21.

If the slope of y = 3x² + ax³ is maximum at x = $\frac{1}{2}$, then the value of a is

• 2

• 1

• - 1

• - 2

If y = 4x - 5 is a tangent to the curve y = px³ + q at (2, 3), then (p + q) is equal to

• - 5

• 5

• - 9

• 9

The point on the curve y = 5 + x - x² at which the normal makes equal intercepts is

• (1, 5)

• (0, - 1)

• (- 1, 3)

• (0, 5)

If the point (a, b) on the curve y = $\sqrt{\mathrm{x}}$ is close to the point (1, 0), then the value of ab is

• $\frac{1}{2}$

• $\frac{\sqrt{2}}{2}$

• $\frac{1}{4}$

• $\frac{\sqrt{2}}{4}$

The image of the interval [- 1, 3] under the mapping f : R $\to$ R given by f(x) = 4x³ - 12x is

• [8, 72]

• [0, 72]

• [- 8, 72]

• [0, 8]

If n(A) = 43, n(B) = 51 and n(A ∪ B) = 75, then n[(A - B) $\cup$ (B - A)] is equal to

• 53

• 45

• 56

• 66

If p and q are non-collinear unit vectors and $\left|\mathrm{p} + \mathrm{q}\right| = \sqrt{3}$, then (2p - 3q) · (3p + q) is equal to

• 0

• $\frac{1}{3}$

• $- \frac{1}{3}$

• $- \frac{1}{2}$

The triangle formed by the three points whose position vectors are 2i + 4j - k, 4i + 5j + k and 3i + 6j - 3k, is

• an equilateral triangle

• a right angled triangle but not isosceles

• an isosceles triangle but not right angled triangle

• a right angled isosceles triangle

If (1, 2, 4) and (2, - 3$\mathrm{\lambda }$, - 3) are the initial and terminal points of the vector i + 5j - 7k, then the value of $\mathrm{\lambda }$ is

• $\frac{7}{3}$

• $\frac{- 7}{3}$

• $\frac{- 5}{3}$

• $\frac{5}{3}$

Let u = 5a + 6b + 7c, v = 7a - 8b + 9c and w = 3 a + 20b + 5c, where a, b and c are non-zero vectors. If u = lv + mw, then the values of l and m respectively are

• $\frac{1}{2}, \frac{1}{2}$

• $\frac{1}{2}, - \frac{1}{2}$

• $- \frac{1}{2}, \frac{1}{2}$

• $\frac{1}{3}, \frac{1}{3}$