A two-digit number is 4 more than 6 times the sum of its digits.

Question: A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The number is 4 more than 6 times the sum of the two digits. Thus, we have $10 y+x=6(x+y)+4$ $\Rightarrow 10 y+x=6 x+6 y+4$ $\Rightarrow 6 x+6 y-10 y-x=-4$ $\Rightarrow 5 x-4 y=-4$ After interchanging the digits, the n...

Draw the graph of equation 2x + 3y = 12.

Question: Draw the graph of equation 2x + 3y = 12. From the graph, find the co-ordinates of the point: (i) whose y-coordinate is 3 (ii) whose x coordinate is -3 Solution: We are given, 2x + 3y = 12 We get, $y=\frac{12-2 x}{3}$ Substituting, x = 0 in $y=\frac{12-2 x}{3}$ We get, $y=\frac{12-2 \times 0}{3}$ y = 12 2 63 y = 0 Thus, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation By plotting the given equation on the graph, we...

A two-digit number is 3 more than 4 times the sum of its digits.

Question: A two-digit number is 3 more than 4 times the sum of its digits. If 8 is added to the number, the digits are reversed. Find the number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The number is 3 more than 4 times the sum of the two digits. Thus, we have $10 y+x=4(x+y)+3$ $\Rightarrow 10 y+x=4 x+4 y+3$ $\Rightarrow 4 x+4 y-10 y-x=-3$ $\Rightarrow 3 x-6 y=-3$ $\Rightarrow 3(x-2 y)=-3$ $\Rightarrow x-2 y=...

Show that

Question: $\int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) e^{2 x} d x$ Solution: $\int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) e^{2 x} d x$ Let $2 x=t \Rightarrow 2 d x=d t$ When $x=1, t=2$ and when $x=2, t=4$ $\therefore \int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) e^{2 x} d x=\frac{1}{2} \int_{2}^{4}\left(\frac{2}{t}-\frac{2}{t^{2}}\right) e^{t} d t$ $=\int_{2}^{+}\left(\frac{1}{t}-\frac{1}{t^{2}}\right) e^{t} d t$ Let $\frac{1}{t}=f(t)$ Then, $f^{\prime}(t)=-\frac{...

The range of the function f(x) = |x − 1| is

Question: The range of the functionf(x) = |x 1| is (a) (, 0) (b) [0, ) (c) (0, ) (d) R Solution: (b) [0, ) $f(x)=|x-1| \geq 0 \forall x \in R$ Thus, range $=[0, \infty 0$...

A two-digit number is 4 times the sum of its digits.

Question: A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The number is 4 times the sum of the two digits. Thus, we have $10 y+x=4(x+y)$ $\Rightarrow 10 y+x=4 x+4 y$ $\Rightarrow 4 x+4 y-10 y-x=0$ $\Rightarrow 3 x-6 y=0$ $\Rightarrow 3(x-2 y)=0$ $\Rightarrow x-2 y=0$ After interchanging the dig...

The range of the function f(x)

Question: The range of the function $f(x)=\frac{x+2}{|x+2|}, x \neq-2$ is (a) {1, 1} (b) {1, 0, 1} (c) {1} (d) (0, ) Solution: (a) {1, 1} $f(x)=\frac{x+2}{|x+2|}, x \neq-2$ Let $y=\frac{x+2}{|x+2|}$ For $|x+2|0$, or $x-2$, $y=\frac{x+2}{x+2}=1$ For $|x+2|0$, or $x-2$ $y=\frac{x+2}{-(x+2)}=-1$ Thus, $y=\{-1,1\}$ or range $f(x)=\{-1,1\}$....

The range of the function f(x)

Question: The range of the function $f(x)=\frac{x+2}{|x+2|}, x \neq-2$ is (a) {1, 1} (b) {1, 0, 1} (c) {1} (d) (0, ) Solution: (a) {1, 1} $f(x)=\frac{x+2}{|x+2|}, x \neq-2$ Let $y=\frac{x+2}{|x+2|}$ For $|x+2|0$, or $x-2$, $y=\frac{x+2}{x+2}=1$ For $|x+2|0$, or $x-2$ $y=\frac{x+2}{-(x+2)}=-1$ Thus, $y=\{-1,1\}$ or range $f(x)=\{-1,1\}$....

If the point (2, -2) lies on the graph of linear equation,

Question: If the point (2, -2) lies on the graph of linear equation, 5x + 4y = 4, find the value of k. Solution: It is given that the point (2,-2) lies on the given equation, 5x + ky = 4 Clearly, the given point is the solution of the given equation. Now, Substituting x = 2 and y = - 2 in the given equation, we get 5x + ky = 4 5 2 + (- 2) k = 4 2k = 10 4 2k = 6 k = 6/2 k = 3...

From the choices given below, choose the equation whose graph is given fig:

Question: From the choices given below, choose the equation whose graph is given fig: (i) y = x + 2 (ii) y = x 2 (iii) y = - x + 2 (iv) x + 2y = 6 Solution: We are given co-ordinates (-1, 3) and (2, 0) as the solution of one of the following equations. We will substitute the value of both co-ordinates in each of the equation and find the equation which satisfies the given co-ordinates. (i)We are given, y=x+2 Substituting x = - 1 and y = 3, We get 3 -1+ 2 L.H.SR.H.S Substituting x = 2 and y = 0, ...

Question: $\int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) e^{2 x} d x$ Solution: $\int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) e^{2 x} d x$ Let $2 x=t \Rightarrow 2 d x=d t$ When $x=1, t=2$ and when $x=2, t=4$ $\therefore \int_{1}^{2}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) e^{2 x} d x=\frac{1}{2} \int_{2}^{4}\left(\frac{2}{t}-\frac{2}{t^{2}}\right) e^{t} d t$ $=\int_{2}^{+}\left(\frac{1}{t}-\frac{1}{t^{2}}\right) e^{t} d t$ Let $\frac{1}{t}=f(t)$ Then, $f^{\prime}(t)=-\frac{...

The sum of a two digit number and the number obtained by reversing the order of its digits is 99

Question: The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number. Solution: Let the digits at units and tens place of the given number be $x$ and $y$ respectively. Thus, the number is $10 y+x$. The two digits of the number are differing by 3 . Thus, we have $x-y=\pm 3$ After interchanging the digits, the number becomes $10 x+y$. The sum of the numbers obtained by interchanging the digits and the original number...

The range of the function f(x)

Question: The range of the function $f(x)=\frac{x}{|x|}$ is (a) R {0} (b) R {1, 1} (c) {1, 1} (d) None of these Solution: (c) {1, 1} $f(x)=\frac{x}{|x|}$ Let $y=\frac{x}{|x|}$ For $x0,|x|=x$ $\Rightarrow y=\frac{x}{x}=1$ For $x0,=-x$ $\Rightarrow y=\frac{x}{-x}=-1$ Thus, r ange of $f(x)$ is $\{-1,1\}$....

The range of the function f(x)

Question: The range of the function $f(x)=\frac{x}{|x|}$ is (a) R {0} (b) R {1, 1} (c) {1, 1} (d) None of these Solution: (c) {1, 1} $f(x)=\frac{x}{|x|}$ Let $y=\frac{x}{|x|}$ For $x0,|x|=x$ $\Rightarrow y=\frac{x}{x}=1$ For $x0,=-x$ $\Rightarrow y=\frac{x}{-x}=-1$ Thus, r ange of $f(x)$ is $\{-1,1\}$....

From the choices given below, choose the equations whose graph is given in fig

Question: From the choices given below, choose the equations whose graph is given in fig (i) y = x (ii) x + y = 0 (iii) y = 2x (iv) 2 + 3y = 7x Solution: We are given co-ordinates (1, - 1) and (-1, 1) as the solution of one of the following equations. We will substitute the value of both co-ordinates in each of the equation and find the equation which satisfies the given co-ordinates. (i) We are given, y = x Substituting x =I and y = - 1, we get; 1-1 L.H.SR.H.S Substituting x = -1 and y = 1, we ...

The domain of the function f(x) =

Question: The domain of the function $f(x)=\sqrt{5|x|-x^{2}-6}$ is (a) (3, 2) (2, 3) (b) [3, 2) [2, 3) (c) [3, 2] [2, 3] (d) None of these Solution: (c) [3, 2] [2, 3] $f(x)=\sqrt{5|x|-x^{2}-6}$ For $f(x)$ to be defined, $5|x|-x^{2}-6 \geq 0$ $\Rightarrow 5|x|-x^{2}-6 \geq 0$ $\Rightarrow x^{2}-5|x|+6 \leq 0$For $x0,|x|=x$ $\Rightarrow x^{2}-5 x+6 \leq 0$ $\Rightarrow(x-2)(x-3) \leq 0$ $\Rightarrow x \in[2,3] \quad \ldots \ldots \ldots(1)$ For $x0,|x|=-x$ $\Rightarrow x^{2}+5 x+6 \leq 0$ $\Righta...

The domain of the function f(x) =

Question: The domain of the function $f(x)=\sqrt{5|x|-x^{2}-6}$ is (a) (3, 2) (2, 3) (b) [3, 2) [2, 3) (c) [3, 2] [2, 3] (d) None of these Solution: (c) [3, 2] [2, 3] $f(x)=\sqrt{5|x|-x^{2}-6}$ For $f(x)$ to be defined, $5|x|-x^{2}-6 \geq 0$ $\Rightarrow 5|x|-x^{2}-6 \geq 0$ $\Rightarrow x^{2}-5|x|+6 \leq 0$For $x0,|x|=x$ $\Rightarrow x^{2}-5 x+6 \leq 0$ $\Rightarrow(x-2)(x-3) \leq 0$ $\Rightarrow x \in[2,3] \quad \ldots \ldots \ldots(1)$ For $x0,|x|=-x$ $\Rightarrow x^{2}+5 x+6 \leq 0$ $\Righta...

Show that

Question: $\int_{-1}^{1} \frac{d x}{x^{2}+2 x+5}$ Solution: $\int_{-1}^{1} \frac{d x}{x^{2}+2 x+5}=\int_{-1}^{1} \frac{d x}{\left(x^{2}+2 x+1\right)+4}=\int_{-1}^{1} \frac{d x}{(x+1)^{2}+(2)^{2}}$ Let $x+1=t \Rightarrow d x=d t$ When $x=-1, t=0$ and when $x=1, t=2$ $\therefore \int_{-1}^{1} \frac{d x}{(x+1)^{2}+(2)^{2}}=\int_{0}^{2} \frac{d t}{t^{2}+2^{2}}$ $=\left[\frac{1}{2} \tan ^{-1} \frac{t}{2}\right]_{0}^{2}$ $=\frac{1}{2} \tan ^{-1} 1-\frac{1}{2} \tan ^{-1} 0$ $=\frac{1}{2}\left(\frac{\pi...

Plot the Points (3, 5) and (-1, 3) on a graph paper and verify that the straight line passing through the points,

Question: Plot the Points (3, 5) and (-1, 3) on a graph paper and verify that the straight line passing through the points, also passes through the point (1, 4) Solution: By plotting the given points (3, 5) and (-1, 3) on a graph paper, we get the line BC. We have already plotted the point A (1, 4) on the given plane by the intersecting lines. Therefore, it is proved that the straight line passing through (3, 5) and (-1, 3) also passes through A (1, 4)....

The sum of two numbers is 1000 and the difference between their squares is 256000.

Question: The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers. Solution: Let the numbers are $x$ and $y .$ One of them must be greater than or equal to the other. Let us assume that $x$ is greater than or equal to $y$. The sum of the two numbers is 1000 . Thus, we have $x+y=1000$ The difference between the squares of the two numbers is 256000 . Thus, we have $x^{2}-y^{2}=256000$ $\Rightarrow(x+y)(x-y)=256000$ $\Rightarrow 1000(x-y)=256000$ $\Righta...

Show that

Question: $\int_{0}^{2} \frac{d x}{x+4-x^{2}}$ Solution: $\int_{0}^{2} \frac{d x}{x+4-x^{2}}=\int_{0}^{2} \frac{d x}{-\left(x^{2}-x-4\right)}$ $=\int_{0}^{2} \frac{d x}{-\left(x^{2}-x+\frac{1}{4}-\frac{1}{4}-4\right)}$ $=\int_{0}^{2} \frac{d x}{-\left[\left(x-\frac{1}{2}\right)^{2}-\frac{17}{4}\right]}$ $=\int_{0}^{2} \frac{d x}{\left(\frac{\sqrt{17}}{2}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}}$ Let $x-\frac{1}{2}=t \Rightarrow d x=d t$ When $x=0, t=-\frac{1}{2}$ and when $x=2, t=\frac{3}{2}$ ...

The Sum of a two digit number and the number obtained by reversing the order of its digits is 121.

Question: The Sum of a two digit number and the number obtained by reversing the order of its digits is 121. If units and tens digit of the number are x and y respectively, then write the linear equation representing the above statement. Solution: The number given to us is in the form of yx, Where y represents the ten's place of the number and x represents the units place of the number Now, the given number is 10y + x Number obtained by reversing the digits of the number is 10x+ y It is given to...

The domain of definition of f(x)

Question: The domain of definition of $f(x)=\sqrt{x-3-2 \sqrt{x-4}}-\sqrt{x-3+2 \sqrt{x-4}}$ is (a) [4, ) (b) (, 4] (c) (4, ) (d) (, 4) Solution: (a) [4, ) $f(x)=\sqrt{x-3-2 \sqrt{x-4}}-\sqrt{x-3+2 \sqrt{x-4}}$ For $f(x)$ to be defined, $x-4 \geq 0$ $\Rightarrow x-4 \geq 0$ $\Rightarrow x \geq 4$ .....(1) Also, $x-3-2 \sqrt{x-4} \geq 0$ $\Rightarrow x-3-2 \sqrt{x-4} \geq 0$ $\Rightarrow x-3 \geq 2 \sqrt{x-4}$ $\Rightarrow(x-3)^{2} \geq(2 \sqrt{x-4})^{2}$ $\Rightarrow x^{2}+9-6 x \geq 4(x-4)$ $\R...

The domain of definition of f(x)

Question: The domain of definition of $f(x)=\sqrt{x-3-2 \sqrt{x-4}}-\sqrt{x-3+2 \sqrt{x-4}}$ is (a) [4, ) (b) (, 4] (c) (4, ) (d) (, 4) Solution: (a) [4, ) $f(x)=\sqrt{x-3-2 \sqrt{x-4}}-\sqrt{x-3+2 \sqrt{x-4}}$ For $f(x)$ to be defined, $x-4 \geq 0$ $\Rightarrow x-4 \geq 0$ $\Rightarrow x \geq 4$ .....(1) Also, $x-3-2 \sqrt{x-4} \geq 0$ $\Rightarrow x-3-2 \sqrt{x-4} \geq 0$ $\Rightarrow x-3 \geq 2 \sqrt{x-4}$ $\Rightarrow(x-3)^{2} \geq(2 \sqrt{x-4})^{2}$ $\Rightarrow x^{2}+9-6 x \geq 4(x-4)$ $\R...

A number is 27 more than the number obtained by reversing its digits.

Question: A number is 27 more than the number obtained by reversing its digits. lf its unit's and ten's digit are x and y respectively, write the linear equation representing the statement. Solution: The number given to us is in the form of yx, Where y represents the ten's place of the number And x represents the unit's place of the number. Now, the given number is 10y + x Number obtained by reversing the digits of the number is 10x + y It is given to us that the original number is 27 more than ...