Ncert Solutions Maths class 12th
Get insights from 2.5k questions on Ncert Solutions Maths class 12th, answered by students, alumni, and experts. You may also ask and answer any question you like about Ncert Solutions Maths class 12th
- 2.5k
Questions
- 0
Discussions
- 0
Active Users
- 65
Followers
Have a question about Ncert Solutions Maths class 12th? Ask our experts
Keep it short & simple. Type complete word. Avoid abusive language.
If (a + 3b) is perpendicular to (7a - 5b) and (a - 4b) is perpendicular to (7a - 2b), then the angle between a and b (in degrees) is…….
If (a + 3b) is perpendicular to (7a - 5b) and (a - 4b) is perpendicular to (7a - 2b), then the angle between a and b (in degrees) is…….
(a+3b). (7a-5b) = 7|a|² - 5ab + 21ab - 15|b|² = 7|a|²+16ab-15|b|²=0.
(a-4b). (7a-2b) = 7|a|² - 2ab - 28ab + 8|b|² = 7|a|²-30ab+8|b|²=0.
Subtracting: 46ab - 23|b|² = 0 ⇒ 2ab = |b|².
Substituting: 7|a|² + 8|b|² - 15|b|² = 0 ⇒ 7|a|² = 7|b|² ⇒ |a|=|b|.
cosθ = ab/ (|a|b|) = ab/|b|² = (1/2)|b|²/|b|² = 1/2.
θ
Upvote 
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible function f: S → S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to ________.
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible function f: S → S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to ________.
f (1)=1.
f (4)=f (2)²=1 or 4.
f (6)=f (2)f (3).
Possible functions determined by values at primes: f (2), f (3), f (5), f (7).
f (2) can be 1 or 2. f (3) can be 1 or 3. f (5)=1,5. f (7)=1,7.
If f (m)=m, f (mn)=mn. One function. f (x)=1 is another.
What if f (2)=1, f (3)=3? f (6)=3.
Upvote If the lines (x-k)/1 = (y-2)/2 = (z-3)/3 and (x+1)/3 = (y+2)/2 = (z+3)/1 are co-planar, then the value of k is…….
If the lines (x-k)/1 = (y-2)/2 = (z-3)/3 and (x+1)/3 = (y+2)/2 = (z+3)/1 are co-planar, then the value of k is…….
Determinant of vectors must be zero. Vector between points on lines: (-1-k, -2-2, -3-3). Vector directions: (1,2,3) and (3,2,1).
| -1-k, -4, -6; 1, 2, 3; 3, 2, 1 | = 0.
(-1-k) (2-6) - (-4) (1-9) + (-6) (2-6) = 0.
4 (1+k) - 32 + 24 = 0.
4+4k - 8 = 0. 4k=4 ⇒ k=1.
Upvote Let a = i + j + k, b and c = j - k be three vectors such that a × b = c and a . b = 1. If the length of projection vector of the vector b on the vector a × c is l, then the value of 3l² is equal to ________.
Let a = i + j + k, b and c = j - k be three vectors such that a × b = c and a . b = 1. If the length of projection vector of the vector b on the vector a × c is l, then the value of 3l² is equal to ________.
a×b=c ⇒ a.c=0, b.c=0.
|c|² = |a|²|b|² - (a.b)² = (3)|b|² - 1. |c|=√2. So |b|²=1, |b|=1.
Projection of b on a×c.
a×c = a× (a×b) = (a.b)a - (a.a)b = a - 3b.
|a-3b|² = |a|²+9|b|²-6 (a.b) = 3+9-6 = 6.
l = |b. (a-3b)|/|a-3b| = | (a.b)-3|b|²|/√6 = |1
Upvote Let a plane P pass through the point (3, 7, -7) and contain the line, (x-2)/(-3) = (y-3)/2 = (z+2)/1. If distance of the plane P from the origin is d, then d² is equal to ________.
Let a plane P pass through the point (3, 7, -7) and contain the line, (x-2)/(-3) = (y-3)/2 = (z+2)/1. If distance of the plane P from the origin is d, then d² is equal to ________.
Vector on plane: (3-2, 7-3, -7- (-2) = (1,4, -5).
Line direction vector (-3,2,1).
Normal to plane n = (1,4, -5)× (-3,2,1) = (14,14,14) or (1,1,1).
Plane: 1 (x-3)+1 (y-7)+1 (z+7)=0 ⇒ x+y+z-3=0.
d = |-3|/√3 = √3. d²=3.
Upvote If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a⁴ + b⁴ + c⁴ is equal to…….
If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a⁴ + b⁴ + c⁴ is equal to…….
(a+b+c)² = a²+b²+c²+2 (ab+bc+ca)
1² = a²+b²+c²+2 (2) ⇒ a²+b²+c² = -3.
a²b²+b²c²+c²a² = (ab+bc+ca)² - 2abc (a+b+c) = 2² - 2 (3) (1) = -2.
a? +b? +c? = (a²+b²+c²)² - 2 (a²b²+b²c²+c²a²) = (-3)² - 2 (-2) = 9+4=13.
Upvote Let f: R → R be a function such that f(2) = 4 and f'(2) = 1. Then the value of lim(x→2) (x²f(2) - 4f(x))/(x-2) is equal to:
Let f: R → R be a function such that f(2) = 4 and f'(2) = 1. Then the value of lim(x→2) (x²f(2) - 4f(x))/(x-2) is equal to:
Using L'Hopital Rule:
lim (x→2) (2xf (2) - 4f' (x)/1 = 2 (2)f (2) - 4f' (2) = 4 (4) - 4 (1) = 12.
Upvote The value of the definite integral ∫₋π/₄π/₄ dx / ((1+e^(xcosx))(sin⁴x + cos⁴x)) is equal to:
The value of the definite integral ∫₋π/₄π/₄ dx / ((1+e^(xcosx))(sin⁴x + cos⁴x)) is equal to:
I = ∫? π/? π/? dx/ (1+e^ (xcosx) (sin? x+cos? x). Using ∫? f (x)dx = ∫? f (a+b-x)dx. a+b=0.
I = ∫? π/? π/? dx/ (1+e? ) (sin? x+cos? x) = ∫? π/? π/? e? dx/ (e? +1) (sin? x+cos? x).
2I = ∫? π/? π/? dx/ (sin? x+cos? x) = 2∫? π/? dx/ (sin? x+cos? x).
I = ∫? π/? sec? xdx/ (tan? x+1). Let t=tanx.
I = ∫? ¹ (t²
Upvote The probability that a randomly selected 2-digit number belongs to the set {n ∈ N : (2ⁿ – 2) is a multiple of 3} is equal to:
The probability that a randomly selected 2-digit number belongs to the set {n ∈ N : (2ⁿ – 2) is a multiple of 3} is equal to:
(2? -2) is multiple of 3.
If n is odd, 2? ≡ (-1)? =-1 (mod 3). 2? -2 ≡ -1-2 = -3 ≡ 0 (mod 3).
If n is even, 2? ≡ (-1)? =1 (mod 3). 2? -2 ≡ 1-2 = -1 (mod 3).
So n must be odd.
2-digit numbers are 10-99 (90 numbers).
Odd numbers are 11,13, .,99. Number of terms = (99-11)/2 + 1 = 45.
Probability = 45/90 =
Upvote Two tangents are drawn from the point P(-1, 1) to the circle x² + y² - 2x – 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to:
<!-- [if !supportLineBreakNewLine]-->
<!--[endif]-->
Two tangents are drawn from the point P(-1, 1) to the circle x² + y² - 2x – 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to:
<!-- [if !supportLineBreakNewLine]-->
<!--[endif]-->
Circle: (x-1)² + (y-3)² = 4. C= (1,3), r=2.
Length of tangent from P (-1,1) is L = √ (-1)²+1²-2 (-1)-6 (1)+6) = √ (1+1+2-6+6) = √4 = 2.
Area of quadrilateral PACB = 2 * Area (PAC) = 2 * (1/2 * L * r) = 2*2=4.
AB is chord of contact. T=0 => -x+y- (x-1)-3 (y-1)+6=0 => -2x-2y+10=0 => x+y=5.
Distance of C f
Upvote