IMPORTANT QUESTIONS FOR CBSE CLASS 12TH 2011-12
INVERSE TRIGONOMETRIC FUNCTIONS
1. Prove that : sin-1 8/17 + sin-1 3/5 = sin-1 77/85= tan-1 (77/36)
2. Prove that : cos-1 12/13 + sin-1 3/5 = sin-1 56/65
3. Prove that: cos-1 4/5 + cos-1 12/13 = cos -1 33/65
4. Prove that : sin-1 3/5 - sin-1 8/17 = cos-1 84/85
5. Prove that :
tan1/2[sin-1 (2x / 1+x2 ) + cos-1 (1- y2 ) / (1+ y2 ) ] = (x+y) / (1-xy),
if |x| < 1, y>0 and xy<1.
6. Prove that : tan-1 Ö x = ½ cos -1 (1-x / 1+x), x € [0, 1]
7. Prove that :
(i) tan-1 {Ö ( 1+cos x) + Ö (1–cos x) / Ö (1+cos x) - Ö(1–cos x)} = p/4 + x/2, 0< p/2
(ii) cot-1 {Ö ( 1+sin x) + Ö (1–sin x) / Ö (1+sin x) - Ö(1–sinx) } = x/2, 0< p/2
8. Prove that :
(i) tan-1 {Ö ( 1+x) - Ö (1–x) / Ö (1+x) + Ö(1–x)} = p/4 - ½ cos-1 x, 0<1
(ii) tan-1 {Ö ( 1+x2) + Ö (1–x2) / Ö (1+x2) - Ö(1–x2) } = p/4 + ½ cos-1 x2, -1<1
9. Prove that :
(i) sin [ cot-1 {cos (tan-1 x)}] = Ö(x2 +1) / (x2 +2)
(ii) cos [ tan-1 {sin (cot-1 x)}] = Ö(x2 +1) / (x2 +2)
10. solve the following equation :
tan-1 2x + tan-1 3x = p/4
11. solve it : tan-1 [2x / (1-x2)] + cot-1 [(1 - x2 ) / 2x ] = 2p/3, x>0
IMPORTANTS QUESTION FOR CBSE CLASS 12TH 2011-12
RELATION
1. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R on the set A = { 1,2,3, … ,13, 14 } defined as
R = { (x,y) : 3x – y = 0 }
(ii) Relation R on the set N of all natural numbers defined as
R = { (x,y) : y = x + 5 and x < 4 }
(iii) Relation R on the set A = {1,2,3,4,5,6 } defined as
R = { (x,y) : y is divisible by x }
(iv) Relation R on the set Z of all integer defined as
R = { (x,y) : x - y is an integer }
2. Show that the relation R on the set R of all real numbers, defined as
Relation = { (a,b) : a £ b2 }
is neither reflexive nor symmetric nor transitive.
3. Show that the relation R on R defined as R = { (a,b) : a £ b },
is reflexive and transitive but not symmetric.
4. Show that the relation R defined on the set A of all triangles in a plane as
R = { (T1, T2 ) : T1 is similar to T2 }
is an equivalence relation.
Consider three right angle triangles T1 with sides 3, 4, 5; T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
5. Show that the relation R on the set A = {1, 2, 3, 4, 5}, given by
R = { (a, b) : | a – b | is even }, is an equivalence relation.
Show all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But, no element of {1, 3, 5} is related to any element of {2, 4}.
6 .Show that the relation R on the set A = { x € Z : 0 £ x £ 12}, given by
R = { (a, b) : | a – b | is a multiple of 4 }
is an equivalence relation. Find the set of all elements related to 1.
7. Prove that the relation R on the set N ´ N defined by
(a, b) R (c, d) Û a + d = b + c for all (a, b), (c, d) € N ´ N
is an equivalence relation.
IMPORTANT QUESTIONS FOR CBSE CLASS 12TH 2011-12
FUNTIONS
1. Show that the function ¦ : N ® N given by ¦ (1) = ¦(2) = 1 and ¦(x) = x – 1 for every x ³ 2, is onto but not one-one.
2. Show that the Signum function ¦: R ® R, given by
1, if x > 0¦(x) = 0, if x = 0
-1, if x < 0
is neither one-one nor onto.
3. Prove that ¦ : R ® R, is given by ¦(x) = 2x, is one-one and onto.
4. Show that the function ¦ : R ® R, defined as ¦(x) = x2 , is neither one-one nor onto.
5. Show that ¦ : R ® R, defined as ¦(x) = x3 , is a bijection.
6. Show that the function ¦ : R0 ® R0 , defined as ¦(x) = 1/x, is one-one onto, where R0 is the set of all non-zero real numbers. Is the result true, if the domain R0 is replaced by N with co-domain being same as R0 ?
7. Show that the modules function ¦ : R ® R , given by ¦(x) = | x | is neither one- one nor onto.
8. Show that the function ¦: R ® R given by ¦(x) = ax + b, where a, b € R, a ¹ 0 is a bijection.
9. Let ¦ : N ® Y be a function defined as ¦(x) = 4x + 3, where
Y : { y € N : y = 4x + 3 for some x € N }. Show that ¦ is invertible. Find its inverse.
10. Let Y = { n2 : n € N } Ì N. Consider ¦ : N ® Y given by ¦(n) = n2 . Show that ¦ is invertible. Find the inverse of ¦.
11. Let ¦ : N ® R be a function defined as ¦(x) = 4x2 + 12x + 15. Show that ¦ : N ® Range (¦) is invertible. Find the inverse of ¦.
12. If ¦(x) = (4x + 3) / (6x – 4), x ¹ 2 / 3, show that ¦o¦ (x ) = x for all x ¹ 2 / 3. What is the inverse of ¦ ?
13. Consider ¦ : R ® R+ ® [4, ¥) given by f(x) = x2 + 4 . Show that ¦ is invertible with inverse ¦ -1 of ¦ given by ¦ -1 (x) = Öx-4 , where R+ is the set of all non-negative real numbers.
14. Consider ¦ : R ® R given by ¦(x) = 4x + 3. Show that ¦ is invertible. Find the inverse of ¦.
15. Consider ¦ : R+ ® [-5, ¥) given by ¦(x) = 9x2 + 6x – 5. Show that ¦ is invertible with ¦ -1 (x) = (Ö (x+6) – 1) / 3.
16. Show that ¦ : [ - 1, 1] ® R , given by ¦(x) = x /(x+2) is one-one. Find the inverse of the function ¦ : [-1, 1] ® Range (¦)
that gof= fog = IR
IMPORTANT QUESTIONS FOR CBSE CLASS 12TH (2011-12)
SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS
1. Solve the following system of equations by matrix method:
(i) x + y + z =3 (ii) x + y +z =3
2x + 3y + z =10 2x – y -z = -1
3x – y - 7z =1 2x + y -3z = -9
(iii) 6x -12y +25z =4 (iv) 3x +4y+7z =14
4x +15y -20z =3 2x – y +3z = 4
2x +18y +15z =10 x +2y-3z = 0
(v) 2/x – 3/y + 3/z =10 (vi) 5x+3y+z =16
1/x + 1/y + 1/z =10 2x+y+3z = 19
3/x – 1/y + 2/z =13 x+2y+4z = 25
(vii) 3x + 4y + 2z =8 (viii) 2x+y+z =2
2y - 3z = 3 x +3y-z = 5
x – 2y + 6z = - 2 3x +y-2z = 6
(ix) 2x+6y = 3 (x) x – y + z =2
3x - z = -8 2x-y = 0
2x –y+z = - 3 2y-z = 1
(xi) 8x+4y+3z =18 (xii) x+y+z =6
2x+y+z =5 x+2z =7
x +2y+z =5 3x+y+z =12
IMPORTANT QUESTIONS FOR CBSE CLASS 12TH (2011-12)
DETERMINANTS
1. Prove that : (b+c)2 a2 a2
b2 (c+a)2 b2 = 2abc(a+b+c)3
c2 c2 (a+b) 2
2. Show that : (b+c)2 ba ca
ab (c+a)2 cb = 2abc(a+b+c)3
ac bc (a+b)2
3. Solve : a+x a-x a-x
a-x a+x a-x = 0
a-x a-x a+x
4. Prove that : a+b+2c a b
` c b+c+2a b = 2(a+b+c)3
C a c+a+2b
5. Prove that : a-b-c 2a 2a
` 2b b-c-a 2b = (a+b+c)3
2c 2c c-a-b
6. Prove that : a+b b+c c+a a b c
b+c c+a a+b = 2 b c a
c+a a+b b+c c a b
7. Prove that : b+c a a
` b c+a b = 4abc
c c a+b
8. Prove that : a b c
` a-b b-c c-a = a3+b3+c3 – 3abc
b+c c+a a+b
9. Prove that : a2 bc ac+c2
` a2 +ab b2 ac = 4a2 b2 c2
ab b2 +bc c2
10. Prove that :
a b-c c-b
a-c b c-a = (a+b-c) (b+c-a) (c+a-b)
a-b b-a c
IMPORTANT QUESTIONS FOR CBSE CLASS 12TH (2011-12)
DERIVATIVE AS A RATE MEASURER
1. A particle moves along the curve, 6y = x3 + 2 . Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
2. A water tank has the slope of an inverted right circular with its axis vertical and vertex lower most. Its semi-vertical angle is tan-1 (0.5). Water is poured into it at a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4m.
3. find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate.
4. The volume of a spherical balloon is increasing at the rate of 25 cm3 / sec. Find the rate of change of its surface area at the instant when radius is 5 cm.
5. The length x of a rectangle is decreasing at the rate of 5 cm / minute and the width y is increasing at the rate of 4 cm/minute. When x=8 cm and y=6cm, find the rates of change of (i) the perimeter (ii) the area of rectangle.
6. A circular disc of radius 3cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2cm.
7. A car starts from a point P at time t=0 seconds and stops at a point Q. The distance x, in meters, covered by it, in t seconds is given by x=t2 (2-t/3). Find the time taken by it to reach Q and also find distance PQ.
8. A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when the radius at 15cm.
9. A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular waves is 10cm, how fast is the enclosed area increasing?
10. The surface area of a spherical bubble is increasing at the rate of 2cm2 /s. when the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing?
IMPORTANT QUESTIONS FOR CBSE CLASS 12TH ( 2011-12)
CONTINUITY
1. Find the value of a if the function f(x) defined by
2x-1, x<2
f(x) = a , x=2 is continuous at x=2.
x+1 , x>2
2. If the function f(x) defined by
Log(1+ax – log(a-bx) / x , if x 0
f(x) =
k , if x = 0
is continuous at x=0, find k.
3. Find the values of a so that the function f(x) defined by:
sin2 ax / x2 , x ¹ 0
f(x) = may be continuous at x=0.
1 , x = 0
4. If the function f(x) given by
3ax + b , if x>1
f(x) = 11 , if x=1
5ax - 2b , if x<1
is continuous at x=1, find the values of a and b.
5. 1-cos4x / x2 , if x<0
let f(x) = a , x = 0
x / (16+ x – 4) , x > 0
Determine the value of a so that f(x) is continuous at x =0
6. Determine the value of the constant k so that the function
kx2 , if x 2
f(x) = 3 , if x > 2 is continuous at x=2.
7. Find the values of a so that the function
ax+5 , if x 2
f(x) = x-1 , if x > 2 is continuous at x=2.
8. Find the value of k if f(x) is continuous at x= /2, where
(K cosx) / (-2x) , x /2
f(x) =
3 , x = /2
9. Determine the values of a,b,c for which the function
[sin(a+1)x + sinx] / x , for x < 0
f(x) = c , for x=0 is continuous at x=0
(x+bx2) - (x / bx3/2 ) , for x > 0
10. Find the value of k for which
(1-cosx) / 8x2 , when x ¹ 0
f(x) = is continuous at x=0;
k , when x = 0
11. For what value of ג is the function
ג (x2 – 2x) , if x 0
f(x) =
4x+1 , if x > 0
Continuous at x=0? What about continuity at x = ± 1 ?
12. For what value of k is the following function continuous at x=2?
2x+1 ; if x < 2
f(x) = k ; x = 2
3x-1 ; x > 2
13. 1-sin3 x / 3cos2 x , if x < p/2
Let f(x) = a , if x= p/2 , if f(x) is continuous at x=p/2, find a and b.
b(1-sinx) / (p-2x)2 , if x > p/2
14. If the functions f(x), defined below is continuous at x=0, find the value of k:
(1-cos2x) / 2x2 , x < 0
f(x) = k , x = 0
x /
x
, x > 0s