For any two sets A and B, A \(\cup\) B = A iff _____.

Which one of the following is not a function

In a ΔABC, if the sides are 7cm, 4\(\sqrt { 3 } \) cm and .\(\sqrt { 13 } \) cm, then the smallest angle is ______.

If ^₁₈C_x = ⁿC₆, then find ⁿC_2.

If in the expansion of (1 +x)ⁿ, the coefficients of fifth, sixth and seventh terms are inA.P. then n is equal to _____.

Let U be the universal set containing 700 elements. If A and B are sub-sets of U such that n(A) = 200, n(B) = 300 and n(A \(\cap\) B) = 100, then n(A' \(\cap\) B') =___.

If the set A has m elements, B has n elements then the number of elements in A x B is ______.

If tan\(\theta\) + cot \(\theta\) = 5 then tan³ \(\theta\) + cot³ \(\theta\) is equal to ______.

If ⁿC_r+ⁿC_r+1=ⁿ⁺¹C_x then x is equal to ______.

If in the expansion of (1 +x)ⁿ, the coefficients of fifth, sixth and seventh terms are inA.P. then n is equal to _____.

If \(A\cap B=B\) then _____.

If the set A has m elements, B has n elements then the number of elements in A x B is ______.

The solution of the equation cos² \(\theta\) -+sin\(\theta\)+ 1=0lies in the interval ______.

In how many ways can a student choose programme of 5 courses, if 9 ourses are available and 2 specific courses are compulsory for every student?

The middle term in the expansion of \(\left( \frac { { 2x }^{ 2 } }{ 3 } +\frac { 3 }{ { 2x }^{ 2 } } \right) ^{ 10 }\)is _____.

For any two sets A and B, (A - B) \(\cup\) (B - A)=_____.

If R is a relation from a finite set A having m elements to a finite set B having n elemen.ts, then the number of relations from A to B is ______.

If sin \(\theta\) + sin \(\phi\)= \(\alpha\) and cos\(\theta\) - cos\(\phi\)= b then tan\({\theta -\phi\over 2}\) is equal to ______.

Prove that

\( \frac {1} {9!}+\frac{1} {10!}+ \frac {1}{11!}=\frac {122} {11!}\)

The coefficient of x⁵y⁸ in the expansion of (x + y)¹³ is _____.

If A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5} then A - B is _____.

If f(x) =\({x+1\over x-1}\) is a real function,\(\neq\) 1, then \(f[f\{f(2)\}]\) is ______.

If A, B, C are in A.P. then \({sinA-sinC\over cosC-cosA}\) is equal to ______.

Find the number of chord that can be drawn through 16 points on a circle

If in the expansion of (a + b)ⁿ and (a + b)^{n + 3}, the ratio of the coefficients of second and third terms and third and fourth terms respectively are equal then n is _____.

If U = {1, 2, 3,4,5,6,7,8,9,10},A = {2, 4, 6, 8, 10},B = {1, 3, 5, 7, 9}, C = {3, 4, 7, 8, 10},find:
B' - A'

Express the complex number in the form a+ib: (1-i)⁴

Find n in the binomial\({ \left( \sqrt [ 3 ]{ 2 } +\frac { 1 }{ \sqrt [ 3 ]{ 3 } } \right) }^{ n }\), if the ratio of 7th term from the beginning to the 7th term from the end is\(\frac { 1 }{ 6 } \).

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.

A line passes through the point (2, 2) and is perpendicular to the line 3x + y = 3. Its y intercept is ______.

If p be the length of the perpendicular from the origin to the line \({x\over a}+{y\over b}=1\) then ______.

The angle between the lines 3x - 2y + 5 = 0 and 2x + 3y - 7 = 0 is ______.

The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centriod is ______.

If the sum of first n even natural numbers is equal to m times the sum of first n is odd natural numbers then m is equal to ______.

The sum to infinity of \(\frac{1}{3}+\frac{3}{9}+\frac{5}{27}+\frac{7}{81}+...+\infty\) is ______.

The sum of the square of (n - 1) natural numbers is ______.

The seen to infinity of the series \(1+2.\frac{1}{2}+3.\frac{1}{2^2}+4.\frac{1}{2^3}+...+\infty\) ______.

The sum of infinity of the G.P. a, ar, ar², ar³, ...... \(\infty\) is ______.

If in the expansion of (a + b)ⁿ and (a + b)^{n + 3}, the ratio of the coefficients of second and third terms and third and fourth terms respectively are equal then n is _____.

If in the expansion of (1 +x)ⁿ, the coefficients of fifth, sixth and seventh terms are inA.P. then n is equal to _____.

If in the expansion of (1 +x)¹⁵, the coefficient of (2x + 3)^th and (r - 1)^th terms are equal then r is equal to _____.

The coefficient of x⁵y⁸ in the expansion of (x + y)¹³ is _____.

Constant term in the expansion of \(\left( x-\frac { 1 }{ x } \right) ^{ 14 }\) _____.

If ⁴⁰C_r+2 = ⁴⁰C_r-2 then r is equal to ______.

If ^mC₂ = ⁿC₁ then ______.

If ⁿC_r+ⁿC_r+1=ⁿ⁺¹C_x then x is equal to ______.

If C₀ + C₁ + C₂ +...+C_n = 256 then ²ⁿC₂ is equal to ______.

If ⁿ⁺¹C₃ = 2.ⁿC₂ then n is equal to ______.

Solve the following linear in equations: 4x - 12 \(\ge\) 0

Solve the following linear in equations: 2x + 8 < 0

Solve the inequalities:\(\frac { x-1 }{ 3 } +4<\frac { x-5 }{ 5 } -2\)

Solve the inequalities: \(\frac { 4+2x }{ 3 } \ge \frac { x }{ 2 } -3\)

Solve the liner inequations x + 7 \(\ge\) - 3x + 19

Find the conjugate of (6 + 5i )²

Simplify the following
(2i)³

Express each of these complex numbers in the form a + ib: \(\frac { 1 }{ 1-cos\theta +2i\quad sin\quad \theta } \)

Find the multiplicative inverse of the following complex number 3+2i

Find the multiplicative inverse of the following complex number cos \(\theta \)+i sin \(\theta \)

Prove that the sum of first n even numbers is n(n+1).

Prove that 2+4+6+8+.....2=n(n+1).

Using principle of mathematical induction, prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) .....\left( 1+\frac { 1 }{ n } \right) =n+1\)

Prove that \(\sum _{ t=1 }^{ n-1 }{ t(t+1) } =\frac { n(n-1)(n+1) }{ 3 } \) , for all natural numbers \(n\ge 2\)

Prove that 2ⁿ>n for all positive integers n.

In a ΔABC if a = 5, b = 6 and c = 5, then ∠B is ______.

If tan\(\theta\) + cot \(\theta\) = 5 then tan³ \(\theta\) + cot³ \(\theta\) is equal to ______.

In a\(\triangle\) ABC, if tan A +tanB+ tan C = o then cot A cot B cot C is equal to______.

The solution of the equation cos² \(\theta\) -+sin\(\theta\)+ 1=0lies in the interval ______.

Find the general solutions of\(\sqrt{3}\) sin x-cos x =2 ______.

Let R be a relation from a set A to B, then ______.

If R is a relation from a finite set A having m elements to a finite set B having n elemen.ts, then the number of relations from A to B is ______.

If \(f(x)={2^x+2^{-x}\over 2}\) then f(x+y) f(x-y)is equal to ______.

If f(x) =\({x+1\over x-1}\) is a real function,\(\neq\) 1, then \(f[f\{f(2)\}]\) is ______.

The domain of the function \(f(x)=\sqrt{x-1}+\sqrt{3-x}\) is ______.

For any two sets A and B, \(A\cap(A\cup B)=\)_____.

If \(A\cap B=B\) then _____.

If A = {1, 2, 3, 4, 5, 6}then the number of proper sub-sets is _____.

If A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5} then A - B is _____.

In a set builder method, the null set is represented by _____.

The number of subsets of a set containing n elements is _____.

For any two sets A and B, \(A\cap(A\cup B)=\)_____.

If \(A\cap B=B\) then _____.

For any two sets A and B, (A - B) \(\cup\) (B - A)=_____.

For any two sets A and B, A \(\cup\) B = A iff _____.

Given the set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set(s) for all the three sets A, B, and C?
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) ф
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1,2, 3, 4, 5, 6, 7, 8}

f f(x) =x² find \(f(1.1)-f(1)\over (1.1-1)\).

Find the general solution for each of the following equations:
sin x + sin 3x + sin 5x = 0

Find sin\({x\over2},cos{x\over2} and \ tan {x\over2}\) in each of the following: sin x =\({1\over4}\),x in quadrant II.

Prove the following by using the principle of mathematical induction for all n ∊ N:1+2+3+... +\(n<\frac { 1 }{ 8 } (2n+1)^{ 2 }\)

In a survey of 400 movie viewers, 150 were listed as liking 'Veer Zaara', 100 were listed as liking 'Aitraaz' and 75 were listed both liking 'Aitraaz' as well as 'Veer zaara'.Find how many people liking neither 'Aitraaz' nor 'Veer Zaara'?

If n(A)=4, n(B)=6, then what can be the minimum number of elements in A\(\cup \) B?

Convert the complex numbers in polar form -1 + i.

The longest side of a triangle is 3 times the shortest and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is atleast 61cm. Find the minimum length of the shortest side.

A flag is in the form of three blocks, each to be coloured differently . If there are 8 different colours to hoose from , then how many flags are possible?

Which of the following are sets? Justify your answer.
The collection of all the months of a year beginning with the letter J.

Draw the Venn diagram of the following:
A' \(\cap\) (B \(\cup\) C)

\(f: R-\{3\} \rightarrow R\) be defined by \(f(x)=\frac{x^{2}-9}{x-3}\) and \(g: R \rightarrow R\) be defined by \(g(x)=x+3\). Find whether f = g or not.

Find the domain and range of f (x) =\(\sqrt{x-2}\) 

Determine the range and domain of the relation: R = {(x, y): y = I x + 1|, x \(\in\) Z, IxI \(\le\)3}.

Let n(U) = 700, n(A) = 200, n(B) = 300 and \(n(A\cap B)\) = 100 .Find \(n({ A }^{ ' }\cap { B }^{ ' })\).

Find the union of each of the following pair of sets A = {1, 3, 5}, B = {1, 2, 3}

Consider the following sets  \(\phi\) , A = {2,5}, B = {1,2,3,4} and C = {1,2,3,4,5} Insert the correct symbol  \(\subset\)  or \(\nsubseteq\)  between the pair set 
A.......C

If A = {3,{4,5},6}, then find the following statement are true? 
\(\phi \subset\) A

Find the intersection of each of the following pairs of sets:
A = {x : x ∊ N, 5 \(\le\) x \(\le\) 10}, B = {x : x ∊ N, 15 \(\le\) x \(\le\) 20}

Find the probability of getting atmost two heads or atleast two tails in a toss of three coins.

In a large metropolitan area, the probabilities are 0.87,0.36,0.30 that a family (randomly chosen for a sample survey) owns a colour television set, a black and white television set, or both kinds of sets. What is the probability that a family owns either anyone or both kinds of sets?

If \(\frac { 5 }{ 14 } \)is the probability of occurrence of an event, find the odd against its occurrence.

A bag contains 6 discs of which 4 red,3 are blue and 2 are yellow.The discs are similar in shape and size.A disc is drawn at random from the bag.Calculate the probability that it will be
either red or blue

In a drawing competition, the odds in favour of competitors A, B, C and D are 1:2,1:3,1:4 and 1:5, respectively. Find the probability that one of them wins the competition.

Find the mean deviation about the median for the data :
36, 72, 46, 42, 60, 45, 53, 46, 51, 49

Find the mean deviation about the mean for the data

Find the mean deviation about median for the following data:

Calculate the mean deviation about median age for the age distribution of 100 persons gives below:

[Hint Convert the given data into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval]

Find the mean and standard deviation of the following distribution:

Identify the quantifiers and write the negation of the following statements
There exists a number which is a multiple of 6 and 9

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.

Give three examples of sentences which are not statements. Give reasons for the answers.

For each of the following compound statement first identify the connecting words and then break it into component statements.
(i) All rational numbers are real and all real numbers are not complex.
(ii) Square of an integer is positive or negative.
(iii) The sand heats up quickly in the sun and does not cool down fast at night.
(iv) x = 2 and x = 3 are the roots of the equation 3x² - x - 10 = 0.

Write the contrapositive and converse of the following statements.
(i) if x is a prime number, then x is odd.
(ii) if the two lines are parallel, then they do not intersect in the same plane.
(iii) Something is cold implies that it has low temperature. 
(iv) You cannot comprehend geometry if you do not know how to reason deductively.
(v) x is an even number implies that x is divisible by 4.

If \(f(x)=\left\{\begin{array}{ll} |x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0 \end{array}\right.\)for what values of a does \(\overset{lim}{x\rightarrow a}\)f(x) exists?

Find the derivative of the following functions from first principle:
(i) - x
(ii) (- x)^-1
(iii) sin (x +1)
(iv) \(\cos\left( x-\frac { \pi }{ 8 } \right) \)

Find the derivative of (x+a) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)

Find the derivative of  \((px+q)\left( \frac { r }{ x } +s \right) \) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)

Find the derivative of \(\frac { ax+b }{ cx+d } \) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)

Find a point in XY plane which is equidistant from three points (2, 0, 3), (0, 3, 2) and (0, 0, 1).

Find the locus of the point which is equidistant from A(3, 4, 0) and B(5, 2, -3).

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.

Three vertices of a parallelogram ABCD are A(3, – 1, 2), B (1, 2, – 4) and C (– 1, 1, 2). Find the coordinates of the fourth vertex.

Show that the coordinates of the centroid of a triangle with vertices A(x₁,x ₂,x₃)., b(y₁,y ₂,y ₃), c(z₁,z₂,z ₃) are \(\left[ \frac { x1+x2+x3 }{ 3 } ,\frac { y1+y2+y3 }{ 3 } ,\frac { z1+z2+z3 }{ 3 } \right] \)

An arch is in the form of a parabola with its vertical. The arch is 10 m high and 5 m wide at the base. How wide it is 2 m from the vertex of the parabola?

The foci of a hyperbola coincide with the faci of the elipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\) find the equation of the hyperbola if its eccentricity is 2.

Find the equation of the hyperbola satisfying the given conditions.
Foci (0, ± \(\sqrt10\) ),passing through (2, 3)

A rod of length 12cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x +y = 16.

The mid-point of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4) find its vertices and also find the centroid of the triangle.

Are the points A(3,6,9), B(10,20,30) and C(25,-41,5), the vertices of a right angled triangle?

Prove that the coordinates of the points which divide the lines joining the vertices of a tetrahedron to the centroid of the opposite faces in the ration 3:1 are same.

Using the section formula, show that the points, (2, -3, 4), (-1, 2, 1) and (0, \(\frac{1}{3}\), 2) are collinear.

Three points A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are forming a triangle . The bisector Ad of

Evaluate \(\lim_{ x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ x^{ 3 } } \)

Evaluate sin ( x + 1)

Evaluate logx²

Evaluate (sin x - cos x)

Evaluate \(\lim_ { x\rightarrow 0 }{ lim } \frac { sin2x+3x }{ 2x+tan3x } \)

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 component statements of the following compound statement and check whether the compound statement is true or false
The sun is a star or Sun is plant

By giving a counter example, shows that the following statement is not true.
p: If all the angles of a triangle are equal, than the triangle is an obtuse angled triangle.

Write down the negation
2 + 4 > 5 or 3 + 4 < 6.

Write down the negation
\(\triangle ABC\) is isosceles, if and only if \(\angle B=\angle C\)

Calculate the mean deviation from the mean of the following distribution.

Find the mean and standard deviation of the following frequency distribution

Find the mean and standard deviation for the following data.

Calculate the mean deviation from the median of the following data

Find the variance and standard deviation for the following data:
65, 68, 58, 44, 48, 45, 60, 62, 60, 50

A bag contains 6 discs of which 4 red,3 are blue and 2 are yellow.The discs are similar in shape and size.A disc is drawn at random from the bag.Calculate the probability that it will be
not blue

One card is drawn from a well-shuffled deck of 52 cards.Calculate the probability that the card will be
a red card

Two dice are thrown find the odd against getting the sum 6.

Find the probability that, when a hand of 5 cards is drawn from a well-shuffled deck of 52 cards, it contains 3 queens.

A card is drawn from an ordinary pack and a gambler bets that it is a diamond or a jack.What are the odds favour in the winning his bet?

Find  the equation of the hyperbola whose one directrix is x + =9, the corresponding focus is (2,2) and eccentricity is 2. 

Find the foci,vertices, eccentricity and length of latusrectum of the hyperbola \(5{ y }^{ 2 }+{ 9x }^{ 2 }=36\)

If a latusrectum of an ellipse subtends a right angle at the centre of the ellipse, then write the eccentricity of the ellipse.

Find the equation of the circle passing through the vertices of a triangle whose sides are represented by the equations x+y=2, 3x-4y=6 and x-y=0

The foci of a hyperbola coincide with the faci of the elipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\) find the equation of the hyperbola if its eccentricity is 2.

If the sum of the distance of a moving point in a plane from the axes is 1, then find the locus of the point.

The perpendicular from the origin to a line meets it at the point (-3, 5), find the equation of the line.

Find the distance of the line 3x - 5y + 8 = 0 from the point (1, 2) along the line 2x - 5y = 0.

Find the value of k if the straight line 2x + 3y + 4 + k (6x - y + 12)= 0 is perpendicular to the line 7x + 5y - 4 = 0.

Find the equation of line passing through the origin and the intersection of the line x - y - 7 = 0 and 2x + y - 2 = 0

The 2nd, 31st and last term of an AP are 7\(\frac { 2 }{ 4 } \) , \(\frac { 1 }{ 2 } \) and - 6 \(\frac { 1 }{ 2 } \), respectively. Find the first term and the number of terms.

Find the number of terms common to the two AP's 3, 7, 11, .... 407 and 2, 9, 16, ....709.

Find  the middle terms in the AP 20, 16, 12, ....., -176.

Find the 15th term from the end of the AP 3, 5, 7, 9, ......, 201.

Three numbers are in AP. If their sum is 27 and the product 648, find the numbers.

Find the value of \(\alpha \) for which the coefficients of the middle terms in the expansions of \({ \left( 1+\alpha x \right) }^{ 4 }\)and \({ \left( 1-\alpha x \right) }^{ 6 }\)are equal.

Find the coefficient of x⁷ in \({ \left( { ax }^{ 2 }+\frac { 1 }{ bx } \right) }^{ 11 }\)and x^-7 in \({ \left( { ax }-\frac { 1 }{ b{ x }^{ 2 } } \right) }^{ 11 }\)and find the relation between a and b so that these coefficients are equal.

Prove that there is no term involving x⁸ in the expansion of\({ \left( { 2x }^{ 2 }-\frac { 3 }{ x } \right) }^{ 11 }\).

Prove that (1+y+y²+...)² = (1+2y+3y²+...)

Show that the ratio of the coefficient of x¹⁰ in (1-x²)¹⁰ and the term independent of x in \(\left( x-\frac { 2 }{ x } \right) ^{ 10 }\) is 1:32.

Three married couples are to be seated in a row having six seats in cinema halls. If spouses are to be seated next to each other, in how many ways can they be seated? Also, find the number of ways of their seating, if all the ladies sit together.

If \(\frac { (2n)! }{ 3!(2n-3)! } \) and \(\frac { n! }{ 2!(n-2)! } \) are in the ration 44:3, find n

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of the digits is allowed?
(ii) repetition of the digits is not allowed?

Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

\(find \ r \ if\quad ^{ 9 }P_{ 5 }+5.\quad ^{ 9 }P_{ 4 }=^{ 10 }P_{ r }\)

Solve graphically \(x-y\le 2;x+2y\le 8;x,y\ge 0\)

Solve the inequalities \(3x+2y\le 12,x\ge 1\ and\ y\ge 2\) graphically.

Solve the inequalities \(x+y\le 9,y>x \ and \ x\ge 1\) graphically. 

Solve the inequality 5x+2y\(\le\)10 graphically

Solve 4x-y>0 graphically.

If a + ib = \(\frac { ({ x }^{ 2 }+1) }{ 2{ x }^{ 2 }+1 } \) , prove that \({ a }^{ 2 }+{ b }^{ 2 }=\frac { ({ x }^{ 2 }+1)^{ 2 } }{ (2{ x }+1)^{ 2 } } \)

Convert the complex numbers in polar form -1 + i.

Convert the complex numbers in polar form -3.

If \(|z+1|=z+2(1+i)\) then find z.

If z = x + iy, w = \(\frac { 1-iz }{ z-i } \) and \(|w|=1\) then show that z is purely real.

Use the principle of mathematical induction to prove that \({ n }^{ 3 }-7n+3\) is divisible by 3, for all natural numbers of n.

If x and y are any two distinct integers, then prove by mathematical induction that \(\left( { x }^{ n }-{ y }^{ n } \right) \) is divisible by (x-y), for all \(n\in N\)

Prove that (2n+7)<(n+3)², for all natural numbers n.

Prove that 2n<(n+2)! for all natural numbers n.

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

Find the angle between the lines \(\sqrt { 3x } \)+y=1 and x+\(\sqrt { 3y } \)=1.

Reduce the following equations into intercept form and find their intercepts on the axes.
(i) 3x + 2y - 12 = 0,
(ii) 4x - 3y = 6,
(iii) 3y + 2 = 0.

Find the area of the triangle formed by the lines y - x = 0, x + y = 0 and x - k = 0.

Find the equation of the line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y + 1 = 0 that has equal intercepts on the axis.

Find the direction in which a straight line must be drawn through the point (-1,2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.

In an A.P. if p^th term is \(\frac { 1 }{ q } \) and q^th term is \(\frac { 1 }{ p } \) prove that the sum of first pq terms is \(\frac { 1 }{ 2 } \)(pq+1), where p \(\neq \)  q.

The sums of n terms of two arithmetic progressions are on the ratio 5n + 4; 9n + 6. Find the ratio of their 18^th terms.

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.

Find the Value of n so that \(\frac { { a }^{ n+1 }+b^{ n+1 } }{ { a }^{ n }+{ b }^{ n } } \) may be the geometric mean between a and b 

Let sum of n, 2n, 3n terms of an A.P. be S₁, S₂ and S₃ respectively, show that S₃ = 3 (S₂ - S₁)

Find the value of \(\left( { a }^{ 2 }+\sqrt { { a }^{ 2 }-1 } \right) ^{ 4 }+({ a }^{ 2 }-\sqrt { { a }^{ 2 }-1)^{ 4 } } \)

Expand using binomial theorem \(\left[ 1+\left( \frac { x }{ 2 } -\frac { 2 }{ x } \right) \right] ^{ 4 },x\neq \)0.

Find the expansion of (3x² - 2ax + 3a²)³ using binomial theorem.

Find the middle terms in the expansion of \(\left( 2x-\frac { { x }^{ 2 } }{ 6 } \right) ^{ 9 }\)

Find the middle terms in the expansions of \(\left( 3-\frac { { x }^{ 3 } }{ 6 } \right) ^{ 7 }\)

Five persons entered in the lift cabin on the ground floor of an 8-floor house. Suppose each of them can leave the cabinindependently at any floor begining with the first. Find the total number of ways in which each of the five persons can leave the cabin
(i) at anyone of the 7 floor.
(ii) at different floors.

How many natural numbers less than 1000 can be formed with the digits 1, 2, 3, 4, and 5, if repetition of digits is allowed?

How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, if
(i) repetition of digits is alllowed?
(ii) repetition of digits is not allowed?

How many different words can be formed with the letters of the word 'HARYANA'? How many of these have H and N together?

Prove that 35! is divisble by 2¹². What is the largest integer n such that 35! is divisible by 2ⁿ?

If sin A=\(\frac { 3 }{ 5 } \), 0\(\frac { \pi }{ 2 } \) and cos B=\(-\frac { 12 }{ 13 } \), \(\pi \) \(\frac { 3\pi }{ 2 } \) , then find the following.
sin (A - B)

If sin A=\(\frac { 3 }{ 5 } \), 0\(\frac { \pi }{ 2 } \) and cos B=\(-\frac { 12 }{ 13 } \), \(\pi \) \(\frac { 3\pi }{ 2 } \) , then find the following.
cos (A + B)

Solve sin2x-sin4x+sin6x=0

The Moon's distance from the Earth is 360000 km and its diameter subtend an angle of 31' at the eye of observer. Find the diameter of the Moon.

Two trees A and B are on the same side of a river. From a point C in the river the distance of trees A and B are 250m and 300m, respectively of the angle C is 45⁰, find the distance between the trees.[use \(\sqrt { 2 } =1.44\)]

If f and g be two real function defined by \(f\left( x \right) =\sqrt { x+1 } \)and \(g\left( x \right) =\sqrt { 9-{ x }^{ 2 } } \) .Then, describe each of the following functions.  \(\frac { f }{ g } \)

If X = {1, 2, 3, 4, 5}, Y = {1, 2, 5, 6, 7, 9, 10, 11, 12, 13, 14} and \(f:X\rightarrow Y\) be defined by f(x) = 2x+3, then find the domainand range of f.

If A = {a,d}, B = {b,c,e} and C = {b,c,f}, then verify that \(A\times (B\cup C)=(A\times B)\cup (A\times C)\)

Find the domain of the function f defined by \(f(x)=\sqrt { 4-x } +\frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \)

If a function \(f:R\rightarrow R\)  be defined by
\(f(x)=\begin{cases} 3x-2,\quad x<0 \\ 1,\quad \quad \quad \quad x=0 \\ 4x+1,\quad x>0 \end{cases}\)
Find f(1), f(-1), f(0), f(2).

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

If n(A)=4, n(B)=6, then what can be the minimum number of elements in A\(\cup \) B?

If X={1,2,3} and n represents any member of X, write the following sets containing all numbers represented by 4n

If X={1,2,3} and n represents any member of X, write the following sets containing all numbers represented by \(\frac { n }{ 2 } \)

If X={1,2,3} and n represents any member of X, write the following sets containing all numbers represented by n-1

A bag contains 4 identical red balls 3 identical black balls. The experiment consists of drawing one ball, then putting it into the bag again drawing a ball. What are the possible outcomes of the experiment?

A and B are two events that P(A) = 0.54, P(B) = 0.69 and \(P(A\cap B)\) = 0.35. Find \(P(A'\cap B')\)

A and B are two events that P(A) = 0.54, P(B) = 0.69 and \(P(A\cap B)\) = 0.35. Find \(P(A\cap B')\)

A die is thrown. Describe the following events
i) A: a number less than 7
ii) B: a number greater than 7
iii) C: a multiple of 3
iv) D: a number less than 4.
v) E: an even number greater than 4.
vi) F: a number not less than3
Also, find \(A\cup B,A\cap B,B\cup C,E\cap F,D\cap E,A-C,D-E,{ F }^{ ' }and\quad E\cap { F }^{ ' }\)

Four cards are drawn at random from pack of 52 playing cards, Find the probability of getting  two red cards and two black cards.

Find the variance and standard deviation for the following data, 6,7,10,12,13,4,8,12.

Given that  \(\bar { x } \) is the mean and \({ \sigma }^{ 2 }\) is the variance of n observations,\({ x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 }+...+{ x }_{ n }\) then prove that the mean and variance of the observations \(a{ x }_{ 1 },a{ x }_{ 2 },...,a{ x }_{ n }\)are a \(\bar { x } \) and \({ a }^{ 2 }\)\({ \sigma }^{ 2 }\), respectively (where,\(a\neq 0\))

Find the mean deviation about the mean for the following data.
38,70,48,40,42,55,63,46,54,44

The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

The standard deviation of some temperature data (\(in^{0}C\)) is 5. Find the variance, if the data were converted into\(^{0}F\)

Write down the truth bvalue of each of the following statements.
(i) Delhi is in India
(ii) Chennai is in Pakistan
(iii) 5 + 6 = 11

Find the conditional statement of 'You will get a sweet dish after the dinner'.

Rewrite the following statement with 'if-then' in five different ways conveying the same meaning 'If a natural is odd, then its square is also odd'.

Write down the compound statements by using 'and','or' of the following statements
p : It is cold 
q : It is raining

Write the negation of the following statements:
(i) Every natural number is greater than 0.
(ii) The sum of 4 and 6 is 10.
(iii) A whole number is an integer.

Evaluate the limits \(\lim _{ x\rightarrow 3 }{ { (4x }^{ 3 }-{ 2x }^{ 2 } } -x+1)\)

Find the value of \(\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}\)

Evaluate \(\lim _{x \rightarrow \pi / 2} \frac{1+\cos 2 x}{(\pi-2 x)^{2}}\)

Find the derivative of a^x from first principle.

Evaluate  \(\lim _{x \rightarrow \frac{1}{2}} \frac{4 x^{2}-1}{2 x-1}\)

A point is on the x-axis. What are its y-coordinate and z-coordinates?

L is the foot of the perpendicular drawn from a point P(5,4,6) on the XY-plane.Find the coordinates of point L

Find the values of x, if the distance between two points (x,-8,4) and (3,-5,4) is 5

Find the distance of point P(3,6,9) from the YZ-plane using distance formula.

Find the equation of the curve formed by the set of all points whose distances from the points (3,4,-5) and (-2,1,4) are equal.

Find the center and radius of each of the following circle.
x² + y² + 6x -4y + 4 = 0

Find the equation of the ellipse, Whose foci are (\(\pm\)3,0) and passing through (4,1)

Find the eccentricity of the hyperbola whose length of latusrectum is 8 and conjugate axis is equal to the half of its distance between the foci.

Find the equation of ellipse with center at the origin, major axis on the y-axis and passing through the point (3,2) and (1,6).

Find the equation of the parabola whose focus is (2, 0) and directrix is x = -2.

In which quadrant, the following points lie?
(-4,1)

Find the new coordinates of point (3,-4), if the origin is shifted to (1,2) by a translation.

If the axes are shifted to the point (-2,3) without rotation, then transform the equation of line y+3x=2 into new axes.

A point moves, so that the sum of its distances from (ae,0) and (-ae,0) is 2a , prove that the equation to its locus is \(\frac { { X }^{ 2 } }{ { a }^{ 2 } } +\frac { { Y }^{ 2 } }{ { b }^{ 2 } } =1\) , where b²=a²(1-e²).

Without using distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (–3, 2) are the vertices of a parallelogram.

The Fibonacci sequence is defined by 1 = a₁ = a₂ and a_n = a_{n - 1} + a_n = a_{n - 2}  n > 2. Find \(\frac { { a }_{ n+1 } }{ { a }_{ n } }\), for n = 1, 2, 3, 4 , 5.

Write the first five terms of each of the following sequence whose nth terms are a_n = \(\frac { n }{ n+1 } \)

Find the indicated terms in each of the sequence, where nth terms are given.
a_n = 5n - 3, a₁₂, a₁₅

Find the indicated terms in each of the sequence, where nth terms are given.
(i) b_n = \({ (-1) }^{ n }({ n }^{ 2 }-1)\), b_7, b₁₃ 
(ii) \(a_{n}=\frac{n^{2}}{2^{n}}, a_{7}\)

Write the first five terms of each of the sequence and obtain the corresponding series. a₁ = 3, a_n = 3a_n-1+2, for all n >1

Prove that \(\overset { n }{ \underset { r=0 }{ \Sigma } } { 3 }^{ r }\quad ^{ n }{ C }_{ r }={ 4 }^{ n }\)

Evaluate the following terms. 5th term from the end in the expansion of \(\left( x-\frac { 1 }{ { x }^{ 2 } } \right) ^{ 12 }.\)

If the third term in the expansion of \(\left( \frac { 1 }{ x } +x^{ log10^{ x } } \right) ^{ 5 }\) is 1000, then find x.

Find the middle term in the expansion of \(\left( \frac { 2x^{ 2 } }{ 3 } +\frac { 3 }{ { 2x }^{ 2 } } \right) ^{ 10 }.\)

Draw the shape of the hyperbola \(\frac { { y }^{ 2 } }{ 9 } -\frac { { x }^{ 2 } }{ 27 } =1\) and find their centre, transverse axis, conjugate axis, value of c, vertices, directrices, foci,eccentricity and latusrectum. 

In how many ways can 6 letters be posted in 5 letter boxes ?

Find the number of different words can be formed from the letters of the word "TRIANGLE", so that
(i) all vowels occur together.
(ii) all vowels do not occur together.

If  \(^{n}{P}{_{r}}\)= 840 and \(^{n}{C}{_{r}}\) =35, find r.

Evaluate the following:  \(^{ 100 }{ C }{ _{ 99 } }\)

It there are 15 persons in a party anf if each two of them shake hands wuth each other.How many hand-sakes happen in the party?

Solve the linear inequality 3x - 5 < x + 7, when x is a natural number

Solve the inequalities 6 \(\le \) - 3(2x - 4) < 12.

Solve the linear inequality 3x - 5 < x + 7, when x is a whole number

Solve the inqualities - 3 \(\le \) 4 - \(\frac { 7x }{ 2 } \) \(\le \) 18.

Solve \(|3x-2|\le \frac { 1 }{ 2 } \)

Find the modulus and the arguments of complex number z= - 1 - i\(\sqrt { 3 } \)

Find the modulus and the arguments of complex number \(z=-\sqrt { 3 } +i\)

Convert the complex numbers in polar form \(\sqrt { 3 } +i\)

If \(x+iy=\frac{(a+i)^2}{2a-i}\) show that \(x^2+y^2=\frac{(a^2+1^2)}{4a^2+1}.\)

Convert the following in the polar form \(\frac { 1+3i }{ 1-2i } \)

Solve the inequalities graphically: 2x +y \(\ge\) 6, 3x +4y \(\le\)12

Solve the inequalities graphically x+y \(\ge\) 4, 2x - y > 0

Solve the inequalities graphically x +y \(\le\) 6, x + y \(\ge\) 4

Solve the inequalities graphically 2x +y \(\ge\) 8,x + 2y \(\ge\) 10

Solve the inequalities graphically - 5x +4y \(\le\) 20, x \(\ge\) 1, y \(\ge\) 2

Prove the following by using the principle of mathematical induction for all n∊ N 1+3+3²+.....+3^n-1 = \(\frac { ({ 3 }^{ n }-1) }{ 2 } \)

Prove the following by using the principle of mathematical induction for all n ∊ N:1.2.3+2.3.4+...+n(n+1)(n+2)= \(\frac { n(n+1)(n+2)(n+3) }{ 4 } \)

Prove the following by using the principle of mathematical induction for all n ∊ N: 1.3+3.5+5.7+....+(2n-1)(2n+1) = \(\frac { n(4{ n }^{ 2 }+6n-1) }{ 3 } \)

Prove the following by using the principle of mathematical induction for all n ∊ N \(\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +....+\frac { 1 }{ 2^{ n } } =1-\frac { 1 }{ 2^{ n } } \)

Prove the following by using the principle of mathematical induction for all n ∊ N:a+ar+ar2+.......+ar^n-1 =\(\frac { a({ r }^{ n }-1) }{ r-1 } \)

The angles of a triangle are in A.P. and the greater angle is double the least angle. Find the angles of the triangle in radians.

Find the values of other five trigonometric functions cos x =\(-{1\over2},\) x lies in third quadrant.

Find the values of other five trigonometric functions cot x = \(3\over4\) ,x lies in third quadrant.

If sin \(\theta\) =\(-5\over 13\) and \(\theta\) lies in third quadrant, find the value of sec \(\theta\) + tan \(\theta\).

Prove the following:
\({sin \ x + sin \ 3x\over cos \ x + cos \ 3x}=\tan 2x\)

If f and g be two real function defined by \(f\left( x \right) =\sqrt { x+1 } \)and \(g\left( x \right) =\sqrt { 9-{ x }^{ 2 } } \) .Then, describe each of the following functions. \(\frac { g }{ f } \)

If A = {a,d}, B = {b,c,e} and C = {b,c,f}, then verify that \(A\times (B\cap C)=(A\times B)\cap (A\times C)\)

If   \(f(x)=\log { \left( 1-x \right) } \)   and , \(g(x)=[x]\) then determine each of the following functions. \(\frac { f }{ g } \). Also, find:\(\left( \frac { f }{ g } \right) \left( \frac { 1 }{ 2 } \right) \)

Let f : R \(\rightarrow\) R defined f(x) = 1 - x² for all x \(\in \) R⁺ . Find its domain and range. Also, draw its graph.

Let A={1,2,3,4,5}.Define a relation R from A to A by \(R={ \{ }(X,Y):Y=x-1.x.y\in A\} \)

If X={1,2,3} and n represents any member of X, write the following sets containing all numbers represented by n-1

Let A={3, 6, 12, 15, 18, 21}, B={4, 8, 12, 16, 20}, C={2, 4, 6, 8, 10, 12, 14, 16} and D= {5, 10, 15, 20}. Find A - B

If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {2, 4, 7}, B = {3, 5, 7, 9, 11} and C= {7, 8, 9, 10, 11}, then compute (A\(\cap \)U)\(\cap \)(B\(\cup \)C)

If X = {a,b,c,d} and Y = {f,b,d,g}, then find 
\(X\cap Y\)

Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
(ii) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) 1⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) ¢ ∈ A
(x) ¢⊂ A
(xi) {¢} ⊂ A.

If p(n): "49ⁿ +16ⁿ +k is divisible by 64 for n\(\epsilon \)N" is true, then find the least negative integral value of K.

Prove by the principle of mathematical induction that \({ 3 }^{ n }>{ 2 }^{ n }\) , for all \(n\in N\) .

If P(n): "3.5²ⁿ⁺¹ +2³ⁿ⁺¹ is divisible by  for all n \(\in\) N" is true, then find the value of \(\lambda \).

Prove that \(2n+1 <{ 2 }^{ n }\),for all natural numbers \(n(n\ge 3)\) by using principle of mathematical induction.

Prove by principle of mathematical induction that, the sum of first n natural numbers is \(\frac { n(n+1) }{ 2 } \)

Find the real values of x and y, if (x⁴ + 2xi) - (3x² + iy) = (3 - 5i) + (1 + 2iy).
(i) Firstly, separate real and imaginary parts of both sides.
(ii) Second, equate the real and imaginary parts of both sides and get equations in terms of x and y.
(iii) Further, solve these equations to get the values of x and y.

If z₁ and z₂ are complex numbers, then prove that Re(z₁ z₂) = Re(z₁) Re(z₂) - Im(z₁) Im(z₂).

Express the following in the form a + ib.
\(\left( -i \right) \left( 3i \right) { \left( -\frac { 1 }{ 6 } i \right) }^{ 3 }\)

Simplify each of the following and put it in the form a + ib.
\({ \left( \frac { 1 }{ 3 } +3i \right) }^{ 3 }\)

Simplify each of the following and put it in the form a + ib.\(\left( 3+\sqrt { -5 } \right) \left( 3-\sqrt { -5 } \right) \)

Convert the following into radians.
520^o

Express the following in radians.
40^o 20'

If cosec \(A=\frac { x }{ y } \), then find the value of cot A.

Prove the following identities.
\(\frac { tan\theta +sec\theta -1 }{ tan\theta -sec\theta +1 } =\frac { 1+sin\theta }{ cos\theta } \)

Prove that \(\frac { { cos15 }^{ 0 }+{ sin15 }^{ 0 } }{ { cos15 }^{ 0 }-{ sin15 }^{ 0 } } =\sqrt { 3 } \)

If A = {a,b} and B = {2,3}, then find the number of relations from A to B.
Number of relations from A to B \(={ 2 }^{ n\left( A \right) \times n\left( B \right) }={ 2 }^{ n\left( A\times B \right) }\)

If n(A) = 3 and B = {2,3,4,6,7,8}, then find the number of relations from A to B.

If A = {1,2}, then find A x A x A

If two functions are defined as \(f(x)=\frac { 1 }{ (x-2) } ,x\neq 2\) and \(g(x)=(x-2)^{ 2 }\) then find  \(\frac { f }{ g } \)

If A x B = {a,1), (b,3), (a,3), (b,1), (a,2), (b,2)}. Then, find A and B.

Let A={1,2,{3,4},5}.Which of the following statements are incorrect and why?
{3,4}\(\subset\)A

If n(A) = 4, n(B) = 5, n(U) = 7 and n(A\(\cap \)B) = 2, then find the value of n(A\(\cup \)B)'

In a class of 60 students, 25 students play cricket, 20 students play Tennis and 10 students play both the gmes. Then, find the number of students who play neither games.

Describe the following set in Roster form. {X: X is positive integer and a divisor of 9}

Find the symmetric difference of sets A={1,3,5,6,7} and B ={3,7,8,9}

Five marbles are drawn from a bag which contains 7 blue marbles and 4 black marbles. What is the probability that 3 will be blue and 2 black?

If A and B are two events associated with a random experiment such that P(A)=0.3.P(B)=0.2 and \(P(A\cap B)=0.1\) ,then find the value of \(P(A\cap B)\)

What is the probability of drawing a 'king' from a well-shuffled deck of 52 cards?

Consider the following experiment of rolling a die.Let A be the event 'getting a prime number' and B be the event'getting an odd number'.Write the sets representing events
A or B

Two dice are rolled. Let E_1, E_2, and E₃ be the events of getting a sum of 4, 5 and respectively.
Which events are elementary events?

The marks obtained by 7 students are 8,9,11 ,13,14,15,21.Find the variance and standard deviation of these marks.

Two plants A and B of a factory show results about the number of workers and wages paid to them.

Which plants A or B is the greater variability in individual wages?

Find the variance of the data 6,5,9,13,12,8 and 10.

Find the standard deviation of first 10 natural numbers

Find the mean deviation about the mean for the following data
36,72,46,60,45,42,53,49,51,46

Check the validity of the following statements
p :20 is a multiple of 4 and 5.

Check the validity of the following statements
r :60 is a multiple of 3 or 5.

By giving a counter example, show that the following statement is false. If x is even integer, then x is a prime.

Using contrapositive method, prove that if n² is an even integer, then n is also an even integer.

Find the component statement of the following compound statements and check whether they are true or false.
100 is divisible by 3,11 and 5.

Evaluate the following limits
\(\lim _{x \rightarrow \sqrt{2}} \frac{x^{4}-4}{x^{2}+3 \sqrt{2} x-8}\)

Evaluate the following limits

\(\lim_ { x\rightarrow 3 }{ lim } \frac { { x }^{ 4 }-81 }{ { 2x }^{ 2 }-5x-3 } \)

Find the derivative of x² – 2 at x = 10.

Find the derivative of 99x at x = 100.

Find the derivate of the following functions from first principle
x^2/3

Show that the three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB

Verify that A(-1, 2, 1) B(1, -2, 5), C(4, -7, 8) and D(2, -3, 4) are the verticles of a parallelofram

Find the ratio in which the line segment joining the points (4, 4, -10) and (-2, 2, 4) is divided by the YZ-plane.

Find the locus of a point which moves such that the sum of its distance from points \(A(0,0,-\propto )\) and \(B(0,0,\propto )\) is constant.

Show that the points (0,4,1), (2,3,-1), (4,5,0) and (2,6,2) are the vertices of a square.

Find the equation of ellipse, if foci are \((\pm 5,0)\) and a=6.

Find the equation of the ellipse, if length of major axis is 22 and foci \((\pm 3,0)\)

Find the equation of the ellipse, where distance between directices is 8 and distance between foci is 2

Find the equation of the circle whose centre is (2,-3) and which passes through the intersection of the lines 3x + 2y + 11 and 2x + 3y = 4

Find the equation of a circle concentric with the circle 2x +2y+8x+10y - 39 = 0 and having its area equals to 16\(\\ \pi \) sq units.

Reduce the following equation into slope intercept form and find their slopes and the y-intercepts y=0.

A line passes through the points A(4,-6) and B(-2,-5). Show that the line AB makes an obtuse angle with the X-axis.

Find the equation of the straight line which bisects the distance between the points A(a, b), B(a^', b^') and also bisects the distance between the point C (- a, b) and D(a^', - b^').

Find the values of \(\theta \)  and p, if the equation xcos \(\theta \) +ysin \(\theta \) =p is the normal form of the line \(​​\sqrt { 3x } \)+y+2=0

If the angle between two lines is \(\pi\over 4\) and slope of one of the lines is 2. Find slope of another line.

In an AP, the pth term is q and the (p+q)th term is 0.Then , find the qth term.

the income of a person is 300000, in the first year and he receives an case of 10000 his income per year for the next 19years .Find the total amount,he received in 20 years.

The 6th and 17th terms of an AP are 19 and 41 respectively, find the 40th term.

Write the first five terms of the sequence whose nth term is 2n² + 3.

Divide 32 into four parts which are in AP. such that the product of extremes is to the product of means is 7:15.

Find the coefficient of x³ in the expansion of \(\left( 3x-\frac { 1 }{ x } \right) ^{ 7 }\)

If P and Q are the sum of odd and even terms in the expansion\(\left( y+b \right) ^{ n }\), then prove that\(\left( y+b \right) ^{ 2n }+\left( y-b \right) ^{ 2n }=2\left( { P }^{ 2 }+Q^{ 2 } \right) \).

Using binomial theorem, expand \( \left( { x+ }\frac { 1 }{ y }  \right) ^{ 6 }\)

using bionominal theorem compute (10.1)⁵.

Find the number of terms in the expansion of (1 + 2x + x²)¹⁴

How many 4-digit numbers are there, when a digit may be repeated any number of times?

How many 3 digit numbers more than 600 can be formed by using the digit 2, 3, 4, 6 and 7.

How many words can be formed out of the letters of the word 'ORIENTAL' so that the vowels occupy the odd places?

How many different words can be formed with the letters of the word SUNDAY? How many of these begin with N? How many begin N and end in Y?

A man wants to select a shirt and trousers from a showroom. If there are 20 shirts and 50 trousers in variety, in how many ways can he choose one shirt and one trousers?

Solve 3x + 8> 2, when x is an integer.

Solve the inequalities : 2(2x + 3) -10< 6 (x - 2) for real x.

Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of atleast 60 marks.

In drilling world's deepest hole it was found that the temperature T in degree Celcius x km below the Earth's surface was given by T (x) = 30 + 25(x - 3), where 3 \(\le \) x \(\le \) 15. At what depth will the temperature be between 155^o C and 205^o C?

Solve the inequalities : 3x - 7 > 5x -1

Find the conjugate of (6 + 5i )²

Find the conjugate and modulus of the complex number (3 - 2i) (3 + 2i) (1 + i).

Find the conjugate and modulus of the complex number \(\frac { 2+3i }{ 3+2i } \)

Express the complex number in the form a+ib: 3(7+i7)+i(7+i7)

Express the complex number in the form a+ib: \(\left( \frac { 1 }{ 5 } +\frac { 2 }{ 5 } i \right) -\left( 4+\frac { 5 }{ 2 } i \right) \)

Prove that the sum of first n even numbers is n(n+1).

Prove that 2+4+6+8+.....2=n(n+1).

Prove that 2+6+18+.....2.3^n-1 = (3ⁿ - 1) for all \(n\in N\).

Prove that 1+2+2²+...+2ⁿ = 2ⁿ⁺¹ 1 for all natural numbers n.

Using principle of mathematical induction, prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) .....\left( 1+\frac { 1 }{ n } \right) =n+1\)

Prove that sin(n+1) x sin(n+2) x + cos(n+1) x. cos(n+2) x = cosx

In any triangle, if angle are in the ratio 1:2:3, then find their corresponding sides.

A railroad curve is to be laid out on a circle. What radius should be used , if the track is to change direction by \(30^{0}\) is a distance of 50m

If \(cos\theta +sin\theta =\sqrt { 2 } cos\theta \)then prove that \(cos\theta -sin\theta =\sqrt { 2 } sin\theta \).

Prove that cos 55⁰+cos65⁰+cos75⁰=2cos40⁰cos35⁰

If U = {1,2,3,4} and R = {(x,y) : y > x for all, x, \(y\in U\)}, then find domain and range of R.

Let\(f:\left[ 2,\infty \right] \rightarrow R\) and \(g:\left[ -2,\infty \right] \rightarrow R\)  be two real functions defined by \(f(x)=\sqrt { x-2 } \) and \(g(x)=\sqrt { x+2 } \). Find \(f+g\) and \(f-g\)

Express R = {(x,y) : x²+ y²= 25, where x,y \(\in\) W} as a set of ordered pairs

Determine the domain and range of the relation R, where R = {(-3,1),(-1,1),(1,0),(3,0)}

The given figure shows a relationship between the sets P and Q.Write this relation

in set builder form

If U={a,b,c,d,e,f}, A={a,b,c}, B={c,d,e,f}, C={c,d,e}, D={d,e,f}, then tabulate the following set A\(\cup \phi \)

In a town with a population of 5000, 3200 people are egg eaters, 2500 meat-eaters and 1500 eat both egg and meat.How many are pure vegetarians?

The members of a group of 400 people speak either Hindi or English or both.If 270 speak Hindi only and 50 speak both Hindi and English, then how many of them speak English only?

Consider the following sets  \(\phi\) , A = {2,5}, B = {1,2,3,4} and C = {1,2,3,4,5} Insert the correct symbol  \(\subset\)  or \(\nsubseteq\)  between the pair set 
A....... B

From the sets given below, select empty set, singleton set, infinite set and equal sets.
D = {2, 4, 6, 8, 10}

A box contains 1 red and 3 black balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

What is the probability that a randomly chosen two-digit positive integer is multiple of 3?

Five marbles are drawn from a bag which contains 7 blue marbles and 4 black marbles. What is the probability that 3 will be blue and 2 black?

Consider the following experiment of rolling a die.Let A be the event 'getting a prime number' and B be the event'getting an odd number'.Write the sets representing events
A but not B

Find the probability that in a random arrangement of the letters of the word 'SOCIAL' vowels come together.

Find the variance of the data 6,5,9,13,12,8 and 10.

Find the standard deviation of first 10 natural numbers

Find the mean, variance and standard deviation of the following data: 62, 65, 57, 56, 69, 51, 62, 60

Find the mean, variance and standard deviation of the following data: 15, 22, 27, 14, 9, 9, 11, 21

Let a,b,c,d and e be the observations with mean m and standard deviation S.Then , find the standard deviation of the observations a+k,d+k,c+k,d+k,e+k.

Given below are two statements
p :100 is a multiple of 5.
q :100 is a multiple of 4.
Write the compound statement connecting these two statements with 'and' and check its validity.

Check the validity of the following statements
p :20 is a multiple of 4 and 5.

By giving a counter example, show that the following statement is false. If x is even integer, then x is a prime.

Using contrapositive method, prove that if n² is an even integer, then n is also an even integer.

Identify the component statements and the connecting word in in the following compound statements.
(i) It is raining or the Sun is shining|
(ii) All primes are either even or odd
(iii) Two lines intersect at a point or they are parallel.

Evaluate the following limit \(\lim_ { x\rightarrow 0 }{ lim } \frac { tanx-sinx }{ x } \)

Evaluate the following limit \(\lim_ { x\rightarrow 0 }{ lim } \frac { tan2x-x }{ 3x-sinx } \)

Evaluate the following limit \(\lim_ { x\rightarrow 0 }{ lim } \frac { { x }^{ 2 }cosx }{ 1-cosx } \)

Evaluate the following limits : \(\lim_{ x\rightarrow 1 }{ lim } \frac { { x }^{ m }-1 }{ { x }^{ n }-1 } \)

Let us consider two functions \(f(x)={ x }^{ 2 }+4\)and g(x) = x-3 such that f(x) and g(x) exist at x=5. Find the limit of the following fuctions at x=5.
f(x) X g(x)

Verify that A(-1, 2, 1) B(1, -2, 5), C(4, -7, 8) and D(2, -3, 4) are the verticles of a parallelofram

Find the ratio in which the line segment joining the points (4, 4, -10) and (-2, 2, 4) is divided by the YZ-plane.

Prove that the triangle formed by joining the three points whose coordinates are A(1,2,3), B(2,3,1) and C(3,1,2), is an equilateral triangle.

Show that D(-1,4,-3) is the circumcentre of \(\Delta ABC\) with vertices A(3,2,-5), B(-3,8,-5) and C(-3,2,1).

A point is on the x-axis. What are its y-coordinate and z-coordinates?

Find the equation of the ellipse with foci at \((\pm 5,0)\) and x=\(\frac { 36 }{ 5 } \) as one of the directrices.

Find the eccentricity of the hyperbola, the length of whose conjugate axis is 3/4 of the length of transverse axis.

Find the equation of the circle whose centre is (2,-3) and which passes through the intersection of the lines 3x + 2y + 11 and 2x + 3y = 4

If lx +my = 1 touches the circle x²+y² = a², then prove that the point (l,m) lies on the circle x²+y²=a^-2.

Find the equation of the circle with circle=(2,3) and radius=5

A line passes through P(1, 2) such that its intercept between the axes is bisected at P. Find the equation of line.

Using slopes, show that the points A(-4,-1), B(-2,-4), C(4,0) and D(2,3) taken in order, are the vertices of a rectangle.

Find the equation of a line with slope -4 and cutting off an intercept of 4 units on negative direction of y-axis.

Find the length of the perpendicular from the point (2, -5) to the line joining the points (1, 4) and (3, 8).

Find the new coordinates of point (3,-4), if the origin is shifted to (1,2) by a translation.

If the sum of n terms of an AP. is 4n² + 7n, then its nth term is ______.

The three geometric means between the numbers 1 and 81 are ______.

In a G.P. if the (m + n)^th terms is p and (m - n)^th terms is q then its mth terms is ______.

If the sum of first n even natural numbers is equal to m times the sum of first n is odd natural numbers then m is equal to ______.

Sum of n terms of ,the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\)..... is ______.

Find the middle term(s) in the given expansion.
\(\left( \frac { x }{ 3 } +9y \right) ^{ 10 }\)

If P and Q are the sum of odd and even terms in the expansion\(\left( y+b \right) ^{ n }\), then prove that\(\left( y+b \right) ^{ 2n }+\left( y-b \right) ^{ 2n }=2\left( { P }^{ 2 }+Q^{ 2 } \right) \).

Find the coefficient of x⁴ in the expansion of (1+x)ⁿ (1-x)ⁿ

Expand the following expansions.
(9-6x)^-3/2

Using binomial theorem, expand the following expressions.
\((2x+3y)^{ 5 }\)

How many 2 digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the digits can be repeated?

How many 4-digit numbers are there, when a digit may be repeated any number of times?

Find the number of triangles thet are formed by choosing the vertices from a set of 10 points 6 of which lie on the same line

How many different signals can be made by 5 flags from 8 flags of different colours?

Find x, if \({1\over 7!}+{1\over 8!}={x\over 9!}\)

Solve 3x + 8> 2, when x is an integer.

Solve \(|3-4x|\ge 9\).

In drilling world's deepest hole it was found that the temperature T in degree Celcius x km below the Earth's surface was given by T (x) = 30 + 25(x - 3), where 3 \(\le \) x \(\le \) 15. At what depth will the temperature be between 155^o C and 205^o C?

Solve the linear inequality 3x - 5 < x + 7, when x is a whole number

A person was not feeling well, so he went to a doctor. Doctor on examination found that his temperature varies between 30^oC to 35^oC.What is the range of temperature in degree Fahrenheit?  Do you think his temperature is normal? If not, what is normal temperature of body in Fahrenheit? Does he need medical attention?
Use conversion formula, F = \(\frac { 9 }{ 5 } \) C + 32

Find the conjugate of the complex number \(\frac { 1-i }{ 1+i }\)

Express \(\frac { 1 }{ 1-cos\theta +2isin\theta } \) in the form a+ib.

Simplify the following
i^-35

Find the value of \(\sqrt { -25 } +3\sqrt { -4 } +2\sqrt { -9 } \)

Express \(\frac { \left( 3+\sqrt { 5i } \right) \left( 3-\sqrt { 5i } \right) }{ \left( \sqrt { 3 } +\sqrt { 2i } \right) -\left( \sqrt { 3 } -\sqrt { 2i } \right) } \)in the form of a+ib

Prove that the sum of first n even numbers is n(n+1).

Using principle of mathematical induction, prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) .....\left( 1+\frac { 1 }{ n } \right) =n+1\)

Prove that \(\sum _{ t=1 }^{ n-1 }{ t(t+1) } =\frac { n(n-1)(n+1) }{ 3 } \) , for all natural numbers \(n\ge 2\)

Prove that 2ⁿ>n for all positive integers n.

If p(n): "49ⁿ +16ⁿ +k is divisible by 64 for n\(\epsilon \)N" is true, then find the least negative integral value of K.

In a ∆ABC, if ∠A = 45^o, ∠B = 60^o and ∠C = 75^o, then the ratio of sides is ______.

In a ΔABC if a = 5, b = 6 and c = 5, then ∠B is ______.

sin⁶\(\theta\) + cos⁶\(\theta\) + 3 sin²\(\theta\) cos²\(\theta\) is equal to ______.

If tan \(\theta={\alpha\over \alpha+1}\) and tan \(\phi ={1\over 2\alpha+1}\) , then (A+ B) is equal to ______.

If the set A has m elements, B has n elements then the number of elements in A x B is ______.

Let R be a relation from a set A to B, then ______.

If f(x) = log\({1+x\over 1-x}\) and g(x) = \({3x+x^3\over 1+3x^2}\) then f(g(x)) is equal to ______.

If f(x) =\({x+1\over x-1}\) is a real function,\(\neq\) 1, then \(f[f\{f(2)\}]\) is ______.

If f: R\(\rightarrow\) R be given by f(x) =\({4^x\over 4^x+2}\) for all x \(\in\) R, then ______.