Class 12 Maths - Important Practice Paper & Exam Pattern 2026

MATHEMATICS
Paper-1
SECTION A

Question numbers 1 to 20 carry 1 mark each.
Question numbers 1 to 10 are multiple choice type questions. Select the correct option.
1. If f and g are two functions from R to R defined as f(x) = | x | + x and
g(x) = | x |  x, then fog (x) for x < 0 is Unit I 1
(A) 4x (B) 2x
(C) 0 (D)  4x
2. The principal value of cot  1 ( 3 ) is Unit I 1

(A) 
6
(B) 
6
(C) 2
3
(D) 5
6
 2 0 0 
 
3. If A = 0  2 0  , then the value of | adj A| is Unit II 1
0 0  2 
(A) 64 (B) 16
(C) 0 (D) 8
4. The maximum value of slope of the curve y =  x + 3x2 + 12x  5 is Unit III
3
1
(A) 15 (B) 12
(C) 9 (D) 0
ex (1 + x)
5.  cos2 (xe x ) dx is equal to Unit III 1

(A) tan (xex) + c (B) cot ( xex) + c
(C) cot (ex) + c (D) ten [ex (1 + x)] + c
3

x 2 d y2   x dy  y 
2
6. The degree of the differential equation is Unit III 1
dx  dx 
(A) 1 (B) 2
(C) 3 (D) 6
7. The value of p for which is a unit vector is Unit IV 1
(A) 0 (B) 1
3
(C) 1 (D) 3
8. The coordinates of the foot of the perpendicular drawn from the point
(2, 8, 7) on the XZ - plane is Unit IV 1
(A) (2, 8, 7) (B) (2, 8, 7)
(C) (2, 0, 7) (D) (0, 8, 0)

1

,9. The feasible region for an LPP is shown below : Unit V 1
Let z = 3x 4y be the objective function. Minimum of z occurs at




(A) (0, 0) (B) (0, 8)
(C) (5, 0) (D) (4, 10)
Fill in the blanks in question numbers 11 to 15.
10. If y = tan1 x + cot1 x, x ∈ R, then dy is equal to ______________. Unit III 1
dx
OR
If cos (xy) = k, where k is a constant and xy  n, n ∈ Z, then dy is dy equal to
dx
_________. Unit III 1
11. The value of  so that the function f defined by Unit III 1

 x, if x  
f(x) = 
cos x, if x > 
is continuous at x =  is _______________.
12. The equation of the tangent to the curve y = sec x at the point (0, 1) is ___________ .
Unit III 1
14. The area of the parallelogram whose diagonals are and is _______ square
units. Unit IV 1
OR
The value of  for which the vectors and are
orthogonal is __________ . Unit IV 1
15. A bag contains 3 black, 4 red and 2 green balls. If three balls are drawn simultaneously at
random, then the probability that the balls are of different colours is ___________ .
Unit VI 1
Question numbers 16 to 20 are very short answer type questions.
16. Construct a 2 × 2 matrix A = [aij] whose elements are given by aij = | (i)2 – j |. Unit II 1
17. Differentiate sin² ( x ) with respect to x. Unit III 1
18. Find the interval in which the function f given by f(x) = 7 – 4x – x² is strictly increasing.
Unit III 1
19. Evaluate: Unit III 1
2
2
| x | dx
OR

2

, Find :
dx
 9  4x 2

20. An unbiased coin is tossed 4 times. Find the probability of getting at least one head.
Unit VI 1
SECTION B
Question numbers 21 to 26 carry 2 marks each.
21. Solve for x: Unit I 2

sin 1 4x + sin 1 3x = 
2
OR
Express in the simplest form.

4  3
Express A = 
 1
22. as a sum of a symmetric and a skew symmetric matrix.
2
Unit II 2
If y² cos   = a², then find
1 dy
23. . Unit III 2
x dx


24. Show that for any two non-zero vectors and , Unit IV 2
iff and are perpendicular vectors.
OR
Show that the vectors and form
the sides of a right-angled triangle. Unit IV 2
25. Find the coordinates of the point where the line through (1, 1,  8) and (5, 2, 10)
crosses the zx-plane. Unit IV 2

26. If A and B are two events such that P(A) = 0.4, P(B) = 0.3 and P(A UB) = 0.6, then find
P(B̕̕∩A). Unit IV 2
SECTION C
Question numbers 27 to 32 carry 4 marks each.
27. Show that the function f : (, 0) → (1, 0) defined by f(x) = x ,
1 | x |
x ∈ (, 0) is one-one and onto. Unit I 4
OR
Show that the relation R in the set A = {1, 2, 3, 4, 5, 6} given by
R = {(a, b) : | a – b | is divisible by 2} is an equivalence relation. Unit I 4
dy
28. If y = x3 (cos x)x + sin1 x , find . Unit III 4
dx

29. Evaluate : Unit III 4


3

, 30. Find the general solution of the differential equation Unit III 4
x²y dx  (x + y ) dy = 0.
3 3

31. Solve the following LPP graphically: Unit V 4
Minimise z = 5x + 7y
subject to the constraints
2x + y ≥ 8
x + 2y ≥ 10
x, y ≥ 0
32. A bag contains two coins, one biased and the other unbiased. When tossed, the biased coin
has a 60% chance of showing heads. One of the coins is selected at random and on tossing
it shows tails. What is the probability it was an unbiased coin? Unit VI 4
OR
The probability distribution of a random variable X, where k is a constant is given below :
0.1 if x=0
 2
x)  
kx , if x=1
P(X = Unit VI 4
kx, if x = 2 or 3
0, otherwise

Determine
(a) the value of k
(b) P(x ≤ 2)
(c) Mean of the variable X.

SECTION D
Question numbers 33 to 36 carry 6 marks each.
33. Solve the following system of equations by matrix method: Unit II 6
x  y + 2z = 7
2x  y + 3z = 12
3x + 2y  z = 5
34. Find the points on the curve 9y2 = x3, where the normal to the curve makes equal
intercepts with both the axes. Also find the equation of the normals. Unit III 6
35. Find the area of the following region using integration: Unit V 6
{(x, y) : y ≤ | x | + 2, y ≥ x )
2

OR
Using integration, find the area of a triangle whose vertices are (1,0), (2,2) and (3,1). 6
36. Show that the lines
x  2 y  2 z  3 x  2 y  3 z  4
  and   intersect.
1 3 1 1 4 2
Also, find the coordinates of the point of intersection. Find the equation of the plane
containing the two lines. Unit VI 6




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