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…)