Solved maths problems — page 4
1518 problems, newest first. Browse by chapter at NCERT solutions by class.
- Evaluate the limit as n approaches infinity of sum from k=1 to n of n / (n² + k²).
(π)/(4)
- Find the local maximum and minimum values of f(x) = x * sqrt(4 - x²).
The local maximum value is 2 at x = √2 and the local minimum value is -2 at x = - √2.
- Find dy/dx if y = (ln(x))^(cos(x)).
(dy)/(dx) = (ln x)^(cos x) ( -sin x · ln(ln x) + (cos x)/(x ln x))
- 18 X +15 Y is equal to 3468 if X is equal to 2 or the value of Y
Y = 228.8
- 12 X +13 Y is equal to 140 FX is equal to 4 what is the value of Y
Y = (92)/(13)
- The radius of a circle is 6 cm what is it circumference and area
The circumference of the circle is 12π cm and the area is 36π cm².
- If sin A + sin B + sin C = cos A + cos B + cos C = 0, prove that sin² A + sin² B + sin² C = cos² A + cos² B + cos² C = 3/2.
sin² A + sin² B + sin² C = (3)/(2) and cos² A + cos² B + cos² C = (3)/(2).
- Evaluate the integral of e^(2x) * cos(3x) dx.
(1)/(13)e²x(2cos(3x) + 3sin(3x)) + C
- सिद्ध कीजिए कि सारणिक |[1, a, a²], [1, b, b²], [1, c, c²]| = (a-b)(b-c)(c-a).
सारणिक 1 & a & a² 1 & b & b² 1 & c & c² = (a-b)(b-c)(c-a) सिद्ध हुआ।
- यदि a = 2i + j + 3k और b = 3i + 5j - 2k तो a x b का परिमाण |a x b| ज्ञात कीजिए।
| a × b| = √507
- यदि sin(A+B) = 1 और cos(A-B) = sqrt(3)/2, जहाँ 0 < A+B <= 90, तो A और B का मान ज्ञात कीजिए।
A = 60^° और B = 30^°
- Solve: x³ - 6x² + 11x - 6 = 0 using the factor theorem.
The roots of the equation x³ - 6x² + 11x - 6 = 0 are x=1, x=2, and x=3.
- If the roots of x² + px + q = 0 are in the ratio 2:3, prove that 6p² = 25q.
The proof shows that if the roots of x² + px + q = 0 are in the ratio 2:3, then 6p² = 25q.
- Find the maximum value of f(x) = 2x³ - 9x² + 12x + 5 on the interval [0, 3].
The maximum value of f(x) on the interval [0, 3] is 14.
- Calculate the mean and variance of the first n natural numbers.
The mean of the first n natural numbers is (n+1)/(2) and the variance is (n²-1)/(12).
- Find the equation of the circle passing through points (1,2), (3,4), and (5,2).
The equation of the circle is x² + y² - 6x - 4y + 9 = 0.
- Find the sum of an infinite geometric series: 1 + 1/3 + 1/9 + 1/27 +...
The sum of the infinite geometric series is (3)/(2).
- If sin(A) = 3/5 and cos(B) = 12/13, find sin(A+B) and cos(A-B).
sin(A+B) = (56)/(65) and cos(A-B) = (63)/(65)
- Find the probability that in a random arrangement of the letters of the word MATHEMATICS, the two A's are not together.
The probability that the two A's are not together is (9)/(11).
- Evaluate the definite integral: integral from 0 to pi/2 of sin^4(x)*cos³(x) dx.
(2)/(35)
- Prove by mathematical induction that 1³ + 2³ + 3³... + n³ = [n(n+1)/2]².
By the principle of mathematical induction, 1³ + 2³ + 3³ + n³ = [(n(n+1))/(2)]² for all positive integers n.
- Find the angle between the vectors a = 2i + 3j + k and b = i - 2j + 3k.
The angle between the vectors is cos^(-1)(-(1)/(14)).
- Evaluate: lim(x->0) (sin(5x) - sin(3x)) / (sin(x)).
2
- Find the equation of the tangent and normal to the curve y = x³ - 3x + 2 at x = 1.
The equation of the tangent is y = 0. The equation of the normal is x = 1.
- Find the local maxima and local minima of the function f(x) = sin(x) + cos(x) on the interval [0, 2*pi].
The function has a local maximum at x = (π)/(4) with value √2, and a local minimum at x = (5π)/(4) with value -√2.
- Evaluate the definite integral of (x² + 1) / (x^4 + 1) dx from 0 to infinity.
(π)/(√2)
- A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle.
The area of the corresponding minor segment of the circle is ((308)/(3) - 49√3) cm².
- Determine the equation of the circle with radius 4 and Centre (-2, 3).
The equation of the circle is (x + 2)² + (y - 3)² = 16.
- 1. Definite Integral Property & Symmetry π ─── 2 √(sin x) Evaluate: ∫ ─────────────────── dx 0 √(sin x) + √(cos x)
I = (π)/(4)
- Emma wants to put a lace ribbon around the edge of a rectangular table. The table has a length of 12 cm and a width of 8 cm.How many centimeters of ribbon does
Emma needs 40 cm of ribbon in total.
- A bag contains 5 red balls and 7 black balls. Two balls are drawn at random one after another without replacement. Find the probability that both drawn balls ar
The probability that both drawn balls are red is (5)/(33).
- Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.
The ratio is 1:1 and the coordinates of the point of division are (-(3)/(2), 0).
- Prove that √5 is irrational and find whether 3/13 is a terminating or non-terminating repeating decimal.
√5 is irrational. (3)/(13) is a non-terminating repeating decimal.
- Evaluate the definite integral ∫_0^((π)/(2)) (√(sin x))/(√(sin x) + √(cos x)) dx.
The value of the definite integral is (π)/(4).
- Prove that (sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ.
(sin θ - 2sin³ θ)/(2cos³ θ - cos θ) = tan θ
- Solve the quadratic equation 3x² - 5x + 2 = 0 using the quadratic formula x = (-b ± √(b² - 4ac))/(2a).
The solutions to the quadratic equation 3x² - 5x + 2 = 0 are x = 1 and x = (2)/(3).
- ind the Taylor series expansion of f(x) = e^(x²) centered at x = 0 and determine its interval of convergence.
The Taylor series expansion of f(x) = e^x² centered at x = 0 is Σ_n=0^(∞) x²nn! = 1 + x² + (x^4)/(2!) + (x^6)/(3!) +. The interval of convergence is (-∞, ∞).
- How to point root 2 on number line
The point where the arc intersects the number line is √2.
- Calculate the area of an isosceles triangle with sides 10,10 and 23
A triangle with sides 10, 10, and 23 units cannot exist.
- Find whether 3/13 is rational or irrational without performing long division
The number (3)/(13) is a rational number.
- Find the area of a circle of radius 6 cm inscribed in a square of side 12 cm. (E2E Verification Run: 1784956549356)
The area of the circle is 36π cm².
- A circle of radius 6 cm is inscribed in a square. Find the area of the shaded region.
The area of the shaded region is (216)/(7) cm².
- A circle of radius 6 cm is inscribed in a Square. Find the area of the shaded region.
The area of the shaded region is (216)/(7) cm ².
- A line y = c reflects the quadratic y = x² - 4x + 7 to yield y = -x² + 4x + 1. Find c and the vertex of the new parabola.
The value of c is 4 and the vertex of the new parabola is (2, 5).
- Solve for x in the exponential equation: 2²x - 9 · 2^x + 8 = 0
The solutions for x are 0 and 3.
- Find the coordinates of a point P(x, y) lying on the line 2x + y = 6 such that the sum of its distances from the fixed points A(1, 1) and B(4, 5) is minimized
The coordinates of point P are ((59)/(20), (1)/(10)).
- In ABC, D is a point on side BC such that (BD)/(DC)=(3)/(2). If the area of ABD is 54 cm² answer the following: 1. Find the area of ACD. 2. Find the area of ABC
1. Area( ACD) = 36 cm² 2. Area( ABC) = 90 cm² 3. BE: BA = 3: 5 4. Area( BDE): Area( BAC) = 9: 25
- A circle of radius 9 cm is inscribed in a square. Find the area of the shaded region. Use π = (22)/(7).
The area of the shaded region is (486)/(7) cm².
- Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 times angle OPQ.
PTQ = 2 OPQ
- A contractor plans to install a slide for children under 5. The top of the slide is at a height of 1.5 m, and it is inclined at an angle of 30° to the ground. W
The length of the slide should be 3 m.
- Two chords AB and CD of a circle intersect at a point P inside the circle. Prove that AP × PB = CP × PD. State the theorem that justifies your answer to part (2
The proof shows that AP × PB = CP × PD. This is justified by the Theorem of Intersecting Chords.
- A circle with centre O has two chords AB and CD such that AB = CD. The chords intersect at a point P inside the circle. It is given that: * AP = 4 cm * PB = 5 c
The problem statement contains inconsistent information because the given segment lengths (AP=4 cm, PB=5 cm, CP=2.5 cm) and the condition AB=CD cannot be simult
- If (1,2), (4,y), (x,6), and (3,5) are the vertices of a parallelogram taken in order, find x and y.
The values are x=6 and y=3.
- A 1.5m tall boy is standing at some distance from a 32m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60°
The distance the boy walked towards the building is (61√3)/(3) m.
- A circle of radius 7 cm is inscribed in a square. Find the area of the shaded region. Use π = (22)/(7).
The area of the shaded region is 42 cm².
- PQ is a chord of length 9cm of a circle of radius 6 cm. The tangents to the circle at points P and Q intersect each other at an external point T. Find the lengt
The length of tangent TP is (18√7)/(7) cm.
- If point A(x, y) is equidistant from B(3, 6) and C(-3, 4), find a direct linear relation connecting x and y.
The linear relation connecting x and y is 3x + y - 5 = 0.
- PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at points P and Q intersect each other at an external point T. Find the leng
The length of tangent TP is (20)/(3) cm.
- The line segment joining the points A(3, 2) and B(6, -7) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line 2x - y + k =
The value of k is -9.
- The line segment joining the points A(2, 1) and B(5, -8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line 2x - y + k =
The value of k is -8.