WBJEE Subjective Mathematics Sample Paper I

Question: 1. The harmonic mean of two numbers is 4. Their arithemetic mean A and the geometric mean G satisfy the relation 2A + G² = 27. find the numbers.
Ans:  (3, 6)
Solution:  Let the number be a, b

Question: 2  If the area of a rectangle is 64 sq. unit, find the minimum value possible for its perimeter.
Ans: 32
Solution: Let the dimesions be a, b
Area = ab
Paimeter = 2(a + b)

Perimeter as function of a

for maxima or minimum

P (8) is minimum
Minimum P (8) = 2 ( 8 + 8) = 32


Question: 3. Find the image of the point (–8, 12) with respect to the line 4x + 7y + 13 = 0
Ans: (–16, –2)
Solution:   



⇒ 4k – 48 = 75 + 56
4k – 7h = 104 ........(ii)
Solving (i) & (ii)
Equation (i) × 7 + Equation (ii) × 4

A′ (– 16, –2) is the image of (–8, 12)


Question:  4. How many triangles can be formed by joining 6 points lying on a circle ?
Ans: 20
Solution:  Number of triangle
 


Question:  5. If r² = x² + y² + z² , then prove that

Ans:   


Question:  6. Determine the sum of imaginary roots of the equation

Ans: 
Solution:  Put 2x² + x = y
⇒ (4 – 1) (24 – 3) = 6, on solving


Question:  7. If cos A + cos B + cos C = 0, prove that  cos 3A + cos 3B + cos 3C = 12 cos A cos B cos C
Ans: 


= 12 cos A . cos B. cos C

Question: 8.  Let IR be the set of real numbers and f : IR → IR  be such that for all x, y ∈IR, |f(x) – f(y)| ≤ | x – y|3 . Prove that f is a constant function.
Ans:  
= |f′ (x) | ≤ | 0 | ⇒ |f′ (x)| = 0
⇒ f (x) = constant


Question: 9. Find the general solution of (x + logy) dy + y dx = 0
Ans. xy + y lny – y = 0
Solution:  x dy + y dx + log y dy = 0

xy + y lny – y = 0
 

Question: 10.  Prove that 
Ans.