[PDF] KBPE Class 12th Maths Important Questions

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KBPE Class 12th Maths Important Questions

[a]F ind f o f (x) [b] Find the inverse of f

Solution:

[a] f o f (x) = f ( f (x)) = f (x / x - 1) = (x / x - 1) / [(x / x - 1) - 1] = x / [x - (x - 1)] = x [b]y = x / (x - 1) xy - x = y xy - y = x y (x - 1) = x f -1 (y) = y / (y - 1) f -1 (x) = x / (x - 1)

Question 2[a]:

Identify the function from the above graph.

[i] tan-1 x [ii] sin-1 x [iii] cos-1 x [iv] cosec-1 x

Answer: [ii]

[b] Find the domain and range of the function represented by the graph.

Solution:

ĺ- ૈ / 2, ૈ / 2]

[c] Prove that tan-1 (1 / 2) + tan-1 (2 / 11) = tan-1 (3 / 4).

Solution:

tan-1 x + tan-1 y = tan-1 [(x + y) / (1 - xy)] tan-1 (1 / 2) + tan-1 (2 / 11) = tan-1 [(1 / 2 + 2 / 11) / (1 - [(1 / 2 * 2 / 11)]] = tan-1 (15 / 20) = tan-1 (3 / 4)

Question 3: Consider the following figure:

[a] Find the point of intersection P of the circle x2 + y2 = 50 and the line y = x.

Solution:

x2 + y2 = 50 x2 + x2 = 50

2x2 = 50

x2 = 50 / 2 x = ± 5 y = ± 5

The point of intersection is P (5, 5).

[b] Find the area of the shaded region.

Solution:

0552 dx

= (x2 / 2)052 + (50 / 2) sin-1 = 25 / 2 + 25ૈ / 2 - (25 / 2) - (25ૈ / 4) = 25ૈ / 4 Question 4[a]: Prove that for any vectors a, b, c, [a + b, b + c, c + a] = 2 [a . b . c].

Solution:

[a + b, b + c, c + a] = (a + b) . [(b + c) x (c + a)] = (a + b) . [b x c + b x a + c x c + c x a] = (a + b) . [b x c + b x a + c x a] = a . (b x c) + a . (b x a) + a . (c x a) + b . (b x c) + b . (b x a) + b . (c x a) = 2 [a . b . c] [b] Show that if [a + b, b + c, c + a] are coplanar, then a, b, c are also coplanar.

Solution:

[a b c] = 0

Hence a, b, c are also coplanar.

Question 5[a]: Find the equation of a plane which makes x, y, z intercepts respectively as 1, 2, 3.

Solution:

x / 1 = y / 2 = z / 3 = 1

6x + 3y + 2z = 6

[b] Find the equation of a plane passing through the point (1, 2, 3) which is parallel to the above plane.

Solution:

A plane parallel to the given plane is 6x + 3y + 2z = k.

Since it passes through (1, 2, 3), k = 18.

The equation of the plane is 6x + 3y + 2z = 18.

Question 6[a]: Prove that the function defined by f (x) = cos (x2) is a continuous function.

Solution:

f (x) = cos x g (x) = x2

Both are continuous functions.

Composition of two continuous functions is continuous. f (g (x)) = f o g (x) = cos (x2) is continuous. [b] [i] [ii] Hence prove that (1 - x2) d2y / dx2 - x (dy / dx) - a2y = 0.

Solution:

[i]

2) dy / dx = -ay

(1 - x2) (dy / dx)2 = a2y2 (1 - x2) 2 (dy / dx) * (d2y / dx2) + (dy / dx)2 * (2x) = 2a2y (dy / dx) (1 - x2) (d2y / dx2) - x (dy / dx) - a2y = 0

Question 7[a]: Find x and y if .

Solution:

2x - y = 10 ---- (1)

3x + y = 5 ---- (2)

_________

5x = 15

x = 15 / 5 x = 3

Substitute the value of x in (1),

2 * 3 - y = 10

6 - y = 10

6 - 10 = y

y = - 4

So, x = 3 and y = -4.

[b] Express the matrix as a sum of symmetric and skew- symmetric matrices.

Solution:

Question 8:

2 + 2x + 2

Solution:

= (- cos mx / m) + c [b] x2 + 2x + 2 = (x + 1)2 + 1

22 + 1

2 + 2x + 2 + c

x / (x + 1) (x + 2) = A / (x + 1) + B / (x + 2)

A = -1, B = 2

= - log (x + 1) + 2 log (x + 2) + c Question 9[a]: Find the angle between the lines x - 2 / 2 = y - 1 / 5 = z + 3 / -3 and x + 2 / -1 = y - 4 / 8 = z - 5 / 4. [b] Find the shortest distance between the pair of lines. r = (i + 2j ڣ r = (4i + 5j + 6k) + ૄ [2i + 3j + k]

Solution:

a1 = 2, b1 = 5, c1 = -3 and a2 = -1, b2 = 8, c2 = 4

ș1a2 + b1b2 + c1c212 + b12 + c1222 + b22 + c22

ș-1

[b] a1 = i + 2j + 3k, b1 = i - 3j + 2k a2 = 4i + 5j + 6k, b2 = 2i + 3j + k Shortest distance = |(a2 - a1) . (b1 x b2) / |b1 x b2|| (a2 - a1) = 3i + 3j + 3k (b1 x b2) = - 9i + 3j + 9k Question 10[a]: Let R be a relation defined on A = {1, 2, 3} by R = [{1, 3} , {3,

1}, {2, 2}]. R is

(a) Reflexive (b) Symmetric (c) Transitive (d) Reflexive but not transitive

Answer: [b]

[b] Find f o g and g o f if f(x) = |x + 1| and g (x) = 2x - 1.

Solution:

f o g (x) = f (g (x)) = f (2x - 1) = |2x - 1 + 1| = |2x| g o f (x) = g (f (x)) = g (x + 1) = 2 |x + 1| - 1 + d). Find the identity element for * if it exists.

Solution:

If (c, d) is the identity element,

(a, b) * (c, d) = (a, b) (a, b) * (c, d) = (a + c, b + d) (a + c, b + d) = (a, b) (c, d) = (0, 0) is not an element of N x N. Question 11[a]: Consider the linear programming problem:

Maximise Z = 50x + 40y

Subject to constraints

[a] Find the feasible region. [b] Find the corner points of the feasible region. [c] Find the maximum value of Z.

Solution:

[a] [b] The corner points are (0, 5), (0, 6), (4, 3). [c] At (0, 5), Z = 200 (0, 6), Z = 240 (4, 3), Z = 320

Z is maximum at (4, 3) and it is 320.

Question 12[a]: The angle between the vectors i + j and j + k is (a) 60o (b) 30o (c) 45o (d) 90o [b] If a, b, c are unit vectors such that a + b + c = 0, find the value of a . b + b . c + c . a.

Solution:

[a] Answer: [a] [b] |a x b x c| = 0 (a + b + c)2 = 0 |a|2 + |b|2 + |c|2 + 2ab + 2bc + 2ca = 0

3 + 2 (a . b + a . c + b . c) = 0

(a . b + a . c + b . c) = - 3 / 2 Question 13[a]: If A = such that A2 = I then a equals to (a) 1 (b) -1 (c) 0 (d) 2

Answer: [c]

[b] Solve the system of equations: x - y + z = 4

2x + y - 3z = 0

x + y + z = 2 by matrix method.

Solution:

Question 14[a]: Area bounded by the curves y = cos x, x = ૈ / 2, x = 0, y = 0 is (a) 1 / 2 (b) 2 / ૈ (c) 1 (d) ૈ / 2 [b] Find the area between the curve y2 = 4ax and x2 = 4ay, a > 0.

Solution:

[a] Answer: (c) [b]

The point of intersection is (4a, 4a).

04a04a (x2 / 4a) dx

3/2 / (3 / 2)]04a - (1 / 4a) (x3 / 3)04a

= [16 / 3] a2 (a) one-one but onto (b) one-one but not onto (c) not one-one not onto (d) onto but not one-one

Answer: [b]

[b]: Find g o f (x), if f (x) = 8x3 and g (x) = x1/3.

Solution:

g o f (x) = g (f ( x)) = g (8x3) = (8x3)ѿ = 2x [c]: Let * be an operation such that a * b = LCM of a and b defined on the set A = {1, 2, 3, 4, 5}. Is * a binary operation? Justify your answer.

Solution:

a * b = LCM of a and b

Let a = 2, b = 3

a * b = 2 * 3 = LCM of 2 and 3 is 6 operation.

2 x dx = (x / 2) + (sin 2x / 4) + c.

Solution:

cos2 x = (1 + cos 2x) / 2 2 = (1 / 2) [x + (sin 2x / 2)] dx 2.

Solution:

2x - x2 = - (x2 - 2x)

= - (x2 - 2x + 1 - 1) = - [(x - 1)2 - 12] = 1 - (x - 1)2 22
= sin-1 (x - 1) + c

Solution:

= x sin x + cos x + c + k) and r = 2i + j - k + ૄ (3i - 5j + 2k).

Solution:

Shortest distance = |[(a2 - a1) . (b1 - b2)] / |[b1 x b2]|| a1 = i + j a2 = 2i + j - k b1 = 2i - j + k b2 = 3i - 5j + 2k (a2 - a1) = i - k [b1 x b2] = 3i - j + 7k |[b1 x b2 Question 18[a]: Equation of the plane with intercepts 2, 3, 4 on the x, y and z- axis respectively is (i) 2x + 3y + 4z = 1 (ii) 2x + 3y + 4z = 12 (iii) 6x + 4y + 3z = 1 (iv) 6x + 4y + 3z = 12

Solution: [iv]

[b] Find the cartesian equation of the plane passing through the points A (2, 5, -3), B (- 2, - 3, 5) and C (5, 3, - 3).

Solution:

The equation is

(x - 2) 16 + (y - 5) 24 + (z + 3) 32 = 0

2x + 3y + 4z = 7

(i) 0.48 (ii) 0.51 (iii) 0.52 (iv) 0.58

Solution: [iv]

[b] A card from a pack of 52 cards is lost. From the remaining cards of the pack, 2 cards are drawn and are found to be diamonds. Find the probability of the lost card being a diamond.

Solution:

E1: The lost card is a diamond.

E2: The lost card is not a diamond.

A: Selecting 2 diamonds from the remaining cards

P (E1) = 13 / 52 = 1 / 4

P (E2) = 39 / 52 = 3 / 4

P (A / E1) = 12C2 / 51C2 = 12 * 11 / 51 * 50

P (A / E2) = 13C2 / 51C2 = 13 * 12 / 51 * 50

P (E1 / A) = [P (E1) * P (A / E1)] / [P (E1) * P (A / E1)] + [P (E2) * P (A / E2)] = [(1 / 4) * [12 * 11 / 51 * 50]] / [(1 / 4) * (12 * 11 / 51 * 50) + (3 / 4) * (13 * 12 /

51 * 50)]

= 132 / 600 = 11 / 50 Question 20: A pair of dice is thrown 4 times. If getting a doublet is considered as a success, [i] Find the probability of getting a doublet. [ii] Find the probability of two successes.

Solution:

[i] 1 / 6 [ii] n = 4 q = 1 - p = 1 - (1 / 6) = 5 / 6

P (X = 2) = 4C2 (5 / 6)2 (1 / 6)2

= 25 / 216 Question 21[a]: Choose the correct statement related to matrices (i) A3 = A, B333 = B (iii) A3 = A, B3 = B (iv) A33

Answer: (iii)

[b] If M = then verify the equation M2 - 10M + 11I2 = 0.

Solution:

[c] Inverse of the matrix .

Solution:

There is the inverse matrix on the right.

Question 22[a]: Prove that = (x + y + z) (x - y) (y - z) (z - x).quotesdbs_dbs47.pdfusesText_47

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