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Published on: 01/06/2021
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Questions + Answers key
Take MCQ Maths Test1.
The number given by the Rolle's theorem for the functlon x3 - 3x2, x ∈ [0, 3] is
1
\(\sqrt { 2 } \)
\(\frac { 3 }{ 2 } \)
2
2.
The function sin4 x + cos4 x is increasing in the interval
\(\left[ \frac { 5\pi }{ 8 } ,\frac { 3\pi }{ 4 } \right] \)
\(\left[ \frac { \pi }{ 2 } ,\frac { 5\pi }{ 8 } \right] \)
\(\left[ \frac { \pi }{ 4 } ,\frac { \pi }{ 2 } \right] \)
\(\left[ 0,\frac { \pi }{ 4 } \right] \)
3.
The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is
\(-4\sqrt { 3 } \)
-4
\(\cfrac { \sqrt { 3 } }{ 12 } \)
\(4\sqrt { 3 } \)
4.
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
-8
-4
-2
0
5.
The volume of a sphere is increasing in volume at the rate of 3 πcm3 / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
3 cm/s
2 cm/s
1 cm/s
\(\cfrac { 1 }{ 2 } cm/s\)
1.
(d)
2
2.
(c)
\(\left[ \frac { \pi }{ 4 } ,\frac { \pi }{ 2 } \right] \)
3.
Given equation of the curve is y = 1 + x3 and the line is x + 12y = 12
Slope of the tangent to the curve
m1 = \(\frac { dy }{ dx } \) = 3x2 and the
Slope of the line = m2
= \(\frac{-1}{2}\) \(\left[ \because m=\frac { co-efficient\quad of\quad x }{ co-efficient\quad of\quad y } \right] \)
Since the slope of the tangent to the curve and the line are orthogonal, m1 m2 = - 1.
∴ 3x2\(\left( \frac { -1 }{ 2 } \right) \) = -1
⇒ \(\frac{x^2}{4}\) = 1
⇒ x2 = 4
⇒ x = ±2
When x = 2, y = 1 + 23 = 9
When x = -2, y = 1+ (-2)3
= 1-8 = -7
∴ Equation of the tangent at (2, 9) is
y-9 = 12(x-2) [∵ m1 = 3x2 = 3(2)2 = 12]
∴ y - 9 = 12x - 24
∴ 12x - y = 15
Equation of the tangent at (-2, -7) is
y + 7= 12(x + 2)
⇒ y + 7 = 12x + 24
⇒ 12x - y = -17
4.
(b)
-4
5.
(a)
3 cm/s
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