Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set 13

ASSERTION & REASON TYPE

Question. Assertion (A): If \( 2x^2 + 3x + 4 = 0 \) and \( ax^2 + bx + c = 0 \) have a common root, then a : b : c = 2 : 3 : 4
Reason(R):For a quadratic equation in x, complex roots occur in conjugate pairs.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): The roots of (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are real.
Reason(R): Discriminent value above Quadratic equation is positive or equal to zero.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. (i) If \( \alpha,\beta \) are the roots of \( ax^2+bx+c=0 \). If K is a real number [\( K \neq 0,1 \)] then the equation whose roots are \( k\alpha, k\beta \) is \( ax^2+kbx+k^2c=0 \).
(ii) 8,2 are roots of \( x^2+ax+\beta=0 \) and 3,3 are the roots of \( x^2+\alpha x+b=0 \) then the roots of \( x^2+ax+b=0 \) are 9, 1.
Which of the above statement is correct
(a) only (i)
(b) only (ii)
(c) Both (i) and (ii)
(d) Neither (i) nor (ii)
Answer: (c) Both (i) and (ii)

Question. Assertion(A): \( x^2+ x+1 \) is greater than zero for all real x.
Reason (R): When \( b^2-4ac<0 \). Then a, \( ax^2+bx+c \) have same sign for all real values of x.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): The curve \( y = x^2 - 6x +9 \) touches the x-axis.
Reason(R): \( y = ax^2+bx+c \) touches the x-axis if \( \Delta = b^2 -4ac =0 \).
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A) If \( x^2-7x+10<0 \) then \( x \in (2, 5) \).
Reason (R): If 'a' and \( ax^2+bx+c \) have opposite signs and \( b^2 - 4ac>0 \) then x lies between the roots of \( ax^2 +bx +c=0 \).
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A): The equation \( x - \frac{2}{x-1} = 1 - \frac{2}{x-1} \) has no root
Reason(R): \( x-1 \neq 0 \), then only above equation defined.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A.

Question. Assertion (A):
\( f(x) = ax^2 + bx + c \), for \( a(\neq 0), b, c \in R \).
When a>0 and \( b^2 -4ac <0 \), \( f(x)= 0 \) has complex roots.
Reason(R): The graph of \( f(x) = ax^2+bx+c \) cuts x-axis.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (c) A is true but R is false

Question. Assertion (A): If \( f(x) \) is a quadratic expression such that \( f(1)+f(2)=0 \), if -1 is a root of \( f(x)=0 \) then the other root is \( \frac{8}{5} \).
Reason (R): If \( f(x) = ax^2 + bx + c \) then \( \alpha + \beta = -\frac{b}{a}, \alpha\beta = \frac{c}{a} \)
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true and R is not the correct explanation of A.
(c) A is true but R is false
(d) A is false but R is true
Answer: (b) Both A and R are true and R is not the correct explanation of A.

Question. Assertion (A):If \( a>0, b>0 \) and \( c>0 \), then \( (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \ge 9 \)
Reason (R): For positive numbers a, b, c, \( AM \ge GM \)
(a) Both A and R are true and R is a consequence of A.
(b) Both A and R are true and R is not a consequence of A.
(c) A is true but R is false
(d) Both A and R are false
Answer: (a) Both A and R are true and R is a consequence of A.

Question. Assertion (A) : If both roots of equation \( x^2 - 2x + a = 0, a \in R \), lie in the interval (-1,1), then \( -2 < a \le \frac{1}{4} \).
Reason (R) : If \( f(x) = 4x^2 - 2x + a \), then \( D \ge 0 \), \( f(-1) > 0 \) and \( f(1) > 0 \)
\( \Rightarrow \) \( -2 < a \le \frac{1}{4} \).
(a) Both A and R are true and R is a consequence of A.
(b) Both A and R are true and R is not a consequence of A.
(c) A is true but R is false
(d) Both A and R are false
Answer: (a) Both A and R are true and R is a consequence of A.

Question. I. If f(x) be a quadratic expression which is positive for all real x and \( g(x) = f(x) + f'(x) + f''(x) \) for any real x, then \( g(x) > 0 \).
II. If the equations \( px^2 - 7x + 3p = 0 \) and \( 2x^2 + qx + 6 = 0 \) have the same roots then \( pq = -14 \). Which of the above statement(s) is(are) true?
(a) only I
(b) Only II
(c) both I and II
(d) neither I nor II
Answer: (c) both I and II

Question. The arrangement of the following quadratic equations in the ascending order of their number of real roots.
A: \( x^2-5x+6=0 \) B: \( x^2-7|x|=0 \)
C: \( x^2-4x+5=0 \) D: \( x^2-5|x|+6=0 \)
(a) C,A,B,D
(b) D,C,B,A
(c) A,D,B,C
(d) D,A,B,C
Answer: (a) C,A,B,D

Question. The arrangement of following quadratic equations in the ascending order of their product of roots.
A: \( 3x^2-5x+7=0 \) B: \( \sqrt{3}x^2 - 10x - 8\sqrt{3} = 0 \)
C: \( 6x^2 - 5x + 1 = 0 \) D: \( x^2 - 5x + 6 = 0 \)
(a) B,C,A,D
(b) C,A,B,D
(c) C,D,B,A
(d) D,A,B,C
Answer: (a) B,C,A,D

INTEGER ANSWER TYPE QUESTIONS

Question. The number of solutions of the equation \( \sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1} \) is _____
Answer: 0

Question. The number of solution of \( 2^{\sin(|x|)} = 4^{|\cos x|} \) in \( [-\pi, \pi] \) is equal to
Answer: 4