Quantification And Numerical Applications - Class 12 Commerce Applied Mathematics - Extra Questions

One side of a triangle measures 8 cm. Of the remaining two sides, one measures 2 cm more than the other. Then, the length of each side of the triangle is greater than  $$x$$ cm. Find $$x$$?



Solve 30x < 160 when x is an integer.



Solve $$\displaystyle \frac{3x-10}{2}\geq \frac{x+1}{6}-1 $$, where $$x$$ is a ntaural no. less than 10.



Solve 30x < 160 when x is a natural number.



Solve $$5x - 7 < 2x + 2$$, when x is a natural number



Solve the inequalities for real $$x$$.
$$4x + 3 < 5x + 7$$



Solve the inequality for real $$x$$.
$${x +\dfrac{x}{2} + \dfrac{x}{3} < 11}$$



Solve  $$-12x > 30$$, when (i) $$x$$ is a natural number. (ii) $$x$$ is an integer



Solve $$5x - 3 < 3x + 1$$ when
(i) $$x$$ is an integer            (ii) $$x$$ is a real number



Solve $$24x < 100$$, when (i) $$x$$ is a natural number. (ii) $$x$$ is an integer



Solve the inequality for real $$x$$.
$$\displaystyle {\frac{x}{3}>\frac{x}{2} +1}$$



Solve the inequality for real $$x$$.
$$3 (2 -x) \geq  2 (1- x)$$



Solve the inequalities for real $$x$$.
$$3x - 7 > 5x -1$$



Solve the inequalities for real $$x$$.
$$3(x - 1) {\leq}  2 (x- 3)$$



Solve, and express the answer graphically.
$$\displaystyle\frac{3x-2}{x+4} < 2$$



Solve the inequality for real $$x$$.
$$2 (2x + 3) -10 < 6 (x -2)$$



Solve the inequality for real $$x$$.
$$37- (3x + 5) \geq 9x -8 (x -3)$$



Solve the inequality for real $$x$$.
$$\displaystyle {\frac{3(x-2)}{5}\leq \frac{5(2-x)}{3}}$$



$$\displaystyle\frac { 2 }{ x+1 } \ge \displaystyle\frac { 1 }{ x-2 } $$ is k,$$\infty$$ what is the value of the k?



Solve, and express the answer in interval form.
$$\displaystyle\frac{2x-8}{x-2} \ge 0$$



Pressure varies inversely with volume while temperatures varies directly with volume. At a time, Volume $$=50m^3$$, Temperature$$=25^o$$K and Pressure$$=1$$ atmosphere. If the volume is increased to $$200m^3$$, then find the temperature in $$^o$$K



Solve the given inequality for real x:
$$\dfrac{x}{3} > \dfrac{x}{2}+1$$



Solve the given inequality for real x :
$$3(x-1) \leq 2(x-3)$$



Solve $$3 - 5x \leq 9$$



Solve $$3(5 - x) > 6$$



Solve the following inequality:
$$\displaystyle\, \frac{3}{x - 2} < 1$$



Solve the following inequality:
$$\displaystyle\, \dfrac{6x - 5}{4x  + 1} < 0$$



Simplify the following expression:
$$\displaystyle\left ( \frac{a + 3b}{(a - b)^2} + \frac{a - 3b}{a^2 - b^2} \right ) : \frac{a^2 + 3b^2}{(a - b)^2}$$



Solve the following inequation, write down the solution set and represent it on the real number line:
$$-2+10x \leq 13x+10 < 24+10x$$, x$$\epsilon$$Z.



Solve the following inequality:
$$\displaystyle\, \frac{2x - 3}{3x - 7} > 0$$



Find all values of $$a$$ for which the inequality $$\displaystyle\, \frac{x - 2a - 3}{x - a + 2} < 0$$ is satisfied for all $$x$$ belonging to the interval $$\displaystyle\, 1\leqslant x\leqslant 2$$.



Solve the following inequality:
$$\displaystyle\, \frac{5x + 8}{4 - x} < 2$$



Solve the following inequality:
$$\displaystyle\, 2 + \frac{3}{x + 1} > \frac{2}{x}$$



Solve the following inequality:
$$\displaystyle\, \frac{|x + 3| + x}{x + 2} > 1$$



Solve the following inequality:
$$\displaystyle\, \left | \frac{2x - 1}{x - 1} \right | > 2$$



Solve the following inequality:
$$\displaystyle\, \frac{|x - 1|}{x + 2} < 1$$



Solve the following inequality:
$$\displaystyle\, \left | \frac{2}{x - 4} \right | > 1$$



Solve the following inequality:
$$\displaystyle\, \frac{14x}{x +1} - \frac{9x - 30}{x - 4} < 0$$



Solve the following inequality:
$$\displaystyle\, \frac{x + 7}{x - 5} + \frac{3x + 1}{2} \geqslant 0$$



Solve the following inequality:
$$\displaystyle\, 1 + \frac{2}{x - 1} > \frac{6}{x}$$



Solve the following inequality:
$$\displaystyle\, \frac{|x - 2|}{x - 2} > 0$$



Solve the following inequalities.
$$\displaystyle x \, - \, 3 \, \sqrt{x \, - \, 3} \, - \, 1 \, > \, 0.$$



Solve the following inequality:
$$\displaystyle \sqrt{\frac{x \, - \, 3}{x \, - \, 2}} \,> \, 0$$



Solve the following inequality:
$$\displaystyle \frac{\sqrt{a \, + \, 4}}{1 \, - \, a} \,  \, < \, 0.$$



Solve the following equation:
$$\displaystyle\, \frac{3}{\sqrt{x + 1} + 1} + 2\sqrt{x + 1} = 5$$



Solve the following inequality:
$$\displaystyle\, \frac{|x + 2| - x}{x} < 2$$



Solve the following inequality:
$$\displaystyle\, \frac{1}{|x| - 3} < \frac{1}{2}$$



Solve the following two- sided inequality.
$$\displaystyle\, 0 < \frac{3x - 1}{2x + 5} < 1$$



Solve the following equation:
$$\displaystyle\, \sqrt{2 - x} + \frac{4}{\sqrt{2 - x} + 3} = 2$$ 



Solve the following inequality:
$$\displaystyle \sqrt{\frac{3x \, - \, 2}{2 \, - \, x}} \,> \, 1$$



Solve the following inequality:
$$\displaystyle \frac{2}{x}  \,+ \, 3 \, \leqslant \, \sqrt{41 \, - \, \frac{16}{x}}.$$



Solve the following inequality:
$$\displaystyle\, \left ( \dfrac{2}{5} \right )^\frac{6 - 5x}{2 + 5x}  < \frac{25}{4}$$



Solve the following inequality:
$$\displaystyle\, 2^x + 2^{-x + 1} - 3 < 0$$



Solve the following inequality:
$$\displaystyle\, 2^{2x + 1} - 21 \cdot \left ( \frac{1}{2} \right )^{2x + 3} + 2 \geqslant 0$$



Solve the following inequality:
$$\displaystyle\, 5^{2x + 1} > 5^x + 4$$



Solve the following systems of inequality:
$$|x| \geqslant  1,     |x - 1| < 3$$ 



Solve the following inequality:

$$\displaystyle\, x^2 \cdot 5^x - 5^(2 + x) < 0$$



Solve the following inequality:
$$\displaystyle\, 4^{-x + 0.5} - 7 \cdot 2^{-x} - 4 < 0$$



Solve the following inequality:
$$\displaystyle\, 2^{3 - 6x} > 1$$



Solve the following inequality:
$$\displaystyle\, 16^x > 0.125$$



Solve the following two- sided inequality.
$$\displaystyle\, 1 \leqslant \frac{2 - x}{x + 1} \leqslant 2$$



Solve the following inequality:
$$\displaystyle\, (0.2)^\dfrac{2x - 3}{x - 2} > 5$$



Solve the following inequality:
$$\displaystyle\, \left ( \dfrac{1}{5} \right )^\frac{2x + 1}{1 - x} > \left ( \frac{1}{5} \right )^{-3}$$



Solve the following inequality:
$$\displaystyle\, \left ( \dfrac{1}{3} \right )^{x + \dfrac{1}{2} - \dfrac{2}{x}} > \dfrac{1}{\sqrt{27}}$$



Solve the following inequality:
$$\displaystyle\, 2^x + 2^{| x| } \geqslant  2\sqrt{2}$$



Solve the following inequality:
$$\displaystyle\, 4^x - 3 \leqslant  2^{x + 1}$$



Solve the following inequalities.
$$\displaystyle\, \frac{1 \, - \, \sqrt{1 \, - \, 8x^2}}{2x} \, < \, 1$$



Solve the following inequality:
$$\displaystyle\, \left ( \dfrac{1}{3} \right )^ \frac{ | x + 2| }{2 - | x| } > 9$$



State which of the following are equation (with variable). Given reasons for your an mention which variable is used from the equation with a variable.
$$8m<17$$



Solve the inequation $$\dfrac{1}{x-2}<0$$ in $$R$$.



If $$x$$ and $$y$$ are integers, where $$12 < x < 18$$ and $$15 < y < 27$$, then the maximum value of $$x + y$$ is 



Write the solution of $$4x+3<6x+7$$ in interval from.



Solve the inequality $$\dfrac {x - 1}{x^{2} - 4x + 3} < 1$$.



Let $$f\left(x\right)=\frac{\left(x-3\right)\left(x+2\right)\left(x+6\right)}{\left(x+1\right)\left(x-5\right)}$$
Find intervals, where$$f\left(x\right)$$ is positive or negative.



Solve: $$\dfrac{x-2}{x+5} > 2$$.



If $$x:y = 6:11$$, find$$(8x-3y) : (3x+2y)$$.



Solve the given inequality for $$x$$:
$$\sqrt{\dfrac{x-2}{1- 2x}}> 1$$



Solve the following inequality: $$2x - 1 < 5$$



Solve:
$$6x^2-7x-3$$.



Yamini and Fatima, two students of class IX of a school, together contributed Rs. $$100$$ towards the Prime Minister's relief fund to help the earthquake victims. Write a linear equation which this data satisfies. Draw the graph of the same.



Solve the given inequality for real x.
$$3x-7>5x-1$$



Solve : $$\dfrac{2x}{3} - \dfrac{x}{2} \le 6.5$$



Solve it :-
If $$A:B=5:6$$ and $$C=8:9$$ find $$A:B:C$$



Solve:$$\dfrac{1}{|x-2|} < 1$$



Solve for real $$x$$ : $$\dfrac{x + 8}{x + 2} > 2$$



Fill in the blank with $$>,\ <$$ or $$=$$ sign
$$(-21)-(-10)$$________$$(-31)+(-11)$$



Solve :- $$18\le 7y+4$$



Solve $$5 x - 3 < 3 x + 1$$ when
i) x is an integer.
ii) $$x$$ is a real number.



Solve :
$$\dfrac{5x+2}{3x-1}-\dfrac{5x-2}{3x+7}<0$$



Solve $$30x < 200$$ 
When
$$(i)x$$ is a natural no.
$$(ii) x$$ is an integer



The least of $$x$$ which satisfies the inequation $$ - 3 \le \dfrac{{4x - 3}}{5} \le 3$$ is



If $$A={4,5,7,9}$$, determine the truth values of the following quantified statements:
(i) $$\exists x \in A$$ such that $$x + 5 \ge 9$$
(ii) $$\forall x \in A,2x \le 17$$



The ratio of two numbers is $$\dfrac{2}{3}$$. If $$2$$ is subtracted from the first and $$8$$ from the second, the ratio becomes the reciprocal of the original ratio. Find the numbers.



See the figure and find the ratio of 
Number of triangle to the number of circle inside the rectangle.
1240542_9dc50951a3b0411cb29e2256eb936931.png



Find the solution set of $$7x - 3 \ge  - 24$$ and $$ - 11x + 10 \ge  - 12$$



The denominator of a fraction exceeds its numerator by $$5$$. If $$2$$ is added to both numerator and the denominator, the fraction becomes $$\dfrac{4}{5}.$$ Find the fraction.



If x : y = 3;5, find the ratio $$3x + 4y : 8x + 5y$$



Solve
$$2-3x > 1$$



If $$\dfrac {a}{b}=\dfrac {c}{d}=\dfrac {e}{f}$$ prove that:
$$\dfrac {a^{3}+c^{3}+e^{3}}{b^{3}+d^{3}+f^{3}}=\dfrac {ace}{bdf}$$



Evaluate the following:
$$\left| x \right| < \,\cfrac{a}{x}$$



Solve
$$14-2x =6$$



Solve $$\dfrac{{\left| {x + 2} \right| - x}}{x} < 2$$



$$3 \le (x - 1) - 2 { \le 4} .$$



Solve $$x > \sqrt { ( 1 - x ) }$$



Express each of the following ratios as a per cent.
$$1:5$$



Find two positive numbers in the ratio $$2:5$$ such that their difference is $$15$$



Determine if the following ratios forms a proportion. Also, write the middle terms and extreme terms if the ratios are in proportion.
$$200\ ml; \; 2.5$$ litre and $$Rs. 4; \; Rs. 50$$



The number of solution (s) of the equation $$\dfrac { 1 } { 2 } \log _ { \sqrt { 3 } } \left( \dfrac { x + 1 } { x + 5 } \right) + \log _ { 9 } ( x + 5 ) ^ { 2 } = 1$$ are/is



Read this proportion, and complete it:
$$\dfrac{8}{2}=\dfrac{20}{x}$$



Complete this proportion:$$8:32$$ as $$9:x$$



$$2\left( {x - 6} \right) < 3x - 7;$$  $$11 - 2x < 6 - x$$.



Rita gave away $$\cfrac{2}{3}$$ of her chocolate to $$6$$ of her friends. What fraction did each of her friends get?



Find the ratio of $$10$$ hours to a day.



$$11 - 5x >  - 4;4x + 13 \le  - 11$$



Solve:
$$\dfrac{1}{x-3}<\dfrac{1}{2}$$



If $$\dfrac {a}{b}=\dfrac {c}{d}=\dfrac {e}{f}$$ prove that:
$$(bdf).\left(\dfrac {a+b}{a}+\dfrac {c+d}{d}+\dfrac {e+f}{f}\right)^{3}=27(a+b)(c+d)(e+f)$$



Complete this proportion $$\dfrac{7}{21}=\dfrac{4}{?}$$



The second third and fourth terms of the proportion are $$25,30$$ and $$150$$. Find the first term. 



If $$A : B = 6 :7$$ , $$B : C = 3:5$$ Find $$ A:B:C$$



Solve the following linear inequations in $$R$$.
$$\dfrac {5x + 8}{4 - x} < 2$$.



Solve the following linear inequations in $$R$$.
Solve : $$12x < 50$$, when $$x\epsilon Z$$



Find the integer values of $$x$$ for which $$x-7\le 4x\le 5+2x$$.



Solve : $$\dfrac{2x+3}{5} > \dfrac{4x-1}{2},\ x\ \epsilon \ W$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {x - 1}{x + 3} > 2$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {5x - 6}{x + 6} < 1$$.



Rationalize the denominator of $$\cfrac {1}{2^\cfrac{1}{3} + 2^\cfrac{-1}{3}}$$ is



A child in danger of drowning in a river is being carried downstream by a current that flows uniformly at a speed of $$2.5  km/h$$. The child is $$0.6  km$$ from shore and $$0.8\ km$$ upstream of a boat landing when a rescue boat sets out. If the boat proceeds at its maximum speed of $$20  km/h$$ with respect to the water, the angle (in degrees) , the boat velocity $$v$$ make with the shore is 37x. The value of x is: 



If $$\left | x-2 \right |< 3$$, then number of integers in the range of $$x$$ is



Solve the inequation $$5(x - 2) > 4(x + 3) - 24$$ and represent its solution on a number line. Given the replacement set is $${-4, -3, -2,-1, 0, 1, 2, 3, 4}.$$



Solve $$\displaystyle \frac{2}{3}(x - 1) + 4 < 10$$ and represent its solution on a number line. Given replacement set is $${-8, -6, -4, 3, 6, 8, 12}$$.



A motorboat going downstream overcame a raft at a point A. One hour later it turned back and after sometime passed the raft at a distance 12 km from the point A. Find the river velocity (in km/hr) assuming the speed of the motorboat to be constant.



Let a and b be positive real numbers such that $$a + b = 1$$. Prove that $$a^ab^b\, +\, a^ab^b\, \leq\, 1$$



Solve 4x + 3 < 6x + 13, where $$x$$ is natural no. less than 3.



Evaluate | 3 | + | -2 - 3 | - 3 - | -7 |



Find $$x$$ from $$1 < | x | < 2$$ and represent it on number line



Solve $$3x + 8 >2,$$ when (i) $$x$$ is an integer. (ii) $$x$$ is a real number



Solve the inequality for real $$x$$.
$$\displaystyle {\frac {x}{4}<\frac{(5x-2)}{3} -\frac{(7x-3)}{5}}$$



Solve $$5x - 3 < 7,$$ when
(i) $$x$$ is an integer.
(ii) $$x$$ is a real number.



Solve the inequality for real $$x$$.
$$\displaystyle \frac{1}{2}\left ( \frac{3x}{5} +4 \right) \geq \frac{1}{3}(x-6)$$



Find the number of values of $$x$$ which satisfies the following inequality.
$$\displaystyle \left | 5x-7 \right |< -18 $$



Find all pairs of consecutive odd positive integers both of which are smaller than $$10$$ such that their sum is more than $$11$$.



Solve the inequality for real $$x$$.
$$\displaystyle {\frac{(2x-1)}{3}\geq \frac{(3x-2)}{4}-\frac{(2-x)}{5}}$$



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 at least $$60$$ marks.



Solve for $$x$$: $$\displaystyle \frac{-2}{x-10} < 0$$. If $$x \in (a,b)$$ then find $$a$$.



Solve, and express the answer in interval notation.
$$\displaystyle\frac{4x-8}{x-7} \ge 0$$



If the solution of the inequation  $$\displaystyle\dfrac { 3 }{ x-3 } \le \displaystyle\dfrac { 2 }{ x+2 } $$ is ($$-\infty$$ ,$$k$$], Find  the value of $$k$$.



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



Solve the inequalities:
$$2\le 3x-4\le 5$$



Solve the inequalities
$$-12<4+\dfrac{3x}{5}\le 2$$ 



Solve the inequalities
$$-3 \le 4-\dfrac{7x}{2}\le 18$$ 



Solve the inequalities
$$-15<\dfrac{3(x-2)}{5}\le 0$$ 



IQ of  a person is given by the formula $$IQ=\dfrac{MA}{CA}\times 100$$
where MA is mental age and CA is chronological age. If $$80 \le IQ \le 140$$ for a group 12 years old children, find the range of their mental age.



Solve the inequalities
$$7<\dfrac{(3x+11)}{2}\le 11$$ 



Solve for $$x:12-2x> 4$$



Solve for $$z:6z+1\le 3(z-2)$$



In a mixture $$60$$ litres, the ratio of milk and water $$2 : 1$$. If this ratio is to be $$1 : 2$$, then the quanity of water to be further added is



Disprove, by finding a suitable counter example, the statement $$x^2>y^2$$ for all $$x>y$$.



If the compound ratio of $$5 : 8$$ and $$3 : 7$$ is $$45 : x$$. Find the value of $$x$$.



If the compound ratio of $$7 : 5$$ and $$8 : x$$ is $$84 : 60$$. Find $$x$$.



Restate the following statements with appropriate conditions, so that they become true statements.
For any $$x, 3x+1>4$$.



The compound ratio of $$3 : 4$$ and the inverse ratio of $$4 : 5$$ is $$45 : x$$. Find $$x$$.



Draw the graph of the following inequation $$2x+3y\le 6$$.



Solve the equations:
$$\dfrac {2x^{3} - 3x^{2} + x + 1}{2x^{3} -3x^{2} - x - 1} = \dfrac {3x^{3} - x^{2} + 5x - 13}{3x^{3} - x^{2} - 5x + 13}$$.



Find four proportionals such that the sum of the extremes is $$21$$, the sum of the means $$19$$, and the sum of the squares of all four numbers is $$442$$.



If $$x + 7 ; 2(x + 14)$$ in the duplicate ratio of $$5 : 8$$, find $$x$$.



$$3(2x+y)=7xy; 3(x+3y)=11xy$$ using elimination method.



Express the following ratio in their simplest form. $$16:20$$.



Find the number of solutions of the equation
$$\sin5x \cos3x =\sin6x \cos2x, \,\, x \in [0, \pi]$$



Solve: $$|(x^2+2x+2)+(3x+7)|<|x^2+2x+2|+|3x+7|$$.



Prove that $${a}^{3}b+a{b}^{3}< {a}^{4}+{b}^{4}$$.



If $$\displaystyle \frac{a}{b} = \frac{c}{d}$$ $$=$$ $$\dfrac{e}{f}$$ and $$\dfrac{2a^4b^2 + 3a^2c^2 - 5e^4f}{2b^6 + 3 b^2d^2 - 5f^5}$$ $$=$$ $$\left (\dfrac{a}{b}\right)^n$$ then find the value of $$ n$$.



Solve the inequality $$24x<100$$ if $$x$$ is a natural number.



Solve that: $$\displaystyle\frac{1}{3}(2x -1)\geq \frac{1}{4}(3x-2)-\frac{1}{5}(2-x)$$ for $$x\epsilon N$$.



Solve: $$\displaystyle \frac{3x-1}{2} < -3$$ and $$\displaystyle \frac{3x+8}{5} + 11 < 0$$



Solve the given inequality for real x :
$$37 - (3x+5) \geq 9x - 8(x-3)$$



Solve the system of inequality:
$$2(3x - 1) < 3 (4x + 1 ) + 16,    4(2 + x) < 3x + 8$$



Solve the following inequality:
$$\left | x\, - \, 2 \, \right | \, \leqslant \, \left | x \, + \, 4 \, \right |$$



Simplify the following expression:
$$\displaystyle\left ( a + \frac{ab}{a - b} \right ) \left ( \frac{ab}{a - b} - a \right ) :\frac{a^2 + b^2}{a^2 - b^2}$$ 



Solve the following inequality:
$$\displaystyle \sqrt{3x \, - \, 10} \, > \, \sqrt{6 \, - \, x}$$



Find natural $$x$$ which satisfy the system of inequality:
$$x + 3 <  4  + 2x,     5x - 3 < 4x - 1$$



Solve the inequality $$\left | x\, + \, 1 \, \right | \, + \, \left | x \, - \, 4 \, \right |\, > \, 7,$$ indicating the least positive integer $$x$$ satisfying the inequality.



Solve the following inequality:
$$\left | 5\, - \, 2x \right | \, < 1$$ 



Solve the following inequality:
$$\left | 3x\, - \, 2.5 \right | \, \geqslant \, 2$$ 



Simplify the following expression:
$$\displaystyle\left ( m + n - \frac{4mn}{m + n}  \right ) :\left ( \frac{m}{m + n} - \frac{n}{n - m} - \frac{2mn}  {m^2 - n^2}\right )$$ 



Solve the following inequality:
$$\left | x\, - \, 4 \, \right | \, < \, x \, - \, 1$$



Solve the following inequality:
$$\displaystyle\, \frac{4^x + 2x  - 4}{x - 1} \leqslant  2$$



Solve the following inequalities.
$$\displaystyle\, 2^{x + 2} - 2^{x + 1} + 2^{x - 1} - 2^{x - 2} \leqslant 9$$



Solve the following two- sided inequality:
$$\displaystyle\, 1< \frac{3x - 1}{2x + 1} < 2$$



Find the domain of definition and the range of the following function:

$$\displaystyle f \, (x) \, = \, \sqrt{2 \, - \, x} \, + \, \sqrt{1 \, + \, x}$$



Solve the following inequality:
$$\displaystyle\, 4^x - 2^{2(x - 1)} + 8^\dfrac{2(x - 2)}{3} > 52$$



Solve the following inequality:
$$\displaystyle\, \frac{2^{x - 1} - 1}{2^{x + 1}  + 1} < 2$$



Solve the following inequalities.
$$\displaystyle\, 25^{-x} - 5^{-x + 1} \geqslant 50$$



Solve the following inequalities.
$$\displaystyle\, 2^{x + 2} - 2^{x + 3} - 2^{x + 4} > 5^{x + 1} - 5^{x + 2}$$



Solve the following inequalities.
$$\displaystyle\, 3^{72} \cdot \left ( \frac{1}{3} \right )^x \cdot \left ( \frac{1}{3} \right )^{\sqrt{x}} > 1$$



Solve the following inequality:

$$\displaystyle\, \left ( \dfrac{1}{3} \right )^\sqrt{x + 2} > 3^{-x}$$



Solve the following inequalities.
$$\displaystyle\, x^22^{2x} + 9 (x + 2) 2^x +  8x^2 \leqslant  (x + 2) 2^{2x} + 9x^22^x + 8x + 16$$



Solve the following inequalities
$$\sqrt{25 - \, x^2} \, \leqslant \, \dfrac{12}{x}.$$



Solve the following inequality:
$$\displaystyle\, 3 \sqrt{x} > 2^a$$



Solve the following inequalities.
$$\displaystyle\, 8 \cdot \frac{3^{x - 2}}{3^x - 2^x} > 1 + \left ( \frac{2}{3} \right )^x$$



Solve the following inequality:
$$\displaystyle\, \sqrt{9^x + 3^x - 2} \geqslant  9 - 3^x$$



Solve the following inequalities.
$$\displaystyle\, \sqrt{9^x - 3^x + 2} > 3^x - 9$$



A person rows a boat across a river making an angle of $$60^o$$ with the downstream. Find the percentage time he would have saved, had he crossed the river in the shortest possible time.



(a) Find all integral values of $$x$$ for which
 $$\begin{matrix} (5x-1)< & { \left( x+1 \right)  }^{ 2 }< & (7x-3) \\ I & II & III \end{matrix}$$.
(b) $$Solve\left| \dfrac { 12x }{ 4{ x }^{ 2 }+9 }  \right| \le 1$$.



Solve: $$\dfrac{(x-1)(x-2)(x-3)}{(x+1)(x+2)(x+3)}>1$$



Let $$a, b, c$$ be positive real numbers, then prove that-
$$\dfrac{1}{a^3(b+ c)} +\dfrac{1}{b^3( a+c)} +\dfrac{1}{c^3(a+b)} \geq \dfrac{3}{2} $$ , if $$abc \leq 1$$.



If the replacement set is the set of real numbers, solve :$$\cfrac { x+3 }{ 8 } <\cfrac { x-3 }{ 5 } $$



A man wants to cut three lengths from a single piece of board of length $$91cm$$. The second length is to be $$3cm$$ longer than the shortest and the third length is to be twice as long as the shortest. Find the length of the shortest piece.



Find the smallest value of x for which $$5 - 2x < 5{1 \over 2} - {5 \over 3}x$$ , where x is an integer.



Solve the inequation :
$$12 + 1{5 \over 6}x \le 5 + 3x$$  and  $$x \in R$$.



If the replacement set is the set of real numbers, 
solve : $$5 + {x \over 4} > {x \over 5} + 9$$



Find the values of $$x$$
$$3{x^2} - 5x + 9 \ge 3x - {x^2}$$



Man of weight $$40$$ kg standing on boat of weight $$15kg$$ and length $$2m$$ man walk one end to other what will be displacement of boat



Solve:
$$\dfrac{x}{4} < \dfrac{5x-2}{3} -\dfrac{(7x-4)}{5}$$.



Solve $$\sqrt {x+y}\ge x; y\ge 2$$



Find the complete set of values of $$x$$ satisfying :
(i) $$\dfrac{5[x]+4}{11[x]+7} \ge 3$$
(ii) $$[x]^{2}-3[x]+2 \le 0$$



If the replacement set is the set of natural numbers, Find the set with maximum elements:
1)  $$x - 5 < 0$$
2)  $$x + 1 \le 7$$
3)  $$3x - 4 > 6$$
4)  $$4x + 1 > 17$$



Mr. Goel left one-third of his property for his son, one fourth to his daughter and the remaining Rs. 2,00,000 to his wife. How much money did he leave?



The compound ratio of $$3:4$$ and the inverse ratio of $$4:5$$ is $$45:x$$. Find $$x$$.



If a, b, c, d are $$4$$-distinct positive numbers, then prove that $$(a+b+c+d)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\right) > 16$$.



Prove that $$\dfrac{3x^2+1}{x}\geq 2\sqrt{3}, x > 0$$.



If x, y, z$$> 0$$, then prove that $$(x+y+z)(x^3+y^3+z^3)\geq 9(xyz)^{\dfrac43}$$.



If x, y, z $$> 0$$, then prove that $$x^3y^3+y^3z^3+z^3x^3 \geq 3x^2y^2z^2$$.



If the air is maintained at $$30^o$$C and the temperature of the body cools from $$80^o$$ C to $$60^o$$ C in $$12$$ minutes, find the temperature of the body after $$24$$ minutes.



If a, b, c $$> 0$$, then prove that $$\dfrac{3a}{b}+\dfrac{4b}{c}+\dfrac{16c}{3a} \geq 12$$.



If a, b, c$$ > 0$$, then prove that $$ \left (a^3+b^3+c^3+\dfrac{1}{a^3b^3c^3} \right )\geq 4 $$.



Solve the following linear programming problem graphically :
Maximize $$z = 60x + 15y$$  subjected to constraints:
$$x + y\leq 50$$
$$3x + y\leq 90$$
$$x ,y\geq 0$$



A motor boat going downstream overcame a raft at a point A; $$\tau = 60 \,min$$ later it turned back and after some time passed the raft at a distance $$l = 6.0km$$ from point A. Find the flow velocity assuming the duty of the engine to be constant.



If $$a:b = 3:7$$ and $$b:c = 3:4$$ then find $$a:b:c$$.
If $$a:b = 3:5$$ and $$b:c = 2:3$$ then find $$a:b:c$$



Solve $$2 < |2x+1| < 5$$



Solve : $$||x - 1| - 5| \ge 2$$



Cost of a table and chair is in the ratio $$2 : 3$$. If the total cost is $$Rs. 2250$$, find the cost table and chair each.



Solve : $$\left | x\ +\ \dfrac{1}{3} \right | > \dfrac{8}{3}$$



$$2 \,km$$ wide river flowing at the rate  $$5 \,km/hr$$. A man can swim is still water $$10 \,km/hr$$. He wants to cross the river along the shorted path. Find-
(a) in which direction should the person swim.
(b) crossing time.



Solve for : $$-12<4+\dfrac{3x}{\left(-5\right)} \le 2$$



Minimize and maximize $$Z=x+2y$$ subject to constraints are $$x+2y\ge 100,2x-y\ge 0,2x+y\le 200$$ and $$x,y\ge 0$$.



Prove that
$$\frac { 1 } { x + 1 } - \frac { 1 } { x } \leq \frac { 1 } { x - 1 } - \frac { 1 } { x - 2 }$$



The present ages of A, B and C are in the ratio of 8: 14:22 respectively. The present ages of B , C and D are in the ratio of 21:33:44 respectively, which of the following represents the ratio of the present ages of A, B, C and D respectively?.



Find three consecutive whole numbers whose sum is more than $$45$$ but less than $$54$$.



The ratio between the prices of a scooter and a refrigerator is $$4 : 1$$. If the scooter costs $$Rs\ 45,000$$ more than the refrigerator, find the price of the refrigerator.



Solve: $$\dfrac{|x + 2| -x}{x} < 2.$$



find range of x:
$$x + 5 > 4 x - 10$$



Solve: $$x + 3 < \dfrac { - 2 } { x + 2 }$$



A factory makes cricket bats and hockey sticks. A bat takes $$1.5$$ hours of machine time and $$2$$ hours of craftsman time. While a hockey stick takes $$2.5$$ hour of a machine time and $$1.5$$ hours of craftsman time. In a day the factory is available up to $$80$$ hours of machine time and $$70$$ hours of craftsman time. Show that in order to make maximum profit the factory should produce only cricket bats given that the profits on a cricket at and a hockey stick are Rs.$$50$$ and Rs. $$35$$ respectively.



Evaluate values of $$x$$:
$$2 ( 3 - x ) \geq \dfrac { x } { 5 } + 4$$



Solve $$\dfrac{{62x + 404}}{{140}} \le \dfrac{{51 + 15x}}{{35}}$$.



Solve : $$9x-7\le 28+4x;$$ $$ x \epsilon R$$



The solution set for $$\left(x+3\right)+4>-2x+5$$ is 



Solve
$$2x \le x-1< 3x+5 $$.



Solve the inequality $$\left( x-11 \right) \left( x-7 \right) \le 1$$. The number of integers satisfy this inequality is  ____ .



Solve for $$x$$:
$$\dfrac { 7 x - 1 } { 2 } < - 3 , \dfrac { 3 x + 8 } { 5 } + 11 < 0$$



Solve for $$x$$ 
$$\dfrac { 1 } { | x | - 3 } < \dfrac { 1 } { 2 }$$



Show that $$\frac { x } { 1 + x } < \ln ( 1 + x ) < x \forall x > 0$$



The first, second and third class railway fares in the ratio between two stations are in the ratio as 6 : 4 : 1 and the number of passenger of the three classes are 2 : 5 :If the sale of the tickets of three classes amount of Rs 12300, find total collection from each class.



If $$a,b$$ are natural numbers, prove that

$$\sqrt { \dfrac { a ^ { 2 } + 2 b ^ { 2 } } { 2 } } > b$$



Solve:
$$5 ^ { \log _ { \sqrt { 5 } } x } - x - 6 < 0$$



Find range of $$x$$
$$\dfrac { 5 x + 8 } { 4 - x } < 2$$



A number is such that it is as much greater than $$65$$ as it is less than $$91$$.Find the number.



A recipe for cookies uses butter and sugar in ratio $$2:3$$. If we are using $$8$$ cups of butter, how many cups of sugar should we use?



Find the ratio in which the line segment joining A (6,4) and B(1,-7) is divided internally by x axis.



the ratio in which YZ plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8)



Find the graphical solution of the following in equations $$3x+4y \le 12$$ and $$x-2y \ge2$$



One kilogram of onions cost $$Rs\ 12.75$$. How much will $$3.5\ kg$$ cost?



Divide $$1500$$ among $$A,B,C$$ in the ratio $$3:5:2$$



Solve:$$9x+3>2$$



If $$3:x::12:20,$$ find the value of $$x?$$



$$\left| \frac { 3 x - 4 } { 2 } \right| \leq \frac { 5 ! } { 12 }$$



Solve:$$2x+5>3x-5$$



A sum of Rs.$$84,000$$ is divided among three persons $$A,B$$ and $$C$$.If $$A$$ gets one-fourth of it and $$B$$ gets one-fifth of it.How much did $$C$$ get?



Solve: $$8x+9\le 3x+2$$



Solve for $$x$$:
$$10 \leq - 5 ( x - 2 ) < 20$$



A bag contains $$187$$ in the form of $$1$$ rupee,$$\,50$$ paise and $$\,10$$ paise coin in the ratio $$3:4:5$$.Find the number of each type of coins?



If $$4x+5:3x+11=13:17$$ find the value of $$x$$



Replace * by '<' or '>' in each of the following to make the statement true:
(i) $$(-6) + (-9) * (-6) - (-9) $$
(ii) $$(-12) - (-12) * (-12) + (-12) $$
(iii) $$(-20)- (-20) * 20 - (65) $$
(iv) $$28- (-10) * (-16)- (-76)$$



Solve the following minimal assignment problem 
JobsJobsJobs
MachinesIIIIII
$$M_1$$$$1$$$$4$$$$5$$
$$M_2$$$$4$$$$2$$$$7$$
$$M_3$$$$7$$$$8$$$$3$$



$$\displaystyle \int \left [ \left ( \dfrac{x}{e} \right )^x + \left ( \dfrac{e}{x}  \right )^x \right ] ln\, x$$ dx



Solve : $$-4x > 30$$, when $$x\epsilon N$$.



Solve the following linear inequations in $$R$$.
Solve : $$12x < 50$$, when $$x\epsilon N$$



Solve the following linear inequations in $$R$$.
$$3x - 7 > x + 1$$.



Solve : $$4x - 2 < 8$$, when $$x\epsilon Z$$.



Solve the following linear inequations in $$R$$.
Solve : $$12x < 50$$, when $$x\epsilon R$$



Solve the following linear inequations in $$R$$.
$$x + 5 > 4x - 10$$.



Solve : $$4x - 2 < 8$$, when $$x\in R$$.



Solve : $$-4x > 30$$, when $$x\in R$$.



Solve : $$4x - 2 < 8$$, when $$x\epsilon N$$.



Solve : $$-4x > 30$$, when $$x\epsilon Z$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {5 - 2x}{3} < \dfrac {x}{6} - 5$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {3x - 2}{5}\leq \dfrac {4x - 3}{2}$$.



Solve the following linear inequations in $$R$$.
$$-(x - 3) + 4 < 5 - 2x$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {2x + 3}{4} - 3 < \dfrac {x - 4}{3} - 2$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {5x}{2} + \dfrac {3x}{4} \geq \dfrac {39}{4}$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {x}{5} < \dfrac {3x - 2}{4} - \dfrac {5x - 3}{5}$$.



Solve the following linear inequations in $$R$$.
$$3x + 9 > -x + 19$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {x - 1}{3} + 4 < \dfrac {x - 5}{5} - 2$$.



Solve the following linear inequations in $$R$$.
$$2(3 - x) \geq \dfrac {x}{5} + 4$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {2(x - 1)}{5}\leq \dfrac {3(2 + x)}{7}$$.



Solve each of the following system of equations in $$R$$.
$$2x - 7 > 5 - x, 11 - 5x \leq 1$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {2x + 3}{5} - 2 < \dfrac {3(x - 2)}{5}$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {x}{x - 5} > \dfrac {1}{2}$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {1}{x - 1} \leq 2$$.



Solve the following linear inequations in $$R$$.
$$x - 2\leq \dfrac {5x + 8}{3}$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {4 + 2x}{3} \geq \dfrac {x}{2} - 3$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {2x - 3}{3x - 7} > 0$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {7x - 5}{8x + 3} > 4$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {3}{x - 2} < 1$$.



Solve the following linear inequations in $$R$$.
$$\dfrac {4x + 3}{2x - 5} < 6$$.



Find the linear inequations for which the solution set is the shaded region given in Fig.
1666922_3df4a29ad8fa49b481ff72c753d2d10e.png



Write the solution set of the inequation $$x + \dfrac {1}{x} \geq 2$$.



Write the solution set of the inequation $$\left |\dfrac {1}{x} - 2\right | < 4$$.



Write the set of values of $$x$$ satisfying the inequation $$(x^{2} - 2x + 1)(x - 4)\geq 0$$.



Write the solution set of the inequation $$|x - 1| \geq |x - 3|$$.



Find the linear inequations for which the shaded ares in Fig. is the solution set. Draw the diagram of the solution set of the linear inequations.
1666917_543cfc80a748423fbc09d436cc0f4bab.png



Write the set of values of $$x$$ satisfying the inequations $$5x + 2 < 3x + 8$$ and $$\dfrac {x + 2}{x - 1} < 4$$.



Write the set of values of $$x$$ satisfying $$|x - 1| \leq 3$$ and $$|x - 1| \leq 1$$.



Solve each of the following system of equations in $$R$$.
$$\dfrac {2x + 1}{7x - 1} > 5, \dfrac {x + 7}{x - 8} > 2$$.



Solve each of the following system of equations in $$R$$.
$$x - 2 < 0, 3x < 18$$.



Classify the given polynomial as linear, quadratic, cubic and biquadratic polynomial.
$$3y$$.



Shikha and Aarushi are sisters. The ratio of their ages is $$3 : 4$$. Their parents give them Rs. $$1400$$ to share in the ratio of their ages. How much does each of them receive?



Fill in the blank.
If $$2x = 5y$$, then $$x : y=$$ ________.



The ratio of copper and zinc in an alloy is $$5 : 3$$. If the weight of copper in the alloy is $$30.5$$g, find the weight of zinc in it.



Fill in the following blanks: $$\cfrac{12}{20}=\cfrac{\Box }{5}=\cfrac{9}{\Box  }$$



The marks (out of $$100$$) obtained by a group of students in science tests are $$85, 76, 90, 84, 39, 48, 56, 95, 81$$ and $$75$$. Find the highest and the lowest marks obtained by the students.



The marks (out of 100) obtained by a group of students in science test are 85, 76, 90, 84, 39, 48, 56, 95, 81 andFind the range of marks obtained.



A boat is traveling upstream in the positive direction of an $$x$$ axis at $$14 km/h$$ with respect to the water of a river. The water is flowing at $$9.0 km/h$$ with respect to the ground. What are the (a) magnitude and (b) direction of the boats velocity with respect to the ground? A child on the boat walks from front to rear at $$6.0 km/h$$ with respect to the boat. What are the (c) magnitude and (d) direction of the childs velocity with respect to the ground?



Solve the following inequation :
$$ \dfrac{2y - 1}{5} \leq 2 , y\,  \varepsilon \, N $$



Solve the following inequation :
$$ -17 \leq 9x - 8, x \, \varepsilon \, Z $$ 



Solve the following inequation:
$$ 4 > 3x - 11 , x \, \varepsilon \, N $$ 



Solve the following inequation:
$$ 3(x - 2) < 2(x - 1 ), x \, \varepsilon \, W $$ 



Solve the following inequation:
$$ -2 ( p + 3) > 5 , p\,  \varepsilon \, I $$



Solve the following inequation:
$$ 2x - 3 < x + 2 , x \, \varepsilon \, N $$ 



Solve the following inequation:
$$ 3 - x\leq 5 - 3x  , x \, \varepsilon \, W $$ 



Solve the following inequation :
$$ \dfrac{2y + 1}{3} + 1 \leq 3 , y\,  \varepsilon \, W $$



Solve the following inequation :
$$ \dfrac{3}{2} - \dfrac {x}{2} > -1  , x \, \varepsilon \, N $$ 



A swimmer can swim with velocity of $$10km/hr$$  w.r.t. the water flowing in a river with velocity of $$5 km /hr$$. In what direction should he swim to reach the point on the other bank just opposite to his starting point? 



Define inequality.



For the following numbers write the ratio of first number to second number in the reduced form.
$$138 , 161$$



A row-boat is observed travelling downstream on a flowing river at a speed of $$8 m/s$$ with respect to the shore. A motor-boat comes from the opposite direction at a speed of $$10 m/s$$ with respect to the water. $$10 seconds $$after meeting each other they are at $$160 meter$$ from each other.
(a) at what speed does the river flow?
(b) Find the speed of row-boat in still water.



A motorboat going downstream overcame a raft at a point $$A$$; $$\tau = 60\:min$$ later it turned back and after some time the raft is at a distance $$l = 6.0\:km$$ from the point $$A$$. Find the flow velocity in $$km/h$$ assuming the duty of the engine to be constant.



Solve the inequation $$ -\left| y \right| +x\sqrt { \left( { x }^{ 2 }+{ y }^{ 2 }-1 \right)  } \ge 1$$. The sum of the ordinate and abscissa of the solution is 



Two boats, $$A$$ and $$B$$, move away from a buoy anchored at the middle of a river along the mutually perpendicular straight lines: the boat $$A$$ along the river, and the boat $$B$$ across the river. Having moved off an equal distance from the buoy the boats returned. The ratio of times of motion of boats $$\displaystyle\frac{\tau_A}{\tau_B}$$ is $$\dfrac{x}{10}$$ if the velocity of each boat with respect to water is $$\eta=1.2$$ times greater than the stream velocity. Find $$x$$ rounded off to the nearest integer.



A child in danger of drowning in a river is being carried downstream by a current that flows uniformly at a speed of $$2.5\ km/h$$. The child is $$0.6\ km$$ from shore and $$0.8\ km$$ upstream of a boat landing when a rescue boat sets out. If the boat proceeds at its maximum speed of $$20\ km/h$$ with respect to the water, how long (in $$min$$)does it take the boat to reach the child? 



Solve the inequality:

$$\displaystyle\frac { 3x+1 }{ x+4 } \le 1$$



Find the ratio compounded of
(1) the ratio 2a, 3b, and the duplicate of $$ 9b^2 : ab . $$
(2) the subduplicate ratio of 64 : 9, and the ratio 27 : 56.
(3) the duplicate ratio of $$ \frac {2a}{b} : \frac {\sqrt{C}a^2}{b^2} , $$ and the ratio 3ax : 2by . 



Solve the following equations: 
Find the least integral value of $$x$$ which satisfies the equation $$\left | x \, - \, 3 \right | \, + \, 2 \, \left | \, x \, + \, 1 \right | \, = \, 4.$$ 



Solve the following equation:
$$\left | x \right | \, - \, \left | \, x \, - \, 2\right | \, = \, 2$$



Solve the following equation:
 $$\left | x \, + \, 2 \, \right | \, = \, 2 \, (3 \, - \, x)$$



Solve the following equation:
$$\left | 3x \, - \, 2 \right | = \, 11$$



Prove the, inequality $$\displaystyle\, n^{n + 1} > (n + 1)^n, n \geqslant 3, n\, \epsilon \, N$$.



Solve the following inequalities.
$$\displaystyle 2^x \, + \, 3^x \, \geqslant \, 2.$$



Solve the following inequalities.
$$\displaystyle\, \frac{1}{x + 2} < \frac{3}{x - 3}$$



Let us consider a boat which moves with a velocity $$v_{bw}$$ = 5 km/h relative to water. At time t=0, the boat passes through a piece of code floating in water while moving downstream. If it turns back at time t=$$t_1$$ when and where does the boat meet the cork again? Assume $$t_1$$, = 30 min.



If a, b, c $$> 0$$, then prove that $$\dfrac{a^3}{b^3}+\dfrac{b^3}{c^3}+\dfrac{c^3}{a^3}\geq 3$$.



If a, b, c $$> 0$$ then prove $$(abc) \left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^3\geq 27$$.



A man can swim at a speed 2 m/s in still water. He starts swimming in a river at an angle 150 to the direction of water flow and reaches the directly the directly opposite point on the opposite bank.
(a) Find the speed of flowing water.
(b) If width od river is 1 km then calculate the time taken to cross the river.



Solve the following system of inequalities : $$\dfrac{2x+1}{7x-1} > 5$$



Solve the following:
$$|x - 1| + |x - 2| + |x - 3| \geq 6.$$



Show that $$\dfrac {a_{1}}{a_{2}} + \dfrac {a_{2}}{a_{3}} + \dfrac {a_{3}}{a_{4}} + ..... + \dfrac {a_{n - 1}}{a_{n}} + \dfrac {a_{n}}{a_{1}} > n$$, where $$a_{1}, a_{2}, ..., a_{n}$$ are different positive integers.



To cross the river in shortest distance, a swimmer should swim making an angle θθ with the upstream. What is the ratio of the time taken to swim across in the shortest time to that in swimming across over shortest distance. [Assume that the speed of swimmer in still water is greater than the speed of river flow]



The ratio of the number of stamps with Rahul to the number of stamps with Sahil is $$4:5$$. If Rahul has $$24$$ stamps, how many stamps does Sahil have?



The different between two whole numbers is $$66$$. The ratio of the the numbers is $$2:5$$. What are the two numbers?



Divide $$Rs.3450$$ among $$A.B$$ and $$C$$ in the ratio $$3:5:7$$.



A woman can row a boat at $$6.40 km/h$$ in still water. (a) If she is crossing a river where the current is $$3.20 km/h$$, in what direction must her boat be headed if she wants to reach a point directly opposite her starting point? (b) If the river is $$6.40 km$$ wide, how long will she take to cross the river? (c) Suppose that instead of crossing the river she rows $$3.20 km$$  down the river and then back to her starting point. How long will she take? (d) How long will she take to row $$3.20 km$$ up the river and then back to her starting point? (e) In what direction should she head the boat if she wants to cross in the shortest possible time, and what is that time? 



A river is flowing due east with a speed $$3m/s$$. A swimmer can swim in still water at a speed of $$4m/s$$ as shown in the figure.
(a) If swimmer starts swimming due north, what will be his resultant velocity (magnitude and direction)?
(b) If he wants to start from point A on the south bank and reaches the opposite point B on the north bank,
(i) which direction should be swim?
(ii) what will be his resultant speed?
(c) From two different cases are mentioned in (a) and (b) above, in which case will he reach opposite bank in the shorter time?
1794436_3e49ad67b63c43418de781f0a04be2ec.png



A boat installed with two engines commutes between two towns P and
Q connected by a river. The boat takes $$ t_1 = 50.00 $$min on its way from P
to Q with only one engine on and $$ t_2 = 25.00  $$min with both the engines
on. Force of water resistance is proportional to velocity of the boat
relative to the water. Assume water flow velocity to be uniform and the
boat always moves with respect to water with its maximum possible
velocity.

(a) What minimum time will the boat take on,its way from Q to P with
one engine on?

(b) What minimum time will the boat take on its way from Q to P with
both the engines on?



Solve the system of inequalities:$$\left| 2x-3 \right| \le 11,\left| x-2 \right| \ge 3$$



A woman who can row a boat at $$6.4 km/h$$ in still water faces a long, straight river with a width of $$6.4 km$$ and a current of $$3.2 km/h$$. Let $$\hat { i } $$ point directly across the river and $$\hat { j } $$point directly downstream. If she rows in a straight line to a point directly opposite her starting position, (a) at what angle to $$\hat { i } $$ must she point the boat and (b) how long will she take? (c) How long will she take if, instead, she rows $$3.2 km$$ down the river and then back to her starting point? (d) How long if she rows $$3.2 km$$ up the river and then back to her starting point? (e) At what angle to $$\hat { i } $$ should she point the boat if she wants to cross the river in the shortest possible time? (f) How long is that shortest time?



A $$200-m$$-wide river has a uniform flow speed of $$1.1 m/s$$ through a jungle and toward the east. An explorer wishes to leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of $$4.0 m/s$$ with respect to the water. There is a clearing on the north bank $$82 m$$ upstream from a point directly opposite the clearing on the south bank. (a) In what direction must the boat be pointed in order to travel in a straight line and land in the clearing on the north bank? (b) How long will the boat take to cross the river and land in the clearing?



Write down the $$3$$ inequalities which define the unshaded region.
1912501_84bd19848393422fbcfd332d816a3947.png



A fisherman sets out upstream on a river. His small boat, powered by an outboard motor, travels at a constant speed $$v$$ in still water. The water flows at a lower constant speed $$v_{w}$$. The fisherman has traveled upstream for $$2.00 km$$ when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another $$15.0 min$$. At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as he returns to his starting point. How fast is the river flowing? Solve this problem in two ways. (a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed $$v - v_{w}$$ and downstream at $$v + v_{w}$$. (b) A second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets.



Two swimmers, Chris and Sarah, start together at the same point on the bank of a wide stream that flows with a speed $$v$$. Both move at the same speed $$c$$ (where $$c > v$$) relative to the water. Chris swims downstream a distance $$L$$ and then upstream the same distance. Sarah swims so that her motion relative to the Earth is perpendicular to the banks of the stream. She swims the distance $$L$$ and then back the same distance, with both swimmers returning to the starting point. In terms of $$L, c$$, and $$v$$, find the time intervals required  (a) for Chriss round trip and (b) for Sarahs round trip.  (c) Explain which swimmer returns first.



Class 12 Commerce Applied Mathematics Extra Questions