1
MCQ (Single Correct Answer)

JEE Main 2016 (Online) 10th April Morning Slot

Let C be a curve given by y(x) = 1 + $$\sqrt {4x - 3} ,x > {3 \over 4}.$$ If P is a point on C, such that the tangent at P has slope $${2 \over 3}$$, then a point through which the normal at P passes, is :
A
(2, 3)
B
(4, $$-$$3)
C
(1, 7)
D
(3, $$-$$ 4),

Explanation

Given,

y = 1 + $$\sqrt {4x - 3} $$

$$ \therefore $$   $${{dy} \over {dx}}$$ = $${1 \over {2\sqrt {4x - 3} }} \times 4 = {2 \over 3}$$

$$ \Rightarrow $$   4x $$-$$ 3 = 9

$$ \Rightarrow $$   x = 3

$$ \therefore $$   y = 1 + $$\sqrt {12 - 3} $$ = 4

$$ \therefore $$   Equation of normal at point P(3,4)

y $$-$$ 4 = $$-$$ $${3 \over 2}$$ (x $$-$$ 3)

$$ \Rightarrow $$   2y $$-$$ 8 = $$-$$ 3x + 9

$$ \Rightarrow $$    3x + 2y $$-$$ 17 = 0
2
MCQ (Single Correct Answer)

JEE Main 2017 (Online) 8th April Morning Slot

The tangent at the point (2, $$-$$2) to the curve, x2y2 $$-$$ 2x = 4(1 $$-$$ y) does not pass through the point :
A
$$\left( {4,{1 \over 3}} \right)$$
B
(8, 5)
C
($$-$$4, $$-$$9)
D
($$-$$2, $$-$$7)

Explanation

As,    $${{dy} \over {dx}}$$ = $$-$$ $$\left[ {{{{{\delta f} \over {\delta x}}} \over {{{\delta f} \over {\delta y}}}}} \right]$$

$${{{\delta f} \over {\delta x}}}$$ = y2 $$ \times $$2x $$-$$ 2

$${{{\delta f} \over {\delta y}}}$$ = x2 $$ \times $$ 2y + 4

$$\therefore\,\,\,$$ $${{dy} \over {dx}}$$ = $$-$$ $$\left( {{{2x{y^2} - 2} \over {2{x^2}y + 4}}} \right)$$

$${\left[ {{{dy} \over {dx}}} \right]_{(2, - 2)}}$$ = $$-$$ $$\left( {{{2 \times 2 \times 4 - 2} \over {2 \times 4 \times ( - 2) + 4}}} \right)$$ = $$-$$ $$\left( {{{14} \over { - 12}}} \right)$$ = $${7 \over 6}$$

$$\therefore\,\,\,$$ Slope of tangent to the curve = $${7 \over 6}$$

Equation of tangent passes through (2, $$-$$ 2) is

y + 2 = $${7 \over 6}$$ (x $$-$$ 2)

$$ \Rightarrow $$$$\,\,\,$$ 7x $$-$$ 6y = 26 . . . . .(1)

Now put each option in equation (1) and see which one does not satisfy the equation.

By verifying each points you can see ($$-$$ 2, $$-$$ 7) does not satisfy the equation.
3
MCQ (Single Correct Answer)

JEE Main 2017 (Online) 9th April Morning Slot

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :
A
$$\left( {{1 \over 3},24} \right)$$
B
$$\left( {{1 \over 2},4} \right)$$
C
$$\left( {2,{1 \over 8}} \right)$$
D
$$\left( {3,{1 \over 28}} \right)$$
4
MCQ (Single Correct Answer)

JEE Main 2017 (Online) 9th April Morning Slot

The function f defined by

f(x) = x3 $$-$$ 3x2 + 5x + 7 , is :
A
increasing in R.
B
decreasing in R.
C
decreasing in (0, $$\infty $$) and increasing in ($$-$$ $$\infty $$, 0)
D
increasing in (0, $$\infty $$) and decreasing in ($$-$$ $$\infty $$, 0)

Questions Asked from Application of Derivatives

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