Correct Answer: D. 3(x + 1)(1 - x)

Explanation

\((x + 2)^2\) - \((2x + 1)^2\)

= \((x^2 + 4x + 4) - (4x^2 + 4x + 1)\)

= \(x^2 \) + 4x + 4 - 4 \(x^2 \) - 4x - 1

= -3 \(x^2 \) + 3

= 3 - 3 \(x^2 \)

= 3(1 - \(x^2 \))

= 3(1 + x)(1 - x)

Explanation

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

Let \(x + 1 = u, x - 1 = u - 2\)

When x = 3, u = 3 + 1 = 4

x = 1, u = 1 + 1 = 2

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

= \(\int_{2}^{4} (\frac{u}{u^{2}} - \frac{2}{u^{2}}) \mathrm {d} u\)

= \(\int_{2}^{4} (\frac{1}{u} - 2u^{-2}) \mathrm {d} u\)

= \([\ln u - \frac{2u^{-1}}{-1}]|_{2}^{4}\)

= \([\ln u + 2u^{-1}]|_{2}^{4}\)

= \([\ln u + \frac{2}{u}]|_{2}^{4}\)

= \((\ln 4 + \frac{2}{4}) - (\ln 2 + \frac{2}{2})\)

= \(\ln 4 - \ln 2 - \frac{1}{2}\)

= \(\ln (\frac{4}{2}) - \frac{1}{2}\)

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

Correct Answer: A. (x + y + 6)(x + y -3)

Correct Answer: B. N44

Correct Answer: (d) 21

Correct Answer: B. 35 units per sec

Correct Answer: A. 26cm

Explanation

Perimeter of a Rhombus = 4L or 2√( d1\(^2\) + d2\(^2\))

P = 2√(12\(^2\) + 5\(^2\))

P = 2√(144 + 25)

P = 2√169 = 2 * 13

P = 26cm

Correct Answer: B. 80

Explanation

The exterior angle theorem states that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

∠M = ∠R + ∠N

→ 141 = 41 + ∠N

∠N = 141 - 41 = 100º

∠QNR = 180 - 100 → 80º

Correct Answer: (a) 1.1

Explanation

(a) (i) No of red discs = \(\frac{1}{4} \times 40 = 10\)

Total no of discs = 40

\(\therefore\) No of white balls = 40 - 10 = 30

(ii) Let the number of red balls to be added = z

\(\therefore \frac{10 + z}{40 + z} = \frac{1}{3}\)

\(3(10 + z) = 40 + z\)

\(30 + 3z = 40 + z \implies 3z - z = 40 - 30\)

\(2z = 10 \implies z = 5\)

Therefore, adding 5 red discs, the probability of picking a red disc = \(\frac{1}{3}\).

(b) CP of a plate = N50.00

Let the number of plates be x

\(\therefore \text{CP of x plates} = N50x\)

\(\text{SP of all plates} = N600\)

\(\frac{100 + 20}{100} \times N50x = N600 \implies N60x = N600\)

\(x = \frac{600}{60} = 10 plates\)