Find the largest possible domain for the real valued function f defined by \(f(x)=\sqrt{x^2-5x+6}.\)

By using the same concept applied in previous example, graphs of y = sin x and y = sin 2x, and also their combined graphs are given figures (a), (b) and (c). The minimum and maximum values of sin x and sin 2x are the same. But they have different x-intercepts. The x-intercepts for y = sin x are \(\pm n\pi\) and for y = sin 2x are \(\pm{1\over 2}n\pi,\ n\in Z.\) ​​​

A model rocket is launched from the ground. The height 'h' reached by the rocket after t seconds from lift off is given by h(t) = -5t² + 100t , \(0\le t\le 20\).  At what time the rocket is 495 feet above the ground?

Solve the following system of linear inequalities 3x - 9 ≥ 0, 4x -10 ≤ 6;

Solve x = \(\sqrt{x+20}\) for x ∈ R

Resolve into partial fractions: \(\frac{x}{(x+3)(x-4)}\)

Solve \(x^{log_3x}=9\)

If sin A = \(\frac{3}{5}\) and cos B = \(\frac{9}{41}\), 0 < A < \(\frac{\pi}{2}\), 0 < B < \(\frac{\pi}{2}\). Find the value of sin (A + B)

Find sin (x - y) given that sin x = \(\frac{8}{17}\) with 0 < x < \(\frac{\pi}{2}\)and cos y = \(-\frac{24}{25}\)with π < y < \(\frac{3\pi}{2}\)

If sin \(\theta\) + cos \(\theta\) = m, show that cos⁶\(\theta\) + sin⁶\(\theta\)  = \(\frac { 4-3({ m }^{ 2 }-1)^{ 2 } }{ 4 } \), where m² \(\le \) 2

Find a quadratic equation whose roots are sin 15^o and cos 15^o

Prove that sin (30^o + \(\theta\)) + cos (60^o + \(\theta\)) = cos θ

If \(\cos { \theta } =\frac { 1 }{ 2 } \left( a+\frac { 1 }{ a } \right) \), show that  \(\cos {3\theta } =\frac { 1 }{ 2 } \left( { a }^{ 3 }+\frac { 1 }{ { a }^{ 3 } } \right) \)

In \(\triangle\)ABC, 60° prove that b + c = 2a cos \(\left( \frac { B-C }{ 2 } \right) \)

In a \(\triangle\)ABC, if a = \(\sqrt { 3 } -1\), b =\(\sqrt { 3 } +1\) and C = 60^o. Find the other side and the other two angles

In any \(\triangle\)ABC, prove that the area \(\triangle\) = \(\frac { { b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 } }{ 4cotA } \)

Find the value of sin 765⁰.

Prove that tan 315° cot (-405°) + cot 495° tan (-585°) = 2.

A circular wire of radius 3 cm is cut and bent so as to lie along the circumference of a sector whose radius is 48 cm, Find in degrees the angle which is subtended at the centre of the sector.

Four children are running a race.
(i) In how many ways can the first two places be filled?
(ii) In how many different ways could they finish the race?

There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees
(i) a particular teacher is included?
(ii) a particular student is excluded?

If an electricity consumer has the consumer number say 238 :110 : 29, then describe the linking and count the number of house connections upto the 29th consumer connection linked to the larger capacity transformer number 238 subject to the condition that each smaller capacity transformer can have a maximal consumer link of say 100.

There are 10 bulbs in a room. Each one of them can be operated independently. Find the number of ways in which the room can be illuminated.

Prove that ¹⁰C₂ + 2 x ¹⁰C₃ + ¹⁰C₄ = ¹²C₄