Solved maths problems — page 12
1518 problems, newest first. Browse by chapter at NCERT solutions by class.
- In an A.P. the p th term is q and the (p+q)^ th term is 0. Then the q th term is (A) -p (B p (C) p+q (D) p-q
The q th term is p.
- आकृति 11.6 में दर्शाए गए वृत्तखंड का क्षेत्रफल ज्ञात कीजिए, यदि वृत्त की त्रिज्या 21 cm है और AOB = 120^° है। [ π = (22)/(7) लीजिए]
वृत्तखंड AYB का क्षेत्रफल 462 - (441√3)/(4) cm² है।
- (b) Given that P = 2 & -1 3 & 4, Q = 5 & 2 7 & 4 and R = 2 & 5 3 & 8, find a matrix S such that PQ - RS is a null matrix. (matrices, matrix multiplication, null
The matrix S is -191 & -110 77 & 44.
- एक समतल जमीन पर खड़ी मीनार की छाया उस स्थिति में 40 m अधिक लंबी हो जाती है जबकि सूर्य का उन्नतांश (altitude) 60^° से घटकर 30^° हो जाता है। मीनार की ऊँचाई ज्ञात
मीनार की ऊँचाई 20 m है।
- For a statistical data x_1, x_2,, x_10 of 10 values, a student obtained the mean as 5.5 and Σ_i=1^(10) x_i² = 371. He later found that he had noted two values i
10.9
- Find the equation of the tangent to the ellipse x²/16 + y²/9 = 1 that passes through the point (8, 0).
The equations of the tangents are y = (√3)/(4)x - 2√3 and y = -(√3)/(4)x + 2√3.
- Discuss the continuity of the function f given by f(x)=x³+x²-1.
The function f(x)=x³+x²-1 is continuous for all real numbers.
- The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observ
The correct mean is 39.9 and the correct standard deviation is 5.09.
- Solve 30 x<200 when (i) x is a natural number, (ii) x is an integer.
(i) x 1, 2, 3, 4, 5, 6 (ii) x, 4, 5, 6
- Check the continuity of the function f given by f(x)=2 x+3 at x=1.
The function f(x) = 2x+3 is continuous at x=1.
- Find the derivative of the function given by f(x)=sin (x²).
f'(x) = 2x cos(x²)
- If sin x=(3)/(5), cos y=-(12)/(13), where x and y both lie in second quadrant, find the value of sin (x+y).
-(56)/(65)
- Write the solution set of the equation x² + x - 2 = 0 in roster form.
1, -2
- Expand (x²+(3)/(x))^4, x ≠ 0
x^8 + 12x^5 + 54x² + (108)/(x) + (81)/(x^4)
- Find an equation of the circle with centre at (0,0) and radius r.
The equation of the circle with centre at (0,0) and radius r is x² + y² = r².
- Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14
The total number of students is 44.
- A fair coin is flipped repeatedly. What is the expected number of flips until the pattern HTH appears?
The expected number of flips until the pattern HTH appears is 10.
- Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle α with the positive x-axis and the equations of its dia
48
- There are 96 fourth-graders at Small Tree School. 43 of them are girls. On Friday, 5 fourth-grade girls and 4 fourth-grade boys were absent. How many fourth gra
There were 49 fourth-grade boys at Small Tree School on Friday.
- Let A=[a_ij]=[ cc log_5 128 & log_4 5 log_5 8 & log_4 25 ]. If A_ij is the cofactor of a_ij, C_ij=Σ_k=1² a_ik A_jk, 1 ≤ i, j ≤ 2, and C=[C_ij], then 8|C| is equ
242
- Sketch the graph of y = x² - 4 and find its roots.
The roots of the equation y = x² - 4 are x = 2 and x = -2. The graph is a parabola opening upwards with its vertex at (0, -4), intersecting the x -axis at (2, 0
- Find the extrema of f(x,y,z) = xyz subject to the constraint x² + 2y² + 3z² = 6 using Lagrange multipliers.
The maximum value is (2√3)/(3) and the minimum value is -(2√3)/(3).
- Let y = y(x) be the solution of the differential equation (xy - 5x² √(1 + x²)) dx + (1 + x²) dy = 0, y(0) = 0. Then y(√3) is equal to
(5√3)/(2)
- Let a = i + 2 j + k and b = 2 i + j - k. Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a. Then such a vector c is:
c = ± (1)/(√2) (- i + k)
- Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.
The pairs of consecutive odd natural numbers are (11, 13), (13, 15), (15, 17), and (17, 19).
- A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and
15
- Find the roots of the quadratic equation x² + 7x + 10 = 0 by factorisation.
The roots of the quadratic equation are x = -2 and x = -5.
- Jason works as a salesperson at a car dealership. He needs to sell 15 cars this month to earn a big bonus. He knows based on historical averages, that for every
Jason needs to make 750 telephone calls to sell 15 cars.
- If the equation a(b - c)x² + b(c - a)x + c(a - b) = 0 has equal roots, where a + c = 15 and b = (36)/(5), then a² + c² is equal to
a² + c² = 144.4
- The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is 30^° than when it is 60^°. Find the height of the tower.
The height of the tower is 20√3 m.
- Two angles of a triangle are 50 and 30. What is the 3rd?
The third angle is 100^°.
- The area of the region (x,y): x-y ≤ y ≤ 4√x is:
The area of the region is (1024)/(3).
- For α,β, R, if _x 0 x²sin(α x)+( -1)e^(x²)sin(2x)-β x=3, then β+ -α is equal to:
7
- Verify that 3, -1, -(1)/(3) are the zeroes of the cubic polynomial p(x) = 3x³ - 5x² - 11x - 3, and then verify the relationship between the zeroes and the coeff
The values 3, -1, -(1)/(3) are verified to be the zeroes of the polynomial p(x) = 3x³ - 5x² - 11x - 3. The relationships between the zeroes and the coefficients
- Find the shortest distance between the skew lines: r⃗₁ = (i + j + k) + λ(2i + 3j + 4k) and r⃗₂ = (2i + 3j + 5k) + μ(i + 2j + 3k).
The shortest distance between the skew lines is (√6)/(6) units.
- Find the value of P such that the difference of the roots of the equation x²-P x+8=0 is 2.
The value of P is ± 6.
- A 1.5m tall boy is standing at some distance from a 30m 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 19√3 m.
- Solve the quadratic equation using factorization: x² - 5x + 6 = 0.
The solutions are x = 2 and x = 3.
- Determine the AP whose 3rd term is 5 and the 7th term is 9.
The Arithmetic Progression (AP) is 3, 4, 5, 6,
- Let m and n be the number of points at which the function f(x)=maxx, x³, x^5,, x^(21), x R is not differentiable and not continuous, respectively. Then m + n is
3
- Find the equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes x + 2y + 3z = 4 and 2x - 3y + 4z = 5.
The equation of the plane is 17x + 2y - 7z - 42 = 0.
- Let A = θ [0, 2π]: 1 + 10 Re ( (2cosθ + isinθ)/(cosθ - 3isinθ)) = 0. Then Σ_θ A θ² is equal to:
(21π²)/(4)
- Discuss the continuity of the function defined by f(x)= r x+2, if x<0 -x+2, if x>0
The function f(x) is continuous for all x R except at x=0.
- Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff, if five different f
320
- If for θ [ -(π)/(3), 0 ], the points (x, y) = ( 3 tan(θ + (π)/(3)), 2 tan(θ + (π)/(6))) lie on xy + α x + β y + = 0, then α² + β² + ² is equal to:
337
- Find the area of a sector of a circle of radius 6 cm with a central angle of 60 degrees.
The area of the sector is 6π cm².
- In quadrilateral ABCD, A is a right angle, AB=(5)/(3), BC=8, CD=6, and AD=5. Are points A, B, C, D concyclic?
No, the points A, B, C, D are not concyclic.
- In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.
Since 2 sin A cos A = 1, the statement is verified.
- Find the value of n such that i) ^nP_5 =42 ^nP_3, n>4 ii) (^nP_4)/(^(n-1)P_4), n>4
(i) n=10, (ii) (n)/(n-4)
- Compute the derivative of 6 x^(100)-x^(55)+x.
600 x^(99) - 55 x^(54) + 1
- Find the arc length of the curve y = ln(sec x) from x = 0 to x = π/4.
The arc length of the curve is ln(√2 + 1).
- In a triangle ABC, if a = 18, b = 24, c = 30 find the value of sin(A/2) and cos(A/2).
sin(A/2) = (√10)/(10) and cos(A/2) = (3√10)/(10).
- Let A=[ cc(1)/(√2) & -2 0 & 1 ] and P=[ cccos θ & -sin θ sin θ & cos θ ], θ>0. If B=PAP^(T), C=P^(T) B^(10) P and the sum of the diagonal elements of C is (m)/(
65
- If the set of all values of a, for which the equation 5x³ - 15x - a = 0 has three distinct real roots, is the interval (α, β), then β - 2α is equal to
30
- Jamaar loves fresh fruit and is headed to the store with 10 he earned mowing lawns. Including tax, peaches and pears are.5 each, apples are.75 each, kiwis are 1
Jamaar can buy 10 plums.
- Let the triangle PQR be the image of the triangle with vertices (1,3), (3,1), and (2,4) in the line x + 2y = 2. If the centroid of PQR is the point (α, β), then
22
- In Tate’s garden pond, there are 4 male guppies, 7 female guppies, 3 male goldfishes, and 5 female goldfishes. He buys 2 male guppies, 1 female guppy, 2 male go
Tate has 5 more female fishes than male fishes.
- The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such num
The two-digit numbers are 42 and 24. There are 2 such numbers.
- Let the domain of the function f(x) = cos^(-1) ( (4x + 5)/(3x - 7)) be [α, β] and the domain of g(x) = log_2(2 - 6log_7(2x + 5)) be [, ]. Then |7(α + β) + 4( +)
| -102 + 2 · 7^(1/3) |
- A candidate has to reach the examination center in time. The probability that he travels by metro, bus, or taxi are 3/10, 1/5, and 1/10 respectively, and 2/5 by
The probability that he travelled by metro, given that he reached the center in time, is (32)/(107).