RS Aggarwal

### Prove that

Solution: Left Hand Limit: $\lim _{x \rightarrow 5-} f(x)=\lim _{x \rightarrow 5-} \frac{x^{2}-25}{x-5}$ $=\lim _{x \rightarrow 5-} \frac{(x+5)(x-5)}{x-5}$ [By middle term splitting]...

### Prove that

Solution: Left Hand Limit: $\lim _{x \rightarrow 3-} f(x)=\lim _{x \rightarrow 3-} \frac{x^{2}-x-6}{x-3}$ $=\lim _{\mathrm{x} \rightarrow 3-} \frac{(\mathrm{x}+2)(\mathrm{x}-3)}{\mathrm{x}-3}$ [By...

### The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...

### The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

Solution: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...

Solution: We need to find: $-x, y, z$, The given set of lines are : $\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 \\ \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\... read more Solution: We need to find:$-x, y, z$The given set of lines are : -$\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\...

### If , find . Using , solve the following system of equations:

Solution: It is given, $A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{array}\right]$ $\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj}(A)$...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x , y , z$ The given set of lines are : $\begin{array}{l} 4 x+3 y+2 z=60 \\ x+2 y+3 z=45 \\ 6 x+2 y+3 z=70 \end{array}$ Now converting the following equations in matrix...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y=3 \\ 2 x+3 y+4 z=17 \\ y+2 z=7 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-2 y+z=0 \\ y-z=2 \\ 2 x-3 z=10 \end{array}$ Now converting the following equations...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 5 x-y=-7 \\ 2 x+3 z=1 \\ 3 y-z=5 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y-2 z=3 \\ x+y=1 \\ x+z=-6 \end{array}$ Now converting the following equations in...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x+2 y+z=4 \\ -x+y+z=0 \\ x-3 y+z=4 \end{array}$ Now converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x+y-z=1 \\ x-y+z=2 \\ 3 x+y-2 z=-1 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x , y , z$ The given set of lines are : - $x + y - z = 1$ $\begin{array}{l} 3 x+y-2 z=3 \\ x-y-z=-1 \end{array}$ Now, converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x , y , z$ The given set of lines are : - $\begin{array}{l} 3 x-4 y+2 z=-1 \\ 2 x+3 y+5 z=7 \\ x+z=2 \end{array}$ Now, converting the following equations in matrix...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, z$ The given set of lines are : - $\begin{array}{l} 6 x-9 y-20 z=-4 \\ 4 x-15 y+10 z=-1 \\ 2 x-3 y-5 z=-1 \end{array}$ Now, converting the...

### Solve each of the following systems of equations using matrix method. ; : .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y+2 z=7 \\ 3 x+4 y-5 z=-5 \\ 2 x-y+3 z=12 \end{array}$ Now, converting the...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 4 x-5 y-11 z=12 \\ x-3 y+z=1 \\ 2 x+3 y-7 z=2 \end{array}$ Now, converting the...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: - $x, y, z$ The given set of lines are : - $\begin{array}{l} 2 x+3 y+3 z=5 \\ x-2 y+z=-4 \\ 3 x-y-2 z=3 \end{array}$ Now, converting the following equations in matrix...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $x+y+z=6$ $\begin{array}{l} x+2 z=7 \\ 3 x+y+z=12 \end{array}$ Now converting following equations in...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+y+z=1 \\ x-2 y+3 z=2 \\ 5 x-3 y+z=3 \end{array}$ Now converting the following...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x-3 y+5 z=11 \\ 3 x+2 y-4 z=-5 \\ x+y-2 z=-3 \end{array}$ Now converting the...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+y+z=4 \\ 2 x-y+z=-1 \\ 2 x+y-3 z=-9 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x-3 y+5 z=16 \\ 3 x+2 y-4 z=-4 \\ x+y-2 z=-3 \end{array}$ Now converting the...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+2 y+z=7 \\ x+3 z=11 \\ 2 x-3 y=1 \end{array}$ Now converting following equations...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 3 x+4 y+7 z=4 \\ 2 x-y+3 z=-3 \\ x+2 y-3 z=8 \end{array}$ Now converting the...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x-y+z=1 \\ 2 x+y-z=2 \\ x-2 y-z=4 \end{array}$ Now converting following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 2 x+8 y+5 z=5 \\ x+y+z=-2 \\ x+2 y-z=2 \end{array}$ Now, converting the following...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 4 x-3 y=3 \\ 3 x-5 y=7 \end{array}$ Now, converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: - $x, y$ The given set of lines are : - $\begin{array}{l} 2 \mathrm{x}-3 \mathrm{y}+1=0 \\ \mathrm{x}+4 y+3=0 \end{array}$ On converting the following equations in matrix...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 5 \mathrm{x}+7 \mathrm{y}+2=0 \\ 4 \mathrm{x}+6 \mathrm{y}+3=0 \end{array}$ On converting the...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} x+2 y=1 \\ 3 x+y=4 \end{array}$ Now converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 3 x+4 y-5=0 \\ x-y+3=0 \end{array}$ On converting the following equations in matrix form,...

### Solve each of the following systems of equations using matrix method. ; .

Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 5 x+2 y=4 \\ 7 x+3 y=5 \end{array}$ On converting the following equations in matrix form,...

### Show that each one of the following systems of equations is inconsistent. ; ; .

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 3 x-y-2 z=2 \\ 2 y-z=-1 \\ 3 x-5 y=3 \end{array}$ Now, converting the following...

### Show that each one of the following systems of equations is inconsistent. ; ; .

Solution: We need to prove: let of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} x+2 y+4 z=12 \\ y+2 z=-1 \\ 3 x+2 y+4 z=4 \end{array}$ Now, converting the following...

### Show that each one of the following systems of equations is inconsistent. ; ; .

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 2 x-y+3 z=1 \\ 3 x-2 y+5 z=-4 \\ 5 x-4 y+9 z=14 \end{array}$ On converting the...

### Show that each one of the following systems of equations is inconsistent. ; ; .

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} x+y-2 z=5 \\ x-2 y+z=-2 \\ -2 x+y+z=4 \end{array}$ Now, converting the following...

### Show that each one of the following systems of equations is inconsistent. ; .

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 6 x+4 y=5 \\ 9 x+6 y=8 \end{array}$ Now converting the following equations in matrix...

### Show that each one of the following systems of equations is inconsistent. ; .

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 4 x-2 y=3 \\ 6 x-3 y=5 \end{array}$ Now, converting the following equations in...

### Show that each one of the following systems of equations is inconsistent, ;

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are: $\begin{array}{l} 2 x+3 y=5 \\ 6 x+9 y=10 \end{array}$ Now, converting the following equations in matrix...

### Show that each one of the following systems of equations is inconsistent,

Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are: - $\begin{array}{l} x+2 y=9 \\ 2 x+4 y=7 \end{array}$ Now, converting the following equations in matrix...

### Find the adjoint of the given matrix and verify in each case that A. .

Solution: Given matrix as $A=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right)$ Find: the adjoint of the...

### Find the adjoint of the given matrix and verify in each case that A.

Solution: Given matrix as $A=\left(\begin{array}{ccc}9 & 7 & 3 \\ 5 & -1 & 4 \\ 6 & 8 & 2\end{array}\right)$. Find: the adjoint of the matrix given. Step: 1 Find the minor...

### Find the adjoint of the given matrix and verify in each case that A.

Solution: Given matrix as $A=\left(\begin{array}{ccc}3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2\end{array}\right)$ Find: the adjoint of the matrix given. Step: 1 Find the minor...

### Find the adjoint of the given matrix and verify in each case that A.

Solution: Given matrix as $A=\left(\begin{array}{ll}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right)$. Find: the adjoint of the matrix given. Step: 1 Find the minor...

### Find the adjoint of the given matrix and verify in each case that A.

Solution: Given matrix as $A=\left(\begin{array}{cc}3 & -5 \\ -1 & 2\end{array}\right)$ Find: the adjoint of the matrix given. Step: 1 Find the minor matrix of $A$....

### Find the adjoint of the given matrix and verify in each case that A. .

Solution: Given matrix as $A=\left(\begin{array}{ll}2 & 3 \\ 5 & 9\end{array}\right)$. Find: the adjoint of the matrix given. Step: 1 Find the minor matrix of $A$....

### On the set of all positive rational numbers, define an operation * on by for all a, Show that (i) is a binary operation on , (ii) * is commutative,Find the identity element in for What is the inverse of

(i) $^{*}$ is an operation as $\mathrm{a}^{*} \mathrm{~b}=\frac{\mathrm{ab}}{2}$ where $\mathrm{a}, \mathrm{b} \in \mathrm{Q}^{+} .$Let $\mathrm{a}=\frac{1}{2}$ and $\mathrm{b}=2 \mathrm{two}$...

### The roots of the equation are(a) real, unequal and rational(b) real, unequal and irrational(c) real and equal(d) imaginary

Answer is (b) real, unequal and irrational $\begin{array}{l} \because D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 2 \times 3 \\ =36-24 \\ =12 \end{array}$ 12 is greater than 0 and it is not a...

### The roots of the equation are(a) real, unequal and rational(b) real, unequal and irrational(c) real and equal(d) imaginary

Answer is (d) imaginary $\begin{array}{l} \because D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 2 \times 7 \\ =36-56 \\ =-20<0 \end{array}$ Thus, the roots of the equation are...

### In the equation , it is given that Then, the roots of the equation are(a) real and equal(b) real and unequal(c) imaginary(d) none of these

Answer is (b) real and unequal We know that when discriminant, $D>0$, the roots of the given quadratic cquation are real and uncqual.

### The roots of are real and unequal, if is(a) (b)=0(c)<0(d) none of these

Answer is $(a)>0$ The roots of the equation are real and unequal when $\left(b^{2}-4 a c\right)>0$.

### Which of the following is not a quadratic equation?(a) (b) (c) (d)

Answer is (c) $(\sqrt{2} x+3)^{2}=2 x^{2}+6$ $\begin{array}{l} \because(\sqrt{2} x+3)^{2}=2 x^{2}+6 \\ \Rightarrow 2 x^{2}+9+6 \sqrt{2} x=2 x^{2}+6 \end{array}$ $\Rightarrow 6 \sqrt{2} x+3=0$, which...

### Which of the following relations are functions? Give reasons. In case of a function, find its domain and range. (i)

Solution: (i) $h = {\{(a, b), (b, c), (c, b), (d, c)}\}$ Here, each of the first set element has different image in second set. $\therefore h$ is a function whose domain = {a, b, c, d} and range (h)...

### Which of the following is a quadratic equation?(a) (b) (c) (d) None of these

Answer is (b) $x^{3}-x^{2}=(x-1)^{3}$ $\because x^{3}-x^{2}=(x-1)^{3}$ $\Rightarrow x^{3}-x^{2}=x^{3}-3 x^{2}+3 x-1$ $\Rightarrow 2 x^{2}-3 x+1=0$, which is a quadratic equation

### Which of the following relations are functions? Give reasons. In case of a function, find its domain and range. (i) (ii)

Solution: For a relation to be a function each element of first set should have different image in the second set(Range) (i) f = {( - 1, 2), (1, 8), (2, 11), (3, 14)} Here, each of the first set...

### Prove that the function is one – one but not onto.

Solution: In the range of $\mathrm{N} \mathrm{f}(\mathrm{x})$ is monotonically increasing. $\therefore f(n)=n^{2}+n+1$ is one one. But Range of $f(n)=[0.75, \infty) \neq N($ codomain $)$ Thus,...

### The length of a rectangle is thrice as long as the side of a square. The side of the square is , more than the width of the rectangle. Their areas being equal, find the dimensions.

Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...

### Let and . Show that each one of and is one one but is not one – one.

Solution: $f: \left[0, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ for given function $\mathrm{f}(\mathrm{x})=\sin$ Recalling the graph for $\sin \mathrm{x}$, we realise that for any two values on...

### A motor boat whose speed in still water is , takes 1 hour more to go upstream than to return to the same spot. Find the speed of the stream.

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. Given: Speed of the boat $=18 \mathrm{~km} / \mathrm{hr}$ $\therefore$ Speed downstream $=(18+x) \mathrm{km} / h r$ Speed upstream...

### A train travels at a certain average speed for a distanced of and then travels a distance of 63 at an average speed of more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

Let the first speed of the train be $x \mathrm{~km} / \mathrm{h}$. Time taken to cover $54 \mathrm{~km}=\frac{54}{x} h .$ New speed of the train $=(x+6) \mathrm{km} / \mathrm{h}$ $\therefore$ Time...

### A train covers a distance of at a uniform speed. If the speed had been less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...

### Define each of the following: (i) bijective function (ii) many – one function Give an example of each type of functions.

Solution: (i)Bijective function: It is, also known as one-one onto function and is a function where for every element of set A, there is exactly one image in set B, such that no element is set B is...

### Define a function. What do you mean by the domain and range of a function? Give examples.

Solution: A function is stated as the relation between the two sets, where there is exactly one element in set B, for every element of set A. A function is represented as f: A → B, which means ‘f’...