# HBSE Class 12th Maths Solved Question Paper 2019

HBSE Class 12th Maths Solved Question Paper 2019

HBSE Class 12 Maths Previous Year Question Paper with Answer. HBSE Board Solved Question Paper Class 12 Maths 2019. HBSE 12th Question Paper Download 2019. HBSE Class 12 Maths Paper Solution 2019. Haryana Board Class 12th Maths Question Paper 2019 Pdf Download with Answer.

SET-A

Q1. Find fog, if f : R → R and g : R → R are given by f(x) = cosx and g(x) = 3x² .

Sol. fog = fo(3x²) = f(3x²) = cos(3x²)

Q2. The value of sin-¹(-½) is :

(A) -π/3

(B) -π/6

(C) -π/4

(D) None of these

Ans. (B) -π/6

sin-¹(-½) = sin-¹(-sin π/6) = sin-¹sin(-π/6) = -π/6

Q3. Construct a 3 × 2 matrix whose elements are given by aij = ½|i-3j| .

Sol.

[a11   a12]      [1     5/2]

A = |a21   a22|  = |1/2      2|

[a31   a32]      [0     3/2

Q4. The value of x for which |3   x| = |3    2| is :

|x   1   |4    1|

(A) ±2√3

(B) ±3√3

(C) ±2√2

(D) None of these

Ans. (C) ±2√2

3(1) – x(x) = 3(1) – 2(4)

3 – x² = 3 – 8

x² = 8

x = √8 = ±2√2

Q5. Differentiate sin(x²+5) w.r.t. x.

Sol. d/dx sin(x²+5) = cos(x²+5).d/dx(x²+5)

= cos(x²+5).2x = 2x.cos(x²+5)

Q6. The rate of change of the area of a circle with respect to its radius r when r = 3 cm  is  :

(A) 6π cm²/sec

(B) 4π cm²/sec

(C) 5π cm²/sec

(D) None of these

Ans. (A) 6π cm²/sec

dA/dr = d/dr(πr²) = 2πr

at r=3,  dA/dr = 2π(3) = 6π cm²/sec

Q7. Find the slope of tangent to the curve  y = x³ – x at x = 2.

Sol. dy/dx = d/dx (x³-x) = 3x²-1

at x=2,  dy/dx = 3(2)²-1 = 3(4)-1 = 12-1 = 11

slope of tangent at x=2 is 11

Q8. Find the value of ∫sin(tan-¹x)/1+x² dx.

Sol. Put tan-¹x = t

1/1+x² dx = dt sin(tan-¹x)/1+x² dx = ∫sint dt

= -cost + c = -cos(tan-¹x) + c

Q9. The value of ∫sin⁵x.cos⁴x.dx is:  (Note: use Limit -1 to 1)

(A) 1

(B) -1

(C) 0

(D) None of these

Ans. (C) 0

f(x) = sin⁵x.cos⁴x

f(-x) = sin⁵(-x).cos⁴(-x) = -sin⁵x.cos⁴x = -f(x)

f1 is odd function

∫sin⁵x.cos⁴x.dx = 0  (Note: use Limit -1 to 1)

Q10. The order of the differential equation 2x²(d²y/dx²) -3(dy/dx) +y = 0  is :

(A) 2

(B) 0

(C) 1

(D) None of these

Ans. (A) 2

Q11. The degree of the differential equation d³y/dx³ +2(d²y/dx²)² -dy/dx +y = 0 is :

(A) 3

(B) 2

(C) 1

(D) None of these

Ans. (C) 1

Q12. If P(E) = 0.6, P(F) = 0.3 and P(E∩F) = 0.2, then find P(E/F).

Sol. P(E/F) = P(E∩F)/P(F) = 0.2/0.3 = 2/3

Q13. If P(A) = 0.3, P(B) = 0.6 and A and B are independent events, then find the value of  P(A and B).

Sol. P(A and B) = P(A∩B) = P(A).P(B) = 0.3 × 0.6 = 0.18

Q14. A bag contains 4 white and 6 black balls. Two balls are drawn at random with replacement. Find the probability that both balls are black.

Sol. P(both black) = 6/10 × 6/10 = 36/100 = 9/25

Q15. Find the unit vector of the sum of the vectors a = 2i + 2j – 5k and b = 2i + j + 3k.

Sol. a + b = 2i + 2j – 5k + 2i +j +3k = 4i + 3j – 2k

|a+b| = √4²+3²+2² = √29

Unit Vector of a+b = (4i+3j-2k)/√29

Q16. If direction ratio’s of a line are 2, −1, −2, then find the direction cosines of the line.

Sol. D.C’s of the line are 2, -1, -2

√2²+1²+2² = √4+1+4 = √9 = 3

D.C’s of the line are 2/3, -1/3, -2/3

SET-B

Q1. Find gof, if f : R → R and g : R → R are given by f(x) = cosx and g(x ) = 3x² .

Sol. gof = go(cosx) = g(cosx) = 3cos²x

Q2. The value of cos-¹(√3/2) is :

(A) π/3

(B) π/4

(C) π/6

(D) None of these

Ans. (C) π/6

cos-¹(√3/2) = cos-¹(cos π/6) = π/6

Q3. Construct a 2 × 2 matrix whose elements are given by aij = (i+j)²/2 .

Sol. A = [a11   a12]  =   [2     9/2]

[a21   a22]       [9/2     8]

Q4. The value of x for which |x     2| = |6     2| is :

|18  x   |18   6|

(A) 0

(B) -5

(C) 7

(D) ±6

Ans. (D) ±6

x(x) – 2(18) = 6(6) – 2(18)

x² – 36 = 36 – 36

x² = 36

x = √36 = ±6

Q5. Differentiate cos(sinx) w.r.t. x.

Sol. d/dx cos(sinx) = –sin(sinx).cosx

= -cosx.sin(sinx)

Q6. The rate of change of the area of a circle with respect to its radius r, when r = 4 cm  is  :

(A) 6π cm²/sec

(B) 8π cm²/sec

(C) 4π cm²/sec

(D) None of these

Ans. (B) 8π cm²/sec

dA/dr = d/dr(πr²) = 2πr

at r = 4,  dA/dr = 2π(4) = 8π cm²/sec

Q7. Find the slope of tangent to the curve  y = 3x⁴ − 4x at x = 4.

Sol. dy/dx = d(3x⁴-4x)/dx = 12x³ – 4

at x = 4, dy/dx = 12(4)³ – 4 = 12(64) – 4 = 768 – 4 = 764

slope of tangent at x=2 is 764.

Q8. Find the value of ∫2x/1+x² dx.

Sol. put 1+x² = t

2x.dx = dt

dx = dt/2x

∫2x/1+x² dx = ∫2x/t.dt/2x = ∫1/t.dt

= logt + c = log(1+x²) + c

Q9. The value of ∫(x³+xcosx+tan⁵x)dx is:  (Note: use Limit -π/2 to π/2)

(A) π/2

(B) -π/2

(C) π

(D) 0

Ans. (D) 0

f(x) = x³+xcosx+tan⁵x

f(-x) = (-x)³+(-x)cos(-x)+tan⁵(-x) = -f(x)

f1 is odd function

∫(x³+xcosx+tan⁵x) dx = 0  (Note: use Limit -π/2 to π/2)

Q10. The order of the differential equation d²y/dx² + y = 0  is :

(A) 1

(B) 0

(C) 2

(D) None of these

Ans. (C) 2

Q11. The degree of the differential equation (dy/dx)² + dy/dx – sin²y = 0 is :

(A) 2

(B) 1

(C) 0

(D) None of these

Ans. (A) 2

Q12. If P(E) = 0.6, P(F) = 0.3 and P(E∩F) = 0.2, then find P(F/E).

Sol. P(F/E) = P(E∩F)/P(E) = 0.2/0.6 = 2/6 = 1/3

Q13. If P(A) = 0.3, P(B) = 0.6 and A and B are independent events, then find the value of  P(A and B not).

Sol. P(B’) = 1 – P(B) = 1 – 0.6 = 0.4

P(A and B not) = P(A and B’) = P(A).P(B’) = 0.3 × 0.4 = 0.12

Q14. A bag contains 4 white and 6 black balls. Two balls are drawn at random with replacement. Find the probability that first ball is white and second ball is black.

Sol. P = 4/10 × 6/10 = 24/100 = 0.24

Q15. Find a vector in the direction of a vector a = i – 2j which has magnitude 7 units.

Sol. a = i – 2j

Magnitude = 7

Vector in the direction of a vector = [(i-2j )/√5] × 7 = (7/√5)i – (2/√5)j

Q16. If a line makes angles 90°, 135° and 45° with the x, y and z-axis, then find the direction cosines of the line.

Sol. D.C’s of the line are cosα, cosβ, cosγ

D.C’s of the line are cos90°, cos135°, cos45°

D.C’s of the line are 0, -1/√2, 1/√2

SET-C

Q1. Find fog if  f(x) = 8x³ and g(x) = x^⅓ .

Sol. fog = fo(x^⅓) = f(x^⅓) = 8(x^⅓)³ = 8x

Q2. The value of tan-¹(- √3) is :

(A) π/3

(B) -π/6

(C) -π/3

(D) None of these

Ans. (C) -π/3

tan-¹(- √3) = tan-¹(- tan π/3) = -π/3

Q3. Construct a 2 × 2 matrix whose elements are given by aij = i/j .

Sol. A = [a11   a12]  =   [1   1/2]

[a21   a22]       [2       1]

Q4. The value of x for which |2      4| = |2x    4| is :

|5     1   |6       x|

(A) ±√3

(B) ±√6

(C) ±√5

(D) None of these

Ans. (A) ±√3

2(1) – 4(5) = 2x(x) – 4(6)

2 – 20 = 2x² – 24

2x² = 24 – 18 = 6

x² = 6/2 = 3

x = ±√3

Q5. Differentiate sin(ax+b) w.r.t. x.

Sol. d/dx sin(ax+b) = cos(ax+b).a

= a.cos(ax+b)

Q6. The rate of change of the area of a circle with respect to its radius r, when r = 5 cm  is  :

(A) 5π cm²/sec

(B) 15π cm²/sec

(C) 10π cm²/sec

(D) None of these

Ans. (C) 10π cm²/sec

dA/dr = d/dr(πr²) = 2πr

at r = 5,  dA/dr = 2π(5) = 10π cm²/sec

Q7. Find the slope of tangent to the curve  y = x³ − x + 1 at x = 2 .

Sol. dy/dx = d/dx (x³ – x + 1) = 3x² – 1

at x = 2,  dy/dx = 3(2)² – 1 = 3(4) – 1 = 12 – 1 = 11

slope of tangent at x=2 is 11.

Q8. Find the value of ∫(logx)²/x dx.

Sol. put logx = t

dt/dx = 1/x

xdt = dx

∫(t²/x) x.dx = ∫t² dt = t³/3 + c = (logx)³/3 + c

Q9. The value of ∫x¹⁰.sin⁷x dx is:       (Note- use Limit -π to π)

(A) π

(B) -π

(C) 1

(D) 0

Ans. (D) 0

f(x) = x¹⁰.sin⁷x

f(-x) = (-x)¹⁰.sin⁷(-x) = -f(x)

f1 is odd function

(-x)¹⁰.sin⁷(-x) dx = 0 (Note: use Limit -π/2 to π/2)

Q10. The order of the differential equation d³y/dx³ + x².(d²y/dx²)³ = 0  is :

(A) 2

(B) 3

(C) 1

(D) None of these

Ans. (B) 3

Q11. The degree of the differential equation (ds/dt)⁴ + 3s.(d²s/dt²) = 0 is :

(A) 4

(B) 2

(C) 1

(D) None of these

Ans. (C) 1

Q12. If P(A) = 0.8, P(B) = 0.5 and P(B/A) = 0.4, then find P(A∩B).

Sol. P(A∩B) P(A).P(B/A) = 0.8 × 0.4 = 0.32

Q13. If P(A) = 0.3, P(B) = 0.6 and A and B are independent events, then find the value of  P(A or B).

Sol.  P(AB) = P(A).P(B),  A & B independent)

P(AB) = P(A) + P(B) – P(AB) = P(A) + P(B) – P(A).P(B) = 0.3 + 0.6 – 0.3 × 0.6 = 0.9 – 0.18 = 0.72

Q14. A bag contains 4 white and 6 black balls. Two balls are drawn at random with replacement. Find the probability that one ball is white and one ball is black.

Sol. P = 4/10 × 6/10 = 24/100 = 0.24

Q15. Find a unit vector in the direction of a vector a = 2i + 3j + k.

Sol. a = 2i + 3j + k

Unit Vector = (2i +3j +k)/√2²+3²+1² = (2i +3j +k)/√14 = (2/√14)i +(3/√14)j +(1/√14)k

Q16. Find the direction cosines of x-axis, y-axis and z-axis .

Sol. D.C’s of the line are cosα, cosβ, cosγ

cos²α + cos²β + cos²γ = 1

3coα = 1    (:. α = β = γ)

cosα = ±1/√3

D.C’s of the line are ±1/√3, ±1/√3, ±1/√3

SET-D

Q1. Find gof if  f(x) = 8x³ and g(x) = x^⅓ .

Sol. gof = go(8x³) = g(8x³) = (8x³)^⅓ = 2x

Q2. The value of cos-¹(- ½) is :

(A) 2π/3

(B) π/4

(C) π/2

(D) None of these

Ans. (A) 2π/3

cos-¹(- ½) cos-¹(- cos π/3) cos-¹[cos (π-π/3)] = cos-¹[cos 2π/3] = 2π/3

Q3. Construct a 2 × 2 matrix whose elements are given by aij = (i+2j)²/2 .

Sol. A = [a11   a12]  =   [9/2   25/2]

[a21   a22]       [8           18]

Q4. The value of x for which |2      3| = |x      3| is :

|4     5   |2x    5|

(A) 1

(B) 2

(C) 3

(D) None of these

Ans. (B) 2

2(5) – 3(4) = x(5) – 3(2x)

10 – 12 = 5x – 6x

2 = -x

x = 2

Q5. Differentiate tan(2x+3) w.r.t. x.

Sol. d[tan(2x+3)]/dx = sec²(2x+3).2 = 2.sec²(2x+3)

Q6. The rate of change of the area of a circle with respect to its radius r, when r = 6 cm  is  :

(A) 6π cm²/sec

(B) 8π cm²/sec

(C) 10π cm²/sec

(D) 12π cm²/sec

Ans. (D) 12π cm²/sec

dA/dr = d(πr²)/dr = 2πr

at r = 6,  dA/dr = 2π(6) = 12π cm²/sec

Q7. Find the slope of tangent to the curve  y = x³ – 3x + 2 at x = 3 .

Sol. dy/dx = d/dx (x³ – 3x + 2) = 3x² – 3

at x=3,  dy/dx = 3(3)² – 3 = 3(9) – 3 = 27 – 3 = 24

slope of tangent at x=3 is 24

Q8. Find the value of ∫1/(x+xlogx) dx.

Sol. ∫1/x(1+logx) dx

put 1+logx = t

dt/dx = 1/x

xdt = dx

∫(1/xt) x.dx = ∫1/t dt = logt + c = log(1+logx) + c

Q9. The value of ∫x³.cosx dx is :  (Note: use Limit -π to π)

(A) π

(B) -π

(C) -1

(D) 0

Ans. (D) 0

f(x) = x³.cosx dx

f(-x) = (-x)³.cos(-x) = -f(x)

f1 is odd function

(-x)³.cos(-x) dx = 0  (Note: use Limit -π to π)

Q10. The order of the differential equation 2x².(d²y/dx²) – 3.(dy/dx) + y = 0  is :

(A) 0

(B) 2

(C) 1

(D) None of these

Ans. (B) 2

Q11. The degree of the differential equation (d²y/dx²)³ + (dy/dx)² + 3y = 0 is :

(A) 1

(B) 2

(C) 3

(D) None of these

Ans. (C) 3

Q12. If P(A) = 7/13, P(B) = 9/13 and P(A∩B) = 4/13, then find P(A/B)

Sol. P(A/B) P(A∩B)/P(B) = (4/13)/(9/13) = 4/9

Q13. If P(A) = 0.3, P(B) = 0.6 and A and B are independent events, then find the value of  P(A’ and B’).

Sol. P(A’) = P(A not) = 1-P(A) = 1-0.3 = 0.7

P(B’) = P(B not) = 1-P(B) = 1-0.6 = 0.4

P(A’B’) = P(A’).P(B’) = 0.7 × 0.4 = 0.28

Q14. Two cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that both cards are aces.

Sol. P = 4/52 × 4/52 = 1/169

Q15. Find a unit vector in the direction of a vector a = i + j + 2k .

Sol. a = i + j + 2k

Unit Vector = (i+j+2k)/√1²+1²+2² = (i+j+2k)/√6 = (1/√6)i +(1/√6)j +(2/√6)k

Q16. Find the direction cosines of a line which makes equal angles with co-ordinate axis.

Sol. D.C’s of the line are cosα, cosβ, cosγ

cos²α + cos²β + cos²γ = 1

3coα = 1    (:. α = β = γ)

cosα = ±1/√3

D.C’s of the line are ±1/√3, ±1/√3, ±1/√3

Other Questions (Coming Soon…)

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