Suppose f(x) = x(x + 3)(x - 2), x ∈ [- 1, 4]. Then, a value of c in (- 1, 4) satisfying f'(c) = 10 is
2
52
3
72
The area included between the parabola y = x24a and the curve y = 8a3x2 + 4a2 is
a22π + 23
a22π - 83
a2π + 43
a22π - 43
D.
For point of intersection, equate both equationsy = x24a and the curve y = 8a3x2 + 4a2⇒ x2x2 + 4a2 = 4a8a3⇒ x4 + 4a2x2 = 32a4⇒ x4 + 4a2x2 - 32a4 = 0⇒ x2x2 + 8a2 - 4a2x2 + 8a2 = 0⇒ x2 + 8a2x2 - 4a2 = 0⇒ x2 - 4a2 = 0 or x2 + 8a2 = 0⇒ x2 = 4a2 or x2 = - 8a2∴ x2 = 4a2
⇒ x = ± 2a ⇒ x = 2aHence, we take limits from 0 to 2a.∴ Area between 2 graphs = 2 × ∫02a8a3x2 + 4a2 - x24adx = 2 × ∫02a8a3x2 + 4a2dx - ∫02ax24adx = 2 × 8a3 × ∫02a1x2 + 4a2dx - 14a∫02ax2dx = 2 × 8a312atan-1x2a02a - 14ax3302a = 2 × 8a32atan-12a2a - tan-102a - 14a2a33 - 033
= 2 × 4a2tan-11 - tan-10 - 14a8a33 - 0 = 2 × 4a2π4 - 0 - 14a8a33 = 2 × 4a2 × π4 - 14a × 8a33 = 2 × a2π - 2a23 = 2a2π - 23 = 2a2π - 23 = a22π - 43
If a, b, c are distinct positive real numbers, then the value of the determinant abcbcacab is
< 0
> 0
0
≥ 0
The equations x - y + 2z = 43x + y + 4z = 6x + y + z = 1 have
unique solution
infinitely many solutions
no solution
two solutions
If x =sin2tan-12 and y = sin12tan-143, then
x > y
x = y
x = 0 = y
x< y
If coshx = 54, then cosh3x = ?
6116
6316
6516
6163
In a∆ABC, if <A = 90°, then cos-1Rr2 + r3 = ?
90°
30°
60°
45°
The value (s) of x for which the function
f(x) = 1 - x, x < 1=1 - x2 - x, 1 ≤ x ≤ 23 - x, x > 2fails to be continuous is (are)
1
all real numbers
If y = log2log2x, then dydx = ?
loge2xlogex
1loge2xx
1xlogexloge2
1xlog2x2
The angle of intersection between the curves y2 + x2 = a22 and x2 - y2 = a2 is
π3
π4
π6
π12