monotonic
(D) odd function
Q.8 For what values of x, the function f(x) = x +
2 x
4
is monotonically decreasing (A) x < 0 (B) x > 2 (C) x < 2 (D) 0 < x < 2 Q.9 If f(x) = 2 x + x 2 for –7
x
7, then f(x) is increasing function of x in the interval(A) [7, 0] (B) (2, 7] (C) [–2, 2] (D) [0, 7] Q.10 The function y = 2 x 1 x
decreases in the interval (A) (–
,
) (B) (–1, 1)(C) (0,
) (D) (–
, –1)Q.11 For which value of x, the function f(x) = x2 –2x is decreasing (A) x > 1 (B) x > 2 (C) x < 1 (D) x < 2 Q.12 Function f(x) = 1 x 2 x
, x
–1 is (A) increasing (B) decreasing (C) not monotonic (D) None of these Q.13 Function f(x) = x3 is(A) increasing in (0,
) and decreasing in (–
, 0) (B) decreasing in (0,
) and increasing in (–
, 0) (C) decreasing throughout(D) increasing throughout Q.14 Function f(x) = x | x | is
(A) monotonic increasing (B) monotonic decreasing (C) not monotonic
(D) None of these
Q.15 If f and g are two decreasing functions such that fog is defined then fog is
(A) decreasing (B) increasing (C) Can't say (D) None of these Q.16 For the function f(x) = | x |, x > 0 is
(A) decreasing (B) increasing (C) constant function (D) None of these
Q.17 In the following, monotonic increasing function is
(A) x + | x | (B) x – | x | (C) | x | (D) x | x |
Q.18 At x = 0, f(x) = 2 x 1 x
is(A) increasing (B) decreasing (C) not monotonic (D) constant
Q.19 If f(x) = 2x3 – 9x2 + 12x – 6, then in which interval f(x) is monotonically increasing
(A) (1, 2) (B) (–
, 1)(C) (2,
) (D) (–
, 1) or (2,
) Q.20 For the function f(x) = x3 – 6x2 – 36x + 7 whichof the following statement is false (A) f(x) is decreasing, if –2 < x < 6 (B) f(x) is increasing, if –3 < x < 5 (C) f(x) is increasing, if x < –2 (D) f(x) is increasing, if x > 6 Q.21 In which interval the function
f(x) = 2 x + x 2 – 6
x
6 (x
0) is decreasing (A) (6, 0) (B) (–2, 2) (C) (2, 6) (D) None of these Q.22 Function f(x) = x2(x –2)2 is (A) increasing in (0, 1)
(2,
) (B) decreasing in (0, 1)
(2,
) (C) decreasing function (D) increasing functionQ.23 For 0
x
1, the function f(x) = |x| + |x – 1| is (A) monotonically increasing(B) monotonically decreasing (C) constant function
(D) identity function
Q.24 If f and g are two increasing function such that fog is defined then fog is ?
(A) increasing (B) decreasing
(C) neither increasing nor decreasing (D) None of these
Question
based on Monotonicity in exponential function Q.25 Function f(x) = ax is monotonically increasing if
(A) a < 0 (B) a > 0 (C) a < 1 (D) a > 1 Q.26 The function f(x) = ex, –1
x < 0 is -(A) decreasing (B) increasing (C) constant function(D) neither increasing, nor decreasing Q.27 Function f(x) = e –1/x (x > 0) is
-(A) increasing (B) decreasing (C) not monotonic (D) None of these
Q.28 Which of the following function is not monotonic -(A) ex – e –x (B) ex + e –x
(C) e –1/x (D) None of these
Q.29 In the following, decreasing function is -(A)n x (B) | x | 1 (C) e1/x (D) None of these
Q.30 For every value of x of the function f(x) = x
5 1
is-(A) decreasing (B) increasing
(C) neither increasing nor decreasing
(D) increasing for x > 0 and decreasing for x < 0 Q.31 The interval in which the function f(x) = xe4–x
decreases is -(A) (–
, 1) (B) (1,
) (C) (0, 4) (D) None of these Q.32 Function 1 e 1 e x 2 x 2
is-(A) increasing (B) decreasing (C) neither increasing nor decreasing (D) even function
Q.33 Which of the following function is a monotonic increasing function for all values of x ?
(A) sin x (B) cos x (C) ex (D) 2 –x
Q.34 Which of the following function is monotonically decreasing for all real values of x ?
(A) e –x (B) x2 (C) tan x (D) | x | Question
based on Monotonicity in lograthmic function Q.35 Function f(x) = 2x2 – log x is increasing when
(A) x
(0, 1/2) (B) x
(1/2,
) (C) x
(–1/2, 1/2)(D) x
(–
, –1/2)
(1/2,
)Q.36 Function f(x) = x – log x decreasing, when (A) x
(0, 1) (B) x
(–1, 1) (C) x
(1,
) (D) None of these Q.37 For x > 0, the function f(x) = log x, x > 0is-(A) decreasing (B) increasing (C) constant function (D) odd function Q.38 Function f(x) = x x log is increasing in (A) (1, 2e) (B) (0, e) (C) (2, 2e) (D) (1/e, 2e)
Q.39 Function f(x) = log sin x is monotonic increasing when
(A) x
(
/2,
) (B) x
(–
/2, 0) (C) x
(0,
) (D) x
(0,
/2)Question
based on Monotonicity in trigonometric function Q.40 Function f(x) = kx + 3 sin x is decreasing
if-(A) k < –3 (B) k > –3 (C) k < 3 (D) k > 3
Q.41 When
x < 3
/2, tan x is(A) increasing (B) decreasing (C) not monotonic (D) constant
Q.42 Function f(x) = x cos 6 x sin 2 x cos 3 x sin
is increasing when -(A)
< 1 (B)
> 1 (C)
< 2 (D)
> 2Q.43 The function f(x) = x + sin x is monotonically increasing for
-(A) x > 0 (B) x < 0
(C) All values of x (D) No value of x Q.44 The function f(x) = x + cos x is
-(A) always monotonically increasing (B) always monotonically decreasing (C) increasing for certain range of x (D) None of these
Q.45 For what value of 'a' the function f(x) = x + cos x – a increases
-(A) 0 (B) 1
(C) –1 (D) Any value Q.46 Function f(x) = cot –1 x + x is increasing
in-(A) (1,
) (B) (–1,
) (C) (–
,
) (D) (0,
)Q.47 For x > 0, which of the following function is not monotonic
-(A) x + | x | (B) ex
(C) log x (D) sin x
Q.48 f(x) =
x – 3cos x is monotonic increasing if (A)
> 3 (B)
> –3LEVEL- 2
Q.1 If f(x) = x5 – 20x3 + 240x, then f(x) is -(A) monotonic increasing everywhere (B) monotonic decreasing only in (0,
) (C) monotonic decreasing everywhere (D) monotonic increasing only in (–
, 0) Q.2 The function y =x log
x
increases in the interval -(A) (–
, 0) (B) (e,
)(C) (0,
) (D) (–
, e)Q.3 For x > 0, which of the following statement is true
(A) x < log (1 + x) (B) x > log (1 + x) (C) x
log (1 + x) (D) None of theseQ.4 The function f(x) = 2log (x – 2) – x2 + 4x + 1 increases in the interval
-(A) (1, 2) (B) (2, 3) (C) (–
, –1) (D) (2, 4)Q.5 If a < 0 then function (eax + e –ax) is monotonic decreasing when
-(A) x < 0 (B) x > 0 (C) x > 1 (D) x < 1
Q.6 Function f(x) = x100 + sin x – 1 is increasing in the interval
(A) (0, 1) (B) (–
/2,
/2) (C) (–1, 1) (D) None of theseQ.7 In which interval f(x) = 2x2 – log | x|, (x
0) is monotonically decreasing -(A) (–1/2, 1/2) (B) (–
, –1/2) (C) (–
, –1/2)
(0, 1/2) (D) (–
, –1/2)
(1/2,
) Q.8 If f(x) = x3 – 10x2 + 200x – 10, then f(x) is-(A) decreasing in (–
, 10] and increasing in (10,
) (B) increasing in (–
, 10] and decreasing in (10,
) (C) increasing for every value of x(D) decreasing for every value of x
Q.9 If the domain of f(x) = sin x is D = {x : 0
x
}, then f(x) is(A) increasing in D (B) decreasing in D (C) decreasing in [0,
/2] & increasing in[
/2,
](D) neither increasing nor decreasing Q.10 Function f(x) = log (1 + x) – x 1 x 2
is monotonic increasing when -(A) x < 0 (B) x > 1 (C) x
R (D) x
R 0 Q.11 f(x) = 2x – tan –1 x – log (x + 1
x2 ) is monotonic increasing when-(A) x > 0 (B) x < 0 (C) x
R (D) x
R 0Q.12 If f '(x) = g(x)(x –
)2 where g(
)
0 and g(x) is continuous at x =
then function f(x) -(A) increasing near to
if g(
) > 0(B) decreasing near to
if g(
) > 0 (C) increasing near to
if g(
) < 0(D) increasing near to
for every value of g(
) Q.13 Functioncos2x + cos2 (
/3 + x) – cos x cos (
/3 + x) for all real values of x will be-(A) increasing (B) constant (C) decreasing (D) None of these Q.14 Let f
(x) > 0 and g
(x) < 0 for all x
Rthen-(A) f{g(x)} > f{g(x + 1)} (B) f{g(x–1)} < f{g(x + 1)} (C) g{f(x–1)} < g{f(x + 1)} (D) g{f(x)} > g{f(x – 1)}
Q.15 The interval of increases of the function given by f(x) = x – ex+ tan (2
/7)is-(A) (0,
) (B) (–
0)(C) (1,
) (D) None of theseQ.16 f(x) = x3 + ax2 + bx + 5 sin2 x is an increasing function in the set of real numbers if a and b satisfy the condition
-(A) a2–3b –15 > 0 (B) a2 –3b + 15 > 0 (C) a2 –3b + 15 < 0 (D) a > 0, b > 0
LEVEL- 3
Q.1 The function ƒ(x) = cos (
/x), (x
0) is increasing in the interval-(A) (2n+1, 2n), n
N (B)
1,2n n 2 1 , n
N (C)
2n 1 1 , 2 n 2 1 , n
N (D) None of theseQ.2 Let y = x2e –x, then the interval in which y increases with respect to x is
-(A) (–
) (B) (–2, 0) (C) (2,
) (D) (0, 2) Q.3 The function y = x3 – 3x2 + 6x – 17(A) increases everywhere (B) decreases everywhere
(C) increases for positive x and decreases for negative x
(D) increases for negative x and decreases for positive x
Q.4 The function ƒ(x) = xx, x > 0decrease on the interval
-(A) (0, e) (B) (0, 1)
(C) (0, 1/e) (D) None of these
Assertion & Reason type Questions
All questions are Assertion & Reason type questions. Each of these questions contains two statements: Statement-I (Assertion) and Statement-II (Reason). Answer these questions from the following four option. (A) If both Statement-I and Statement- II are true,
and Statement - II is the correct explanation of Statement– I.
(B) If both Statement - I and Statement - II are true but Statement - II is not the correct explanation of Statement – I.
(C) If Statement - I is true but Statement - II is false. (D) If Statement - I is false but Statement - II is true.
Q.5 Statement I : Both sin x and cos x are decreasing function in (
/2,
).Statement II : If a differentiable function decreasing in an interval (a, b) then it's derivative also decreases in (a, b).
Q.6 Statement I :If f(x) and g(x) are monotonically (or strictly) increasing (or decreasing) functions on [a, b] then gof(x) is always monotonically (or strictly) increasing function in [a, b].
Statement II : If one of the two functions f(x) and g(x) is strictly (or monotonically) increasing and other strictly (monotonically) decreasing, then (gof) (x) is sometimes strictly (monotonically) decreasing on [a, b].
Q.7 Statement I : f(x) = ) x e log( ) x log(
is increasing on
,e e .Statement II : x log x is increasing for x > 1/e.
Passage Based Questions
Passage :
If f(x) = | x – 2| + |x – 4| + |x – 6|
On the basis of above information, answer the following
questions-Q.8 Find the set of values of x such that f(x) is
increases-(A) [2, 4] (B) [4, 6]
(C) [4,
) (D) None of theseQ.9 Find the set of values of x such that f(x) is decreases–
(A) (–
) (B) (–
] (C) (–
, 6] (D) None of theseQ.10 If f(x) is symmetrical about the line x = K,
then-(A) K = 2 (B) K = 4
(C) K = 6 (D) None of these
Q.11 Find the set of values of x such that f(x) is invertible–
(A) [4,
) (B) (–
, 4]Q.12 Find the set of values of a such that equation f(x) – a = 0 has no
solution-(A) [4, 5] (B) [2,
)(C) (–
, 4) (D) None of theseColumn Matching Questions
Match the entry in Column 1 with the entry in Column 2. Q.13 Match the interval of monotonicity
-Column 1 Column 2
(A) y = x – ex (P) increase on (–
,
) (B) y = log (x
1
x2) (Q) increase on (0, 3)(C) y = x 4x
x2 (R) increases on (–
, 0) and decreases on (0,
) (D) y = x 6 x 9 x 4 10 23
(S) increases on (1/2, 1) and on (–
, 0)
(0, 1/2)
(1,
) decreasingQ.14 Column-I Column-II
(A) The function f(x) = 2x3 – 9x2– 24x + 7 increases on (P) (4,
) (B) f(x) = 4x3 – 21x2+ 18x + 20 increases on (Q) (–
, – 1) (C) f(x) = 2 x 5 4 x 3 1
increases on (R) (–
, 0) (D) f(x) = (x2 – 2x) log x – 2 3 x2+ 4x increases on (S) (e,
)ANSWER KEY (Monotonicity)
LEVEL- 1
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ans. A A B B C B C D B D C A D A B B D A D B Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Ans. B A C A D B A B C A B A C A B A B B D A Q.No. 41 42 43 44 45 46 47 48 Ans. A B C A D C D ALEVEL- 2
Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Ans. A B B B A A C C D B C A B A B CLEVEL- 3
Q.No. 1 2 3 4 5 6 7 8 9 10 1 1 1 2 A ns. C D A C C C D C B B D C Q.13 A
R ; B
P ; C
Q ; D
S Q.14A
P,Q ; B
P,Q,R ; C
Q,R ; D
P,Q,R,SLEVEL-1
(Maxima – Minima)
Question
based on Maxima & Minima of function Q.1 f(c) is a maximum value of f(x) if -(A) f
(c) = 0, f
(c) > 0 (B) f
(c) = 0, f
(c) < 0 (C) f
(c)
0, f
(c) = 0 (D) f
(c) < 0, f
(c) > 0 Q.2 f(c) is a minimum value of f(x) if -(A) f
(c) = 0, f
(c) > 0 (B) f
(c) = 0, f
(c) < 0 (C) f
(c)
0, f
(c) = 0 (D) f
(c) < 0, f
(c) > 0Q.3 f(c) is a maximum value of f(x) when at x = c-(A) f
(x) changes sign from +ve to –ve(B) f
(x) changes sign from –ve to + ve (C) f
(x) does not change sign(D) f
(x) is zeroQ.4 f(c) is a minimum value of f(x) when at x = c-(A) f
(x) changes sign +ve to –ve(B) f
(x) changes sign from –ve to + ve (C) f
(x) does not change sign(D) f
(x) is zeroQ.5 The correct statement is
-(A) f(c) is an extreme value of f(x) if f
(c) = 0 (B) If f(c) is an extreme value of f(x) then f
(c) = 0 (C) If f
(c) = 0 then f(c) is an extreme value of f(x) (D) All the above statements are incorrect Q.6 Which of the following function has maximumvalue at x = 0
(A) x2 (B) –x2 (C) | x | (D) [x]
Q.7 The point of maxima of sec x is -(A) x = 0 (B) x =
/2 (C) x =
(D) x = 3
/2 Q.8 x3 – 3x + 4 is minimum at-(A) x = 1 (B) x = –1 (C) x = 0 (D) No where
Q.9 The maximum value of 2x3 – 9x2 + 100 is -(A) 0 (B) 100 (C) 3 (D) 30
Q.10 If f(x) = x3 – kx + 7 is maximum at x = –1, then the value of k is
-(A) 3 (B) 6 (C) –3 (D) –6
Q.11 Which of the following function has no extreme
point-(A) 2x (B) [x]
(C) log10x (D) All these functions
Q.12 Function x – sin x has -(A) a maxima
(B) a minima
(C) a maxima and a minima (D) no maxima and no minima Q.13 Let f(x) = | x |, then
-(A) f
(0) = 0(B) f(x) has a maximum at x = 0 (C) f(x) has a minimum at x = 0
(D) f(x) has no maximum and no minimum Q.14 The function f(x) =
5 1 K 2 ) K x ( assumes minimum value for x given by(A) 5 (B) 3 (C) 5/2 (D) 2 Q.15 If f(x) = x3 – 3x2 + 3x + 7, then
-(A) f(x) has a maximum at x = 1 (B) f(x) has a minimum at x = 1
(C) f(x) has a point of inflexion at x = 1 (D) None of these
Q.16 In [0, 2] the point of maxima of 3x4 – 2x3 – 6x2 + 6x + 1 is – (A) x = 0 (B) x = 1
(C) x = 1/2 (D) Does not exist
Q.17 If f
(c) changes sign from negative to positive as x passes through c, then-(A) f(c) is neither a maximum nor a minimum value of f(x)
(B) f(c) is a maximum value of f(x) (C) f(c) is a minimum value of f(x)
(D) f(c) is either a maximum or a minimum value of f(x)
