Ncert Solutions Maths class 12th
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The negation of the Boolean expression p∨(¬p∧q) is equivalent to:
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The negation of the Boolean expression p∨(¬p∧q) is equivalent to:
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p∨ (¬p∧q) = (p∨¬p)∧ (p∨q)=T∧ (p∨q)=p∨q. Negation is ¬p∧¬q.
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If {p} denotes the fractional part of the number p, then {3²⁰⁰/8}, is equal to
If {p} denotes the fractional part of the number p, then {3²⁰⁰/8}, is equal to
(80+1)? /8 = (¹C?80? +.+¹C? )/8. Remainder is 1/8.
Upvote If the value of (1 + 2/3 + 6/3² + 10/3³ + .upto ∞) / (log?.?(1/3 + 1/3² + 1/3³ + .upto ∞)) is l, then l² is equal to……….
If the value of (1 + 2/3 + 6/3² + 10/3³ + .upto ∞) / (log?.?(1/3 + 1/3² + 1/3³ + .upto ∞)) is l, then l² is equal to……….
The expression inside log is a GP sum: (1/3)/ (1-1/3) = 1/2.
log? (1/2) = 1/2.
The series S = 1+2/3+6/3²+.
S - S/3 = 1 + 4/3² + 4/3³ + . = 1 + (4/9)/ (1-1/3) = 1+2/3=5/3.
(2/3)S = 5/3 ⇒ S=5/2.
l = (5/2)/ (1/2)=5. l²=25.
Upvote Let S = {n ∈ N | [n,i;0,1][a,b;c,d] = [a,b;c,d] ∀ a,b,c,d ∈ R}, where i = √-1. Then the number of 2-digit numbers in the set S is …….
Let S = {n ∈ N | [n,i;0,1][a,b;c,d] = [a,b;c,d] ∀ a,b,c,d ∈ R}, where i = √-1. Then the number of 2-digit numbers in the set S is …….
S = {n ∈ N | [n i; 0 1] [a b; c d] = [a b; c d] ∀a, b, c, d ∈ R}
[na+ic, nb+id; c, d] = [a, b;c, d]
This must be an identity matrix. n=1. The question seems to have typos. Let's follow the solution logic.
The solution implies the matrix must be [1 0; 0 1] or [-1 0; 0 -1]. And n must be a multiple of 8
Upvote Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:
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Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:
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Required probability = (? C? +? C? )/¹¹C? = 25/165=5/33.
Upvote The area (in sq. units) of the region A={(x,y): |x|+|y|≤1, 2y²≥|x|} is:
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The area (in sq. units) of the region A={(x,y): |x|+|y|≤1, 2y²≥|x|} is:
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Required area = 4 [∫? ¹/² 2y²dy + ½×½×1] = 5/6.
Upvote If α and β be two roots of the equation x² - 64x + 256 = 0. Then the value of (α³/β⁵)¹/⁸ + (β³/α⁵)¹/⁸ is:
If α and β be two roots of the equation x² - 64x + 256 = 0. Then the value of (α³/β⁵)¹/⁸ + (β³/α⁵)¹/⁸ is:
α+β=64; αβ=256=2?
(α³/β? )¹/? + (β³/α? )¹/? = (α+β)/ (αβ)? /? = 64/32=2.
Upvote Let the foot of perpendicular from a point P(1,2,-1) to the straight line L: x/1 = y/0 = (z-1)/-1 be N. Let a line be drawn from P parallel to the plane x+y+2z = 0 which meets L at point Q. If α is the acute angle between the lines PN and PQ, then cosα is equal to……
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Let the foot of perpendicular from a point P(1,2,-1) to the straight line L: x/1 = y/0 = (z-1)/-1 be N. Let a line be drawn from P parallel to the plane x+y+2z = 0 which meets L at point Q. If α is the acute angle between the lines PN and PQ, then cosα is equal to……
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L: x/1=y/0=z/-1. Let a point on L be (r,0, -r).
Direction ratio of PN is (r-1, -2, -r+1).
PN is perpendicular to L, so (r-1) (1)+ (-2) (0)+ (-r+1) (-1)=0 ⇒ 2r-2=0 ⇒ r=1. N (1,0, -1).
Let Q be (s,0, -s). Direction ratio of PQ is (s-1, -2, -s+1).
PQ is parallel to plane x+y+2z=0, so normal to plane is per
Upvote The number of real roots of the equation e? - e? - 2e³? - 12e²? + e? + 1 = 0 is:
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The number of real roots of the equation e? - e? - 2e³? - 12e²? + e? + 1 = 0 is:
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e? - e? - 2e³? - 12e²? + e? + 1 = 0
e³? - 2 + e? ³? - e? [e³? + 12e? - 1] = 0
Let y=e? y? -y? -2y³-12y²+y+1=0.
The solution breaks the equation into (e³? - 4e? - 1) (e³? - 1 + 3e? ) = 0
Either e³? = 4e? + 1 (One Solution)
OR e³? = 1 - 3e? (One Solution)
∴ the equation has total 2 solutions.
Upvote Let 9 distinct balls be distributed among 4 boxes, B?, B?, B? and B?. If the probability that B? contains exactly 3 balls is k(3/4)? then k lies in the set:
Let 9 distinct balls be distributed among 4 boxes, B?, B?, B? and B?. If the probability that B? contains exactly 3 balls is k(3/4)? then k lies in the set:
Three balls can be given to B? in? C? ways. Now remaining 6 balls can be distributed into 3 boxes in 3? ways.
Total no. of favourable events =? C? × 3? = 28 × 3?
Total no. of events = 9 balls distributed into 4 boxes in 4? ways.
probability = 28 × 3? /4? = 28/9 × (3/4)? ⇒ k = 28/9
k ∈ |x-3|<1
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