Correct Answer: C. \(\frac{\sqrt{2}}{2}\)

Explanation

Given tan x = 1

x = tan^-1(1)

x = 45^o

Now, \(\frac{1 - ( \frac{1}{\sqrt{2} )^2}}{\frac{1}{\sqrt{2}}}\)

= \(\frac{1 - \frac{1}{2}}{\frac{1}{2}}\)

= \(\frac{1}{2} + \frac{1}{\sqrt{2}}\)

= \(\frac{1}{2} \times \frac{1}{\sqrt{2}}\)

= \(\frac{\sqrt{2}}{2}\)

Correct Answer: D. 2

Explanation

\(S_{n} = \frac{n}{2}(2a + (n - 1)d \)

\(\frac{n}{2}(2a + (n - 1) d = n^{2} + 2n\)

\(n(2a + (n - 1) d = 2n^{2} + 4n\)

\(2an + n^{2}d - nd = 2n^{2} + 4n\)

Comparing the equations, d = 2 (coefficient of \(n^{2}\)).

Explanation

(a) \(y = 6 - x - x^{2}\)

When x = 0, y = 6.

When y = 0, \(x^{2} + x - 6 = 0\)

\(x^{2} + 3x - 2x - 6 = 0 \implies x(x + 3) - 2(x + 3) = 0\)

\(x = -3 ; x = 2\)

Graph passes through (0, 6), (-3, 0) (2, 0).

Turning point : \(\frac{\mathrm d y}{\mathrm d x} = - 1 - 2x\)

\(-1 - 2x = 0 \implies x = \frac{-1}{2}\)

When \(x = \frac{-1}{2}\), \(y = 6 - (-\frac{1}{2}) - (\frac{-1}{2})^{2} = 6\frac{1}{4}\).

Turning point : \((-\frac{1}{2}, 6\frac{1}{4})\).

\(y = 3x^{2} - 2x + 3\)

When x = 0, y = 3.

\(b^{2} - 4ac = (-2)^{2} - 4(3)(3) = 4 - 36 < 0\)

Graph does not cut x - axis.

\(\frac{\mathrm d y}{\mathrm d x} = 6x - 2\)

At turning point, \(6x - 2 = 0 \implies x = \frac{2}{6} = \frac{1}{3}\)

When \(x = \frac{1}{3}, y = 3(\frac{1}{3})^{2} - 2(\frac{1}{3}) + 3 = 2\frac{2}{3}\)

Turning point : \((\frac{1}{3}, 2\frac{2}{3})\).

(b) For the x- coordinates, we use the equation.

\(3x^{2} - 2x + 3 = 6 - x - x^{2}\)

\(4x^{2} - x - 3 = 0 \implies 4x^{2} - 4x + 3x - 3 = 0\)

\(4x(x - 1) + 3(x - 1) = 0 \implies (4x + 3)(x - 1) = 0\)

\(x = -\frac{3}{4} ; 1\)

(c) Area of shaded portion :

\(\int_{-\frac{3}{4}} ^{1} [(6 - x - x^{2}) - (3x^{2} - 2x + 3)] \mathrm {d} x\)

= \(\int_{-\frac{3}{4}} ^{1} (3 + x - 4x^{2}) \mathrm {d} x\)

= \([3x + \frac{x^{2}}{2} - \frac{4}{3}x^{3}]|_{-\frac{3}{4}} ^{1}\)

= \((3(1) + \frac{1}{2} - \frac{4}{3}) - (3(-\frac{3}{4}) + \frac{(\frac{-3}{4})^{2}}{2} - \frac{4}{3} (-\frac{3}{4})^{3}\)

= \(\frac{13}{6} - (-\frac{45}{32})\)

= \(\frac{343}{96} sq. units\)

Correct Answer: A. P∩Q = ∅

Correct Answer: C. 4

Explanation

(x\(^2\) - 4x + p)

Use the coefficient of the middle variable(-4x)

= (\(\frac{-4}{2}\))\(^2\)

= (-2)\(^2\)

= 4

Correct Answer: D. 3x - 2y

Correct Answer: C. 36°

Explanation

The sum of angles in a triangle = 180º

(180 - 108)º = 72º

Isosceles triangle has both two equal sides and two equal angles.

∠OSN = ∠SON = \(\frac{72}{2}\)

= 36º

Correct Answer: B. x > 3/2

Correct Answer: D. 1 + \(\sqrt{6}\)

Explanation

x*y = x\(^2\) - y\(^2\) + xy

(\(\sqrt{3}\))*(\(\sqrt{2}\)) = (\(\sqrt{3}\))\(^2\) - (\(\sqrt{2}\))\(^2\) + \(\sqrt{3}\) x \(\sqrt{2}\)

= 3 - 2 + \(\sqrt{6}\)

= 1 + \(\sqrt{6}\)

Correct Answer: C. 2

Explanation

Given: \(f(x + 1) = x^{3} + 3x^{2} - 4x + 2\).

\(f(2) = f(x + 1) \implies x + 1 = 2; x = 1\)

\(f(2) = 1^{3} + 3(1)^{2} - 4(1) + 2 = 1 + 3 - 4 + 2 = 2\)