WBJEE Subjective Mathematics Sample Paper II | WBJEE Sample Papers, Section Wise & Full Mock Tests 2025 PDF Download

Question: 1. Prove that the equation cos²x + a sinx = 2a – 7 possesses a solution if  
Ans: ⇒ cos²x + a sinx = 2a – 7
⇒ 2sin²x – asinx + (2a – 8) = 0
Since sin ∈IR  
∴ Given equation has solution of 

Question: 2. Find the values of x, (–S < x < S, x ≠ 0) satisfying the equation, 
Ans: 


Question: 3.  Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is 
Ans: Let centre be c(h, h + 1) , 0(0, 0)


Question: 4. Prove by induction that for all n∈ N, n² + n is an even integer (n ≥ 1)
Ans:  x = 1, x² + x = 2 is an even integer
Let for n = k, k² + k is even
Now for n = k + 1, (k + 1)² + (k + 1) – (k² + k)
= k² + 2k + 1 + k + 1 – k² – k = 2k + 2 which is even integer also k² + k is even integer
Hence (k + 1)² + (k + 1) is also an even integer
Hence n² + n is even integer for all n∈ N.
 

Question: 5. If A, B are two square matrices such that AB = A and BA = B, then prove that B² = B
Ans:  B² = B.B = (BA)B = B (AB) = B(A) = BA = B (Proved)
 

Question: 6. 

Ans:  


Question: 7. Use the formula 
Ans:  


Question: 8.    prove that,     where A is constant
Ans:  


Question: 9. Evaluate the following integral  
Ans:   



Question: 10. 
Ans:   
= g'(a) f(a) – g(a) f'(a)
= (2)(2) – (–1) (1) = 4 + 1 = 5