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\(\int\) cos4 x dx let u = 4x \(\frac{dy}{dx}\) = 4 dx = \(\frac{dy}{4}\) \(\int\)cos u. \(\frac{dy}{4}\) = \(\frac{1}{4}\)\(\int\)cos u du = \(\frac{1}{4}\) sin u + k = \(\frac{1}{4}\) sin4x + k

Let the numbers be a, a + 1, a + 2, a + 3 a + a + 1 + a + 2 + a + 3 = 34 4a = 34 - 6 4a = 28 a = \(\frac{28}{4}\) = 7 The least of these numbers is a = 7

Median = \(\sum f\) = 23+1 = 24Th = \(\frac{24}{2}\) = 12TH position is 8 = 3 Range = 5 - 0 = 5 (3, 5)

\(\begin{array}{c|c}Class Interval & 3 - 3 & 6 - 8 & 9 - 11 \\ x & 4 & 7 & 10 \\ f & 2 & 2 & 2 \\ f - x & 8 & 14 & 20 \\ |x - \bar{x}|^2 & 9 & 0 & 9 \\ |x - \bar{x}|^2 & 18 0 & 18 \end{array}\) \(\bar{x}\) = \(\frac {\sum fx}{\sum f}\) = \(\frac {8 + 14 + 20}{2 + 2 + 2}\) = \(\frac{42}{6}\) \(\bar{x}\) = 7 S.D = \(\sqrt\frac{\sum f(x - \bar{x})^2}{\sum f}\) = \(\sqrt\frac{18 + 0 + 18}{6}\) = \(\sqrt\frac{36}{6}\) = \(\sqrt {6}\)

ELATION Since there are 7 letters. The first letter can be arranged in 7 ways, , the second letter in 6 ways, the third letter in 5 ways, the 4th letter in four ways, the 3rd letter in three ways, the 2nd letter in 2 ways and the last in one way. therefore, 7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! ways

The first person will sit down and the remaining will join. i.e. (n - 1)! = (5 - 1)! = 4! = 24 ways

Prime numbers = (43,47,53,59) N = (43, 44, 45,..., 60) The universal set contains 18 numbers. The prime numbers between 43 and 60 are 4 Probability of picking a prime number = \(\frac{4}{18}\) = \(\frac{2}{9}\)

\(\frac {\sin\theta}{\cos\theta}\) \(\frac{\cos \theta {\frac{d(\sin \theta)}{d \theta}} - \sin \theta {\frac{d(\cos \theta)}{d \theta}}}{\cos^2 \theta}\) \(\frac{\cos \theta. \cos \theta - \sin \theta (-\sin \theta)}{cos^2\theta}\) \(\frac{cos^2\theta + \sin^2 \theta}{cos^2\theta}\) Recall that sin2 \(\theta\) + cos2 \(\theta\) = 1 \(\frac{1}{\cos^2\theta}\) = sec2 \(\theta\)

No expanded working has been attached for this question yet. The verified answer is Option A.

\(\frac{90}{360} \times 100 = \frac{100}{4}\) = 25

45 + 60 + 30 + 45 = 180 students.

D14.95 = N112.00 D1.00 = \(\frac{N112}{D14.95} \times\) D1.00 = 7.49

\(\frac{3}{5}\)(2x - 1) = \(\frac{1}{4}\)(5x - 3) \(\frac{6x}{5} - \frac{3}{5} = \frac{5x}{4} - \frac{3}{4}\) \(\frac{6x}{5} - \frac{5x}{4} = \frac{3}{5} - \frac{3}{4}\) \(\frac{24x - 25x}{20} = \frac{12 - 15}{20}\) \(\frac{-x}{20} = \frac{-3}{20}\) -20x = -60 x = \(\frac{-60}{-20}\) x = 3

cos xo = \(\frac{1}{r}\); \(\sqrt{r^2 - 1}\) By Pythagoras r2 = 12 + x2 - 1 x = \(\sqrt{r^2 - 1}\) tan xo = \(\sqrt{r^2 - 1}\) = \(\sqrt{r^2 - 1}\)

3x - 2y = 7; Solve by elimination method x + 2y = -3 3x - 2y = 7.....(i) -3x + 6y = -9.....(ii) -8y = 16 y = \(\frac{16}{-8} = -2\) Put y = -2 into equation (i); 3x - 2y = 7 3x - 2(-2) = 7 3x + 4 = 7 3x = 7 - 4 3x = 3 x = \(\frac{3}{3}\) = 1