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New answer posted

a year ago

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P
Payal Gupta

Contributor-Level 10

This is Other Questions as classified in NCERT Exemplar

G i v e n     t h a t         z = 2 − i ( 1 − 2 i ) 2 = 2 − i 1 + 4 i 2 − 4 i = 2 − i 1 − 4 − 4 i                                                             = 2 − i − 3 − 4 i = 2 − i − 3 − 4 i * − 3 + 4 i − 3 + 4 i                                                             = − 6 + 8 i + 3 i − 4 i 2 ( − 3 ) 2 − ( 4 i ) 2 = − 6 + 1 1 i + 4 9 − 1 6 i 2                                                             = − 2 + 1 1 i 9 + 1 6 = − 2 2 5 + 1 1 2 5 i ∴                                               z ¯ = − 2 2 5 − 1 1 2 5 i H e n c e ,                       z ¯ = − 2 2 5 − 1 1 2 5 i

New answer posted

a year ago

0 Follower 2 Views

A
alok kumar singh

Contributor-Level 10

x‾=10

⇒x? =63+a+b8=10

⇒a+b=17

Since, variance is independent of origin.

So, we subtract 10 from each observation.

So,  σ2=13.5=79+ (a-10)2+ (b-10)28

⇒a2+b2-20 (a+b)=-171

⇒a2+b2=169

From (1) and (2) ; a=12 and b=5

New answer posted

a year ago

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P
Payal Gupta

Contributor-Level 10

This is a True or False Type Questions as classified in NCERT Exemplar

( i ) C o m p a r i s o n     o f     t w o     p u r e l y     i m a g i n a r y     c o m p l e x     n u m b e r s     i s     n o t     p o s s i b l e .     H o w e v e r ,     t h e     t w o               p u r e l y     i m a g i n a r y     r e a l     c o m p l e x     n u m b e r s     c a n     b e     c o m p a r e d .               S o ,     i t     i s     ' F a l s e ' . ( i i ) L e t     z = x + y i                 z . i = ( x + y i ) i = x i − y     w h i c h     r o t a t e s     a t     a n g l e     o f     1 8 0 0               S o ,     i t     i s     ' F a l s e ' . ( i i i ) L e t     z = x + y i                 ∴ | z | + | z + 1 | = x 2 + y 2 + ( x − 1 ) 2 + y 2                 T h e     v a l u e     o f     | z | + | z + 1 |     i s     m i n i m u m     w h e n     x = 0 ,     y = 0     i . e . ,     1 .               H e n c e ,     i t     i s     ' T r u e ' . ( i v ) L e t     z = x + y i                 G i v e n     t h a t :                       | z − 1 | = | z − i |                 t h e n                                     | x + y i − 1 | = | x + y i − i | ⇒                                                     | ( x − 1 ) + y i | = | x − ( 1 − y ) i | ⇒                                           ( x − 1 ) 2 + y 2 = x 2 + ( 1 − y ) 2 ⇒                                           ( x − 1 ) 2 + y 2 = x 2 + ( 1 − y ) 2 ⇒                             x 2 − 2 x + 1 + y 2 = x 2 + 1 + y 2 − 2 y ⇒                                                     − 2 x + 2 y = 0 ⇒                                                                       x − y = 0     w h i c h     i s     a     s t r a i g h t     l i n e .                 S l o p e = 1         Now  equation  of  a  line  through  the  point  (1,0)  and  (0,1)                                                                                 y − 0 = 1 − 0 0 − 1 ( x − 1 ) ⇒                                                                                     y = − x + 1     w h o s e     s l o p e = − 1 .                 N o w     t h e     m u l t i p l i c a t i o n     o f     t h e     s l o p e s     o f     t w o     l i n e s = − 1 * 1 = − 1 ,                 S o     t h e y     a r e     p e r p e n d i c u l a r .               H e n c e ,     i t     i s     ' T r u e ' .

( v ) L e t     z = x + y i ,     z ≠ 0     a n d     R e ( z ) = 0         Since  real  part  is  0  ⇒x=0               ∴ z = 0 + y i = y i             ∴ I m ( z 2 ) = y 2 i 2 = − y 2     w h i c h     i s     r e a l .               H e n c e ,     i t     i s     ' F a l s e ' . ( v i ) G i v e n     t h a t :         | z − 4 | < | z − 2 |                 L e t     z = x + y i ⇒                                                     | x + y i − 4 | < | x + y i − 2 | ⇒                                             | ( x − 4 ) + y i | < | ( x − 2 ) + y i | ⇒                                   ( x − 4 ) 2 + y 2 < ( x − 2 ) 2 + y 2 ⇒                                           ( x − 4 ) 2 + y 2 < ( x − 2 ) 2 + y 2 ⇒                                                               ( x − 4 ) 2 < ( x − 2 ) 2 ⇒                                               x 2 − 8 x + 1 6 < x 2 − 4 x + 4 ⇒                                                         − 8 x + 4 x < − 1 6 + 4 ⇒                                                                             − 4 x < − 1 2 ⇒                                                                                             x > 3               H e n c e ,     i t     i s     ' T r u e ' . ( v i i ) L e t     z 1 = x 1 + y 1 i     a n d       z 2 = x 2 + y 2 i ⇒                                                     | z 1 + z 2 | = | z 1 | + | z

New answer posted

a year ago

0 Follower 3 Views

P
Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

G i v e n     t h a t     e a c h     e d g e     o f     t h e     c u b o i d     i s     2     u n i t s . ∴     C o o r d i n a t e s     o f     t h e     v e r t i c e s     a r e             A ( 2 , 0 , 0 ) ,     B ( 2 , 2 , 0 ) ,     C ( 0 , 2 , 0 ) ,     D ( 0 , 2 , 2 ) ,     E ( 0 , 0 , 2 ) ,     F ( 2 , 0 , 2 ) ,     G ( 2 , 2 , 2 )     a n d     O ( 0 , 0 , 0 ) .

New answer posted

a year ago

0 Follower 14 Views

A
alok kumar singh

Contributor-Level 10

x2a2+y2b2=1 (ab)2b2a=10⇒b2=5a

Now,  ? (t)=512+t-t2=812-t-122
? (t)max=812=23=e⇒e2=1-b2a2=49

⇒a2=81  (From (i) and (ii)

So,  a2+b2=81+45=126

New answer posted

a year ago

0 Follower 3 Views

A
alok kumar singh

Contributor-Level 10

f (2)=8, f' (2)=5, f' (x)≥1, f'' (x)≥4, ∀x∈ (1,6)

Using LMVT

f'' (x)=f' (5)-f' (2)5-2≥4⇒f' (5)≥17

f' (x)=f (5)-f (2)5-2≥1⇒f (5)≥11

Therefore f' (5)+f (5)≥28

New answer posted

a year ago

0 Follower 10 Views

P
Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

Let  the  given  points  are  A(0,−1,−7),  B(2,1,−9),  C(6,5,−13)            AB=(2−0)2+(1+1)2+(−9+7)2=4+4+4=12=23            BC=(6−2)2+(5−1)2+(−13+9)2=16+16+16=48=43            AC=(6−0)2+(5+1)2+(−13+7)2=36+36+36=108=63             23+43=63i.e.,          AB+BC=AC∴                AB:AC=23:63=1:3Hence,  point  A  divides  B  and  C  in  1:3  externally.

New answer posted

a year ago

0 Follower 2 Views

A
alok kumar singh

Contributor-Level 10

∫03? g (x)-f (x)=∫03? ||x-2|-2|dx-∫03? |x-2|dx

=12*2*2+1+12*1*1-12*2*2+12*1*1

=2+1+12-2+12=1

New answer posted

a year ago

0 Follower 7 Views

P
Payal Gupta

Contributor-Level 10

P (En) = n/36 for n = 1, 2, 3, …., 8

P (A)=Any  possible  sum  of   (1, 2, 3, ........., 8) (=α say)36


α36≥45

∴a≥29

If one of the number from {1, 2, ….8} is left then total ≥ 29 by 3 ways

Similarly by leaving terms more 2 or 3 we get 16 more combinations

∴ Total number of different set a possible is 16 + 3

= 19

New answer posted

a year ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

The direction of ratios of the lines,   x−3−3=y−22k=z−32&x−13k=y−11=z−6−5 , are −3, 2k, 2and3k, 1, −5 respectively.

It is known that two lines with direction ratios,   a1,  b1,  c1 and a2,  b2, c2 , are perpendicular, if  a1a2 + b1b2 + c1c2 =0

∴−3 (3k)+2k*1+2 (−5)=0⇒−9k+2k−10=0⇒7k=−10⇒k=−107

Therefore, for k= -10/7, the given lines are perpendicular to each other.

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