CBSE Class 12 Mathematics HOTs All Chapters Set 03

Relations and Functions
6 Marks Questions

1. Let f:[0,∞) → R be a function defined by f(x)=9x²+6x-5.Prove that f is not invertible. Modify, only the codomain of f to make it invertible and then find f¹

2. If f: A→ B such that A=R-{3}, B=R-{1} and f(x)= x-2/x-3 show that f is one-one and onto. Hence find f1.

3. Let ∗ be a binary defined on RxR by (a,b)∗(c,d)=(a+c,b+d) is a binary operation or not.If a binary, is it commutative and associative too? Find its identity element if it exists, Also find the inverse of the element (2,-3) in A.

4. Let ∗ be a binary on QxQ defined by (a,b)∗(c,d) = (ac,b+ad).Determine, whether ∗ is commutative and associative .Find the identity element for ∗ and the invertible elements of QxQ.

5. Show that the relation R in the set A ={x∈ Z : 0 ≤ x ≤ 10}, given by R = {(a,b) : |a - b| is a multiple of 3} is an equivalence relation. Write all its equivalence classes. 

Inverse Trigonometric Functions
4 Marks Questions 

Continuity and Differentiability
1 Mark Questions 

Integrals - 1 Mark Questions

Differential Equations - 1 Mark Questions

2 Marks Questions

1. Verify that y-cosy=x is a solution of the differential equation (ysiny+cosy+x)y/=y.

2. Form the diff. eqn. representing the family of curves y=asin(x+b).

3. Find the general solution of the differential equation: ylogydx– xdy = 0

4. Find the general solution of the differential equation: e^x tany dx + (1- e^x)sec²y dy = 0

Vectors and 3-D Geometry - 1 Mark Questions 

2 Marks Questions

1. Find the distance between the planes 2x-y+2z +5=0 and 4x-2y +4z+16=0.

2. Write the vector eqn. of a line passing through the point (2,-1,3) and perpendicular to the plane 3x-5y+2z=6.

3. Find the angle between the lines: 3+x/3 = 1-y/-5 = z+2/4 and x+1/1 = y-4/1 = 5-3z/-6

4. Write the eqn. of plane 2x-3y+4z+6=0 in intercept form.

5. Find the equation of plane bisecting the line segment joining the points (3,2,5) and (5,4,-1) at right angles.

6. Write the eqn. of plane on which the foot of perpendicular from origin is (5,2,1).

7. The line r̅ = 2î - 3ĵ + pk̂+ t (î - ĵ + k̂) lies in the plane r̅. (3î + ĵ - k̂) - 4 = 0 ,then find the values of p and q.

Probability
1 or 2 mark(s) questions

1. Mother, father and son line up at random for a family picture. If E: son on one end F: father in the middle, determine P(E/F).

2. Given that the two numbers appearing on throwing two dice are different, find the probability of the event `the sum of numbers on the dice is 4`.

3. P(A)= 6/11, P(B) = 5/11 and P(A U B) = 7/11 find P(B/A)

4. An unbiased die is thrown twice.Let the event A be `odd number on the first throw` and B the event ` odd number on the second throw`. Check the independence of the events A and B.

5. If A and B are two independent events, then show that the probability of occurrence of atleast one of A and B is given by 1-P(A/)P(B/).

6. A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. Given that there is atleast one girl on the committee, calculate the probability that there are exactly 2 girls on the committee.

7. If A and B are independent events, P(A)=p and P(B)=2p and P(exactly one of A,B)= 5/9 , then find p.