Solved maths problems — page 5
1518 problems, newest first. Browse by chapter at NCERT solutions by class.
- The vertices of a triangle are A(-3, 5), B(7, 1), and C(3, 9). 1. Find the coordinates of the midpoint M of BC. 2. Show that AM is not a median of an equilatera
1. The coordinates of the midpoint M of BC are (5, 5). 2. The lengths of the sides are AB = √116, BC = √80, and AC = √52. Since the side lengths are not equal,
- The vertices of a triangle are A(5, -1), B(-3, -2), and C(1, 8). Find the length of the median drawn from vertex A to side BC, and determine the coordinates of
The length of the median drawn from vertex A to side BC is 2√13 units, and the coordinates of the centroid of ABC are (1, (5)/(3)).
- In ABC, AD is the bisector of A, meeting side BC at D. If AB = 12 cm, AC = 18 cm, and BC = 10 cm, find the lengths of segments BD and DC.
The length of segment BD is 4 cm and the length of segment DC is 6 cm.
- If the points A(x, y), B(2, 4), and C(4, 2) are vertices of a triangle whose area is 9 sq. units, show that: x+y=15 or x+y=-3
The relationships are x+y=15 or x+y=-3.
- In ABC, AD is the bisector of A, meeting side BC at D. If AB = 10 cm, AC = 14 cm, and BC = 6 cm, find the lengths of segments BD and DC.
The length of segment BD is 2.5 cm and the length of segment DC is 3.5 cm.
- If the points A(x, y), B(1, 3), and C(3, 1) are vertices of a triangle whose area is 8 sq. units, show that: x+y=12 or x+y=-4
The relationship between x and y is x+y=12 or x+y=-4.
- If the points A(x, y), B(2, 3), and C(3, 2) are vertices of a triangle whose area is 7 sq. units, show that: x+y=19 or x+y=-9
The relationship between x and y is x+y=19 or x+y=-9.
- In Δ ABC, AD is the bisector of A, meeting side BC at D. If AB = 10 cm, AC = 14 cm, and BC = 6 cm, find the lengths of segments BD and D
The length of segment BD is 2.5 cm and the length of segment DC is 3.5 cm.
- If the points A(x, y), B(1, 2), and C(2, 1) are vertices of a triangle whose area is 6 sq. units, show that: x+y=15 or x+y=-9
x+y=15 or x+y=-9
- A train covered a certain distance at a uniform speed. If the train had been 10 km/h faster, it would have taken 2 hours less than the scheduled time. If the tr
The distance covered by the train is 600 km.
- The angle of elevation of a cloud from a point h meters above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the height
The height of the cloud is (h(tan β + tan α))/(tan β - tan α)
- Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed
The canal will irrigate an area of 562500 m² in 30 minutes.
- If the sum of the first m terms of an AP is the same as the sum of its first n terms (m ≠ n), show that the sum of its first (m + n) terms is: S_m+n = 0
S_m+n = 0
- Find the coordinates of the point dividing the line segment joining A(3, 5) and B(9, 11) internally in the ratio 2: 1.
The coordinates of the point dividing the line segment are (7, 9).
- A vertical pole 12 m high casts a shadow 5√3 m long. Find the angle of elevation of the Sun.
The angle of elevation of the Sun is tan^(-1)((4√3)/(5)).
- Heights and Distances From a point on the ground, the angle of elevation of the top of a tower is 45°. If the point is 30 m from the base of the tower, find the
The height of the tower is 30 m.
- Trigonometry If sin θ = (5)/(13) and θ is acute, find: * cos θ * tan θ
cos θ = (12)/(13), tan θ = (5)/(12)
- 36. Find the number of ways in which six people can ride a toboggan if one of a subset of three must drive. 37. (a) Find the number of ways in which five person
The number of ways in which six people can ride a toboggan if one of a subset of three must drive is 360.
- In an obtuse-angled ABC (obtuse at B), AD is perpendicular to CB produced. Prove that AC² = AB² + BC² + 2BC × BD.
AC² = AB² + BC² + 2BC × BD
- Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
The length of the chord of the larger circle which touches the smaller circle is 8 cm.
- A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengt
The lengths of the sides are AB = 15 cm and AC = 13 cm.
- Construct a triangle ABC with AB = 5 cm, BC = 6 cm and angle B = 60° and find its area
The area of the triangle ABC is 7.5√3 cm ².
- Find the mean of the following frequency distribution: class intervals 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6.
The mean of the given frequency distribution is 27.
- Find the zeroes of the polynomial p(x) = x² - 5x + 6 and verify the relationship between the zeroes and the coefficients.
The zeroes of the polynomial p(x) = x² - 5x + 6 are x=2 and x=3. The relationship between the zeroes and coefficients is verified as: Sum of zeroes = 2+3=5 and
- A shopkeeper buys a book for ₹250 and sells it at a profit of 20%. A customer buys 3 such books. How much does the customer pay in total, in ₹?
The customer pays ₹900 in total.
- Prove that (1 + tan² A) / (1 + cot² A) = tan² A.
(1 + tan² A)/(1 + cot² A) = tan² A
- Find the roots of the quadratic equation 2x² - 7x + 3 = 0 using the quadratic formula, and state the value of the discriminant.
The roots of the quadratic equation are x = 3 and x = (1)/(2). The discriminant is 25.
- Prove that √5 is irrational and hence show that 3 + 2 √5 is irrational.
Therefore, 3 + 2√5 is irrational.
- Area of circle is 32 cm² what were the radius of the circle be?
The radius of the circle is √((32)/(π)) cm or approximately 3.19 cm.
- From an external point P a tangent PT is drawn to a circle of radius 8 cm. If the distance OP from the centre is 17 cm, find the length of the tangent PT.
The length of the tangent PT is 15 cm.
- The radius of a circle is 12 cm long what would its circumference be
The circumference of the circle is 24π cm.
- Q1: Check whether the following are quadratic equations: (i) (x+1)² = 2(x-3) Sol: x²+1+2x = 2x-6 x²+1+2x-2x-6=0 x²+7=0 It is in the form of ax²+bx+c=0. Hence th
Both given equations are quadratic equations.
- A toy is in the shape of a cone mounted on a hemisphere, both of radius 7 cm. The total height of the toy is 21 cm. Find the volume of the toy. (Use π = 22/7.)
The volume of the toy is (4312)/(3) cm³ or approximately 1437.33 cm³.
- Find the coordinates of the point that divides the line segment joining P(1, 2) and Q(7, 8) internally in the ratio 1: 2.
The coordinates of the point are (3, 4).
- Divide x⁴ + 2x³ + 2x² + 2x + 2 by (x² − 1), and write down the quotient and the remainder.
Quotient: x² + 2x + 3, Remainder: 4x + 5
- Prove the identity: (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ) = 2 cosec θ.
The identity (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ) = 2 cosec θ is proven.
- Find the 20th term of the AP 7, 11, 15, 19,...
The 20th term of the AP is 83.
- Solve using the quadratic formula: 2x² + 5x − 3 = 0.
The solutions are x = (1)/(2) and x = -3.
- Prove that √3 is irrational.
√3 is irrational.
- Find the compound interest on ₹10,000 at 10% per annum for 2 years, compounded annually.
The compound interest is ₹2,100.
- Quadratic Equation Any equation is in the form of ax² + bx + c = 0 where a ≠ 0 is a quadratic equation. Q1: Check whether the following are quadratic equations:
Yes, the given equation (x+1)² = 2(x-3) is a quadratic equation.
- A stadium has 48750 seats. During a concert 3/5 of the seats were filled, and another 6200 people entered. How many seats remained empty?
13300 seats remained empty.
- A stadium has 48750 seats. During a concert 3/5 of the seats were filled in the first hour, and another 6200 people entered in the second hour. How many seats r
13300 seats remained empty after the second hour.
- Find the distance between the points A(5, 9) and B(3, 11).
The distance between the points A(5, 9) and B(3, 11) is 2√2 units.
- Calculate the area of a rectangle with a length of 12 cm and a width of 4 cm.
The area of the rectangle is 48 cm².
- 4. [HCF/LCM reasoning] Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 a.m., at what time will they next ri
The bells will next ring together at 10:12 a.m.
- 1. A stadium has 48,750 seats. During a concert 3/5 of the seats were filled in the first hour, and another 6,200 people entered in the second hour. How many se
13,300 seats remained empty after the second hour.
- 2. Calculate the area of a rectangle with a length of 12 cm and a width of 4 cm.
The area of the rectangle is 48 cm ².
- Two rectangles have the same perimeter of 24 cm. One of them is 8 cm by 4 cm. Give the dimensions of a different rectangle with the same perimeter, and compare
A different rectangle with the same perimeter of 24 cm could have dimensions of 10 cm by 2 cm. The area of the first rectangle is 32 cm ², and the area of the s
- हिंदी में जवाब दो: Priya buys items costing ₹45.75, ₹120.50 and ₹8.25. She pays with a ₹200 note. How much change should she receive?
प्रिया को ₹25.50 का बदलाव मिलना चाहिए।
- हिंदी में जवाब दो: In a bag, red and blue marbles are in the ratio 2: 3. If there are 12 red marbles, how many blue marbles are there and how many marbles are t
नीले कंचों की संख्या 18 है और कुल कंचों की संख्या 30 है।
- In a bag, red and blue marbles are in the ratio 2: 3. If there are 12 red marbles, how many blue marbles are there and how many marbles are there in total?
There are 18 blue marbles and a total of 30 marbles.
- हिंदी में जवाब दो: A submarine is at a depth of 120 m below sea level. It rises by 45 m and then descends by 30 m. At what depth is it now?
The submarine is now at a depth of 105 m below sea level.
- हिंदी में जवाब दो: Find all the common factors of 24 and 36. Which of these is the highest common factor?
The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12. The highest common factor is 12.
- हिंदी में जवाब दो: The floor of a room measures 6 m by 4 m. Square tiles of side 50 cm are used to cover it. How many tiles are needed?
96 tiles are needed.
- हिंदी में जवाब दो: Can a triangle have two right angles? Explain your reasoning.
No, a triangle cannot have two right angles.
- हिंदी में जवाब दो: A tank holds 500 litres and is 3/5 full. If 90 litres more are added, how many more litres are still needed to fill the tank completely?
टैंक को पूरी तरह से भरने के लिए 110 लीटर पानी की और आवश्यकता होगी।
- A tank holds 500 litres and is 3/5 full. If 90 litres more are added, how many more litres are still needed to fill the tank completely?
110 litres are still needed to fill the tank completely.
- हिंदी में जवाब दो: Mark the following on a number line and state which is the greatest: −4, 2, −1, 3 and 0.
The greatest number is 3.
- A chocolate bar is divided into 12 equal squares. Shade 3/4 of the bar. How many squares are shaded?
9 squares are shaded.