JEE Mathematics Application of dy dx MCQs

Choose the most appropriate option (a, b, c or d).

Question. If m be the slope of tangent to the curve \( e^y = 1 + x^2 \) then
(a) \( |m| > 1 \)
(b) \( m < 1 \)
(c) \( |m| < 1 \)
(d) \( |m| \leq 1 \)
Answer: (d) \( |m| \leq 1 \)

Question. If at each point of the curve \( y = x^3 - ax^2 + x + 1 \) the tangent is inclined at an acute angle with the positive direction of the x-axis then
(a) \( a > 0 \)
(b) \( a \leq \sqrt{3} \)
(c) \( -\sqrt{3} \leq a \leq \sqrt{3} \)
(d) None of the options
Answer: (c) \( -\sqrt{3} \leq a \leq \sqrt{3} \)

Question. The slope of the tangent to the curve \( y = x^2 - x \) at the point where the line \( y = 2 \) cuts the curve in the first quadrant is
(a) 2
(b) 3
(c) -3
(d) None of the options
Answer: (b) 3

Question. The slope of the tangent to the curve \( y = \sqrt{4 - x^2} \) at the point where the ordinate the abscissa are equal, is
(a) -1
(b) 1
(c) 0
(d) None of the options
Answer: (a) -1

Question. The slope of the tangent to the locus \( y = \cos^{-1}(\cos x) \) at \( x = -\frac{\pi}{4} \) is
(a) 1
(b) 0
(c) 2
(d) -1
Answer: (d) -1

Question. The slope of the tangent to the curve \( y = \int_{0}^{x} \frac{dt}{1 + t^3} \) at the point where \( x = 1 \) is
(a) \( \frac{1}{2} \)
(b) 1
(c) \( \frac{1}{4} \)
(d) None of the options
Answer: (a) \( \frac{1}{2} \)

Question. The equation of the curve is given by \( x = e^t \sin t \), \( y = e^t \cos t \). The inclination of the tangent to the curve at the point \( t = \frac{\pi}{4} \) is
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{2} \)
(d) 0
Answer: (d) 0

Question. The curve given by \( x + y = e^{xy} \) has a tangent parallel to the y-axis at the point
(a) (0, 1)
(b) (1, 0)
(c) (1, 1)
(d) None of the options
Answer: (b) (1, 0)

Question. \( P(2, 2) \) and \( Q(\frac{1}{2}, -1) \) are two points on the parabola \( y^2 = 2x \). The coordinates of the point R on the parabola, where the tangent to the curve is parallel to the chord PQ, is
(a) \( (\frac{5}{4}, \frac{5}{2}) \)
(b) (2, -1)
(c) \( (\frac{1}{8}, \frac{1}{2}) \)
(d) None of the options
Answer: (c) \( (\frac{1}{8}, \frac{1}{2}) \)

Question. The number of tangent to the curve \( x^{3/2} + y^{3/2} = a^{3/2} \), where the tangents are equally inclined to the axes, is
(a) 2
(b) 1
(c) 0
(d) 4
Answer: (b) 1

Question. The point on the curve \( \sqrt{x} + \sqrt{y} = 2a^2 \), where the tangent is equally inclined to the axes, is
(a) \( (a^4, a^4) \)
(b) \( (0, 4a^4) \)
(c) \( (4a^4, 0) \)
(d) None of the options
Answer: (a) \( (a^4, a^4) \)

Question. The parabola \( x^2 = 5 - 4y \) and \( y = x^2 \) cut at the point (1, 1) at an angle
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) None of the options
Answer: (a) \( \frac{\pi}{2} \)

Question. The angle between two tangents to the ellipse \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) at the points where the line \( y = 1 \) cuts the curve is
(a) \( \frac{\pi}{4} \)
(b) \( \tan^{-1} \frac{6\sqrt{2}}{7} \)
(c) \( \frac{\pi}{2} \)
(d) None of the options
Answer: (b) \( \tan^{-1} \frac{6\sqrt{2}}{7} \)

Question. The number of tangents to the curve \( y^2 - 2x^3 - 4y + 8 = 0 \) that pass through (1, 2) is
(a) 3
(b) 1
(c) 2
(d) 6
Answer: (c) 2

Question. The equation of a curve is \( y = f(x) \). The tangents at (1, f(1)), (2, f(2)) and (3, f(3)) make angles \( \frac{\pi}{6}, \frac{\pi}{3} \) and \( \frac{\pi}{4} \) respectively with the positive direction of the x-axis. Then the value of \( \int_{2}^{3} f'(x)f''(x)dx + \int_{1}^{3} f''(x)dx \) is equal to
(a) \( -\frac{1}{\sqrt{3}} \)
(b) \( \frac{1}{\sqrt{3}} \)
(c) 0
(d) None of the options
Answer: (a) \( -\frac{1}{\sqrt{3}} \)

Question. The equation of the tangent to the curve \( y = e^{-|x|} \) at the point where the curve cuts the line \( x = 1 \) is
(a) \( x + y = e \)
(b) \( e(x + y) = 1 \)
(c) \( y + ex = 1 \)
(d) None of the options
Answer: (d) None of the options

Question. The equation of the tangent to the curve \( y = be^{-x/a} \) where it cuts the y-axis is
(a) \( \frac{x}{a} + \frac{y}{b} = 1 \)
(b) \( \frac{x}{a} + \frac{y}{b} = -1 \)
(c) \( \frac{x}{a} - \frac{y}{b} = 1 \)
(d) None of the options
Answer: (a) \( \frac{x}{a} + \frac{y}{b} = 1 \)

Question. If the tangent to the curve \( \sqrt{x} + \sqrt{y} = \sqrt{a} \) at any point on it cuts the axes OX and OY at P and Q respectively then \( OP + OQ \) is
(a) 2a
(b) a
(c) \( \frac{1}{2}a \)
(d) None of the options
Answer: (b) a

Question. If the line joining the point (0, 3) and (5, -2) is a tangent to the curve \( y = \frac{c}{x + 1} \) then the value of c is
(a) 1
(b) -2
(c) 4
(d) None of the options
Answer: (c) 4

Question. The curve \( \frac{x^n}{a^n} + \frac{y^n}{b^n} = 2 \) touches the line \( \frac{x}{a} + \frac{y}{b} = 2 \) at the point
(a) (b, a)
(b) (a, b)
(c) (1, 1)
(d) \( (\frac{1}{b}, \frac{1}{a}) \)
Answer: (b) (a, b)

Question. The sum of the intercepts made on the axes of coordinates by any tangent to the curve \( \sqrt{x} + \sqrt{y} = 2 \) is equal to
(a) 4
(b) 2
(c) 8
(d) None of the options
Answer: (a) 4

Question. The area bounded by the axes of reference and the normal to \( y = \log_e x \) at the point (1, 0) is
(a) 1 unit\(^2\)
(b) 2 unit\(^2\)
(c) \( \frac{1}{2} \) unit\(^2\)
(d) None of the options
Answer: (c) \( \frac{1}{2} \) unit\(^2\)

Question. The normal to the curve \( 2x^2 + y^2 = 12 \) at the point (2, 2) cuts the curve again at
(a) \( (-\frac{22}{9}, -\frac{2}{9}) \)
(b) \( (\frac{22}{9}, \frac{2}{9}) \)
(c) (-2, -2)
(d) None of the options
Answer: (a) \( (-\frac{22}{9}, -\frac{2}{9}) \)

Question. Two cyclists start from the junction of two perpendicular roads, their velocities being 3v metres/minute and 4v metres/minute. The rate at which the two cyclists are separating is
(a) \( \frac{7}{2}v \) m/min
(b) 5v m/min
(c) v m/min
(d) None of the options
Answer: (b) 5v m/min

Question. A stick of length a cm rests against a vertical wall and the horizontal floor. If the foot of the stick slides with a constant velocity of b cm/s then the magnitude of the velocity of the middle point of the stick when it is equally inclined with the floor and the wall, is
(a) \( \frac{b}{\sqrt{2}} \) cm/s
(b) \( \frac{b}{2} \) cm/s
(c) \( \frac{ab}{2} \) cm/s
(d) None of the options
Answer: (a) \( \frac{b}{\sqrt{2}} \) cm/s

Question. If \( y = \int_{0}^{x} \frac{t^2}{\sqrt{t^2 + 1}} dt \) then rate of change of y with respect to x when x = 1, is
(a) \( \sqrt{2} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{\sqrt{2}} \)
(d) None of the options
Answer: (c) \( \frac{1}{\sqrt{2}} \)

Question. On the curve \( x^3 = 12y \) the abscissa change 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 the options
Answer: (a) (-2, 2)

Question. A balloon is pumped at the rate of a cm\(^3\)/minute. The rate of increase of its surface area when the radius is b cm, is
(a) \( \frac{2a^2}{b^4} \) cm\(^2\)/min
(b) \( \frac{a}{2b} \) cm\(^2\)/min
(c) \( \frac{2a}{b} \) cm\(^2\)/min
(d) None of the options
Answer: (c) \( \frac{2a}{b} \) cm\(^2\)/min

Question. x and y are the sides of two squares such that \( y = x - x^2 \). The rate of change of the area of the second square respect to that of the first square is
(a) \( 2(1 - x^2)x \)
(b) \( 2x^2 - 3x + 1 \)
(c) \( 2(2x^2 - 3x + 1) \)
(d) None of the options
Answer: (b) \( 2x^2 - 3x + 1 \)

Question. Let the equation of a curve be \( x = a(\theta + \sin \theta) \), \( y = a(1 - \cos \theta) \). If \( \theta \) changes at a constant rate k then the rate of change of the slope of the tangent to the curve at \( \theta = \frac{\pi}{3} \) is
(a) \( \frac{2k}{\sqrt{3}} \)
(b) \( \frac{k}{\sqrt{3}} \)
(c) k
(d) None of the options
Answer: (d) None of the options

Question. If there is an error of k% in measuring the edge of a cube then the per cent error in estimating its volume is
(a) k
(b) 3k
(c) \( \frac{k}{3} \)
(d) None of the options
Answer: (b) 3k

Question. If \( 1^\circ = \alpha \) radius then the approximate value of \( \cos 60^\circ 1' \) is
(a) \( \frac{1}{2} + \frac{\alpha\sqrt{3}}{120} \)
(b) \( \frac{1}{2} - \frac{\alpha}{120} \)
(c) \( \frac{1}{2} - \frac{\alpha\sqrt{3}}{120} \)
(d) None of the options
Answer: (c) \( \frac{1}{2} - \frac{\alpha\sqrt{3}}{120} \)

Choose the correct options. One or more options may be correct.

Question. Let the parabola \( y = x^2 + ax + b \) and \( y = x(c - x) \) touches each other at the point (1, 0). Then
(a) a = -3
(b) b = 1
(c) c = 2
(d) b + c = 3
Answer: (a) a = -3, (d) b + c = 3

Question. Let \( y = f(x) \) be the equation of a parabola which is touched by the line \( y = x \) at the point where x = 1. Then
(a) \( f'(0) = f'(1) \)
(b) \( f'(1) = 1 \)
(c) \( f(0) + f'(0) + f''(0) = 1 \)
(d) \( 2f(0) = 1 - f'(0) \)
Answer: (b) \( f'(1) = 1 \), (d) \( 2f(0) = 1 - f'(0) \)

Question. A point on the ellipse \( 4x^2 + 9y^2 = 36 \) where the tangent is equally inclined to the axes is
(a) \( (\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}}) \)
(b) \( (-\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}}) \)
(c) \( (\frac{9}{\sqrt{13}}, -\frac{4}{\sqrt{13}}) \)
(d) None of the options
Answer: (a) \( (\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}}) \), (b) \( (-\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}}) \), (c) \( (\frac{9}{\sqrt{13}}, -\frac{4}{\sqrt{13}}) \)

Question. If the line \( ax + by + c = 0 \) is a normal to the rectangular hyperbola \( xy = 1 \) then
(a) a > 0, b > 0
(b) a > 0, b < 0
(c) a < 0, b > 0
(d) a < 0, b < 0
Answer: (b) a > 0, b < 0, (c) a < 0, b > 0

Question. A tangent to the curve \( y = \int_{0}^{x} |t| dt \), which is parallel to the line \( y = x \), cuts off an intercept from the y-axis equal to
(a) 1
(b) \( -\frac{1}{2} \)
(c) \( \frac{1}{2} \)
(d) -1
Answer: (b) \( -\frac{1}{2} \), (c) \( \frac{1}{2} \)