Please refer to Class 12 Mathematics Sample Paper Set M with solutions below. The following CBSE Sample Paper for Class 12 Mathematics has been prepared as per the latest pattern and examination guidelines issued by CBSE. By practicing the Mathematics Sample Paper for Class 12 students will be able to improve their understanding of the subject and get more marks.
1. If the point of intersection of the lines
is (x, y, z), then y + x is.
(a) 9
(b) –8
(c) 8
(d) 1
Answer
C
2. If |a̅| = √6 and |b̅| = 5 , then [(a̅ × b̅) × b̅] × b̅ is equal to
(a) 5(b̅ × a̅)
(b) 5(a̅ × b̅)
(c) 6(a̅ × b̅)
(d) 6(a̅ × b̅)
Answer
A
3. A bag contains 4 balls of unknown colours. A ball is drawn at random from it and is found to be white. The probability that all the balls in the bag are white is
(a) 4/5
(b) 1/5
(c) 3/5
(d) 2/5
Answer
D
4. A natural number is selected at random from the first 100 natural numbers. Let A, B and C denote the events of selection of even number, a multiple of 3 and a multiple of 5, respectively. Then
(a) P(A ∩ B) = 4/25
(b) P(B ∩ C) = 3/50
(c) P(C ∩ A) = 1/10
(d) All of these
Answer
D
5. Find the distance between the following parallel planes
2x – y + 2z + 3 = 0 and 4x – 2y + 4z + 5 = 0
(a) 2/3
(b) 4
(c) 1/6
(d) 2/5
Answer
C
6. Find the angle between the pair of lines given by
r̅ = 3î + 2ĵ − 4k̂ + λ(î + 2ĵ + 2k̂)
and r̅ = 5î − 2ĵ + m(3î + 2ĵ + 6k̂)
(a) cos−1 (19/21)
(b) cos−1 (21/19)
(c) cos (19/20)
(d) cos−1 (20/21)
Answer
A
7. Solve the following differential equation
(a) tan x = cot y + C
(b) cot x = tan y + C
(c) cot y = tan x + C
(d) tan y = cot x + C
Answer
D
8. Solve the following differential equation
(1 + x2)dy + 2xy dx = cot x dx, (x ≠ 0).
(a) sin x + C
(c) 1 + x2 + C
(d) log|x| + C
Answer
B
9. Find the area bounded by curves
{(x, y) : y ≥ x2 and y ≤ |x|}.
(a) 3 sq. units
(b) 5 sq. units
(c) 2 sq. units
(d) 1/3 sq. units
Answer
D
10. Find the area bounded by the curves y = sin x between x = 0 and x = 2π.
(a) 4 sq. units
(b) 6 sq. units
(c) 2 sq. units
(d) 8 sq. units
Answer
A
11.
(a) odd
(b) even
(c) periodic
(d) None of these
Answer
A
12.
(a) –1
(b) 0
(c) 1
(d) 2
Answer
B
13. If fn(x) = efn–1(x) ∀ n ∈ N and f0(x) = x, then f n′ (x) equals
Answer
B
14. The function f(x) = sin (π/x) is strictly decreasing in the interval
(a) (2n + 3, 2n + 5), n ∈ I
(d) None of these
Answer
C
15. For x > 1, y = ln x satisfies the inequality
(a) y < x – 1
(b) y < 1 – 1/x
(c) y < x2 – 1
(d) None of these
Answer
A,C
16. For any function f(x), if
f ′(a) = f ′′(a) = …… = f (n –1)(a) = 0 but f (n)(a) ≠ 0,
then f(x) has a minima at x = a if
(a) n is even and f (n)(a) > 0
(b) n is even and f (n)(a) < 0
(c) n is odd and f (n)(a) > 0
(d) n is odd and f (n)(a) < 0
Answer
A
17. The largest term in the sequence an = n2 /n3 + 200, is
(a) a1
(b) a7
(c) a8
(d) None of these
Answer
B
18. The value of ∫π0 [tan x]dx ([⋅] denotes integral part), is equal to
(a) –π/2
(b) 0
(c) –1
(d) None of these
Answer
A
19. If w is an imaginary cube root of unity, then the value of
(a) –2
(b) –1
(c) 1
(d) 0
Answer
B
20. If (2 + i)(2 + 2i) (2 + 3i) …… (2 + ni) = x + iy, then 5. 8. 13. …… (4 + n2) is equal to
(a) x2 – y2
(b) x2 + y2
(c) x4 – y4
(d) x4 + y4
Answer
B
21. The value of the expression
where ω is an imaginary cube root of unity is
(a) n(n2+ 3)/3
(b) n(n2+ 2)/3
(c) n(n2+ 1)/3
(d) None of these
Answer
B
22. If a straight line through the point P(3, 4) makes an angle π/6 with x-axis and meets the line 12x + 5y + 10 = 0 at Q, find the length of PQ.
(a) 132
(b) 12√3 + 5
(c) −132/12√3 + 5
(d) 5
Answer
C
23. Find the orthocentre of the triangle whose sides have equations x – 2 = 0, y – 5 = 0 and 5x + 2y – 10 = 0.
(a) (5, 2)
(b) (0, 2)
(c) (5, 0)
(d) (2, 5)
Answer
D
24. The lengths of the tangents from any point on the circle 15x2 + 15y2 – 48x + 64y = 0 to the two circles 5x2 + 5y2 – 24x + 32y + 75 = 0 and 5x2 + 5y2 – 48x + 64y + 300 = 0 are in the ratio
(a) 1 : 2
(b) 2 : 3
(c) 3 : 4
(d) None of these
Answer
A
25. If the point P(4, –2) is the one end of the focal chord PQ of the parabola y2 = x, then the slope of the tangent at Q is
(a) – 1/4
(b) 1/4
(c) 4
(d) – 4
Answer
C
26. The focus of the parabola y2 – x – 2y + 2 = 0 is
(a) (1/4,0)
(b) (1, 2)
(c) (3/4,1)
(d) (5/4,1)
Answer
D
27. If P is a variable point on the ellipse x2/a2 + y2/b2 = 1 with AA′ as the major axis. Find the maximum value of the area of the triangle APA′.
(a) b
(b) a2
(c) ab
(d) a
Answer
C
28. If e and e′ be the eccentricities of a hyperbola and its conjugate, then 1/e2 + 1/e’2 is equal to
(a) 0
(b) 1
(c) 2
(d) None of these
Answer
B
29. The number of solutions of the system of equations 3x – 2y + z = 5, 6x – 4y + 2z = 10 and 9x – 6y + 3z = 15 is
(a) 0
(b) 1
(c) 2
(d) infinite.
Answer
D
30. If A is a square matrix satisfying the equation
A2 – 4A– 5I = 0, then A–1 =
(a) A – 4I
(b) 1/3 (A – 4I)
(c) 1/4 (A – 4I)
(d) 1/5 (A – 4I)
Answer
D
31. If p : It rains today, q : I go to the school, r : I shall meet my friends and s : I shall go for a movie, then which of the following is the proposition :
If does not rain or if I do not go to school, then I shall meet my friend and go for a movie.
(a) ~ (p ∧ q) ⇒ (r ∧ s)
(b) (~ p ∧ ~ q) ⇒ (r ∧ s)
(c) ~ (p ∨ q) ⇒ (r ∨ s)
(d) None of these
Answer
A
32. If p : Ajay is tall
q : Ajay is intelligent
then the symbolic statement ~ p ∨ q means
(a) Ajay is not tall or he is intelligent.
(b) Ajay is tall or he is intelligent.
(c) Ajay is not tall and he is intelligent.
(d) Ajay is not tall then he is intelligent.
Answer
A
33. (~(~p)) ∧ q is equal to
(a) ~ p ∧ q
(b) p ∧ q
(c) ~ p ∧ ~ q
(d) p ∧ ~ q
Answer
B
34. Negation of the compound proposition If the Examination is difficult, then I shall pass if I study hard.
(a) The Examination is difficult and I study hard but I shall not pass.
(b) The Examination is not difficult and I study hard and I shall pass.
(c) The Examination is difficult and I study hard and I shall pass.
(d) None of these.
Answer
A
35. If x2 + y2 = 1, then
(a) yy′′ – (2y′)2 + 1 = 0
(b) yy′′ + (y′)2 + 1 = 0
(c) yy′′ – (y′)2 – 1 = 0
(d) yy′′ + 2(y′)2 + 1 = 0
Answer
B
36. The differential equation of all parabolas whose axes are parallel to y-axis, is
Answer
A
37. The integrating factor of the differential equation (y log y) dx = (log y – x) dy is
(a) 1/log y
(b) log(log y)
(c) 1 + log y
(d) log y
Answer
D
38. If the vectors a̅ = (2, log3 x, a) and b̅ = (–3, a log3 x, log3 x) are inclined at an acute angle, then
(a) a = 0
(b) a < 0
(c) a > 0
(d) None of these
Answer
D
39. If a̅, b̅, c̅ are linearly independent vectors and
(a) Δ = 0
(b) Δ = 1
(c) Δ = any non-zero value
(d) None of these
Answer
C
40. The p.c of the centre of the sphere |r̅|2+ r̅ .(î + ĵ − k̂) − 9 = 0 is
(a) î + ĵ − k̂
(b) 1/2 (î + ĵ − k̂)
(c) − 1/2 (î + ĵ − k̂)
(d) −î + ĵ + k̂
Answer
C
