Solved maths problems — page 11
1518 problems, newest first. Browse by chapter at NCERT solutions by class.
- Prove that the Gaussian integral ∫₋∞^∞ e^(-x²) dx = √π using polar coordinates.
∫_-∞^(∞) e^-x² dx = √(π)
- Prove that 3 sin (π)/(6) sec (π)/(3)-4 sin (5 π)/(6) cot (π)/(4)=1
The given equation is proven to be true, as the left-hand side simplifies to 1.
- Consider the identity function I_N: N → N defined as I_N(x)=x x N. Show that although I_N is onto but I_N+I_N: N → N defined as (I_N+I_N)(x)=I_N(x)+I_N(x)=x+x=2
The identity function I_N(x) = x is onto because its range is N, which is equal to its codomain. However, the function (I_N+I_N)(x) = 2x is not onto because its
- Let X₁, X₂,..., Xₙ be iid exponential random variables with parameter λ. Find the distribution of max(X₁, X₂,..., Xₙ).
The distribution of max(X_1, X_2,, X_n) has the following cumulative distribution function (CDF) and probability density function (PDF): F_Y(y) = (1 - e^(- y))^
- Let the function f(x)=(x²+1)|x²-ax+2|+cos|x| be not differentiable at the two points x=α=2 and x=β. Then the distance of the point (α, β) from the line 12x+5y+1
3
- If the set of all a R, for which the equation 2x²+(a-5)x+15=3a has no real root, is the interval (α, β), and X=x Z: α < x < β, then Σ_x X x² is equal to:
2139
- The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by (21)/(2).
4
- Find the solution to the heat equation ∂u/∂t = α²∂²u/∂x² with boundary conditions u(0,t) = u(L,t) = 0 and initial condition u(x,0) = sin(πx/L).
u(x,t) = sin((π x)/(L)) e^(-α² (π/L)² t)
- For some n ≠ 10, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of (1 + x)^(n+4) be in A.P. Then the largest coefficient in the ex
189
- If cos x=-(3)/(5), x lies in the third quadrant, find the values of other five trigonometric functions.
The values of the other five trigonometric functions are: sin x = -(4)/(5), tan x = (4)/(3), csc x = -(5)/(4), sec x = -(5)/(3), and cot x = (3)/(4).
- MODIFIED: Given that variable_91, variable_91, variable_91 are roots of x³ - 3x² + 4 = 0, find the value of variable_91⁴ + variable_91⁴ + variable_91⁴.
The value of α^4 + β^4 + ^4 is 33.
- Solve (5-2 x)/(3) ≤ (x)/(6)-5.
x ≥ 8
- If the function f(x) = (2)/(x) sin (k_1 + 1) x + sin (k_2 - 1) x, & x < 0 4, & x = 0 (2)/(x) log_e ( (2 + k_1 x)/(2 + k_2 x)), & x > 0 is continuous at x = 0, t
10
- A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tow
The height of the tower is 15√3 m.
- If the domain of the function f(x) = (1)/(√(10 + 3x - x²)) + (1)/(√(x + |x|)) is (a, b), then (1 + a)² + b² is equal to:
26
- Let for two distinct values of p the lines y = x + p touch the ellipse E: (x²)/(4)+(y²)/(3)=1 at the points A and B. Let the line y = x intersect E at the point
(20√6)/(7)
- Find (d y)/(d x), if x^((2)/(3))+y^((2)/(3))=a^((2)/(3))
(dy)/(dx) = -((y)/(x))^((1)/(3))
- Find the derivative of f(x)=(x+1)/(x)
f'(x) = -(1)/(x²)
- If the domain of the function log_5(18x-x²-77) is (α, β) and the domain of the function log_(α-1)((2x²+3x-2)/(x²-3x-4)) is (,), then α²+β²+ ² is equal to:
171
- Let A(6,8), B(10cosα, -10sinα) and C(-10sinα, 10cosα), be the vertices of a triangle. If L(a,9) and G(h,k) be its orthocenter and centroid respectively, then (5
145
- If the probability that the random variable X takes the value x is given by P(X=x)=k (x+1)3^(-x), x=0,1,2,, where k is a constant, then P(X 3) is equal to:
(1)/(9)
- Observe Fig. 6.30 and then find P. [Figure: ABC with AB = 3.8, BC = 6, CA = 3√3, A = 80°, B = 60°; and PQR with PQ = 12, QR = 7.6, RP = 6√3.]
P = 40^(°)
- Let V = a, e, i, o, u and B = a, i, k, u. Find V - B and B - V.
V - B = e, o and B - V = k
- A cylindrical tank with radius 2 m and height 5 m is filled with water. Find the work required to pump all the water to the top of the tank.
490000π J
- Considering the principal values of the inverse trigonometric functions, sin^(-1) ((√3)/(2)x + (1)/(2)√(1-x²)), -(1)/(2) < x < (1)/(√2) is equal to:
(π)/(6) + sin^(-1)x
- The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.
The sum of the first sixteen terms of the AP is 76 or 20.
- The largest n N such that 3^n divides 50! is:
22
- Prove that the relation R in the set A = 1, 2, 3, 4, 5 given by R = (a, b): |a - b| is even is an equivalence relation.
The relation R is an equivalence relation.
- In a right angled triangle, what are possible values of the other two angles?
The sum of the other two angles must be 90^°, and each of these angles must be greater than 0^° and less than 90^°.
- The marks distribution of 30 students in a mathematics examination are given in the table below. Find the mode of this data. |l|c|c| Class interval & Number of
The mode of the given data is 52.
- Gissela, Gordy, and Gary are truck drivers. Gissela has a truck large enough to haul 4,000 pounds of gravel. Gordy's truck can haul 800 pounds more than Gissela
Gary's truck can carry 2800 pounds of gravel.
- आकृति 6.31 में, OA · OB = OC · OD है। दर्शाइए कि A = C और B = D है।
चूंकि AOD COB, इसलिए A = C और B = D हैं।
- Solve the wave equation ∂²u/∂t² = c²∂²u/∂x² with initial conditions u(x,0) = f(x), ∂u/∂t(x,0) = g(x).
u(x,t) = (1)/(2)[f(x-ct) + f(x+ct)] + (1)/(2c)∫_x-ct^(x+ct) g() d
- Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
The equation of the line joining A(1,3) and B(0,0) is 3x - y = 0. The values of k for which the area of triangle ABD is 3 sq units are k=2 or k=-2.
- How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed?
3024
- Convert 40^(°) 20^() into radian measure.
(121π)/(540) radians
- Find the sum of the first 20 terms of the AP: 5, 8, 11, 14, and so on.
The sum of the first 20 terms of the given AP is 670.
- 30. If (sin x)^y = y^(cos x), then find (dy)/(dx). (differentiation, implicit differentiation, logarithmic differentiation, chain rule, trigonometric functions)
The derivative (dy)/(dx) is (-y(sin x ln y + y cot x))/(y ln(sin x) - cos x).
- The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is ((b+c-2 a)(c+a))/(2(b-a)).
The sum of the A.P. is ((b+c-2 a)(c+a))/(2(b-a)).
- If α > β > > 0, then the expression cot^(-1) β + ((1 + β^5))/((α - β)) + cot^(-1) + ((1 + ²))/((β -)) + cot^(-1) α + ((1 + α²))/(( - α)) is equal to:
0
- Let A=[a_ij] be a matrix of order 3 × 3, with a_ij=(√2)^(i+j). If the sum of all the elements in the third row of A² is α+β√2, α, β Z, then α+β is equal to:
224
- The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is
100
- The product of the last two digits of (1919)^(1919) is _____
63
- Let f(x) = log_e x and g(x) = (x^4 - 2x³ + 3x² - 2x + 2)/(2x² - 2x + 1). Then the domain of f ° g is
The domain of f ° g is (-∞, ∞) or R.
- One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will (i) be an ace, (ii) not be an ace.
(i) The probability that the card will be an ace is (1)/(13). (ii) The probability that the card will not be an ace is (12)/(13).
- Find the natural number a for which Σ_k=1^(n) f(a+k)=16(2^(n)-1), where the function f satisfies f(x+y)=f(x). f(y) for all natural numbers x, y and further f(1)
The natural number a is 3.
- The function f is defined by f(x)= 1-x, & x<0 1, & x=0 x+1, & x>0 Draw the graph of f(x).
The graph of f(x) consists of three parts: a line y=1-x for x<0, a point (0,1) for x=0, and a line y=x+1 for x>0. All three parts connect at the point (0,1).
- Evaluate the double integral ∬∫∫_D e^(x²+y²) dx dy where D is the disk x² + y² ≤ 4.
π (e^4 - 1)
- The pair of equations x = 3 and y = - 2 graphically represent lines which are:
Perpendicular to each other
- A rod of length eight units moves such that its ends A and B always lie on the lines x-y+2=0 and y+2=0, respectively. If the locus of the point P, that divides
23
- Let X = Ram, Geeta, Akbar be the set of students of Class XI, who are in school hockey team. Let Y = Geeta, David, Ashok be the set of students from Class XI wh
X Y = Ram, Geeta, Akbar, David, Ashok
- If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (-4,0), find the greatest and least values of |z+1|.
The greatest value of |z+1| is 6 and the least value is 0.
- If the system of linear equations 3x + y + β z = 3, 2x + α y - z = -3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β - 9α is:
31
- A family of parents and a child go to the cinema. The cost of an adult ticket is 12 and a child ticket is 8. Then they buy 2 popcorns for 3 each. How many dolla
They pay 38 in total.
- Find the value of 2 x^(4)+5 x³+7 x²-x+41, when x=-2-√3 i
6
- Find the area of the region bounded by the curves y² = 4x and x² = 4y.
The area of the region bounded by the curves y² = 4x and x² = 4y is (16)/(3) square units.
- Find the value of the definite integral: ∫_0^(π) (x sin(x))/(1 + cos²(x)) dx
(π²)/(4)
- Let A = 2 & 2 + p & 2 + p + q 4 & 6 + 2p & 8 + 3p + 2q 6 & 12 + 3p & 20 + 6p + 3q. If det(adj(adj(3A))) = 2^m · 3^n, m, n N, then m + n is equal to:
24
- Express (-√3+√(-2))(2 √3-i) in the form of a+i b
(-√3+√(-2))(2 √3-i) = (-6+√2) + i(√3+2√6)
- The distance of the line (x-2)/(2)=(y-6)/(3)=(z-3)/(4) from the point (1,4,0) along the line (x)/(4)=(y-2)/(2)=(z+3)/(3) is:
The distance is (√4221)/(8).