Vectors:Class 12th:DPP

Vectors
Q1 Solve the following-                                          
Show that the vectors

4i+13j-18k,

 i-2j+3k,

2i+3j-4k are coplaner                          (3)

Q2 if a,b,c are non co-planer vectors then prove that the vectors a+b+c ,a+b-3c, a+4b-9c are collinear.                                (3)     
Q3 Prove the section formula for internal division  (4)
Q4 If the three points A(3,2,p),B(q,8,-10) and c(-2,-3,1) are collinear then find 
i) the ratio in which point C divides line segment AB
ii) the values of p and q                                              (4)
Q5 Prove that the medians of triangle are concurrent. (3)
Q6 Prove that the angle subtended on a semicircle is a right angle. (3)

Pair of Straight lines: Class12th:DPP

Pair of Straight lines: 
Q1 Find the separate equations of lines represented by.    (8)

                               I.            x²-4y² = 0

                              II.            3x²-7xy+4y² = 0

                             III.            x²+2xy-y² = 0

                             IV.            5x²-3y²=0

Q2 Find the joint equation of pair of lines passing through origin and perpendicular to the lines represented by ax²+2hxy+by²=0                                                                   (3)
Q3 Find the value of k ,if the slope of one of lines is given by 4x²+kxy+y²=0 is four times the other                                   (3)
Q4 Find p and q if the equation px²- 8xy +3y²+14x+2y+q=0 represents pair of perpendicular lines.                                   (4)
Q5 Find the acute angle between the lines  x²-4xy+y²=0                                                                                                                  (2)

Trigonometric functions:class 12th: DPP

Trigonometric Functions

Q 1 Answer the following questions                                 (5M)

a) The solutions of a trigonometric equation which are not generalized by using its periodicity are known as _____.

i) General Solution  ii) Principle Solution iii) Normal Solution iv) Ordinary Solution

b) Which of the following is not a trigonometric equation.

i) sinx = ⎷3/2      ii) secӨ = cosec x   iii) ⎷3/x = 17  iv)  cosx + Sin 3x + tan 5x = 0

d) If cos Ө = 0 then Ө = _______

i) nπ ii) (2n+1)π/2   iii) 2nπ  iv) (2n+1)π

e) For Cos x = ½ then the general solution of x is

i)0  ii) π/3,5π/3  iii) π/3,5π/3,13π/3….. iv) (n+1) π/3

Q2 Prove the following equations (any three)         (6M)

a) If tan ²x =  tan ²β then x = nπ 士 β

b) Find the Principle solution of cos3x = cos2x

c) Find the general Solution of tan 3x = -1

d) Find the general solution of ⎷3cosec x = 2

Q3 Find the general Solution of equation            (9M)

a) cos5x = sin3x

b) cosx - sinx = 1

c) 2tanx - cotx + 1 = 0
Q4)  Show that                                                       (6M)

a) 2tan^-1x = cos^-1(1-x²/1+x²) For x>0

b) 2tan^-1x =  tan^-1(2x/1-x²) For -1

Q5) Prove the following properties                      (6M)

a) sin^-1(x) + cos^-1(x) = π/2  for x∈[-1,1]

b) tan^-1(x) + tan^-1(y) = tan^-1((x+y)/(1-xy)),if x,y>0 and xy≠1

Q6) Prove the property                                         (4M)

a) cosec^-1(-x) = - cosec^-1(x)

b) cot^-1(-x) = π - cot^-1(x)

Q7) Show that                                                          (4M)

tan^-1x + tan^-1(2x/1+x²) = tan^-1(3x - x³/1-3x²).     for |x|

Matrices:12th:DPP

Matrices

Mathematical logic:Class 12th

Mathematical logic
Q1. State the truth value of following statements.   (5)

a) Every real no. is complex no.

b) The sum of interior triangle is 180°.

c) The square of every real number is positive.

d) Square of an odd no. is odd.

e) Every set is a finite set.

Q2. Express following statements in symbolic form.(5)
A. Paris is in England and Rome is in Italy.
B. 2+3=5 or √2+√3=√5
C. 2 is rational number or √2 is irrational number.
D. P M will visit to either Pune or Mumbai.
E. Either we play football or go for cycling.
Q3. Find the truth value of following statements.    (2)
1. (~q ⋀p) ⋀ q
2. [P ⋀ (p⟶q)]⟶q
Q4. Construct a truth table for the following logic expression.                                                                    (3)
 p ⋀ q =~(p⟶~q)
Q5. Find the truth value                                             (4)
If p and q are the true and r and s are false statements, Find the truth value of the following

1. (p ⋀ q) V r
2. p ⋀ (r⟶s)
3. (p V s) ↔️ (q ⋀ r)
4. ~(p ⋀~r) V (~q V s)

Topicwise test on Application of derivative

 Test on Application of derivative

Topicwise Test on Integration

Test ~ Integration ~ 25 marks ~ 50 min