HBSE Class 12th Maths Solved Question Paper 2020

HBSE Class 12th Maths Solved Question Paper 2020 

HBSE Class 12 Maths Previous Year Question Paper with Answer. HBSE Board Solved Question Paper Class 12 Maths 2020. HBSE 12th Question Paper Download 2020. HBSE Class 12 Maths Paper Solution 2020. Haryana Board Class 12th Maths Question Paper 2020 Pdf Download with Answer.

Subjective Questions (Coming Soon…) 

SET-A 

Q1. Let f : R → R is defined as f(x) = x³ then f is : 

(A) One-one, into    

(B) One-one, onto   

(C) Many-one, onto    

(D) Many-one, into  

Ans. (B) One-one, onto 

Q2. The principal value of tan-¹x is : 

(A) [0, π/2] 

(B) [0, π] 

(C) [-π/2, π/2] 

(D) None of these  

Ans. (D) None of these (-π/2, π/2)

Q3. If X + Y = [5    2]  and X – Y = [3     6] then matrix X: 

                        [0    9]                      [-2   1] 

(A) [4      4]   

       [-1    5]

(B) [8      8] 

      [-2  10] 

(C) [1    -2] 

      [1      4] 

(D) None of these 

Ans. (A) [4      4]   

                [-1    5]

Q4. If det. |2     4| = |2x   4| then the value of x is : 

                  |5     1|    |6     x|

(A) 6 

(B) ±6 

(C) -6 

(D) None of these  

Ans. (D) None of these (x = ± √3 ) 

2(1) – 5(4) = 2x(x) – 6(4) 

2 – 20 = 2x² – 24 

-18 = 2x² -24 

2x² -24 = -18 

2x² = -18 + 24 = 6 

x² = 6/2 = 3 

x = ± √3 

Q5. Differentiate sec(tan√x) with respect to x. 

Sol. sec(tan√x).tan(tan√x).sec²√x.1/2√x 

Q6. f(x) = x³ – 3x + 4  has a maxima at x is equal to : 

(A) -1 

(B) 1 

(C) 0 

(D) None of these 

Ans. (A) -1 

f'(x) = 3x² – 3 = 0,    x = ±1 

f”(x) = 6x at x = -1,  f”(x)<0 

Maxima at x = -1 

Q7. f(x) = log(sinx) is strictly decreasing in interval : 

(A) (0, π/2) 

(B) (π/2, π) 

(C) (0, π) 

(D) None of these 

Ans. (B)  (π/2, π) 

f'(x) = cosx/sinx = cotx < 0  in (π/2, π).  

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

Ans. (tan-¹x)²/2 + c 

Q9. Evaluate ∫sin³x.cos² dx. (Note- use Limit -π/2 to π/2) 

Sol. 

Q10. Find the degree and order of the differential equation x³(d²y/dx²)³ + (dy/dx)² + x.dy/dx + y = 0. 

Ans. Degree = 3, Order = 2 

Q11. Solve the differential equation : 

(1+x²)dy/dx = 1+y² 

Sol. dy/1+y² = dx/1+x² 

tan-¹y = tan-¹x + c 

Q12. A bag contains 4 white and 6 black balls. Two balls are drawn at random with replacement. Find the probability both the balls are black.  

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

Q13. A and B are independent event such that P(A) = 0.3 and P(B) = 0.4, find the P(A/B ).  

Sol. P(A/B) = P(A) = 0.3 (A & B independent)

Q14. A random variable X has the following probability distribution :

Find k. 

Sol. ∑P(x) = 1 

0 + k + 2k + 2k + 3k + k² + 2k² + 7k² + k = 1 

10k² + 9k = 1 

10k² + 9k – 1 = 0 

(10k – 1)(k + 1) = 0 

k = 1/10 or k ≠ -1 

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

Sol. a + b = 2i + 2j – 5k + j – k = 2i + 3j – 6k 

|a + b| = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 

unit vector = 2/7i + 3/7j – 6/7k 

Q16. Write the equation of line passing through the point with position vector i + 2j + 3k and in the direction 3i + 2j – 2k in vector form.  

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

r = i + 2j + 3k + λ(3i + 2j – 2k) 

SET-B 

Q1. Let f : R → R+ is defined as f(x) = x⁴ then f is : 

(A) One-one, onto    

(B) One-one, into   

(C) Many-one, onto    

(D) Many-one, into  

Ans. (C) Many-one, onto

Q2. The principal value of cos-¹x is : 

(A) [0, π] 

(B) [-π/2, π/2] 

(C) (-π/2, π/2) 

(D) None of these 

Ans. (A) [0, π]

Q3. If 2X + Y = [1     0] & 2X – Y = [3    4] then matrix X: 

                           [-3   2]                   [-1  2] 

(A) [4      4]   

       [-4    4]

(B) [1      1] 

      [-1     1] 

(C) [-1    -2] 

      [-1      0] 

(D) None of these 

Ans. (B) [1      1] 

               [-1     1] 

Q4. If det. |3x    5|  =  |2     0| then the value of x is : 

                  |1      x|       |-1   2| 

(A) +2/3  

(B) 2  

(C) ±√3  

(D) 0 

Ans. (C) ±√3  

3x(x) – 5(1) = 2(2) – 0(-1) 

3x² – 5 = 4 – 0 

3x² = 4 + 5 

3x² = 9 

x² = 9/3 = 3 

x = ±√3  

Q5. Differentiate log(sec√x) with respect to x. 

Sol. d/dx log(sec√x) = (1/sec√x).sec√x.tan√x.(1/2√x) = (tan√x)/(2√x)

Q6. f(x) = sinx + cosx  has a local maxima at x is equal to : 

(A) 0 

(B) π/6  

(C) π/4  

(D) π/2 

Ans. (C) π/4  

f(x) = sinx + cosx 

f'(x) = cosx – sinx = 0 

sinx = cosx 

sinx/cosx = tanx = 1 

x = π/4, 5π/4 

f”(x) = – sinx – cosx 

f”(x) is (-ve) at x = π/4 (maxima) 

Q7.  f(x) = log(sinx) is strictly increasing in the interval : 

(A) (0, π/2) 

(B) (π/2, π) 

(C) (0, π) 

(D) None of these 

Ans. (A) (0, π/2) 

f'(x) = cosx/sinx = cotx > 0 for x ∈ (0, π/2).  

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

Ans. (sin-¹x)²/2 + c 

Q9. Evaluate ∫x³/1+x² dx. (Note- use Limit -1 to +1) 

Ans. 0, odd function 

Q10. Find the degree and order of the differential equation d⁴y/dx⁴ -5(dy/dx)² -6y = logx.

Ans. Degree = 1, Order = 4  

Q11. Solve the differential equation : 

dy/dx = y tanx 

Sol. dy/y = tanxdx  (integrating) 

logy = log(secx) + logc 

y = c secx

Q12. A bag contains 4 white and 6 black balls. Two balls are drawn at random one after the other without replacement. Find the probability that both the balls are black.  

Sol. P(BB) = 6/10 × 5/9 = 30/90 = 1/3  

Q13. If P(A) = 3/5, P(B) = 1/5 and P(A∩B) = 1/10, find P(B/A).

Sol. P(B/A) = P(A∩B)/P(A) = (1/10)/(3/5)  

                     = 1/10 × 5/3 = 5/30 = 1/6 

Q14. A random variable X has the following probability distribution : 

Find k. 

Sol. ∑P(x) = 1 

0.1 + k + 2k + 2k + k = 1 

6k + 0.1 = 1  

6k = 1 – 0.1 = 0.9  

k = 0.9/6 = 0.15 

Q15. Find a unit vector in the direction of PQ, where points P and Q are (1, 2, 3) and (4, 5, 6) respectively. 

Sol. Vector PQ = P.V. of Q – P.V. of P 

= (4i + 5j + 6k) – (1i + 2j + 3k) = 3i + 3j + 3k 

Unit Vector = 3(i + j + k)/√(3² + 3² + 3²) 

= 3(i + j + k)/√27 = 3(i + j + k)/3√3  

Q16. Find the equation of line passing through the point (-3, 5, -6) and parallel to the direction 2i + 4j + 2k. 

Sol. r = -3i + 5j – 6k, b = 2i + 4j + 2k 

r = -3i + 5j – 6k + λ(2i + 4j + 2k) 

SET-C 

Q1. Let f : N → N is defined as f(x) = x³ then f is : 

(A) One-one, onto    

(B) One-one, into   

(C) Many-one, onto    

(D) Many-one, into  

Ans. (B) One-one, into   

Q2. The principal value of sin-¹x is : 

(A) [0, π] 

(B) [-π/2, π/2] 

(C) [0, 2π] 

(D) None of these 

Ans. (B) [-π/2, π/2] 

Q3. If 2X + 3Y = [3  2] & 2X – 3Y = [1   0] then matrix Y: 

                            [1  4]                      [-3 2] 

(A) 1/5 [4      2]   

             [-2     6]

(B) [2      2] 

      [4      2] 

(C) 1/6 [2    2] 

             [4    2] 

(D) None of these 

Ans. (C) 1/6 [2    2] 

                      [4     2] 

Q4. If det. |x     2|  =  |6     2| then the value of x is : 

                  |18   x|      |18   6|

(A) 6 

(B) -6 

(C) ±6 

(D) 0 

Ans. (C) ±6 

x(x) – 2(18) = 6(6) – 2(18) 

x² – 36 = 36 – 36 

x² – 36 = 0 

x² = 36 

x = √36 =  ±√6   

Q5. Differentiate e^sin√x with respect to x. 

Sol. d/dx(e^sin√x) = e^sin√x.cos√x.(1/2√x) 

Q6. The function f(x) = sinx – cosx, 0 < x < 2π has a local maxima at x is equal to : 

(A) π/4 

(B) 3π/4  

(C) 5π/4  

(D) None of these 

Ans. (B) 3π/4 

f(x) = sinx – cosx 

f'(x) = cosx + sinx = 0 

sinx = – cosx 

sinx/cosx = -1 

tanx = -1 

x = 3π/4, 7π/4 

f”(x) = – sinx + cosx < 0 at x=3π/4 

Q7. f(x) = log(sinx) is strictly decreasing in interval : 

(A) (0, π/2) 

(B) (π/2, π) 

(C) (0, π) 

(D) None of these 

Ans. (B) (π/2, π)

f'(x) = cosx/sinx = cotx  decreasing  for x ∈ (π/2, π). 

Q8. Find the value of ∫ e^tan-¹x/1+x² dx . 

Ans. e^tan-¹x + c 

Q9. Evaluate ∫tan³x dx.      (Note- use Limit -π/4 to π/4) 

Ans. 0, odd function

Q10. Find the degree and order of the differential equation d²y/dx² + (dy/dx)² + 2y = 0.

Ans. Degree = 1, Order = 2 

Q11. Solve the differential equation : 

dy/dx = -4xy²  

Sol. dy/y² = -4xdx 

-1/y = -2x² + c 

1/y = 2x² – c  

Q12. Find the probability of getting an even prime number on each die, when a pair of dice is rolled. 

Sol. P(Even Prime) = 1/6 × 1/6 = 1/36  

Q13. If E and F be two events such that P(E) = 0.6, P(F) = 0.3 and P(E∩F) = 0.2, find P(E/F).  

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

Q14. A random variable X has the following probability distribution : 

Determine the Value of k. 

Sol. ∑P(x) = 1 

0.2 + k + 2k + 2k + 3k + k + 0.1 = 1 

9k + 0.3 = 1 

9k = 1 – 0.3 = 0.7 

k = 0.7/9 = 7/90 

       

Q15. Find a unit vector in the direction of the difference of the vectors a = 2i + 2j – 5k and b = j – k . 

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

Unit Vector = (2i + j – 4k)/√(2² + 1² + 4²) 

= (2i + j – 4k)/√21 

Q16. Find the equation of line passing through the point (5, 2, -4) and parallel to the vector 3i + 2j – 8k. 

Sol. a = 5i + 2j – 4k,  b = 3i + 2j – 8k 

r = 5i + 2j – 4k + λ(3i + 2j – 8k) 

(x-5)/3 = (y-2)/2 = (z+4)/-8

SET-D 

Q1. Let f : R → R is defined as f(x) = 3x then f is : 

(A) One-one, onto    

(B) Many-one, onto  

(C) One-one, into 

(D) Many-one, into  

Ans. (A) One-one, onto   

Q2. The principal value of tan-¹x is : 

(A) [-π/2, π/2] 

(B) (-π/2, π/2) 

(C) [0, π] 

(D) (0, π/2) 

Ans. (B) (-π/2, π/2) 

Q3. If 2X + 3Y = [2    3]  and Y = [3     2] then matrix X: 

                            [4    0]                 [1    4] 

(A) [-4     -3]   

      [1    -12]

(B) 1/2 [-7   -1] 

             [2    -8] 

(C) 1/2 [-7    -3] 

             [1   -12] 

(D) None of these 

Ans. (C) 1/2 [-7    -3] 

                       [1   -12] 

Q4. If det. |3     x|  =  |3   2| then the value of x is : 

                  |x     1|      |4   1|

(A) 2  

(B) 4  

(C) ±2√2  

(D) None of these 

Ans. (C) ±2√2  

3(1) – x(x) = 3(1) – 2(4) 

3 – x² = 3 – 8  

3 – x² = -5 

x² = 3 + 5 = 8 

x = √8 = ±2√2   

Q5. Differentiate e^sin√x with respect to x. 

Sol. d/dx(e^sin√x) = e^sin√x.cos√x.(1/2√x) 

Q6. The function f(x) = cosx – sinx  has a local maxima at x is equal to : 

(A) π/4 

(B) 3π/4  

(C) 5π/4  

(D) 7π/4 

Ans. (D) 7π/4 

f(x) = cosx – sinx  

f'(x) = – sinx – cosx = 0 

– sinx = cosx 

tanx = -1 

x = 3π/4, 7π/4 

f”(x) = – cosx + sinx < 0 for x=7π/4 

Q7.  f(x) = log(cosx) is strictly increasing in interval : 

(A) (0, π/2) 

(B) (0, π) 

(C) (-π/2, π/2) 

(D) None of these 

Ans. (D) None of these

f'(x) = -sinx/cosx = – tanx > 0 

tanx < 0 ,  x ∈ (π/2, π) ∪ (3π/2, 2π) 

Q8. Evaluate ∫ cosx/√1+sinx dx . 

Ans. 2√1+sinx  + c 

Q9. Evaluate ∫ sin⁵x dx.      (Note- use Limit -π/2 to π/2) 

Ans. 0, odd function 

Q10. Find the degree and order of the differential equation (dy/dx)⁴ + 3y(d²y/dx²) = 0.

Ans. Degree = 1, Order = 2 

Q11. Solve the differential equation : 

dy/dx = (1+y²)/(1+x²)   

Sol. dy/1+y² = dx/1+x² 

tan-¹y = tan-¹x + c  

Q12. A bag contains 10 white and 15 black balls. Two balls are drawn one by one without replacement. Find the probability of one white and one black ball.  

Sol. P(WB or BW) = (10/25 × 15/24) + (15/25 × 10/24) = 1/2 

Q13. If P(A) = 5/26, P(B) = 5/13, A and B are independent, find P(A/B). 

Sol. P(A/B) = P(A) = 5/26,  A & B are independent 

Q14. Find the probability distribution of number of heads in two tosses of a coin. 

Sol. X = 0,  P(X) = 1/4 

X = 1,  P(X) = 1/2 

X = 2,  P(X) = 1/4 

Q15. If a and b are two vectors such that a = 5i + 2j – 4k and b = 3i + 2j – 4k. Find the unit vector parallel to a + b. 

Sol. a + b = (5i + 2j – 4k) + (3i + 2j – 4k) = 8i + 4j -8k 

Unit Vector parallel a + b = (8i + 4j -8k)/√8²+4²+8² 

= 4(2i +j -2k)/4√36 = (2i +j -2k)/6 

Q16. Find the equation of the line passing through the point (1, 2, 3) and parallel to the vector i + 2j – k.

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

r = i + 2j + 3k + λ(i + 2j – k) 

(x-1)/1 = (y-2)/2 = (z-3)/-1