MCQ Test of Class 10, Maths Pynomial - Study Material
Question 2 :
The degree of the remainder is always less than the degree of the divisor.
Question 4 :
If $a\ne 2$, which of the following is equal to $\cfrac { b\left( { a }^{ 2 }-4 \right) }{ ab-2b } $?
Question 6 :
If $\alpha , \beta$ are the zeros of the polynomials $f(x) = x^2+x+1 $ then $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=$________.
Question 7 :
If a polynomial $p(x)$ is divided by $x - a$ then remainder is<br/>
Question 8 :
State whether True or False.Divide : $a^2 +7a + 12 $ by $  a + 4 $, then the answer is $a+3$.<br/>
Question 9 :
If the quotient of $\displaystyle x^4 - 11x^3 + 44x^2 - 76x +48$. When divided by $(x^2 - 7x +12)$ is $Ax^2 + Bx + C$, then the descending order of A, B, C is
Question 11 :
What is $\dfrac {x^{2} - 3x + 2}{x^{2} - 5x + 6} \div \dfrac {x^{2} - 5x + 4}{x^{2} - 7x + 12}$ equal to
Question 12 :
$\alpha $ and $\beta $ are zeroes of polynomial $x^{2}-2x+1,$ then product of zeroes of a polynomial having zeroes $\dfrac{1}{\alpha }$  and    $\dfrac{1}{\beta }$ is
Question 13 :
If $P=\dfrac {{x}^{2}-36}{{x}^{2}-49}$ and $Q=\dfrac {x+6}{x+7}$ then the value of $\dfrac {P}{Q}$ is:
Question 14 :
If $\alpha , \beta$ are the roots of equation $x^2 \, - \, px \, + \, q \, = \, 0,$ then find the equation the roots of which are $\left ( \alpha ^2  \, \beta ^2 \right )  \,  and  \,  \,  \alpha \, + \,\beta $.
Question 18 :
Factorise the expressions and divide them as directed.$4yz(z^2 + 6z-  16)\div  2y(z + 8)$<br/>
Question 20 :
Divide the following and write your answer in lowest terms: $\dfrac{x^2-4x-5}{x^2-25}\div \dfrac{x^2-3x-10}{x^2+7x+10}$
Question 21 :
Find all values of a for which the equation $x^4+(a−1)x^3+x^2+(a−1)x+1=0$ possesses at least two distinct negative roots.
Question 22 :
What must be added to $f(x)=4x^4+2x^3+2x^2+x-1$ so that the resulting polynomial is divisible by $g(x)=x^2+2x-3$<br>
Question 23 :
The roots of the equation $x^2 + kx -12=0$ will differ by unity only, when
Question 26 :
(64x$^3$ + y$^3$) $\div$ (16x$^2$ - 4xy + y$^2$) is equal to
Question 28 :
$\left[2x\right]-2\left[x\right]=\lambda$ where $\left[.\right]$ represents greatest integer function and $\left\{.\right\}$ represents fractional part of a real number then 
Question 30 :
Evaluate :$\displaystyle \frac { 50xyz\left( x+y \right) \left( y+z \right) \left( z+x \right) }{ 100xy\left( x+y \right) \left( y+z \right) }$
Question 31 :
Divide: $(6a^{5}+ 8a^{4}+ 8a^{3} +2a^{2}+26a +35)$ by $(2a^{2} + 3a +5)$<br/>Answer: $3a^{3} - 3a^{2} + a +7$
Question 32 :
$mx^2+(m-1)x +2=0$ has roots on either side of x=1 the m $\in$
Question 33 :
Find the value of $p$ in the equation, where the roots are real,<br/>$\displaystyle 5{ x }^{ 2 }+3x-p=0$<br/>
Question 34 :
If the roots of the equation, $ax^2+bx+c=0$, are of the form $\alpha / (\alpha -1)$ and $(\alpha +1)/\alpha$, then the value of $(a+b+c)^2$ is
Question 35 :
Let $ p $ and $ q $ be real numbers such that $ p \neq 0, p^{3} \neq q $ and $ p^{3} $ $ \neq-q . $ If $ \alpha $ and $ \beta $ are non-zero complex numbers satisfying and $ \alpha+\beta=-p $ and $ \alpha^{3}+\beta^{3}=q, $ then a quadratic equation having $ \dfrac{\alpha}{\beta} $ and $ \dfrac{\beta}{\alpha} $ as its roots is
Question 36 :
If $p, q$ are the distinct roots of the equation $x^2 + px + q = 0$, then
Question 37 :
Which of the following equations has the sum of its roots as 3?<br/>
Question 38 :
If $\alpha$ and $\beta$ are the roots of $x^2-pX +1=0$ and $\gamma$ is a root of $X^2+pX+1=0$, then $(\alpha+\gamma)(\beta+\gamma)$ is
Question 40 :
If $\alpha, \beta$ are the roots of the quadratic equation $ax^2+bx+c=0$ and $3b^2=16ac$ then
Question 41 :
If $3{p}^{2}=5p+2$ and $3{q}^{2}=5q+2$, where $p\ne q$, then $pq$ is equal to
Question 42 :
When ${ x }^{ 2 }-2x+k$ divided the polynomial ${ x }^{ 2 }-{ 6x }^{ 3 }+16{ x }^{ 2 }-25x+10$ the reminder is (x+a), the value of is
Question 43 :
Divide the following and write your answer in lowest terms: $\dfrac{3x^2-x-4}{9x^2-16}\div \dfrac {4x^2-4}{3x^2-2x-1}$
Question 44 :
If $\alpha$ and $\beta$ are the zeros of the polynomial $f(x)=6x^2-3-7x$, then $(\alpha+1)(\beta+1)$ is equal to<br/>
Question 45 :
If$\displaystyle \alpha ,\beta$ are the roots of the equation$\displaystyle { x }^{ 2 }-x-4=0$, find the value of$\displaystyle \frac { 1 }{ \alpha } +\frac { 1 }{ \beta } -\alpha \beta$.
Question 46 :
The equation$ \displaystyle \frac{\left ( x+2 \right )\left ( x-5 \right )}{\left ( x-3 \right )\left ( x+6 \right )}= \frac{x-2}{x+4} $ has
Question 47 :
Find the product of roots if the quadratic equation $ax^2+bx+c=0$ has exactly one non-zero root.
Question 49 :
The sum of reciprocals of the roots of the equation $ax^2+bx+c=0$, $a, b,$ $c\ne0$, is _______.
Question 50 :
The difference of the roots of$\displaystyle 2y^{2}-ky+16=0$ is 1/3 Find k
Question 52 :
$x_1$ and $x_2$ are the real roots of $ax^2+bx+c=0$ and $x_1x_2 < 0$. The roots of $x_1(x-x_2)^2+x_2(x-x_1)^2=0$ are<br/>
Question 53 :
The equation $\displaystyle x^{2}+Bx+C=0$ has 5 as the sum of its roots and 15 as the sum of the square of its roots. The value of C is
Question 54 :
Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $
Question 56 :
If $\alpha, \beta$ be the roots $x^2+px-q=0$ and $\gamma, \delta$ be the roots of $x^2+px+r=0$, then $\dfrac{(\alpha -\gamma)(\alpha -\delta)}{(\beta -\gamma )(\beta -\delta)}=$
Question 57 :
Suppose $\alpha ,\beta .\gamma $ are roots of ${ x }^{ 3 }+{ x }^{ 2 }+2x+3=0$. If $f(x)=0$ is a cubic polynomial equation whose roots are $\alpha +\beta ,\beta +\gamma ,\gamma +\alpha $ then $f(x)=$
Question 58 :
If $\cos{\cfrac{\pi}{7}},\cos{\cfrac{3\pi}{7}},\cos{\cfrac{5\pi}{7}}$ are the roots of the equation $8{x}^{3}-4{x}^{2}-4x+1=0$<br>The value of $\sec{\cfrac{\pi}{7}}+\sec{\cfrac{3\pi}{7}}+\sec{\cfrac{5\pi}{7}}=$
Question 59 :
If$\alpha ,\beta $ are roots of the equation $2x^{2}+6x+b=0$ where $b<0$, then find least integral value of$\displaystyle \left ( \dfrac{\alpha ^{2}}{\beta }+\dfrac{\beta ^{2}}{\alpha } \right )$.<br>
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