Find the values of cos 75^o

In an examination, a student has to answer 4 questions out of 5 questions, questions 1 and 2 are compulsory. Determine the number of ways in which the student can make the choice.

Find the inclination of the lines x - y + 3 = 0 with the positive direction of X-axis

Find the derivative of x⁵(3-6x^-9)

Find the coordinates of the point which divides the line segment joining the points (1, -2, -1) and
(1,5, -8) in the ratio
(i) 3:4 internally (ii) 3:4 externally

Which of the following pairs of sets are equal? Justify your answer.
A= {x : x is a letter of the word "LOYAL"},
B= {x : x is a letter of the word "ALLOY"}  

Let A be the set of first 10 natural numbers and let R be a relation on A defined by \(\left( x,y \right) \in R\Leftrightarrow x+2y=10\)  i.e  R = \(\left\{ \left( x,y \right) :x\in A,y\in A\quad and\quad x+2y=10 \right\} \). Express R and \({ R }^{ -1 }\) as sets of ordered pair. Also, determine
(i) domain of R and  \({ R }^{ -1 }\)
(ii)Range of R and  \({ R }^{ -1 }\)

Evaluate \(\underset { x\longrightarrow 0 }{ lim } \cfrac { { tan\quad x }^{ 0 } }{ { x }^{ 0 } } \) 

Evaluate:\(\lim_ { x\rightarrow 0 } \frac { { e }^{ x }-1-x-{ x }^{ 2 } }{ { x }^{ 2 } } \)

Prove that \({ 1 }^{ 2 }{ +2 }^{ 2 }+....+{ n }^{ 2 }>\frac { { n }^{ 3 } }{ 3 } ,n\in N\) .

How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5, if the repetition of the digits is not allowed?

Show that the following statement is true. p:For any real numbers x,y if x = y, then 2x + a = 2y + a when a \(\in\) Z.

Write the relation between set P and Q given by an arrow diagram in
(i) roster form     (ii) set-builder form

Evaluate (sin x - cos x)

If A = {3, 5, 7,9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17};find:
(l) A \(\cap\) B
(ii) B \(\cap\) C
(iii) A \(\cap\) C \(\cap\) D
(iv) A \(\cap\) C
(v) B \(\cap\) D
(vi)A \(\cap\) (B\(\cup\) C)
(vii) A \(\cap\) D
(viil) A \(\cap\) (B \(\cup\) D)
(ix) (A \(\cap\)B) \(\cap\) (B \(\cup\) C)
(x) (A \(\cup\) D) \(\cap\) (B \(\cup\) C)

How many numbers are there between 100 and 1000 such that at least one of their digits is 7?

Find the square root of \(3-4\sqrt 7i\).

Which of the following sentences are statements? Give reasons for your answer.
(i)There are 35 days in a month.
(ii)Mathematics is difficult.
(iii)The sum of 5 and 7 is greater than 10.
(iv) The square of a number is an even number.
(v) The sides of a quadrilateral have equal length.
(vi) Answer this question.
(vii)The product of (-1) and 8 is 8.
(viii) The sum of all interior angles of a triangle is 180°.
(ix) Today is a windy day.
(x) All real numbers are complex numbers.

The difference between any two consecutive interior angles of a polgon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.

Given that P(5, 4, -2), Q (7, 6, -4) and R (11, 10, -8) are collinear points. Find the ratio in which Q divides PR.