Most Important Questions for CHSE 2026
(5 Mark)
MATHEMATICS Booklet
Class - 12
Relations & Functions 8. Determine the matrices A and B, where
1 2 0 2 −1 5
𝐴 + 2𝐵 = [ 6 −3 3] and 2𝐴 − 𝐵 = [2 −1 6]
1. Let R be a relation on the set ℝ of real numbers such −5 3 1 0 1 2
that aRb if a – b is an integer. Test whether R is an
equivalence relation. If so find the equivalence class Determinants
of 1 and ½.
9. Solve the system of equations by matrix inversion
2. Test whether the relation R defined by method.
𝑚
𝑅 = {(𝑚, 𝑛): is a power of 5} on the set Z−{0} x+y+z=4
𝑛
2x – y + 3z = 1
3. If f(x) = cos[𝜋 2 ]𝑥 + cos[−𝜋 2 ]𝑥 ,where [x] stands 3x + 2y - z = 1
for the greatest integer function,
𝜋 𝜋 10. Solve the system of equations by Cramer’s Rule
then evaluate f( ), f( ), f(𝜋) , f(−𝜋).
2 4 x+y+z=4
2x - y + 3z = 1
4. Examine the injectivity and surjectivity 3x + 2y – z = 1
1
(i) f : R+ → R+ , f(x) = x + , Vectors
𝑥
where R+ = {x ∈ R : x >0} 11. Prove by vector method 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
(ii) f: R →R , f(x) = x3 + 1 12. Prove by vector method: a = b cos C + c cos B
Inverse Trigonometric Functions 13. If 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗ are three vectors such that
|𝑎⃗| = 5, |𝑏⃗⃗| = 12, |𝑐⃗| = 13 and 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0
⃗⃗, then
5. Show that ⃗⃗ ⃗⃗
find the value of 𝑎⃗. 𝑏 + 𝑏. 𝑐⃗ + 𝑐⃗. 𝑎⃗.
𝜋 1 𝑎 𝜋 1 𝑎 2𝑏 14. Find a unit vector perpendicular to each of the
tan ( + cos −1 ) + tan ( − cos −1 ) =
4 2 𝑏 4 2 𝑏 𝑎
vectors 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗, where 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and
6. Prove that 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂ .
𝑏+𝑎cos𝑥 𝑎−𝑏 𝑥
Three Dimensional Geometry
cos −1 ( ) = 2tan−1 (√ tan )
𝑎+𝑏cos𝑥 𝑎+𝑏 2
15. Find the shortest distance between the following two
𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−4 𝑧−5
Matrices lines = = and = =
2 3 4 3 4 5
Find also the equation of the line of shortest distance.
7. Verify that, (𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 where,
16. Prove that the straight lines
1 2 3 1 2 3 𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−3 𝑧−4
= = and = = intersect
𝐴 = [6 7 8] , 𝐵 = [ 3 4 2] 2 3 4 3 4 5
each other. Find their point of intersection.
6 −3 4 5 6 1
, 2
17. Find the distance of the point (1, −1, −10) from the 25. An urn contains 4 white balls and 6 red balls. Four
𝑥−4 𝑦+3 𝑧+1 balls are drawn at random from the urn. Find the
line = = measured parallel to the line
1 −4 7
𝑥+2 𝑦−3 𝑧−4 probability distribution of the number of white balls.
= = .
2 −3 8
26. A random variable 'x' has following probability
Linear Programming distributions
18. Maximize: Z = 50 x + 60y X 0 1 2 3 4 5
Subject to: x+y ≤5 P(x) 0 R 2R 3R 3R R
x + 2y ≥ 4
x,y≥0 Find (i) R (ii) P(x < 4) (iii) P(x≥2) (iv) P(2< x< 5)
19. Maximise: Z = 4x +3y 27. There are 25 girls and 15 boys in class XI and 30 boys
Subject to: x + y ≤ 50 and 20 girls in Class XII. If a student from a Class
x + 2y ≤ 80 selected at random, happens to be a boy, find the
2x + y ≥ 20, probability that he has been chosen from Class XII.
x,y ≥0
Indefinite Integration
20. Maximise: Z = 4x + 6y
𝑑𝑥
Subject to: 5x +10y ≥ 30 28. Integrate: ∫
(1+𝑥)√1−𝑥 2
6x +3y ≥ 18
x+y≤6 𝜋 𝑥
x,y ≥0 29. Evaluate: ∫0 𝑑𝑥
1+ 𝑠𝑖𝑛𝑥
𝑑𝑥
Continuity & Differentiability 30. Integrate: ∫
cos𝑥(1+2sin𝑥)
𝑎𝑥 2 + 𝑏, 𝑖𝑓 𝑥<1 2sin𝑥+3cos𝑥
31. Integrate: ∫
21. If 𝑓(𝑥) = { 1, 𝑖𝑓 𝑥 = 1 is continuous at 3sin𝑥+4cos𝑥
2𝑎𝑥 − 𝑏, 𝑖𝑓 𝑥>1
x=1, the find a and b.
Area Under The Curve
𝑑𝑦 1/x x
22. Find , if y = x + (sin x) 𝑥2 𝑦2
𝑑𝑥
32. Find the area of the ellipse + =1
9 16
−1
23. If 𝑦 = 𝑒 𝑚cos 𝑥 ,
then show that (1 − 𝑥 2 )𝑦2 − 𝑥𝑦1 = 𝑚2 𝑦 Differential Equations
𝑑𝑦 1 𝑦 𝑦2
Probability 33. Solve: = ( + )
𝑑𝑥 2 𝑥 𝑥2
24. Four cards are drawn successively with replacement 𝑑𝑦 𝜋
from a well shuffled pack of 52 cards. What is the 34. Solve: + 2𝑦tan𝑥 = sin𝑥, 𝑦 ( ) = 0
𝑑𝑥 3
probability that:
(i) All the four cards are diamond ? 𝑑𝑦
35. Solve: (1 + 𝑥 2 ) + 2𝑥𝑦 = cos𝑥.
(ii) Only two cards are diamonds ? 𝑑𝑥
(iii) None of the cards is a diamond ?
PW Web/App - https://smart.link/7wwosivoicgd4
Library- https://smart.link/sdfez8ejd80if
, Most Important Questions for CHSE 2026
(2 & 3 Mark)
MATHEMATICS Booklet
Class - 12
Vector 12. Find the coordinates of the foot of the perpendicular
from the point (1, 0, -2) on the line joining (-2, 4, -2)
1. If 𝑎⃗ = 2𝑖̂ + 3𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ − 3𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 3𝑗̂ − 4𝑘̂, and (1, 5, 10).
then determine the DC's of 𝑎⃗ − 𝑏⃗⃗ + 2𝑐⃗.
13. If a 3iˆ ˆj 2k
ˆ and b 2iˆ 3jˆ 4k,
ˆ then verify
2. Find the values of m and n if (𝑚 − 1)𝑖̂ + (𝑛 + 2)𝑗̂ + that a b is perpendicular to both a and b .
4𝑘̂ and (𝑚 + 1)𝑖̂ + (𝑛 − 2)𝑗̂ + 8𝑘̂ are parallel to
each other.
14. If 𝑎⃗ = (2 , −2 , 1) , 𝑏⃗⃗ = (2,3,6) 𝑎𝑛𝑑 𝑐⃗ = (−1,0,2)
3. If the position vectors of the points P, Q, R and S are then find the magnitude and direction of 𝑎⃗ + 𝑏⃗⃗ − 𝑐⃗.
4𝑖̂ + 3𝑗̂ − 𝑘̂ , 5𝑖̂ + 2𝑗̂ + 2𝑘̂ , 2𝑖̂ − 2𝑗̂ − 3𝑘̂ and 4𝑖̂ −
4𝑗̂ + 3𝑘̂ respectively, then show that ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ are
𝑃𝑄 and 𝑅𝑆 15. Find a vector 𝑏⃗⃗ such that 𝑎⃗ × ⃗⃗⃗⃗ 𝑏 = 𝑐⃗ and 𝑎⃗. 𝑏⃗⃗ = 3
parallel. where 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ , 𝑐⃗ = 𝑗̂ − 𝑘̂ .
4. Find the vector in the direction of the vector 16. ABCD is a parallelogram . Using vector method
𝑖̂ − 2𝑗̂ + 2𝑘̂ that has magnitude 9 units. prove that the line joining A and the mid point of BC
intersect the diagonal BD in the ratio 1 : 2.
5. If sum of the two unit vectors is a unit vector, then
show that the magnitude of their difference is √3 . 17. Find the scalar projection of the vector
𝑎⃗ = 3𝑖̂ + 6𝑗̂ + 9𝑘̂ on 𝑏⃗⃗ = 2𝑖̂ + 2𝑗̂ − 𝑘̂.
6. Show that 𝑎⃗ × (𝑏⃗⃗ + 𝑐⃗) + 𝑏⃗⃗ × (𝑐⃗ + 𝑎⃗) + 𝑐⃗ × (𝑎⃗ +
18. Find a unit vector perpendicular to each of the vectors
𝑏⃗⃗)
𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗, where 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ +
2𝑗̂ + 3𝑘̂ .
7. Find the angle between two vectors 𝑎⃗ and 𝑏⃗⃗ with
magnitudes 2 and 1 respectively, such that 𝑎⃗ ⋅ 𝑏⃗⃗ = √3. 19. Find the area of the parallelogram whose diagonals
are represented by the vectors 𝑎⃗ = 2𝑖̂ − 3𝑗̂ + 4𝑘̂ and
8. Find the shortest distance between the lines whose
vector equations are 𝑏⃗⃗ = 2𝑖̂ − 𝑗̂ + 2𝑘̂ .
𝑟⃗ = (𝑖̂ + 𝑗̂) + 𝜆(2𝑖̂ − 𝑗̂ + 𝑘̂ ) 20. Find the angle between 𝑎⃗ and 𝑏⃗⃗ if |𝑎⃗ × 𝑏⃗⃗| = |𝑎⃗ ⋅ 𝑏⃗⃗|.
𝑟⃗ = (2𝑖̂ + 𝑗̂ − k̂) + 𝜇(3𝑖̂ − 5𝑗̂ + 2𝑘̂ )
3D Geometry
9. Find the component of the vector 𝑏⃗⃗ = 8𝑖̂ + 𝑗̂ in the
21. If a line makes angles 90°, 60° and 30° with the
direction of the vector 𝑎⃗ = 𝑖̂ + 2𝑗̂ − 2𝑘̂ . positive directions of x, y and z-axes respectively,
then find its direction cosines.
10. Find the value of ‘m’ if the vectors
(𝑚 + 1)𝑖̂ + 𝑚2 𝑗̂ − 𝑚𝑘̂ and (𝑚2 − 𝑚 + 1)𝑖̂ − 𝑚𝑗̂ + 𝑘̂ are 22. Find the direction cosines of a line whose direction
orthogonal. ratios are 2 , -6 , 3.
11. If a b c 0, | a | 3 , | b | 5 and | c | 7 ,then find 23. Find the angle between the pair of lines
𝑥+3
=
𝑦−1
=
3 5
the angle between a and b . 𝑧+3 𝑥+1 𝑦−4 𝑧−5
and = = .
4 1 1 2
(5 Mark)
MATHEMATICS Booklet
Class - 12
Relations & Functions 8. Determine the matrices A and B, where
1 2 0 2 −1 5
𝐴 + 2𝐵 = [ 6 −3 3] and 2𝐴 − 𝐵 = [2 −1 6]
1. Let R be a relation on the set ℝ of real numbers such −5 3 1 0 1 2
that aRb if a – b is an integer. Test whether R is an
equivalence relation. If so find the equivalence class Determinants
of 1 and ½.
9. Solve the system of equations by matrix inversion
2. Test whether the relation R defined by method.
𝑚
𝑅 = {(𝑚, 𝑛): is a power of 5} on the set Z−{0} x+y+z=4
𝑛
2x – y + 3z = 1
3. If f(x) = cos[𝜋 2 ]𝑥 + cos[−𝜋 2 ]𝑥 ,where [x] stands 3x + 2y - z = 1
for the greatest integer function,
𝜋 𝜋 10. Solve the system of equations by Cramer’s Rule
then evaluate f( ), f( ), f(𝜋) , f(−𝜋).
2 4 x+y+z=4
2x - y + 3z = 1
4. Examine the injectivity and surjectivity 3x + 2y – z = 1
1
(i) f : R+ → R+ , f(x) = x + , Vectors
𝑥
where R+ = {x ∈ R : x >0} 11. Prove by vector method 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
(ii) f: R →R , f(x) = x3 + 1 12. Prove by vector method: a = b cos C + c cos B
Inverse Trigonometric Functions 13. If 𝑎⃗, 𝑏⃗⃗ and 𝑐⃗ are three vectors such that
|𝑎⃗| = 5, |𝑏⃗⃗| = 12, |𝑐⃗| = 13 and 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0
⃗⃗, then
5. Show that ⃗⃗ ⃗⃗
find the value of 𝑎⃗. 𝑏 + 𝑏. 𝑐⃗ + 𝑐⃗. 𝑎⃗.
𝜋 1 𝑎 𝜋 1 𝑎 2𝑏 14. Find a unit vector perpendicular to each of the
tan ( + cos −1 ) + tan ( − cos −1 ) =
4 2 𝑏 4 2 𝑏 𝑎
vectors 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗, where 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and
6. Prove that 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂ .
𝑏+𝑎cos𝑥 𝑎−𝑏 𝑥
Three Dimensional Geometry
cos −1 ( ) = 2tan−1 (√ tan )
𝑎+𝑏cos𝑥 𝑎+𝑏 2
15. Find the shortest distance between the following two
𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−4 𝑧−5
Matrices lines = = and = =
2 3 4 3 4 5
Find also the equation of the line of shortest distance.
7. Verify that, (𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 where,
16. Prove that the straight lines
1 2 3 1 2 3 𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−3 𝑧−4
= = and = = intersect
𝐴 = [6 7 8] , 𝐵 = [ 3 4 2] 2 3 4 3 4 5
each other. Find their point of intersection.
6 −3 4 5 6 1
, 2
17. Find the distance of the point (1, −1, −10) from the 25. An urn contains 4 white balls and 6 red balls. Four
𝑥−4 𝑦+3 𝑧+1 balls are drawn at random from the urn. Find the
line = = measured parallel to the line
1 −4 7
𝑥+2 𝑦−3 𝑧−4 probability distribution of the number of white balls.
= = .
2 −3 8
26. A random variable 'x' has following probability
Linear Programming distributions
18. Maximize: Z = 50 x + 60y X 0 1 2 3 4 5
Subject to: x+y ≤5 P(x) 0 R 2R 3R 3R R
x + 2y ≥ 4
x,y≥0 Find (i) R (ii) P(x < 4) (iii) P(x≥2) (iv) P(2< x< 5)
19. Maximise: Z = 4x +3y 27. There are 25 girls and 15 boys in class XI and 30 boys
Subject to: x + y ≤ 50 and 20 girls in Class XII. If a student from a Class
x + 2y ≤ 80 selected at random, happens to be a boy, find the
2x + y ≥ 20, probability that he has been chosen from Class XII.
x,y ≥0
Indefinite Integration
20. Maximise: Z = 4x + 6y
𝑑𝑥
Subject to: 5x +10y ≥ 30 28. Integrate: ∫
(1+𝑥)√1−𝑥 2
6x +3y ≥ 18
x+y≤6 𝜋 𝑥
x,y ≥0 29. Evaluate: ∫0 𝑑𝑥
1+ 𝑠𝑖𝑛𝑥
𝑑𝑥
Continuity & Differentiability 30. Integrate: ∫
cos𝑥(1+2sin𝑥)
𝑎𝑥 2 + 𝑏, 𝑖𝑓 𝑥<1 2sin𝑥+3cos𝑥
31. Integrate: ∫
21. If 𝑓(𝑥) = { 1, 𝑖𝑓 𝑥 = 1 is continuous at 3sin𝑥+4cos𝑥
2𝑎𝑥 − 𝑏, 𝑖𝑓 𝑥>1
x=1, the find a and b.
Area Under The Curve
𝑑𝑦 1/x x
22. Find , if y = x + (sin x) 𝑥2 𝑦2
𝑑𝑥
32. Find the area of the ellipse + =1
9 16
−1
23. If 𝑦 = 𝑒 𝑚cos 𝑥 ,
then show that (1 − 𝑥 2 )𝑦2 − 𝑥𝑦1 = 𝑚2 𝑦 Differential Equations
𝑑𝑦 1 𝑦 𝑦2
Probability 33. Solve: = ( + )
𝑑𝑥 2 𝑥 𝑥2
24. Four cards are drawn successively with replacement 𝑑𝑦 𝜋
from a well shuffled pack of 52 cards. What is the 34. Solve: + 2𝑦tan𝑥 = sin𝑥, 𝑦 ( ) = 0
𝑑𝑥 3
probability that:
(i) All the four cards are diamond ? 𝑑𝑦
35. Solve: (1 + 𝑥 2 ) + 2𝑥𝑦 = cos𝑥.
(ii) Only two cards are diamonds ? 𝑑𝑥
(iii) None of the cards is a diamond ?
PW Web/App - https://smart.link/7wwosivoicgd4
Library- https://smart.link/sdfez8ejd80if
, Most Important Questions for CHSE 2026
(2 & 3 Mark)
MATHEMATICS Booklet
Class - 12
Vector 12. Find the coordinates of the foot of the perpendicular
from the point (1, 0, -2) on the line joining (-2, 4, -2)
1. If 𝑎⃗ = 2𝑖̂ + 3𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ − 3𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 3𝑗̂ − 4𝑘̂, and (1, 5, 10).
then determine the DC's of 𝑎⃗ − 𝑏⃗⃗ + 2𝑐⃗.
13. If a 3iˆ ˆj 2k
ˆ and b 2iˆ 3jˆ 4k,
ˆ then verify
2. Find the values of m and n if (𝑚 − 1)𝑖̂ + (𝑛 + 2)𝑗̂ + that a b is perpendicular to both a and b .
4𝑘̂ and (𝑚 + 1)𝑖̂ + (𝑛 − 2)𝑗̂ + 8𝑘̂ are parallel to
each other.
14. If 𝑎⃗ = (2 , −2 , 1) , 𝑏⃗⃗ = (2,3,6) 𝑎𝑛𝑑 𝑐⃗ = (−1,0,2)
3. If the position vectors of the points P, Q, R and S are then find the magnitude and direction of 𝑎⃗ + 𝑏⃗⃗ − 𝑐⃗.
4𝑖̂ + 3𝑗̂ − 𝑘̂ , 5𝑖̂ + 2𝑗̂ + 2𝑘̂ , 2𝑖̂ − 2𝑗̂ − 3𝑘̂ and 4𝑖̂ −
4𝑗̂ + 3𝑘̂ respectively, then show that ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ are
𝑃𝑄 and 𝑅𝑆 15. Find a vector 𝑏⃗⃗ such that 𝑎⃗ × ⃗⃗⃗⃗ 𝑏 = 𝑐⃗ and 𝑎⃗. 𝑏⃗⃗ = 3
parallel. where 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ , 𝑐⃗ = 𝑗̂ − 𝑘̂ .
4. Find the vector in the direction of the vector 16. ABCD is a parallelogram . Using vector method
𝑖̂ − 2𝑗̂ + 2𝑘̂ that has magnitude 9 units. prove that the line joining A and the mid point of BC
intersect the diagonal BD in the ratio 1 : 2.
5. If sum of the two unit vectors is a unit vector, then
show that the magnitude of their difference is √3 . 17. Find the scalar projection of the vector
𝑎⃗ = 3𝑖̂ + 6𝑗̂ + 9𝑘̂ on 𝑏⃗⃗ = 2𝑖̂ + 2𝑗̂ − 𝑘̂.
6. Show that 𝑎⃗ × (𝑏⃗⃗ + 𝑐⃗) + 𝑏⃗⃗ × (𝑐⃗ + 𝑎⃗) + 𝑐⃗ × (𝑎⃗ +
18. Find a unit vector perpendicular to each of the vectors
𝑏⃗⃗)
𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗, where 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ +
2𝑗̂ + 3𝑘̂ .
7. Find the angle between two vectors 𝑎⃗ and 𝑏⃗⃗ with
magnitudes 2 and 1 respectively, such that 𝑎⃗ ⋅ 𝑏⃗⃗ = √3. 19. Find the area of the parallelogram whose diagonals
are represented by the vectors 𝑎⃗ = 2𝑖̂ − 3𝑗̂ + 4𝑘̂ and
8. Find the shortest distance between the lines whose
vector equations are 𝑏⃗⃗ = 2𝑖̂ − 𝑗̂ + 2𝑘̂ .
𝑟⃗ = (𝑖̂ + 𝑗̂) + 𝜆(2𝑖̂ − 𝑗̂ + 𝑘̂ ) 20. Find the angle between 𝑎⃗ and 𝑏⃗⃗ if |𝑎⃗ × 𝑏⃗⃗| = |𝑎⃗ ⋅ 𝑏⃗⃗|.
𝑟⃗ = (2𝑖̂ + 𝑗̂ − k̂) + 𝜇(3𝑖̂ − 5𝑗̂ + 2𝑘̂ )
3D Geometry
9. Find the component of the vector 𝑏⃗⃗ = 8𝑖̂ + 𝑗̂ in the
21. If a line makes angles 90°, 60° and 30° with the
direction of the vector 𝑎⃗ = 𝑖̂ + 2𝑗̂ − 2𝑘̂ . positive directions of x, y and z-axes respectively,
then find its direction cosines.
10. Find the value of ‘m’ if the vectors
(𝑚 + 1)𝑖̂ + 𝑚2 𝑗̂ − 𝑚𝑘̂ and (𝑚2 − 𝑚 + 1)𝑖̂ − 𝑚𝑗̂ + 𝑘̂ are 22. Find the direction cosines of a line whose direction
orthogonal. ratios are 2 , -6 , 3.
11. If a b c 0, | a | 3 , | b | 5 and | c | 7 ,then find 23. Find the angle between the pair of lines
𝑥+3
=
𝑦−1
=
3 5
the angle between a and b . 𝑧+3 𝑥+1 𝑦−4 𝑧−5
and = = .
4 1 1 2
