# HBSE Class 12th Maths Solved Question Paper 2021

HBSE Class 12th Maths Solved Question Paper 2021

HBSE Class 12 Maths Previous Year Question Paper with Answer. HBSE Board Solved Question Paper Class 12 Maths 2021. HBSE 12th Question Paper Download 2021. HBSE Class 12 Maths Paper Solution 2021. Haryana Board Class 12th Maths Question Paper 2021 Pdf Download with Answer.

Objective Questions

Q1. The relation on R defined R = {(a,b) : a ≤ b} is :

(A)  Reflexive and Symmetric

(B)  Symmetric and Transitive

(C)  Reflexive and Transitive

(D)  None of these

Ans. (C)  Reflexive and Transitive

Q2. If f : R → R defined  by f(x) = 3x, then f is :

(A)  One-one  onto

(B)  Many-one,  onto

(C)  One-one  not  onto

(D)  Neither  one-one  nor  onto

Ans. (A)  One-one  onto

Q3. If  a  binary  operation  *  on  N  defined  as a * b = a² + b², choose the correct answer  :

(A)  Associative  and  Commutative

(B)  Commutative  but  not  Associative

(C)  Associative  but  not  Commutative

(D)  Neither  Associative  nor  Commutative

Ans. (A)  Associative  and  Commutative

Q4. tan-¹(1) + cos-¹(-½) is equal to :

(A)  – π/12

(B)  7π/12

(C)  11π/12

(D)  5π/12

Ans. (B)  7π/12

tan-¹(1) + cos-¹(½) = π/4 + π/3 = 7π/12

(Using Formula : cos-¹(-x)=cos-¹x )

Q5. sin-¹ (√1-x²), |x| < 1 is equal to :

(A)  sin-¹x

(B)  cos-¹x

(C)  tan-¹x

(D)  None of these

Ans. (B)  cos-¹x

Let  x = cosθ

sin-¹√1 – cos²θ = sin-¹√sin²θ = sin-¹sinθ θ = cos-¹x

Q6. If A is a matrix of order 2 × 3 and B is a matrix of order 3 × 2, then AB is of order :

(A)  2 × 2

(B)  3 × 3

(C)  Not defined

(D)  None of these

Ans. (A)  2 × 2

Q7. If A and B are two invertible matrices of some order, which of the following is always true ?

(A)  (AB)-¹ = B-¹ A-¹

(B)  (AB)-¹ = A-¹ B-¹

(C)  (A+B)-¹ = A-¹ + B-¹

(D)  (A-B)-¹ = A-¹ – B-¹

Ans. (A)  (AB)-¹ = B-¹ A-¹

Q8. If  |x      6| = |6    -3|

|8    2x|    |8     4, then the value of x is :

(A)  6

(B)  2

(C)  0

(D)  None of these

Ans. (D)  None of these

2x(x) – 8(6) = 6(4) – 8(-3)

2x² – 48 = 24 + 24

2x² – 48 = 48

2x² = 48 + 48

2x² = 96

x² = 96/2 = 48

x = √48 = ±4√3

Q9. The function f(x) = ax + 3, x ≤ 5

= 18,        x > 5

is a continuous function at 5=x, then the value of a is :

(A)  5

(B)  3

(C)  1

(D)  None of these

Ans. (B)  3

ax + 3 = 18

5a + 3 = 18

5a = 18 -3 = 15

a = 15/5 = 3

Q10. If  y = log(cos e^x),  then dy/dx is equal to :

(A)  sec(e^x)

(B)  – sec(e^x)ex

(C)  – tan(e^x)ex

(D)  None of these

Ans. (C)  – tan(e^x)ex

y = log(cos e^x)

dy/dx = y’ = (1/cos e^x) × (-sin e^x) × ex

dy/dx = y’ = – e^x tanx

Q11. The slope of the tangent to the curve y = √(4x-3) – 1 at (3, 2) is :

(A)  2/3

(B)  3/2

(C)  1/6

(D)  None of these

Ans. (A)  2/3

y = √(4x-3) – 1

y’ = 4/2√(4x-3) = 2/√4x-3

At  x = 3,

y’ = 2/√4×3-3 = 2/√12-3

y’ = 2/√9 = 2/3

Q12. ∫tan-¹x/(1+x²) is equal to  :

(A)  tan-¹x + c

(B)  (tan-¹x)² + c

(C)  1/2(tan-¹x)² + c

(D)  None of these

Ans. (C)  1/2(tan-¹x)² + c

Put  x = tanθ

dx = sec²θ dθ

= ∫ (tan-¹tanθ)/(1+tan²θ).sec²θ dθ

= ∫ θ.sec²θ/sec²θ = ∫ θ dθ

= θ²/2 + c = 1/2(tan-¹x)² + c

Q13. ∫ 1/√(1-x²) dx is :     (Note- use Limit 0 to 1)

(A)  π/2

(B)  π/4

(C)  – π/2

(D)  0

Ans. (A)  π/2

Put  x = sinθ

dx = cosθ dθ

= ∫ cosθ/√(1-sin²θ) dθ

= dθ = [ θ ]

(Note:  use Limit 0 to 1)

= [ sin-¹x ] = [ sin-¹1 – sin-¹0 ]

= (π/2 – 0) = π/2

Q14. Area bounded by the curve y² = x, above x-axis and x = 0 to x = 1 is  :

(A)  1/3

(B)  2/3

(C)  1

(D)  3/2

Ans. (B)  2/3

y² = x

y = √x

∫y dx = ∫√x dx = (2/3) × X^3/2 + c

= 2/3  (Note: use Limit 0 to 1)

Q15. The differential equation of the family  of curves y = a sin(x+b),  where a and b are arbitrary is :

(A)  d²y/dx² – y = 0

(B)  d²y/dx + ay = 0

(C)  d²y/dx² + y = 0

(D) None of these

Ans. (C)  d²y/dx² + y = 0

y = a sin(x+b)

dy/dx = y’ = a cos(x+b)

d²y/dx² = y” = – a sin(x+b)

y” = – y

y” + y = 0

d²y/dx² + y = 0

Q16. If E and F are independent events, then which of the following is not true ?

(A)  P(E∩F) = P(E).P(F)

(B)  P(E/F) = P(E)

(C)  P(E/F) = P(F)

(D)  P(E/F).P(F) = P(E∩F)

Ans. (D)  P(E/F).P(F) = P(E∩F)

Q17. The value of  λ for which the vectors a = 2i – j + λk is perpendicular to the vector b = i – 3j – 5k, is :

(A)  0

(B)  1

(C)  –5

(D)  None of these

Ans. (B)  1

a.b = 0

2 × 1 + (-1) × (-3) + λ × (-5) = 0

2 + 3 – 5λ = 0

5 = 5λ

λ = 1

Q18. If a line makes angle 60⁰ with x-axis 45⁰ with y-axis, then this line will make angle  with z-axis is :

(A)  30⁰

(B)  45⁰

(C)  60⁰

(D)  90⁰

Ans. (B)  45⁰

cos²α + cos²β + cos²γ = 1

cos²60 + cos²45 + cos²γ = 1

1/4 + 1/2 + cos²γ = 1

cos²γ = 1 – 1/4 – 1/2 = 1/2

cosγ = √1/2 = cos45°

γ = 45°

Q19. If  f(x) = x²  and  g(x) = sin x where f : R→R  and  g : R→[-1, 1], then  fog(x) = ……………  ( sinx², sin²x, x²sinx )

Ans. sin²x

Q20. cos[2 sin-¹(- 1/2)] = …………….    ( √3/2, 1/2, -1/2, -√3/2 )

Sol. cos[2 sin-¹(- 1/2)]

cos[2 sin-¹(210°)]

cos[2 × 210°]

= cos (420°) = cos (360° + 60°)

= cos 60° = 1/2

Q21. cos(tan-¹x) is equal to ………………      ( x/√1+x², 1/√1+x², 1/√1-x² )

Ans. 1/√1+x²

Q22. If A is a square matrix of order 3 with |A| = 5, then  det(Adj A) = |Adj A| = ……………..     ( 5, 25, 125, 1/5 )

Sol. |Adj A| = |A| n-¹ = 5³-¹ = 5² = 25

Q23. If A and B are inverse of each other then which of the following is true ?

(A)  AB = BA

(B)  AB = BA = 0

(C)  AB = 0, BA = I

(D)  AB = BA = I

Ans. (D)  AB = BA = I

Q24. If  x = 2at², y = 4at,  then dy/dx =

(A)  1/t

(B)  – 1/t

(C)  1/t²

(D)  None of these

Ans. (A)  1/t

x = 2at²,  dx/dt = 4at

y = 4at,   dy/dt = 4a

dy/dx = (dy/dt)/(dx/dt) = 4a/4at = 1/t

Q25. If  f(x) = tan3x/x,  x ≠ 0  x is in radiang

= k            ,  x = 0

and f(x) is continuous at x = 0, then the value of k is …………….

Sol. f(x) = tan3x/x = tan3x/x × 3/3

= tan3x/3x × 3 = 1 × 3  (Using Formula : tanx/x = 1)

k = 3

Q26. ∫ cot²x dx  is :

(A)  cot²x + x + c

(B)  -cotx – x +c

(C)  tanx – x + c

(D)  None of these

Ans. (B)  -cotx – x +c

∫cot²x dx ∫(cosec²x – 1)dx

= ∫cosec²x dx – ∫1dx = – cotx – x + c

Q27. ∫e^x(tanx + sec²x) dx = …………….

Sol. ∫e^x(tanx + sec²x) dx e^x tanx + c

(Formula:  ∫e^x[f(x) + f'(x)] dx = e^x f(x) + c)

Q28. A fair die is rolled. Consider the events E = {1, 3, 5}, F = {2, 3}, find the P(F/E) .

Sol. P(E) = 3/6,  P(E∩F)

P(F/E) = P(E∩F)/P(E) = 1/6 ÷ 3/6 = 1/3

Q29. If  f : [-1, 1] → R, is given by f(x) = x/x+2, then find  f-¹(x).

Sol. f(x) = x/x+2,  Let  x/x+2 = y

x = yx + 2y

x – yx = 2y

x(1-y) = 2y

x = 2y/1-y

so,  f-¹(x) = 2x/1-x

Q30. Find the value of  tan-¹(1/4) + tan-¹(3/4).

Sol. tan-¹(1/4) + tan-¹(3/5)

= tan-¹ [(1/4 + 3/5) ÷ (1 – 1/4×3/5)]

(Formula:  tan-¹x + tan-¹y = tan-¹[(x+y)/(1-xy)] )

= tan-¹ [ (17/20) ÷ (17/20) ]

= tan-¹ (1) = π/4

Q31. If A is a square matrix of order 3 and |A|=4,  then find  det|2A|.

Sol. det|2A| = 2^n|A| = 2³ × 4 = 32

Q32. If  x³ + y³ + 3axy = 0,  then fin dy/dx.

Sol. x³ + y³ + 3axy = 0

3x² + 3y²y’ + 3ay + 3ay’ = 0

3y²y’ + 3axy’ = – 3x² – 3ay

3y'(y² + 3ax) = 3(-x² – ay)

dy/dx = y’ = (-x²-ay)/(y²+3ax)

Q33. The maximum value of  sin x – cos x  in the interval [0, π] is ………………

Ans. √2

Q34. ∫cos²x/1+sinx dx equal to ……………….

Sol. cos²x/1+sinx dx

∫(1-sin²x)/(1+sinx) dx

∫(1-sinx)dx = x + cosx + c

Q35. Evaluate  ∫1/x²+4 dx

Sol. ∫1/x²+4 dx

Put  x = 2tanθ

dx = 2 sec²θ dθ

∫2sec²θ/(4tan²θ+4) dθ

∫sec²θ/4(tan²θ+1) dθ

∫sec²θ/2sec²θ dθ

∫1/2 dθ = 1/2θ + c

= 1/2 tan-¹x/2 + c

Q36. Find the area of a quadrant of an ellipse 4x² + y² = 4.

Sol. 4x² + y² = 4

4x²/4 + y²/4 = 4/4

x²/(1)² + y²/(2)² = 1

(Using Formula : x²/a² + y²/b² = 1)

Here  a = 1,  b = 2

Area = πab = π×1×2 = 2π

Q37. The order of differential equation (d²y/dx²)⁴ – (dy/dx)³ = 0 is :

(A)  4

(B)  3

(C)  2

(D)  1

Ans. (C)  2

Q38. The probabilities of two independent events A and B are 1/2 and 1/3 respectively. Find the probability of P(A∪B).

Sol. P(A) = 1/2,  P(B) = 1/3,  P(A∩B) = 1/6

P(A∪B) = P(A) + P(B) – P(A∩B)

P(A∪B) = 1/2 + 1/3 – 1/6 = 4/6 = 2/3

Q39. If vector  a = i – 7j + 7k  and  b = 3i – 2j + 2k,  then find a×b.

Sol.

|  i       j      k  |

a × b =  |  1     -7     7 |

|  3     -2     2 |

= i(-14+14) – j(2-21) + k(-2+21)

= 0i + 19j – 19k

Q40. If  |a| = 2,  |b| = 1/√2  and  a×b  is a unit vector, then find angle between vector a and b.

Sol.  |a| = 2 ,   |b| = 1/√2 ,  |a×b| = 1

|a × b| = |a| |b| sinθ

1 = 2 × 1/√2 × sinθ

sinθ = √2/2 = 1/√2 = sin45°

θ = π/4

Subjective Questions (Coming soon…)

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