Solution: Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3}...

### If and be three points such that area of a is 4 sq units, find the value of .

Solution: Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3}...

### Find the value of for which the area of a ABC having vertices and is 35 sq units.

Solution: Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3}...

### Find the value of for which the points and are collinear.

### Find the value of for which the points and are collinear.

### Find the value of for which the points and are collinear.

### Use determinants to show that the following points are collinear. and

### Use determinants to show that the following points are collinear. and

### Use determinants to show that the following points are collinear. and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### Find the area of the triangle whose vertices are: and

### The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Determine their present ages.

let the present ages of father and his son be \[~x\text{ }years\text{ }and\text{ }\left( 45\text{ }\text{ }x \right)\text{ }years\] hence five years ago, Father’s age \[=\text{ }\left( x\text{...

### The sum S of first n even natural numbers is given by the relation S = n(n + 1). Find n, if the sum is 420.

According to ques, \[S\text{ }=\text{ }n\left( n\text{ }+\text{ }1 \right)\] Also, \[S\text{ }=\text{ }420\] So, \[n\left( n\text{ }+\text{ }1 \right)\text{ }=\text{ }420\] Or, \[{{n}^{2}}~+\text{...

### Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels 5 km/hr faster than the second train. If after 2 hours, they are 50 km apart, find the speed of each train.

Let the speed of the second train be \[~x\text{ }km/hr.\] Then, the speed of the first train is \[\left( x\text{ }+\text{ }5 \right)\text{ }km/hr\] Let O be the position of the railway station,...

### A plane left 30 minutes later than the schedule time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

Let the usual speed of the plane to be \[x\text{ }km/hr\] The distance to travel \[=\text{ }1500km\] since, Time = Distance/ Speed As the ques suggests, \[{{x}^{2}}~+\text{ }250x\text{ }\text{...

### Rs 6500 was divided equally among a certain number of persons. Had there been 15 persons more, each would have got Rs 30 less. Find the original number of persons.

let the original number of persons to be x. According to ques, Total money which was divided is \[=\text{ }Rs\text{ }6500\] Each person’s share is \[=\text{ }Rs\text{ }6500/x\] Then, as the question...

### An aeroplane travelled a distance of 400 km at an average speed of x km/hr. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for: (i) the onward journey; (ii) the return journey. If the return journey took 30 minutes less than the onward journey, write down an equation in x and find its value.

According to ques, Distance \[=\text{ }400\text{ }km\] Average speed of the airplane \[=\text{ }x\text{ }km/hr\] Also, speed while returning \[=\text{ }\left( x\text{ }+\text{ }40 \right)\text{...

### A hotel bill for a number of people for overnight stay is Rs 4800. If there were 4 people more, the bill each person had to pay, would have reduced by Rs 200. Find the number of people staying overnight.

Let the number of people staying overnight as x. According to ques, total hotel bill \[~=\text{ }Rs\text{ }4800\] Now,hotel bill for each person \[=\text{ }Rs\text{ }4800/x\] therefore,...

### A trader buys x articles for a total cost of Rs 600. (i) Write down the cost of one article in terms of x. If the cost per article were Rs 5 more, the number of articles that can be bought for Rs 600 would be four less. (ii) Write down the equation in x for the above situation and solve it for x.

According to ques, Number of articles \[=\text{ }x\] And, the total cost of articles \[=\text{ }Rs\text{ }600\] Again, (i) Cost of one article \[=\text{ }Rs\text{ }600/x\] (ii) also,...

### The distance by road between two towns A and B is 216 km, and by rail it is 208 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. Calculate: (iii) If the train takes 2 hours less than the car, to reach town B, obtain an equation in x and solve it. (iv) Hence, find the speed of the train.

(iii) According to the question, \[4x\text{ }+\text{ }1728\text{ }=\text{ }{{x}^{2}}~+\text{ }16x\] Or, \[{{x}^{2}}~+\text{ }12x\text{ }\text{ }1728\text{ }=\text{ }0\] Or, \[{{x}^{2}}~+\text{...

### The distance by road between two towns A and B is 216 km, and by rail it is 208 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. Calculate: (i) the time taken by the car to reach town B from A, in terms of x; (ii) the time taken by the train to reach town B from A, in terms of x.

According to ques, Speed of car = \[x\text{ }km/hr\] Speed of train = \[\left( x\text{ }+\text{ }16 \right)\text{ }km/hr\] Time = \[Distance/\text{ }Speed\] (i)Time taken by the car to reach town B...

### The age of the father is twice the square of the age of his son. Eight years hence, the age of the father will be 4 years more than three times the age of the son. Find their present ages.

We should expect the current age of the child to be x years. Thus, the current age of the dad = \[2x2\text{ }years\] Eight years henceforth, Child's age = \[\left( x\text{ }+\text{ }8 \right)\] a...

### One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.

ACCORDING TO QUES, How about we think about the current age of the child to be x years. Along these lines, the current age of the man \[=\text{ }x2\text{ }years\] One year prior, Child's age...

### The ages of two sisters are 11 years and 14 years. In how many years’ time will the product of their ages be 304?

Given, the ages of two sisters are 11 years and 14 years. Leave x alone the quantity of years some other time when their result of their ages become 304. In this way, \[\left( 11\text{ }+\text{ }x...

### The product of the digits of a two digit number is 24. If its unit’s digit exceeds twice its ten’s digit by 2; find the number.

How about we accept the ten's and unit's digit of the necessary number to be x and y individually. Then, at that point, from the inquiry we have \[x\text{ }\times \text{ }y\text{ }=\text{ }24\]...

### A stone is thrown vertically downwards and the formula d = 16t2 + 4t gives the distance, d metres, that it falls in t seconds. How long does it take to fall 420 metres?

As indicated by the inquiry, \[16t2\text{ }+\text{ }4t\text{ }=\text{ }420\] \[4t2\text{ }+\text{ }t\text{ }\text{ }105\text{ }=\text{ }0\] \[4t2\text{ }\text{ }20t\text{ }+\text{ }21t\text{ }\text{...

### The sum S of n successive odd numbers starting from 3 is given by the relation: S = n(n + 2). Determine n, if the sum is 168.

From the inquiry, we have \[n\left( n\text{ }+\text{ }2 \right)\text{ }=\text{ }168\] \[n2\text{ }+\text{ }2n\text{ }\text{ }168\text{ }=\text{ }0\] \[n2\text{ }+\text{ }14n\text{ }\text{ }12n\text{...

### A girl goes to her friend’s house, which is at a distance of 12 km. She covers half of the distance at a speed of x km/hr and the remaining distance at a speed of (x + 2) km/hr. If she takes 2 hrs 30 minutes to cover the whole distance, find ‘x’.

Given, The young lady covers a distance of 6 km at a speed x km/hr. Along these lines, the time taken to cover initial 6 km \[=\text{ }6/x\text{ }hr\] [Since, Time = Distance/Speed] Likewise given,...

### A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

We should accept x km/h to be the first speed of the vehicle. We realize that, Time = Distance/Speed From the inquiry, The time taken by the vehicle to finish 400 km = \[400/x\text{ }hrs\]...

### If the speed of an aeroplane is reduced by 40 km/hr, it takes 20 minutes more to cover 1200 km. Find the speed of the aeroplane.

How about we think about the first speed of the plane to be x km/hr. Presently, the time taken to cover a distance of \[1200\text{ }km\text{ }=\text{ }1200/x\text{ }hrs\] [Since, Time =...

### The speed of an ordinary train is x km per hr and that of an express train is (x + 25) km per hr. (i) Find the time taken by each train to cover 300 km. (ii) If the ordinary train takes 2 hrs more than the express train; calculate speed of the express train.

(i) Given, Speed of the conventional train \[=\text{ }x\text{ }km/hr\] Speed of the express train \[=\text{ }\left( x\text{ }+\text{ }25 \right)\text{ }km/hr\] Distance \[=\text{ }300\text{ }km\] We...

### The perimeter of a rectangle is 104 m and its area is 640 m2. Find its length and breadth.

We should take the length and the expansiveness of the square shape be x m and y m. Thus, the edge \[=\text{ }2\left( x\text{ }+\text{ }y \right)\text{ }m\] \[104\text{ }=\text{ }2\left( x\text{...

### The diagonal of a rectangle is 60 m more than its shorter side and the larger side is 30 m more than the shorter side. Find the sides of the rectangle.

How about we think about the more limited side of the square shape to be x m. Then, at that point, the length of the opposite side \[=\text{ }\left( x\text{ }+\text{ }30 \right)\text{ }m\] Length of...

### The hypotenuse of a right-angled triangle exceeds one side by 1 cm and the other side by 18 cm; find the lengths of the sides of the triangle.

Leave the hypotenuse of the right triangle alone x cm. From the inquiry, we have Length of one side \[=\text{ }\left( x\text{ }\text{ }1 \right)\text{ }cm\] Length of opposite side \[=\text{ }\left(...

### The sides of a right-angled triangle are (x – 1) cm, 3x cm and (3x + 1) cm. Find: (i) the value of x, (ii) the lengths of its sides, (iii) its area.

Given, The more drawn out side = \[Hypotenuse\text{ }=\text{ }\left( 3x\text{ }+\text{ }1 \right)\text{ }cm\] Furthermore, the lengths of other different sides are\[\left( x\text{ }\text{ }1...

### The hypotenuse of a right-angled triangle is 26 cm and the sum of other two sides is 34 cm. Find the lengths of its sides

Given, a right triangle \[Hypotenuse\text{ }=\text{ }26\text{ }cm\] and the amount of other different sides is \[34\text{ }cm.\] Presently, let believe the other different sides to be \[x\text{...

### The sides of a right-angled triangle containing the right angle are 4x cm and (2x – 1) cm. If the area of the triangle is 30 cm2; calculate the lengths of its sides.

Given, the space of triangle \[=\text{ }30\text{ }cm2\] As, x can't be negative, just \[x\text{ }=\text{ }3\] is substantial. Thus, we have \[AB\text{ }=\text{ }4\text{ }\times \text{ }3\text{...

### The sum of the squares of two consecutive positive even numbers is 52. Find the numbers.

Leave the two back to back certain even numbers alone taken as \[x\text{ }and\text{ }x\text{ }+\text{ }2.\] Now, \[x2\text{ }+\text{ }\left( x\text{ }+\text{ }2 \right)2\text{ }=\text{ }52\]...

### The sum of the squares of two positive integers is 208. If the square of larger number is 18 times the smaller number, find the numbers.

How about we expect the two numbers to be x and y, y being the bigger of the two numbers. Then, at that point, from the inquiry \[x2\text{ }+\text{ }y2\text{ }=\text{ }208\text{ }\ldots \text{...

### Divide 15 into two parts such that the sum of their reciprocals is 3/10

We should expect the two sections to be \[x\text{ }and\text{ }15\text{ }\text{ }x.\] now, \[150\text{ }=\text{ }45x\text{ }\text{ }3x2\] \[3x2\text{ }\text{ }45x\text{ }+\text{ }150\text{ }=\text{...

### Two natural numbers differ by 3. Find the numbers, if the sum of their reciprocals is 7/10.

How about we believe the two normal numbers to be\[x\text{ }and\text{ }x\text{ }+\text{ }3\] . (As they vary by 3) now, \[20x\text{ }+\text{ }30\text{ }=\text{ }7x2\text{ }+\text{ }21x\] \[7x2\text{...

### The sum of a number and its reciprocal is 4.25. Find the number.

Leave the number alone x. In this way, its complementary is 1/x now, \[4x2\text{ }\text{ }17x\text{ }+\text{ }4\text{ }=\text{ }0\] \[4x2\text{ }\text{ }16x\text{ }\text{ }x\text{ }+\text{...

### Find the two natural numbers which differ by 5 and the sum of whose squares is 97.

We should expect the two normal numbers to be\[x\text{ }and\text{ }x\text{ }+\text{ }5\] . (As given they contrast by 5) So from the inquiry, \[x2\text{ }+\text{ }\left( x\text{ }+\text{ }5...

### The sum of the squares of two consecutive natural numbers is 41. Find the numbers.

Allow us to take the two successive regular numbers as x and x + 1. So from the inquiry, \[x2\text{ }+\text{ }\left( x\text{ }+\text{ }1 \right)2\text{ }=\text{ }41\] \[2x2\text{ }+\text{ }2x\text{...

### The product of two consecutive integers is 56. Find the integers.

let two continuous numbers to be $$ \[x\text{ }and\text{ }x\text{ }+\text{ }1.\] So from the inquiry, \[x\left( x\text{ }+\text{ }1 \right)\text{ }=\text{ }56\] \[x2\text{ }+\text{ }x\text{...