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The most general solution of tan θ = 1 and cos theta = – 1 / √2 is

tan θ = 1 and cos theta = – 1 / √2\(\begin{array}{l}\cos \theta=-1 / \sqrt{2}=-\cos (\pi / 4) \\ =\cos... View Article

Find the derivative of cos √x with respect to x using the first principle.

f (x) = cos √x\(\begin{array}{l}f(x)=Lt_{h \rightarrow 0} \frac{\cos \sqrt{x+h}-\cos \sqrt{x}}{h}=\operatorname{Lt}_{h \rightarrow 0} \frac{-2}{h} \sin \frac{\sqrt{x+h}+\sqrt{x}}{2} \sin \frac{\sqrt{x+h}-\sqrt{x}}{2}\\ =L t_{h \rightarrow... View Article

The graph of y = log x cuts

y = logb x x = byThe graph cuts the x-axis at x = 1.

A room is 6 m long, 5 m broad and 4 m in height. If all its walls are to be covered with paper 50 cm wide, the length of the paper is

The room is in the shape of a cuboid.Area of cuboid = 2bh + 2hl= (2 × 20) + (2... View Article

Show that 107 is of the form 4q + 3 for any integer q.

107 = 4 * 26 + 3This is of the form a = bq + r.

A rectangle measures 40 cm at its length and 20 cm at its width. Find the ratio of the length to the width.

Length of the rectangle = 40 cmWidth of the rectangle = 20 cmLength / Width = 40 / 20= 2... View Article

log (70 / 33) + log (22 / 135) – log (7 / 18)

log (70 / 33) + log (22 / 135) – log (7 / 18)= log (70 / 33) * (22... View Article

If 3y = y + 2, calculate the value of 2y.

The given equation is\(\begin{array}{l}\Rightarrow 3 y=y+2 \\ \Rightarrow 3 y-y=2 \\ \Rightarrow 2 y=2 \\ \Rightarrow y=\frac{2}{2}=1\\ 2 \mathrm{y}=2 \times... View Article

A number from 1 to 11 is chosen at random. What is the probability of choosing an odd number?

The numbers from 1 to 11 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. The probability... View Article

If cos θ + cos 2θ + cos 3θ = 0, then the general value of θ is:

cos θ + cos 2θ + cos 3θ = 0\(\begin{array}{l}\Rightarrow (\cos 3 \theta+\cos \theta)+\cos 2 \theta=0\\ \Rightarrow 2 \cos 2... View Article

If 79 ⊕ 86 = 6897 then the value of 53 ⊕ 47 =

79 ⊕ 86 = 689753 ⊕ 47 = 7435

Two adjacent angles of a parallelogram are equal. What is the measure of each angle?

The adjacent angles of a parallelogram are equal.The sum of adjacent angles of parallelogram = 180. Assume the adjacent angles... View Article

(√11 – √7) / (√11 + √7) = a – √77 b. Find a and b.

On rationalizing the LHS,\(\begin{array}{l}\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}\\ =\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}} \times \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}-\sqrt{7}} \\ =\frac{(\sqrt{11}-\sqrt{7})^{2}}{(\sqrt{11}+\sqrt{7})(\sqrt{11}-\sqrt{7})} \\ =\frac{(\sqrt{11})^{2}+(\sqrt{7})^{2}-2(\sqrt{11})(\sqrt{7})}{(\sqrt{11})^{2}-(\sqrt{7})^{2}} \\ =\frac{11+7-2 \sqrt{77}}{11-7} \\ =\frac{18-2 \sqrt{77}}{4} \\ =\frac{18}{4}-\frac{2}{4} \sqrt{77}... View Article

A cube whose volume is 1 / 8 cubic centimetre is placed on top of a cube whose volume is 1 cm³. The two cubes are then placed on top of a third cube whose volume is 8 cm³. The height of stacked cubes is

Let the volume of the three cubes be V1, V2 and V3 and a1, a2 and a3 be the sides... View Article

The sum of the fractions 1 / 2 + 1 / 6 + 1 / 12 + 1 / 20 + 1 / 30 + 1 / 42 + 1 / 56 + 1 / 72 + 1 / 90 + 1 / 110 + 1 / 132 is

The sum of the fractions 1 / 2 + 1 / 6 + 1 / 12 + 1 / 20... View Article

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