Solved maths problems — page 31
1829 problems, newest first. Browse by chapter at NCERT solutions by class.
- Let a_n be the n th term of an A.P. If S_n = a_1 + a_2 + + a_n = 700 for some n, a_6 = 7 and S_7 = 7, then a_n is equal to:
64
- The number of complex numbers z, satisfying |z| = 1 and |(z)/(2) + (2)/(z)| = 1, is:
0
- If the system of equations 2x + y + 3z = 5 3x + 2y - z = 7 4x + 5y + z = 9 has infinitely many solutions, then ( ² + ²) is equal to:
29
- Let the values of for which the shortest distance between the lines (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and (x -)/(3) = (y - 4)/(4) = (z - 5)/(5) is (1)/(√6) be _
(5√2)/(3)
- Find all positive integers n such that φ(n) = n/3, where φ is Euler's totient function.
All positive integers n such that (n) = n/3 are of the form n = 2^a · 3^b, where a and b are positive integers (a ≥ 1, b ≥ 1).
- If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of
757
- Let the length of a latus rectum of an ellipse (x²)/(a²) + (y²)/(b²) = 1 be 10. If its eccentricity is the minimum value of the function f(t) = t² + t + (11)/(1
126
- A manufacturer has 600 litres of a 12 % solution of acid. How many litres of a 30 % acid solution must be added to it so that acid content in the resulting mixt
The manufacturer must add more than 120 litres but less than 300 litres of the 30% acid solution.
- On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits (i) A before B? (iii) A first and B last? (
(i) 1/2 (ii) 1/6 (iii) 1/12 (iv) 1/2 (v) 1/4
- Let the system of equations: 2x + 3y + 5z &= 9, 7x + 3y - 2z &= 8, 12x + 3y - (4 +)z &= 16 - have infinitely many solutions. Then the radius of the circle centr
The radius of the circle is (7)/(5).
- Find the derivative of f(x)=x².
The derivative of f(x)=x² is f'(x)=2x.
- Consider the hyperbola (x²)/(a²) - (y²)/(b²) = 1 having one of its foci at P = (-3, 0). If the latus rectum through its other focus subtends a right angle at P
α = 810, β = 1134
- 38. A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below: Based on give
(i) y = ±(3)/(4)√(16 - x²) (ii) (3)/(4)[(x)/(2)√(16 - x²) + 8sin^(-1)((x)/(4))] + C (iii) (a) The area of the region enclosed within the elliptical ground exclu
- OPQ में, जिसका कोण P समकोण है, OP = 7 cm और OQ - PQ = 1 cm, sin Q और cos Q के मान ज्ञात कीजिए।
sin Q = (7)/(25) और cos Q = (24)/(25)
- Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i + 2 j + k, i + 3 j - 2 k and 2 i + j - k respectively. The altitude from the vert
The position vector of E is (13)/(9) i + 2 j - (11)/(9) k.
- Show that f: N → N, given by f(x)= & x+1, if x is odd, & x-1, if x is even is both one-one and onto. The image of 1 and -1 under f is 1. Fig 1.4 Solution Suppos
The function f: N → N given by f(x)= x+1, & if x is odd x-1, & if x is even is both one-one and onto.
- Find the distance between the points (3, 4) and (-3, -4).
The distance between the points (3, 4) and (-3, -4) is 10 units.
- If ∫_-(π)/(2)^((π)/(2)) (25 x² cos² x)/(2(x + e)²) dx = π (α π² + β), α, β Z, then (α + β)² equals
The problem statement appears to be incorrect or have a typo, as the integral does not simplify to the given form using standard techniques.
- Find the equation of the circle with centre (-3,2) and radius 4.
(x+3)² + (y-2)² = 16
- Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.
The value of x is 4.
- If Σ_r=1^(30) r³ (^(30)C_r)²^(30)C_r-1 = α × 2^(29), then α is equal to:
α = 13080
- Let a sample space be S= _1, _2,, _6. Which of the following assignments of probabilities to each outcome are valid? ccccccc Outcomes & _1 & _2 & _3 & _4 & _5 &
Assignments (a) and (b) are valid.
- Roy has saved 40% more in money earned by chores than his brother Anthony. Anthony has saved 10.00 more than their sister Eva. Eva has saved 20.00. How much mon
Roy has 42.00.
- Let A = cosθ & 0 & -sinθ 0 & 1 & 0 sinθ & 0 & cosθ. If for some θ (0,π), A² = A T, then the sum of the diagonal elements of the matrix (A + I)³ + (A - I)³ - 6A
6
- Define the real valued function f: R-0 → R defined by f(x)=(1)/(x), x R-0. Complete the Table given below using this definition. What is the domain and range of
The completed table is: |l|c|c|c|c|c|c|c|c|c| x & -2 & -1.5 & -1 & -0.5 & 0.25 & 0.5 & 1 & 1.5 & 2 y=(1)/(x) & -0.5 & -0.67 & -1 & -2 & 4 & 2 & 1 & 0.67 & 0.5 T
- Find all the points of discontinuity of the function f defined by f(x)= rr x+2, & if x<1 0, & if x=1 x-2, & if x>1
The function f(x) is discontinuous at x=1.
- Find the distance between the parallel lines 3x + 4y - 7 = 0 and 3x + 4y + 23 = 0.
The distance between the parallel lines is 6 units.
- Find the radius of curvature for the curve y = e x at the point (0,1).
The radius of curvature for the curve y = e^x at the point (0,1) is 2 √2.
- Let ⌊·⌋ denote the greatest integer function. If ∫_0^e³ Bigl (1)/(e^(x-1)) Bigr dx = α - ln 2, then α³ is equal to:
8