By taking suitable sets A, B, C, verify the following results:
C-(B-A) = (C\(\cap \) A) \(\cup \) (C\(\cap \)B')

Discuss the following relations for reflexivity, symmetricity and transitivity:
Let A be the set consisting of all the members of a family. The relation R defined by "aRb if a is not a sister of b".

Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

Draw the curves of
(i) y = x² + 1
(ii) Y = (x + 1)² by using the graph of curve y = x.

Let A = {0,1, 2, 3}. Construct relations on A of the following types:
(i) reflexive, not symmetric, not transitive.
(ii) reflexive, not symmetric, transitive.

Simplify and hence find the value of n: \(3^{2 n} 9^{2} 3^{-n} / 3^{3 n}=27\)

Solve \(\frac { 1 }{ \left| 2x-1 \right| } <6\) and express the solution using the interval notation.

Represent the following inequalities in the interval notation:
\(-2x>0\) or \(3x-4<11\)

Discuss the nature of roots of -x² + 3x + 1 = 0

Resolve into partial function \(\frac{2}{x^2-1}\).

Identify the quadrant in which an angle of each given measure lies; 328⁰

Find the principal value of cosec^-1(-1)

Find the degree measure corresponding to the following radian measure; \(\frac { 2\pi }{ 5 } \)

Find the degree measure corresponding to the following radian measure; \(\frac { 7\pi }{ 3 } \)

Show that tan (45^o + A) = \(\frac { 1+\tan { A } }{ 1-\tan { A } } \)

Find the principal solution and general solutions of the following cot\(\theta\) = \(\sqrt { 3 } \)

Find the values of other five trigonometric functions for the following
Sec \(\theta\) = \(\frac { 13 }{ 5 },\) \(\theta\) lies in the IV quadrant

Evaluate sin\(\left( \frac { -11\pi }{ 3 } \right) \).

Express each of the following as a product.
sin 75^o - sin 35^o

Express each of the following as a product.
cos 65^o + cos 15^o

Convert : 18° to radians.

Find the principal value of sin^-1\(({\sqrt{3}\over2})\)

Find the principal value of cosec^-1\(({2\over\sqrt{3}})\)

Simplify: cos A + cos (120° + A) + cos (120° - A)

Find the values of cos(300°).

Find the number of ways of distributing 12 distinct prizes to 10 students?

Evaluate \(\frac { n! }{ r!(n-r)! } \) For any n when r = 2

If ^(n-1)P₃ :ⁿ P₄ = 1 : 10, find n

If \(\frac { 6! }{ n! } \) = 6, then find the value of n.

Evaluate: ⁵P₃.

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has atleast one boy and one girl

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

Write the first 6 terms of the sequences whose n^th term a_n given below
\({ a }_{ n }=\begin{cases} n+1\quad if\quad n\quad is\quad odd \\ n\quad \quad if\quad n\quad is\quad even \end{cases}\)

If a, b, c are in A.P., show that (a-c)² = 4(b² - ac).

Find the middle term in the expansion of (x +y)⁶.

Find the equation of the line perpendicular to x-axis and having intercept -2 on x-axis.

If p is the length of the perpendicular from the origin to the line \(\frac{x}{a}+\frac{y}{b}=1\), then prove that \(\frac{1}{p_2}=\frac{1}{a^2}+\frac{1}{b^2}\)

Find the equation of the line through (1, 2) and which is perpendicular to the line joining (2, -3) (-1, 5)..

Find the combined equation of the straight lines through the origin one of which is parallel to and the other is perpendicular to the straight line 3x + y + 5 = 0.

A line passing through the points (a, 2a) and (-2, 3) is perpendicular to the line 4x+3y+ 5 = 0, find the value of a.

Suppose that a matrix has 12 elements. What are the possible orders it can have? What if it has 7 elements?

Evaluate :\(\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}\) 

Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3).

Determine the values of a so that the following matrices are singular: A =\(\begin{bmatrix} 7& 3 \\ -2 & a \end{bmatrix}\)

Determine the values of b so that the following matrices are singular:\(\begin{bmatrix}b-1 &2 &3 \\3 & 1 & 2 \\ 1 & -2 &4 \end{bmatrix}\)

For what value of x the matrix A =\(\left[ \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right] \)  is singular.

Find the direction cosines of a vector whose direction ratios are 3, -1, 3

Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\) - 3\(\hat{k}\) + 4\(\hat{j}\)

Write two different vectors having same magnitude.

Show that the vectors \(2\hat{i}-3\hat{j}+4\hat{k}\) are \({-4\hat{i}+6\hat{j}-8\hat{k}}\) are collinear.

If \(\vec { P } =-3\vec { i } +4\vec { j } -7\vec { k } \) and \(\vec { q } =6\vec { i } +2\vec { j } -3\vec { k } \) then find \(\vec { p } \times \vec { q } \) .Verify that \(\vec { p } \) and \(\vec { p } \times \vec { q } \) are perpendicular to each other and also verify that \(\vec { q } \) and \(\vec { p } \times \vec { q } \) are perpendicular to each other.

Complete the table using calculator and use the result to estimate the limit.
\(lim_{x\rightarrow{2}}{x-2\over x^2-4}\)

In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow{-3}}{\sqrt{1-x}-2\over x+3}\)

Compute\(lim_{x\rightarrow-2}(-{3\over 2}x)\)

Compute \(lim_{x\rightarrow{0}}[{x^2+x\over x}+4x^3+3]\) .

Evaluate the following limits :\(lim_{x\rightarrow \infty}(1+{1\over x})^{7x} \)

State how continuity is destroyed at x = x _o for each of the following graphs.

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { { e }^{ 5x }-1 }{ x } \)

If \(f(x)=\{ \begin{matrix} 2x+3, & x\le 0 \\ 3(x+1), & x>0 \end{matrix}\) .Find \(\underset { x\rightarrow 0 }{ lim } \underset { x\rightarrow 1 }{ lim } f(x)\) and \(\underset { x\rightarrow 1 }{ lim } f(x)\)

Differentiate the following with respect to x : \(y={cos \ x \over x^3}\)

Find the derivatives of the following functions with respect to corresponding independent variables : \(y={tan \ x \over x}\) .

Differentiate 2^x.

Differentiate the following: y = 4 sec 5x

Find the derivation : sin 5 + log₁₀ x + 2 sec x

Integrate the following with respect to x : \({x^{24}\over x^{25}}\)

Integrate the following with respect to x : \({1\over (5-4x)}\)

Integrate the following with respect to x : \(e^{2x}-1\over e^{x}\)

Integrate the following with respect to x : e^2x sin x

Integrate the function with respect to x : \(\sqrt { 169-\left( 3x+1 \right) ^{ 2 } } \)

If an experiment has exactly the three possible mutually exclusive outcomes A, B, and C, check in each case whether the assignment of probability is permissible.
P(A) = 0.421, P(B) = 0.527  P(C) = 0.042

A single card is drawn from a pack of 52 cards. What is the probability that
The card will be 6 or smaller?

A single card is drawn from a pack of 52 cards. What is the probability that
The card is either a queen or 9?

If \(P(A)=0.6, P(B)=0.5\), and \(P(A \cap B)=0.2\) Find \( P(\bar{A} / B)\)

Nine coins are tossed once, find the probability to get at least two heads

Events A and B are such that P(A) = \(\frac { 1 }{ 2 } \) , P(B) = \(\frac { 7 }{ 12 } \) and P(not A or not B) = \(\frac { 1 }{ 4 } \). State whether A and B are independent? 

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1) (y, 2) (z, 1) are in A\(\times\)B, find A and B, where x, y, z are distinct elements.

Find the quotient of the identity function by the modulus function

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is reflexive

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is transitive

On the set of natural number let R be the relation defined by aRb if a + b \(\le\) 6. Write down the relation by listing all the pairs. Check whether it is symmetric

If \({{a}\over{x}}+{{y}\over{b}}=1\) and show that \({{b}\over{y}}+{{z}\over{c}}=1.\)

Determine the region in the Plane determined by the inequalities.
\(3x+5y\ge 45,\ x\ge 0,\ y\ge 0\)

If  \(\frac { log \ x }{ y-z } =\frac { log \ y }{ z-x } =\frac { log \ z }{ x-y } \) , then prove that xyz = 1

Find the real roots of x⁴ = 16

Our monthly electricity bill contains a basic charge, that is independent of units consumed and a charge that depends on the units consumed. Let us say Electricity board charges Rs. 110 as basic charge and charges Rs. 4 for each unit we use. If a person wants to keep his electricity bill below Rs. 250, then what should be his electricity usage?

Solve x = \(\sqrt{x+20}\) for x ∈ R

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 cos (A - B)

Find cos(x - y), given that cos x = \(-\frac{4}{5}\) with \(\pi<x<{{3\pi}\over{2}}\) and \(sin \ y = -{{24}\over{25}}\) with \(\pi<x<{{3\pi}\over{2}}\). 

For each given Angle, find a coterminal angle with a measure of \(\theta\) such that \(0^o\le \theta \le 360°\) 
1150⁰ 

For each given Angle, find a coterminal angle with a measure of \(\theta\) such that \(0^o\le \theta \le 360°\) 
-450⁰ 

If \(\theta\) is an acute angle, then find \(\sin { \left( \frac { \pi }{ 4 } -\frac { \theta }{ 2 } \right) } \), when \(\sin { \theta } =\frac { 1 }{ 25 } \)

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 \(\left( \frac { B-C }{ 2 } \right) \)

The angles of a triangle ABC, are in arithmetic progression and if b:c = \(\sqrt { 3 } :\sqrt { 2 } \) , find \(\angle A.\)

Two slopes leave a port at the same time one goes 24 km/hr in the direction N 45^o E and other travels 32 km/hr in the direction S 75^o E. Find the distance between the ships at the end of 3 hours.

Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 15°.

Let \(\alpha,\beta\) be such that \(\pi<\alpha-\beta<3\pi.\)If \(sin\alpha+sin\beta=-\frac{21}{65}\ and\ cos\alpha+cos\beta=-\frac{27}{65}\) then find the value of \(cos\frac{\alpha-\beta}{2}\) is

In ∆ABC, if tan \(\frac{A}{2}=\frac{5}{6}\) and tan \(\frac{C}{2}=\frac{2}{5}\), then show that a, b, c, are in A.P.

How many strings can be formed from the letters of the word ARTICLE, so that vowels occupy the even Places?

If ⁿP_r = 720. If ⁿC_r = 120, find n, r = ?

A coin is tossed 8 times,
(i) How many different sequences of heads and tails are possible?
(ii) How many different sequences containing six heads and two tails are possible?

If \({n!\over 3!(n-4)!}and {n!\over 5!(n-5)!}\) are in the ratio 5 : 3 find the value of n.

Compute 99⁴

Write the first 4 terms of the logarithmic series of log (1 - 2x). Find the intervals on which the expansions are valid

If the sum of the coefficients in the expansion of (x+y)ⁿ is 4096. Then find the greatest coefficient in the expansion.

Find the value of k and b, if the points P(-3, 1) and Q(2, b) lie on the locus of x² - 5x + ky = 0.

If the points P(6, 2) and Q(-2, 1) and R are the vertices of a Δ PQR and R is the point on the locus of y = x²- 3x + 4, then find the equation of the locus of centroid of Δ PQR

A straight line cuts intercepts from the axes of co-ordinates the sum of whose reciprocals is a constant. Show that it always passes through a fixed point.

If θ is a parameter, find the equation of the locus of a moving point, whose coordinates are (a sec θ, b tan θ)

Find the equation of the line which passes through the point (- 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5: 3 by this point.

A fruit shop keeper prepares 3 different varieties of gift packages. Pack-I contains 6 apples, 3 oranges, and 3 pomegranates. Pack-II contains 5 apples, 4 oranges and 4 pomegranates and Pack –III contains 6 apples, 6 oranges and 6 pomegranates. The cost of an apple, an orange and a pomegranate respectively are Rs. 30, Rs. 15 and Rs. 45. What is the cost of preparing each package of fruits?

Without expanding the determinant, prove that \(\begin{vmatrix} s & a^2 & b^2+c^2 \\ s & b^2 &c^2+a^2 \\ s & c^2 & a^2+b^2 \end{vmatrix}=0\)

Write the general form of a 3 \(\times\) 3 skew-symmetric matrix and prove that its determinant is 0.

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

Let \(\vec a\) and \(\vec b\) be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are \(​​​​\frac{\vec{a}+2 \vec{b}}{3} \text { and } \frac{\vec{b}+2 \vec{a}}{3} \text {. }\)

For any vector \(\overrightarrow{r}\) prove that \(\overrightarrow{r}\) = (\(\overrightarrow{r}.\hat{i}\)) \(\hat{i}\) + (\(\overrightarrow{r}.\hat{j}\)) \(\hat{j}\) + (\(\overrightarrow{r}.\hat{k}\)) \(\hat{k}\).

The vertices of a triangle have position vectors \(4\hat { i } +5\hat { j } +6\hat { k } ,5\hat { i } +6\hat { j } +4\hat { k } ,6\hat { i } +4\hat { j } +5\hat { k } \) Prove that the triangle is equilateral.

Find the relation between a and b if \(lim_{x\rightarrow3}f(x)\) exists where \(f(x)= \begin{cases}a x+b & \text { if } x>3 \\ 3 a x-4 b+1 & \text { if } x<3\end{cases}\)

Evaluate the following limits :
\(lim_{x\rightarrow2}{{1\over x}-{1\over2}\over x-2}\)

Evaluate the following limits \(lim_{x\rightarrow{3}}{x^2-9\over x^2(x^2-6x+9)}\)

Evaluate the following limits :\(\)\(lim_{x \rightarrow \infty}\{ x[log(x+a)-log(x)]\}\)

Find the points of discontinuity of the function f, where
f(x) = {\(\begin{matrix} x+2, & if\quad x\ge 2 \\ { x }^{ 2 }, & if\quad x<2 \end{matrix}\)

Evaluate \(\lim _{ x\rightarrow \pi }{ \frac { \sin { x } }{ x-\pi } } \)

Let \(f\left( x \right) =\begin{cases} 2x,\quad x<2 \\ 2,\quad x=2 \\ { x }^{ 2 },\quad x>2 \end{cases}.\) Prove that 2 is a removable discontinuity of f.

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
\(f(x)=\sqrt{1-x^2}\)

Find the derivatives of the following functions with respect to corresponding independent variables : y = x sin x cos x

Find f '(x) if f(x) = \({1\over 3\sqrt{x^2+x+1}}\)

Differentiate the following: \(y=5^{\frac{-1}{x}}\)

Find \({dy\over dx}\) if x² + y² = 1

If x = \(a\sec ^{ 3 }{ \theta }\) and \(y=a\tan ^{ 3 }{ \theta }\) find \(\frac { dy }{ dx }\) at \(\theta =\frac { \pi }{ 3 }\)

Integrate the following functions with respect to x : \(e^{xlog a}e^x\)

Integrate the following with respect to x : \({x^2\over 1+x^6}\)

Evaluate the following integrals : \(\int e^{-5x}sin 3x \ dx\)

Evaluate the following integrals : \(\int {1\over x^2-2x+5}dx\)

Integrate the following functions with respect to x : \(\sqrt{81+(2x+1)^2}\)

Evaluate \(\int { \frac { { sin }^{ 6 }x+cos^{ 6 }x }{ sin^{ 2 }xcos^{ 2 }x } } \)

If A and B are two independent events such that P(A\(\cup \)B) = 0.6, P(A) = 0.2,  find P(B).

A main road in a City has 4 crossroads with traffic lights. Each traffic light opens or closes the traffic with the probability of 0.4 and 0.6 respectively. Determine the probability of
(i) a car crossing the first crossroad without stopping
(ii) a car crossing first two crossroads without stopping
(iii) a car crossing all the crossroads, stopping at third cross.
(iv) a car crossing all the crossroads, stopping at exactly one cross.

Let the matrix M = \(\left[ \begin{matrix} x & y \\ z & 1 \end{matrix} \right] \), If x,y and z are chosen at random from the set {1, 2,3, } and repetition is allowed (i.e., x = y = z ), what is the probability that the given matrix M is a singular matrix?

A town has 2 fire engines operating independently. The probability that a fire engine is available when needed is 0.96.
(i) What is the probability that a fire engine is available when needed?
(ii) What is the probability that neither is available when needed?

For a sports meet, a winners’ stand comprising of three wooden blocks is in the form as shown in figure. There are six different colours available to choose from and three of the wooden blocks is to be painted such that no two of them has the same colour. Find the probability that the smallest block is to be painted in red, where red is one of the six colours.

A man has 2 ten rupee notes, 4 hundred rupee notes and 6 five hundred rupee notes in his pocket. If 2 notes are taken at random, what are the odds in favour of both notes being of hundred rupee denomination and also its probability?

A persons has undertaken a construction job. The probabilities are 0.65, that there will be strike, 0.80 that the construction job will be completed on time, if there is no strike and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time.

For the given curve, \(y=x^{1\over 3}\)given in  figure draw
(i) \(y=-x^{ \left( \frac { 1 }{ 3 } \right) }\)
(ii) ​\(y=x^{ \left( \frac { 1 }{ 3 } \right) }+1\)
(iii)​ \(y=x^{ \left( \frac { 1 }{ 3 } \right) }-1\)
(iii) ​\(y=(x+1)^{1\over 3}\)​

Write the values of f at -4, 1, -2, 7, 0 if
\(f(x)=\left\{ \begin{matrix} -x+4& if -\infty <x\leq -3\\ x+4& if -3<x<-2\\ x^{2}-x& if -2\leq x < 1 \\ x-x^{2}& if 1\leq x<7\\ 0& otherwise\\ \end{matrix}\right.\)

Find the sum and difference of the identity function and the modulus function?

Write the values of f at -3, 5, 2, -1, 0 if
\(f(x)=\begin{cases} x^2+x-5\quad if\ x \in(-\infty, 0) \\x^2+3x-2\quad if\ x\in(3,\infty) \\x^2\quad \quad \quad \quad \quad if\ x\ \in(0,2) \\x^2-3 \quad \quad \quad otherwise \end{cases}\)

If f:R \(\rightarrow\) R is defined by f(x) = 3x - 5, prove that f is a bijection and find its inverse.

Solve : \({ log }_{ 2 }x-3{ log }_{ \frac { 1 }{ 2 } }x=6\)

Find all values of x for which \({{x^3(x-1)}\over{x-2}}>0.\)

Resolve the following rational expressions into partial fractions.
\({{x}\over{(x^2+1)(x-1)(x+2)}}\)

Find the condition that one of the roots of ax²+ bx + c may be thrice the other.

Find the condition that one of the roots of ax²+bx+c may be reciprocal of the other.

Forensic Scientists use h = 61.4+2.3F to predict the height h in centimeters for a female whose thigh bone (femur) measures F cm. If the height of the female lies between 160 to 170 cm find the range of values for the length of the thigh bone?

A plane is 1 km from one landmark and 2 km from another. From the planes point of view the land between them subtends an angle of 45⁰. How far apart are the land marks?

Prove that \(sin\frac { \theta }{ 2 } sin\frac { 7\theta }{ 2 } +sin\frac { 3\theta }{ 2 } sin\frac { 11\theta }{ 2 } =sin2\theta sin5\theta \)

If A + B + C = 180⁰, prove that \(sinA+sinB+sinC=4cos\frac { A }{ 2 } cos\frac { B }{ 2 } cos\frac { C }{ 2 } \)

If A + B + C = 180⁰, prove that sin(B + C - A) + sin(C + A - B) + sin(A + B - C)= 4 sin A sin B sin C

Find the values of \(\tan { \left( \alpha +\beta \right) } \), given that \(\cot { \alpha } =\frac { 1 }{ 2 } ,\alpha \epsilon \left( \pi ,\frac { 3\pi }{ 2 } \right) and \ sec { \beta } =-\frac { 5 }{ 3 } ,\beta \epsilon \left( \frac { \pi }{ 2 } ,\pi \right) \)

Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the eastern direction are 30⁰ and 45⁰ respectively. If A and B stand 5 km apart, find the distance of the intruder from B

Express \( tan ^{ -1 }{ \left( \frac { \cos { x } }{ 1-\sin { x } } \right) } ,\frac { \pi }{ 2 }\) in the simplest form.

Find the values of sin 72°.

In a triangle ABC, prove that \({a^2+b^2\over a ^2+c^2}={1+cos(A-B)cosC\over1+cos(A-C)cosB}\)

Find x from the equation cosec (90° + A) + x cos A cot (90° + A) = sin (90° + A).

Using the mathematical induction, show that for any natural number n,
\({1\over 1.2.3}+{1\over 2.3.4}+{1\over 3.4.5}+..+{1\over n(n+1)(n+2)}+={n(n+3)\over 4(n+1)(n+2)}\)

Prove that ²ⁿC_{n =}  \(\frac { { 2 }^{ n }\times 1\times3\times ...(2n-1) }{ n! } \)

Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee.

By the principle of mathematical induction, prove that, for all integers n ≥ 1,
1² + 2² + 3²+...n² = \(\frac { n(n+1)(2n+1) }{ 6 } \).

2n < (n + 2)! for all natural number n.

Compute the sum of first n terms of the following series 6 + 66 + 666 + .......

Prove that \(\sqrt [ 3 ]{ { x }^{ 3 }+6 } -\sqrt [ 3 ]{ { x }^{ 3 }+3 } \) is approximately equal to \(\frac { 1 }{ { x }^{ 2 } } \) when x is sufficiently large.

If p - q is small compared to either p or q, then show that \(n\sqrt { \frac { p }{ q } } =\frac { \left( n+1 \right) p+\left( n-1 \right) q }{ \left( n-1 \right) p+\left( n+1 \right) q } \)
Hence find \(8\sqrt { \frac { 15 }{ 16 } } \)

Find the coefficient of x in the expansion of \(log(\frac{1}{1-5x+6x^2})\).

Find \(\sum_{k=1}^{n}{1\over k(k+1)}.\)

If R is any point on the x - axis and Q is any point on the y- axis and P is a variable point on RQ with RP = b, PQ = a. then find the equation of locus of P

A family is using Liquefied petroleum gas (LPG) of weight 14.2 kg for consumption. (Full weight 29.5kg includes the empty cylinders tare weight of 15.3kg.). If it is use with constant rate then it lasts for 24 days. Then the new cylinder is replaced.
(i) Find the equation relating the quantity of gas in the cylinder to the days.
(ii) Draw the graph for first 96 days.

The slope of one of the straight lines ax² + 2hxy + by² = 0 is three times the other, show that 3h² = 4ab.

If the equation λx² - 10xy + 12y²+ 5x -16y - 3 = 0 represents a pair of straight lines, find
(i) the value of λ and the separate equations of the lines
(ii) point of intersection of the lines
(iii) angle between the lines.

Find the equation of the line passing through the point of intersection 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x +4y = 7.

Express the equation √3x - y + 4 = 0 in the following equivalent form Intercept form,

Show that f(x) f(y) = f(x + y), where f(x) =\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

Prove that \(\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix}=4a^2b^2c^2\)

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of \(\begin{vmatrix} a & b & c \\ p & q & r \\ 1& 1 &1 \end{vmatrix}\)

If a, b, c are all positive and are p^th, q^th and r^th terms of a G.P., show that \(\begin{vmatrix} log \ a & p & 1 \\ log\ b & q & 1 \\ log\ c & r & 1 \end{vmatrix}=0.\)

Prove that \(\begin{vmatrix} 1 &x^2 &x^3 \\ 1 & y^2 &y^3 \\1 &z^2 &z^3 \end{vmatrix}\) = (x - y)(y - z)(z - x)(xy + yz + zx).

If \(A=\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \), Show that k so that A² -4A- 51 = 0

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians.

The position vectors of the vertices of a triangle are \(\hat{i}+2\hat{j}+3\hat{k};3\hat{i}-4\hat{j}+5\hat{k}\) and\(-2\hat{i}+3\hat{j}-7\hat{k}\).Find the perimeter of the triangle.

Let \(\overrightarrow { a } =\hat { i } +\hat { j } +2\hat { k } \) and \(\overrightarrow { b } =\hat { i } +2\hat { j } +\hat { k } \) and \(\overrightarrow { c } \)  be a unit vectorin the plane determined by \(\overrightarrow { a } \) and \(\overrightarrow { b } \). If \(\overrightarrow { c } \) is perpendicular to the vector \(\hat { i } +\hat { j } +\hat { k } \) and makes an obtuse angle with \(\overrightarrow { a } \), then prove that \(\overrightarrow { c } =\frac { \hat { j } -\hat { k } }{ \sqrt { 2 } } \)

Let \(\vec{a}=2\hat{i}+\hat{j}-2\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}\) . Let \(\vec{c}\) be a vector such that \(\vec { a } .\vec { c } =\left| \vec { c } \right| ,\left| \vec { c } -\vec { a } \right| =2\sqrt { 2 } \) and the angle between  and  is 30^o.Then find the value of \(\left| (\vec { a } \times \vec { b } )\times \vec { c } \right| \)

Sketch the graph of f, then identify the values of x₀ for which \(lim_{x\rightarrow{x_o}}f(x)\) exists.
​\(f(x)=\begin{cases} { x^ 2 } ,\quad x \le 2 \\ 8-{ 2x } ,\quad 2 < x < 4 \\ { 4 } ,\quad x\ge 4 \end{cases}\)​

Sketch the graph of a function f that satisfies the given values :
f(- 2) = 0
f(2) = 0
\(lim_{x\rightarrow -2}f(x)=0\)
\(lim_{x\rightarrow -2}f(x)\) does not exist.

A tomato wholesaler finds that the price of a newly harvested tomatoes is Rs. 0.16 per kg if he purchases fewer than 100 kgs each day. However, if he purchases at least 100 kgs daily, the price drops to Rs. 0.14 per kg. Find the total cost function and discuss the cost when the purchase is 100 kgs.

Find the constant b that makes g continuous on \((-\infty,\infty)\)
\(g(x)= \begin{cases}x^{2}-b^{2} & \text { if } x<4 \\ b x+20 & \text { if } x \geq 4\end{cases}\)

Evaluate \(\lim _{ x\rightarrow \frac { 1 }{ \sqrt { 2 } } }{ \frac { x-\cos { (\sin ^{ -1 }{ (x) } ) } }{ 1-\tan { (\sin ^{ -1 }{ x } ) } } } \)

Show that the following functions are not differentiable at the indicated value of x.
\(f(x)=\left\{\begin{array}{ll} -x+2, & x \leq 2 \\ 2 x-4, & x>2 \end{array} ; \quad x=2\right.\)

Differentiate: \(y={x^{3\over4}\sqrt{x^2+1}\over (3x+2)^5}\)

Find \({d^2y\over dx^2}\) if x² + y² = 4.

Find the derivatives of the following : y = x^logx + (log x)^x

Find the derivatives of the following : \(x=\frac{1-t^2}{1+t^2}, y=\frac{2 t}{1+t^2}\)

If \(\log { ({ x }^{ 2 }+{ y }^{ 2 }) } =2\tan ^{ -1 }{ \frac { y }{ x } , } \) Show that \(\frac { dy }{ dx } =\frac { x+y }{ x-y } .\)

If f'(x) = 3x² - 4x + 5 and f(1) = 3, then find f(x).

Integrate the following with respect to x: : \({x sin^{-1}\over \sqrt{1-x^2}}\)

Integrate the following with respect to x : \(e^{tan^{-1}x}({1+x+x^2\over 1+x^2})\)

Evaluate the following integrals : \(\int {x+1\over x^2-3x+1}dx\)

Integrate the following with respect to x : \({2x-3\over x^2+4x-12}\)

Integrate the function with respect to x
e^2x sin 3x dx

A problem in Mathematics is given to three students whose chances of solving  \(\frac { 1 }{ 3 } ,\frac { 1 }{ 4 } \) and \(\frac { 1 }{ 5 } \) (i) What is the probability that the problem is solved? (ii) What is the probability that exactly one of them will solve it?

Two cards are drawn from a pack of 52 cards in succession. Find the probability that both are Jack when the first drawn card is (i) replaced (ii) not replaced.

Suppose ten coins are tossed. Find the probability to get (i) exactly two heads (ii) at most two heads (iii) at least two heads.

X speaks truth in 70 percent of cases, and Y in 90 percent of cases. What is the probability that they likely to contradict each other in stating the same fact?

The probability that a new railway bridge will get an award for its design is 0.48, the probability that it will get an award for the efficient use of materials is 0.36, and that it will get both awards is 0.2. What is the probability, that (i) it will get at least one of the two awards (ii) it will get only one of the awards.

p(A) = 0.3, P(B) = 0.6 and \(P(A\cap B)=0.25\) .Find
(i) \(P(A\cup B)\) 
(ii) P(A/B)
(iii) \(P(B/\bar { A } )\) 
(iv) \(P(\bar { A } /B)\) 
(v) \(P(\bar { A } /\bar { B } )\)