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If sin α + sin β = a and cos α + cos β = b, then prove that cos (α − β) = [a² + b² – 2] / 2.

On squaring sin α + sin β = a and cos α + cos β = b and adding them,... View Article

Find the sum of all natural numbers which are multiples of 7 or 3 or both and lie between 200 and 500.

The sum is calculated in the following way. \(\begin{array}{l}\mathbf{S}=\mathbf{S}_{3}+\mathbf{S}_{7}-\mathbf{S}_{21}\\ \mathbf{S}_{3}=\text { Sum of all the number between } \mathbf{2 0... View Article

Find the reciprocal of 3 / 5, 9 / 10, 3 / 9, 7 / 4 and 9 / 2 also classify them as proper, improper fractions.

The given fractions are \(\begin{array}{l}\begin{array}{l} \begin{array}{l} \frac{3}{5} \longrightarrow \frac{5}{3} \quad \text { improper } \\ \frac{9}{10} \rightarrow \frac{10}{9} \text {... View Article

(d / dx) [(1 + cot x) / (1 – cot x)] =

To find the derivative of the given function. \(\begin{array}{l}\frac{d}{d x}\left(\frac{1+\cot x}{1-\cot x}\right)=\frac{\frac{d}{d x}(1+\cot x)-\frac{d}{d x}(1-\cot x)}{(1-\cot x)^{2}} \\ =\frac{-\csc ^{2}... View Article

Fact 1: All chickens are birds. Fact 2: Some chickens are hens. Fact 3: Female birds lay eggs. If the first three statements are facts, which of the following statements must also be a fact? I: All birds lay eggs. II: Some Hens are birds. III. Some chickens are not hens.

The first statement is false because only female birds lay eggs. The second statement is true because few hens are... View Article

How many pairs of successive numbers have a difference of 2 in the following sequence? 6, 4, 1, 2, 2, 8, 7, 4, 2, 7, 5, 3, 8, 6, 2, 1, 7, 0, 4, 1, 3, 2, 8, 6

64, 42, 75, 53, 86, 13 and 86 are 7 such pairs of successive numbers have a difference of 2... View Article

The modulus of (1 + i) (1 + 2i) (1 + 3i) is equal to

The modulus is calculated as follows. \(\begin{array}{l}\begin{array}{l} |(\mathbf{1}+\mathbf{i})(\mathbf{1}+\mathbf{2 i})(\mathbf{1}+\mathbf{3 i})|\\=|\mathbf{1}+\mathbf{i}||\mathbf{1}+\mathbf{2} \mathbf{i}||\mathbf{1}+\mathbf{3 i}| \quad\left(\left|\mathbf{Z}_{1} \mathbf{Z}_{2}\right|=\left|\mathbf{Z}_{1} \| \mathbf{Z}_{2}\right|\right) \\ =\sqrt{\mathbf{2}} \cdot \sqrt{\mathbf{5}}... View Article

What is the number whose 20% is 30% of 40?

Let the number be x. \(\begin{array}{l}\text \ 20 \ \% \ of \ x=30 \% \text \ of \ 40\\... View Article

2, 15, 4, 12, 6, 7, ?, ?

Assume the missing terms of the series to be x1 and x2. The sequence 2, 15, 4, 12, 6, 7,... View Article

The expansion of (3x – 2y)³ is

The given expression is of the form \(\begin{array}{l}\begin{array}{l} (a-b)^{3}=a^{3}-b^{3}-3 a^{2} b+3 a b^{2} \\ (3 x-2 y)^{3}=27 x^{3}-8 y^{3}-54 x^{2}... View Article

A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads (ii) at most 4 heads

A coin is tossed 5 times. \(\begin{array}{l}\begin{array}{l} \text { (i) Probability (at least } 4 \text { heads }) \Rightarrow... View Article

Find the area of a quadrant of a circle whose radius is 14 cm.

r = 14 cm Area = πr2 = (22 / 7) * 142 = 4 * 154 cm2 Area of... View Article

Find the sum of even numbers between 1 and 25.

Sn = [2a + (n – 1) d] ………… (1) The even numbers between 1 and 25 are 2, 4,... View Article

A cylindrical tub of radius 16 cm contains water to a depth of 30 cm. A spherical iron ball is dropped into the tub and thus level of water is raised by 9 cm. What is the radius of the ball?

Let r be the radius of the iron ball. Volume of iron ball = Volume of water raised in the... View Article

State whether the following statement is true or false: Some real numbers are rational numbers.

The numbers which can be expressed in the form p / q are called rational numbers. The numbers which are... View Article

In the series: 7 8 9 7 6 5 3 4 2 8 9 7 2 4 5 9 2 9 7 6 4 7 Which figures have an equal frequency?

2 occurs 3 times. 3 occurs once. 4 occurs 3 times. 5 occurs 2 times. 6 occurs 2 times. 7... View Article

Show that [latex]left|begin{array}{cc} sin 10^{circ} & -cos 10^{circ} \ sin 80^{circ} & cos 80^{circ} end{array}right|=1[/latex].

The given determinant is \(\begin{array}{l}\begin{array}{l} \left|\begin{array}{cc} \sin 10^{\circ} & -\cos 10^{\circ} \\ \sin 80^{\circ} & \cos 80^{\circ} \end{array}\right| \\ =\left|\begin{array}{cc}... View Article

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