A plane left 40 minutes late due to bad weather and in order to reach its destination,
Question: A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane. Solution: Let the usual speed of plane be $x \mathrm{~km} / \mathrm{hr}$. Then, Increased speed of the plane $=(x+400) \mathrm{km} / \mathrm{hr}$ Time taken by the plane under usual speed to cover $1600 \mathrm{~km}=\frac{1600}{x} \mathrm{hr}$ Time taken by the plane under increase...
Read More →The shaded part of the number line in given figure can also be represented as
Question: The shaded part of the number line in given figure can also be represented as (a) $x \in\left(\frac{9}{2}, \infty\right)$ (b) $x \in\left[\frac{9}{2}, \infty\right)$ (c) $x \in\left[-\infty, \frac{9}{2}\right)$ (d) $x \in\left(-\infty, \frac{9}{2}\right]$ Solution: Since shaded part includes $\frac{9}{2}$ and every point after $\frac{9}{2}$ $\therefore$ Shaded region can be represented by $\left[\frac{9}{2}, \infty\right)$ Hence,thecorrectanswerisoptionB....
Read More →Factorise
Question: Factorise $32 x^{4}-500 x$ Solution: $32 x^{4}-500 x=4 x\left(8 x^{3}-125\right)=4 x\left((2 x)^{3}-5^{3}\right)$ we know $a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)$ $a=2 x, b=5$ $32 x^{4}-500 x=4 x\left((2 x)^{3}-5^{3}\right)=4 x(2 x-5)\left(4 x^{2}+25+10 x\right)$...
Read More →Solution of a linear inequality in variable x is represented on the number line as shown in the given figure.
Question: Solution of a linear inequality in variablexis represented on the number line as shown in the given figure. The solution can also be described as (a) $x \in(-\infty, 5)$ (b) $x \in(-\infty, 5]$ (c) $x \in[5, \infty)$ (d) $x \in(5, \infty)$ Solution: Since, number line representing solution does not include 5. But, include every value after 5 Solution can be described by x (5, ) Hence, the correct answer is option D....
Read More →Solution of a linear inequality in variable x is represented on the number line as shown in the given figure.
Question: Solution of a linear inequality in variablexis represented on the number line as shown in the given figure. The solution can also be described as (a) $x \in(-\infty, 5)$ (b) $x \in(-\infty, 5]$ (c) $x \in[5, \infty)$ (d) $x \in(5, \infty)$ Solution: Since, number line representing solution does not include 5. But, include every value after 5 Solution can be described by x (5, ) Hence, the correct answer is option D....
Read More →The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey.
Question: The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/hr more than the speed of going, what was the speed per hour in each direction? Solution: Let the ongoing speed of person be $x \mathrm{~km} / \mathrm{hr}$. Then, Returning speed of the person is $=(x+10) \mathrm{km} / \mathrm{hr}$. Time taken by the person in going direction to cover $150 \mathrm{~km}=\frac{150}{x} \mathrm{hr}$ Time taken by the per...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $|x+3| \geq 10$, then (a) $x \in(-13,7]$ (b) $x \in(-13,7)$ (c) $x \in(-\infty,-13) \cup(7, \infty)$ (d) $x \in(-\infty,-13] \cup[7, \infty)$ Solution: $|x+3| \geq 10$ $\Rightarrow x+3 \geq 10$ or $x+3 \leq-10$ $\Rightarrow x \geq 10-3$ or $x \leq-10-3$ $\Rightarrow x \geq 7$ or $x \leq-13$ $\Rightarrow x \in(-\infty,-13] \cup[7, \infty)$ Hence, the correct option is (d)....
Read More →Factorize:
Question: Factorize: $x-8 x y^{3}$ Solution: $x-8 x y^{3}=x\left(1-8 y^{3}\right)$ $=x\left[1^{3}-(2 y)^{3}\right]$ $=x(1-2 y)\left(1^{2}+1 \times 2 y+(2 y)^{2}\right)$ $=x(1-2 y)\left(1+2 y+4 y^{2}\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $|x+3| \geq 10$, then (a) $x \in(-13,7]$ (b) $x \in(-13,7)$ (c) $x \in(-\infty,-13) \cup(7, \infty)$ (d) $x \in(-\infty,-13] \cup[7, \infty)$ Solution: $|x+3| \geq 10$ $\Rightarrow x+3 \geq 10$ or $x+3 \leq-10$ $\Rightarrow x \geq 10-3$ or $x \leq-10-3$ $\Rightarrow x \geq 7$ or $x \leq-13$ $\Rightarrow x \in(-\infty,-13] \cup[7, \infty)$ Hence, the correct option is (d)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $\frac{|x-2|}{x-2} \geq 0$, then (a) $x \in[2, \infty)$ (b) $x \in(2, \infty)$ (c) $x \in(-\infty, 2)$ (d) $x \in(-\infty, 2]$ Solution: $\frac{|x-2|}{x-2} \geq 0$ $\Rightarrow x-20$ $\Rightarrow x2$ $\Rightarrow x \in(2, \infty)$ Hence, the correct option is (b)....
Read More →Factorise
Question: Factorise $\frac{x^{3}}{216}-8 y^{3}$ Solution: We know $a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)$ We have, $\frac{x^{3}}{216}-8 y^{3}=\left(\frac{x}{6}\right)^{3}-(2 y)^{3}$ So, $a=\frac{x}{6}, b=2 y$ $\frac{x^{3}}{216}-8 y^{3}=\left(\frac{x}{6}-2 y\right)\left(\left(\frac{x}{6}\right)^{2}+\frac{x}{6} \times 2 y+(2 y)^{2}\right)=\left(\frac{x}{6}-2 y\right)\left(\frac{x^{2}}{36}+\frac{x y}{3}+4 y^{2}\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $\frac{|x-2|}{x-2} \geq 0$, then (a) $x \in[2, \infty)$ (b) $x \in(2, \infty)$ (c) $x \in(-\infty, 2)$ (d) $x \in(-\infty, 2]$ Solution: $\frac{|x-2|}{x-2} \geq 0$ $\Rightarrow x-20$ $\Rightarrow x2$ $\Rightarrow x \in(2, \infty)$ Hence, the correct option is (b)....
Read More →If $f(x)=\sqrt{x+3}$ and $g(x)=x^{2}+1$ be two real functions,
[question] Question. If $f(x)=\sqrt{x+3}$ and $g(x)=x^{2}+1$ be two real functions, then find fog and gof. [/question] [solution] Solution: $f(x)=\sqrt{x+3}$ For domain, $x+3 \geq 0$ $\Rightarrow x \geq-3$ Domain of $f=[-3, \infty)$ Since $f$ is a square root function, range of $f=[0, \infty)$ $f:[-3, \infty) \rightarrow[0, \infty)$ $g(x)=x^{2}+1$ is a polynomial. $\Rightarrow g: R \rightarrow R$ Computation of fog: Range of $g$ is not a subset of the domain of $f$. and domain $(f o g)=\{x: x \i...
Read More →A passenger train takes one hour less for a journey of 150 km
Question: A passenger train takes one hour less for a journey of 150 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train. Solution: Let the usual speed of train be $x \mathrm{~km} / \mathrm{hr}$ then Increased speed of the train $=(x+5) \mathrm{km} / \mathrm{hr}$ Time taken by the train under usual speed to cover $150 \mathrm{~km}=\frac{150}{x} \mathrm{hr}$ Time taken by the train under increased speed to cover $150 \mathrm{~km}=\frac{150}{(x+5)} \math...
Read More →Factorize:
Question: Factorize: $8 x^{3}-\frac{1}{27 y^{3}}$ Solution: $8 x^{3}-\frac{1}{27 y^{3}}=(2 x)^{3}-\left(\frac{1}{3 y}\right)^{3}$ $=\left(2 x-\frac{1}{3 y}\right)\left[(2 x)^{2}+2 x \times \frac{1}{3 y}+\left(\frac{1}{3 y}\right)^{2}\right]$ $=\left(2 x-\frac{1}{3 y}\right)\left(4 x^{2}+\frac{2 x}{3 y}+\frac{1}{9 y^{2}}\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: The solution set of the inequation $|x+2| \leq 5$ is (a) $(-7,5)$ (b) $[-7,3]$ (c) $[-5,5]$ (d) $(-7,3)$ Solution: $|x+2| \leq 5$ $\Rightarrow-5 \leq x+2 \leq 5$ $\Rightarrow-5-2 \leq x+2-2 \leq 5-2$ $\Rightarrow-7 \leq x \leq 3$ $\Rightarrow x \in[-7,3]$ Hence, the correct option is (b)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: The solution set of the inequation $|x+2| \leq 5$ is (a) $(-7,5)$ (b) $[-7,3]$ (c) $[-5,5]$ (d) $(-7,3)$ Solution: $|x+2| \leq 5$ $\Rightarrow-5 \leq x+2 \leq 5$ $\Rightarrow-5-2 \leq x+2-2 \leq 5-2$ $\Rightarrow-7 \leq x \leq 3$ $\Rightarrow x \in[-7,3]$ Hence, the correct option is (b)....
Read More →Factorize:
Question: Factorize: $a^{3}-0.064$ Solution: $a^{3}-0.064=(a)^{3}-(0.4)^{3}$ $=(a-0.4)\left[a^{2}+a \times(0.4)+(0.4)^{2}\right]$ $=(a-0.4)\left(a^{2}+0.4 a+0.16\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: The linear inequality representing the solution set given in Fig. 15.44 is (a) $|x|5$ (b) $|x|5$ (c) $|x| \geq 5$ (d) $|x| \leq 5$ Solution: As according to the graph, $x$ lies between $(-\infty,-5]$ and $[5, \infty)$ $\Rightarrow x \geq 5$ or $x \leq-5$ $\Rightarrow|x| \geq 5$ Hence, the correct option is (c)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: The linear inequality representing the solution set given in Fig. 15.44 is (a) $|x|5$ (b) $|x|5$ (c) $|x| \geq 5$ (d) $|x| \leq 5$ Solution: As according to the graph, $x$ lies between $(-\infty,-5]$ and $[5, \infty)$ $\Rightarrow x \geq 5$ or $x \leq-5$ $\Rightarrow|x| \geq 5$ Hence, the correct option is (c)....
Read More →Factorize:
Question: Factorize: $x^{3}-512$ Solution: $x^{3}-512=x^{3}-8^{3}$ $=(x-8)\left(x^{2}+8 x+8^{2}\right)$ $=(x-8)\left(x^{2}+8 x+64\right)$...
Read More →A fast train takes one hour less than a slow train for a journey of 200 km.
Question: A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speed of the two trains. Solution: Let the speed of the fast train be $x \mathrm{~km} / \mathrm{hr}$ then the speed of the slow train be $=(x-10) \mathrm{km} / \mathrm{hr}$ Time taken by the fast train to cover $200 \mathrm{~km}=\frac{200}{x} \mathrm{hr}$ Time taken by the slow train to cover $200 \mathrm{~km}=\frac{200}{(x-10...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: The inequality representing the following graph is (a) $|x|3$ (b) $|x| \leq 3$ (c) $|x|3$ (d) $|x| \geq 3$ Solution: As according to the graph, $x$ lies between $-3$ and 3 $\Rightarrow-3 \leq x \leq 3$ $\Rightarrow|x| \leq 3$ Hence, the correct option is (b)....
Read More →Factorize:
Question: Factorize: $64 a^{3}-343$ Solution: $64 a^{3}-343=(4 a)^{3}-(7)^{3}$ $=(4 a-7)\left(16 a^{2}+4 a \times 7+7^{2}\right)$ $=(4 a-7)\left(16 a^{2}+28 a+49\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: The inequality representing the following graph is (a) $|x|3$ (b) $|x| \leq 3$ (c) $|x|3$ (d) $|x| \geq 3$ Solution: As according to the graph, $x$ lies between $-3$ and 3 $\Rightarrow-3 \leq x \leq 3$ $\Rightarrow|x| \leq 3$ Hence, the correct option is (b)....
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