MATHEMATICS PRACTICE TEST (MCQs WITH CORRECT ANSWERS AND QUESTIONS.

MATHEMATICS PRACTICE TEST (MCQs WITH CORRECT ANSWERS
AND QUESTIONS.



1. If ( f(x) = x^3 - 3x + 1 ), how many real roots does ( f(x) = 0 ) have?
A. 1
B. 2
C. 3
D. 0
Correct Answer: C
Explanation: The cubic function has turning points and changes sign,
indicating three real roots. Graphical or derivative analysis confirms
this.



2. Evaluate: ( \lim_{x \to 0} \frac{\sin 5x}{x} )
A. 1
B. 5
C. 0
D. Does not exist
Correct Answer: B
Explanation: Using standard limit ( \lim_{x \to 0} \frac{\sin ax}{x} = a ),
so the result is 5.



3. If ( \log_2(x-1) + \log_2(x+1) = 3 ), find x.
A. 2
B. 3
C. √5
D. 5
Correct Answer: B
Explanation: Combine logs: ( \log_2[(x-1)(x+1)] = 3 \Rightarrow x^2 -1
= 8 \Rightarrow x^2 = 9 \Rightarrow x=3 ).



4. The derivative of ( y = e^{2x} \sin x ) is:
A. ( 2e^{2x}\sin x )
B. ( e^{2x}(2\sin x + \cos x) )
C. ( e^{2x}(2\cos x + \sin x) )
D. ( e^{2x}(\sin x + \cos x) )

, Correct Answer: B
Explanation: Product rule: derivative = ( e^{2x}(2\sin x) + e^{2x}\cos x
).



5. Find the determinant of matrix ( \begin{pmatrix} 2 & 3 \ 1 & 4
\end{pmatrix} ).
A. 5
B. 6
C. 7
D. 8
Correct Answer: A
Explanation: Determinant = ( (2×4 - 3×1) = 8 - 3 = 5 ).



6. Solve: ( x^2 - 6x + 9 = 0 )
A. x = 3
B. x = -3
C. x = 0
D. x = 6
Correct Answer: A
Explanation: Perfect square: ( (x-3)^2 = 0 \Rightarrow x=3 ).



7. The sum of an infinite geometric series with ( a=4 ), ( r=1/2 ) is:
A. 6
B. 8
C. 10
D. 12
Correct Answer: B
Explanation: ( S = \frac{a}{1-r} = \frac{4}{1-1/2} = 8 ).



8. If vectors ( \vec{a} = (1,2,3) ), ( \vec{b} = (4,5,6) ), find ( \vec{a} \cdot
\vec{b} ).
A. 20
B. 30
C. 32
D. 28

, Correct Answer: C
Explanation: Dot product = ( 1×4 + 2×5 + 3×6 = 4+10+18=32 ).



9. Solve: ( \int x e^x dx )
A. ( e^x )
B. ( xe^x - e^x + C )
C. ( xe^x + C )
D. ( e^x(x+1) )
Correct Answer: B
Explanation: Integration by parts gives ( xe^x - e^x + C ).



10.The probability of getting exactly 2 heads in 3 tosses is:
A. 1/2
B. 3/8
C. 1/4
D. 1/8
Correct Answer: B
Explanation: Binomial probability: ( \binom{3}{2}(1/2)^3 = 3/8 ).



11.Solve: ( 2^x = 16 )
A. 2
B. 3
C. 4
D. 5
Correct Answer: C
Explanation: ( 16 = 2^4 \Rightarrow x=4 ).



12.The slope of the line passing through (1,2) and (3,6) is:
A. 1
B. 2
C. 3
D. 4
Correct Answer: B
Explanation: Slope = ( (6-2)/(3-1)=4/2=2 ).

, 13.If ( A \cap B = \emptyset ), then A and B are:
A. Independent
B. Mutually exclusive
C. Equal
D. Complementary
Correct Answer: B
Explanation: Disjoint sets have no common elements.



14.Find the derivative of ( \ln x ).
A. x
B. 1/x
C. ln x
D. e^x
Correct Answer: B
Explanation: Standard derivative rule.



15.Solve: ( \sqrt{x+5} = 3 )
A. 4
B. 9
C. 14
D. 16
Correct Answer: A
Explanation: Square both sides: ( x+5=9 \Rightarrow x=4 ).



16.The variance of a constant is:
A. 1
B. 0
C. Undefined
D. Infinity
Correct Answer: B
Explanation: No variability means zero variance.



17.Find ( \cos 60^\circ ).
A. 0
B. 1/2
C. √3/2