Correct Answer: B. 2/3

Explanation

Substitute x by (x + h) to have f(x + h) = 3(x + h)\(^2\) + 2x + 2h - 1 - 3x\(^2\) - 2x + 1

= 6\(x\)h + 3h\(^2\) + 2h

Divided through by h and obtained \(\frac{f(x + h) - f(x)}{h} = \frac{6 \times h + 3h^2 + 2h}{h} = 6x + 3h + 2\)

Taking limit as h tends to zero, lim\(_{h \to o}\) \(\frac{f(x+h) -f(x)}{h}\)

= lim\(_{h \to o}\) (6x + 3h + 2) so that f\(^1\)(x) = 6x + 2

Correct Answer: C. 35.26°

Explanation

6 sin 2θ tan θ = 4, where 0º < θ < 90º

sin 2θ = 2sin θ cos θ and tanθ = \(\frac{sinθ}{cosθ}\)

= 6 x 2sin θ cos θ x \(\frac{sin θ}{cos θ} = 4\)

= \(sin^2 θ = 4\)

= \(sin^2 θ = \frac{4}{12}=\frac{1}{3}\)

=\(sin θ = \frac{\sqrt1}{3}=\frac{1}{\sqrt3}\)

= \(θ = sin^{-1}(\frac{1}{\sqrt3})\)

∴ θ = 35.26º

Correct Answer: B. 2%

Correct Answer: A. 6 days

Explanation

12 men working 8 hours a day can complete the work in 4 days

12 men working1 hour a day will complete the same work in (8 x 4) days [working less hours a day means they have to work for more days]

1 man working 1 hour a day will complete same work in (8 x 4 x 12) days [fewer number of men working means the remaining ones will have to work for more days]

So, it will take1 man working1 hour a day to complete a piece of work in(8 x 4 x 12) days = 384 days

Now,

1 man working16 hours a day will complete the piece of work in (384/16) days = 24 days [working more hours a day means less days to complete the work]

∴4 men working 16 hours a day will complete the piece of work in (24/4) days = 6 days [more men working means fewer days to complete the work]

ALTERNATIVELY

12 x 8 x 4 = 4 x 16 x d

⇒ d = \(\frac{12 \times 8 \times 4}{4 \times 16}\)

⇒ d = \(\frac{384}{64}\)

⇒ d = 6

∴ 4 men working 16 hours a day will complete the piece of work in 6 days

Correct Answer: B. \(\frac{3}{8}\)

Explanation

Let the probability of getting a head = p = \(\frac{1}{2}\) and that of tail = q = \(\frac{1}{2}\)

\((p + q)^{3} = p^{3} + 3p^{2}q + 3pq^{2} + q^{3}\)

In the equation above, \(p^{3}\) and \(q^{3}\) are the probabilities of 3 heads and 3 tails respectively

while, \(p^{2}q\) and \(pq^{2}\) are the probabilities of 2 heads and one tail and 2 tails and one head respectively.

Probability of exactly 2 heads = \(3p^{2}q = 3(\frac{1}{2})^{2}(\frac{1}{2})\)

= \(\frac{3}{8}\)

Correct Answer: C. \(\begin{pmatrix} 2 & 7 \\ 1 & 8 \end{pmatrix}\)

Explanation

\(\begin{pmatrix} 1 & 2 \\ 5 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 3 \end{pmatrix}\)

= \(\begin{pmatrix} 1\times 0 + 2\times 1 & 1\times 1 + 2\times3 \\ 5\times0 + 1\times1 & 5\times1 + 1\times 3 \end{pmatrix}\)

= \(\begin{pmatrix} 2 & 7 \\ 1 & 8 \end{pmatrix}\)

Correct Answer: B. 16 : 35

Explanation

m:n = \(2\frac{1}{3} : 1\frac{1}{5}\) = m : n = \(\frac{7}{3} : \frac{6}{5}\)

\(\frac{7}{3} : \frac{6}{5}\) = \(\frac{7}{3} \div \frac{6}{5}\)

\(\frac{m}{n}\) = \(\frac{7}{3} \times \frac{5}{6}\)

\(\frac{m}{n}\) = \(\frac{35}{18}\) = m = \(\frac{35n}{18}\)

n : q = \(1\frac{1}{2} : 1\frac{1}{3}\) = \(\frac{3}{2} : \frac{4}{3}\)

\(\frac{n}{q}\) = \(\frac{3}{2} \times\frac{3}{4}\)

\(\frac{n}{q}\) = \(\frac{9}{8}\) = q = \(\frac{8n}{9}\)

q : m = \(\frac{8n}{9}\) : \(\frac{35n}{18}\)

\(\frac{q}{m}\) = \(\frac{8n}{9} \div \frac{35n}{18}\)

\(\frac{q}{m}\) = \(\frac{8n}{9}\times\frac{18}{35n}\)

=\(\frac{q}{m} = \frac{16}{35}\) = q : m = 16 : 35

Correct Answer: D) f (x ) = 4x + 8

Explanation

only the f(x)=4x+8 satisfies the linear equation when proven; f(0)=4(0)+8 f(0)=8 f(1)=4(1)+8 f(1)=4+8 F(1)=12

Correct Answer: D. 90 < x < 180

Explanation

obtuse angle \(\to\) 90^o < x < 180^o