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Solve for \(u \) and \(v \) by changing into linear equations \(2(3uv)=5uv, 2 (u+3v) = 5uv. \)
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..(1)
..(2)
Multiplying (1) by 3 and (2) by 1
….(3)
…(4)
Adding eqn (3) and (4)
For what values of \(a \) and \(b \) does the following pair of linear equations have an infinite number of solution : \(2x+3y=7 ; a(x+y)b(xy)=3a+b2\)
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Also
If \(4 \) times the area of a smaller square is subtracted from the area of a larger square, the result is \(144\ m^2\). The sum of the areas of the two squares is \(464\ m^2\). Determine the sides of the two square.
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Let be the area of smaller square and for larger
A/Q …(1)
….(2)
side of first square
side of larger square
Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test ?
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Let number of correct answer be and incorrect answer be
A/Q …(1)
…(2)
Subtracting (2) from (3) we get
Total number of equation
Six years hence a man’s age will be three times his son’s age and three years ago, he was nine times as old as his son. Find their present ages.
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Let age of man be and son be
A/Q
…(1)
Also
…(2)
years
Father’s age years and son’s age years
A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km downstream in 6 hours 30 minutes. Find the speed of the boat in still water.
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Let speed of boat in still water be km/hr and speed of stream km/hr
…(1)
Also …(2)
Let
on solving
on solving
Speed of stream = 4 km/hr, speed of boat is 10 km/hr
A person travles 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes 6 hours 30 minutes. But if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the car and that of the train.
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Let speed of train be km/hr and car be km/hr
A/Q …(1)
…(2)
eqn (2) …(3)
subtracting (3) from (2) we get
speed of train = 100 km/hr and speed of car = 80 km/hr
A lending library has a fixed charge for first three days and an additional charge for each day there after. Bhavya paid Rs. 27 for a book kept for seven days, while Vrinda paid Rs. 21 for a book kept for five days. Find the fixed charge and the charge for each extra day.
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Let fixed charge be and additional charge be
…(1)
…(2)
solving eqn (1) and (2)
So fixed charge = ₹ 15 and additional = ₹ 3
Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars ?
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Let speed of first car be km/hr and speed of second car be km/hr
A/Q
…(1)
Also …(2)
From (1)
speed of first car = 60 km/hr, second car = 20 km/hr
Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers.
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Let the numbers be and
…(1)
… (2)
eqn (2)
eqn (1)
solving two equation
Solve : \(\dfrac{b}{a}x+\dfrac{a}{b}y=a^2+b^2;x+y=2ab \)
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Multiplying (2) by
…(3)
Subtracting (3) from (1) we get
If \((x+3) \) is a factor of \(x^3+ax^2bx+6 \) and \(a+b=7 \), find the values of \( a\) and \(b \).
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is a factor
…(1)
….(2)
eqn (2)
If \((x+1) \) is a factor of \(2x^3+ax^2+2bx+1 \), then find the values of \(a \) and \(b \) given that \(2a+3b=4 \)
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..(1)
Also …(2)
eqn (1)
Sovle for \(x \) and \(y \) : \(\dfrac{x+y}{xy}=2,\dfrac{xy}{xy}=6 \)
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Let and
…(3)
…(4)
Adding equation (3) and (4)
Solve : \(\dfrac{2}{3x+2y}+\dfrac{3}{3x2y}=\dfrac{17}{5};\dfrac{5}{3x+2y}+\dfrac{1}{3x2y}=2 \)
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Let and
….(1)
…(2)
Multiplying eqn (2) by (3) and subtracting from (1) we get
and
on solving
\(\dfrac{x}{a}+\dfrac{y}{b}=a+b,\dfrac{x}{a^2}+\dfrac{y}{b^2}=2,a,b\ne0 \)
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….(1)
….(2)
Multiplying eqn (1) by
….(3)
subtracting eqn (3) from (2)
Putting in eqn (1)
\(ax+by=1,bx+ay=\dfrac{2ab}{a^2+b^2} \)
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….(1)
…(2)
Multiplying eqn (1) by
….(3)
Multiplying eqn (2) by
….(4)
subtracting eqn (4) from (3) we get
A two digit number is obtained by either multiplying the sum of the digits by 8 and adding 1, or by multiplying the difference of the digits by 13 and adding 2. Find the number. How many such numbers are there?
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Let unit digit be and ten’s digit be
Number
A/Q Two digit number
So
…(1)
Also
on solving eqn (1) and (2)
Required number = 41
A man wished to give Rs 12 to each person and found that he fell short of Rs 6 when he wanted to give to all persons. He therefore, distributed Rs 9 to each person and found that Rs 9 were left over. How much money Ldid he have and how many persons were there?
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Let number of persons be and the money the man has
A/Q …(1)
Also …(2)
There were 5 persons and man had Rs.
The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
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Let length be and breadth be
Area
A/Q
…(1)
Also
…(2)
on solving eqn (1) and (2)
so length of rectangle = 17 units and breadth = 9 units
A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from station A to B cots Rs 2530. Also, one reserved first class ticket and one reserved first class half ticket from A to B cost Rs 3810. Find the full first class fare from station A to B and also the reservation charges for a ticket.
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Let full fare and reservation charges
…(1)
Also
So full fare is Rs. 2500 and half fare is Rs. 30
It takes 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. How long would it take for each pipe to fill the pool separately?
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Let pipe A takes hours and pipe B takes hours
A/Q ….(1)
…(2)
Let
and
on solving we get
pipe A would take 20 hours and pipe B take 30 hours
Ages of two friends A and B differ by 3 years. A’s father D is twice as old as A, and B is twice as old as his sister C. Ages of C and D differ by 30 years. Find the ages of A and B.
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Let present age of A be years and B be years
…(1)
…(2)
Present age of A = 19 years , B = 16 years
In a rectangle, if the length is increased and breadth reduced each by 2 meters, the area is reduced by 28 sq. m. If the length is reduced by 1 m and the breadth is increased by 2 m, the area increases by 33 sq. m. Find the length and the breadth of the rectangle.
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Let length be and breadth be
original area
A/Q …(1)
…(2)
subtracting (1) from (2) we get
Length and breadth are 23 m and 11 m respectively
There are two classrooms A and B containing students. If 5 students are shifted from room A to room B, the resulting number of students in the two rooms become equal. If 5 students are shifted from room B to room A, the resulting number of students in room A becomes double the number of students left in room B. Find the original number of students in the two rooms.
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Let the number of students in the classroom A be and B be
…(1)
…(2)
on solving (1) and (2) we get
Number of students in room A = 35 and room B = 25
Students of a class are made to stand in rows. If four students are extra in each row, there would be two rows less. If 4 students are less in each row, there would be 4 more rows. Find the number of students in the class.
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Let be the number of students in a row and be the number of rows
A/Q …(1)
…(2)
Equation (1) becomes …(3)
Equation (2) becomes …(4)
solving eqn (3) and (4)
Number of children
A person invested some amount @ 12% simple interest and some other amount @ 10% simple interest. He received an yearly interest of Rs. 13000. But if he had interchanged the invested amounts, he would have received Rs. 400 more as interest. How much amount did he invest at different rates?
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Let the person invested ₹ at 12% simple interest and ₹ 10% simple interest
…(1)
Also
…(2)
2 men and 5 women can together finish a piece of work in 4 days, while 3 men and 6 women can finish it in 3 days. Find the time taken by 1 man alone to finish the work, and also that taken by 1 woman alone.
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Let man takes days and woman takes days
…(1)
…(2)
equation (1)
equation (2)
Draw the graphs of the lines x = 2 and y = 3. Write the vertices of the figure formed by these lines, the xaxis and the yaxis. Also, find the area of the figure.
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vertices are (0,0) (2, 3) (0, 3) (2, 0)
The figure formed is an rectangle
1 = 2 units
b = 3 units
Area of rectangle
Use a single graph paper to draw the graphs of 2y – x = 8, 5y – x = 14 and y – 2x = 1. Obtain the vertices of the triangle so obtained.
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and
or
or
or
The coordinates of the vertices of PQR are P(4, 2), Q(2, 5) and R(1, 3)
Solve graphically the following system of equations : x + 2y = 5, 2x – 3y = 4.
Also, find the points where the lines meet the xaxis.
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The system of equations have a unique solution, x = 1, y = 2
The line x + 2y = 5 meets the xaxis at (5, 0) and 2x – 3y = 4 meets the xaxis at (2, 0)
Draw the graph of the pair of equations 2x + y = 4 and 2x – y = 4. Write the vertices of the triangle formed by these lines and the yaxis. Also shade this triangle.
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Area sq units
Draw the graphs of the equations 4x – y – 8 = 0 and 2x – 3y + 6 = 0. Shade the region between two lines and xaxis. Also, find the coordinates of the vaetices of the triangle formed by these liens and the xaxis.
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vertices of
Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system: 2x – y = 2, 4x – y = 8.
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Now,
when , we have
when , we have
we have,
when , we have
when we have
Solve graphically the pair of linear equations : 3x + y + 1 = 0; 2x – 3y + 8 = 0
Write the coordinates of the vertices of the triangle formed by these lines with xaxis.
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Clearly two lines intersect at (1, 2).
Hence, is the solution of the given system of equations
A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km downstream in \(6\dfrac{1}{2} \) hours. Find the speed of boat in still water and also the speed of the stream.
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Let speed of boat in still water is km/hr and speed of stream is km/hr
A/Q …(1)
…(2)
Let
on solving
on solving
Speed of stream = 4 km/hr, speed of boat is 10 km/hr
8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in 14 days. Find the time taken by one man alone and that by one boy alone to finish the work.
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Let the taken by one man be and one boy be
A/Q and
Let
…(1) and …(2)
Multiplying eqn (1) by 3 and (2) by 4
one man can finish in 140 days and one boy in 280 days
Solve graphically the pair of linear equations : x – y = 1 and 2x + y – 10 = 0. Also find the area of the region bounded by these lines and xaxis.
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The vertices of triangle are (5, 0), (3, 4), (1, 0)
Solve for x and y : 6(ax + by) = 3a + 2b, 6(bx – ay) = 3b – 2a.
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…(1)
…(2)
Multiplying (1) by and (2) by
on subtracting in eqn (1) we get
Solve : \(\dfrac{44}{x+y}+\dfrac{30}{xy}=10;\dfrac{55}{x+y}+\dfrac{40}{xy}=13 \)
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Let
…(1)
…(2)
Multiplying eqn (1) by 4 and eqn (2) by 3
..(3)
…(4)
and on solving
Solve : \(\dfrac{1}{2(2x+3y)}+\dfrac{12}{7(3x+2y)}=\dfrac{1}{2};\dfrac{7}{(2x+3y)}+\dfrac{4}{3x+2y}=2 \), where \((2x+3y) \ \ne\ 0 \) and \((3x+2y)\ \ne\ 0 \)
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Let
…(1)
….(2)
Subtracting eqn (2) from (1)