Answer is (c) $16 \mathrm{~m}$ Let the length and breadth of the rectangle be $l$ and $b$. Perimeter of the rectangle $=82 \mathrm{~m}$ $\begin{array}{l} \Rightarrow 2 \times(l+b)=82 \\ \Rightarrow...
The perimeter of a rectangle is and its area is . The breadth of the rectangle is
If the equation has no real roots then
(a)
(b)
(c)
(d) None of these
Answer is c) $-2<\mathrm{k}<2$ It is given that the equation $x^{2}-k x+1=0$ has no real roots. $\begin{array}{l} \therefore\left(b^{2}-4 a c\right)<0 \\ \Rightarrow(-k)^{2}-4 \times 1...
If the roots of are real and distinct then
(a)
(b) only
(c)
d) either or
Answer is (d) either $k>2 \sqrt{5}$ or $k<-2 \sqrt{5}$ It is given that the roots of the equation $\left(5 x^{2}-k+1=0\right)$ are real and distinct. $\begin{array}{l} \therefore\left(b^{2}-4...
The roots of are real and unequal, if is
(a)
(b)=0
(c)<0(d) none of these
Answer is $(a)>0$ The roots of the equation are real and unequal when $\left(b^{2}-4 a c\right)>0$.
If one root of be the reciprocal of the other root then the value of is
(a) 0
(b) 1
(c) 2
(d) 5
Answer is (d)5 Let the roots of the equation $\frac{-2}{3}$ be $\alpha$ and $\frac{1}{\alpha}$. $\therefore$ Product of the roots $=\frac{c}{a}$ $\Rightarrow \alpha \times...
The length of a rectangular field is three times its breadth. If the area of the field be 147 sq meters, find the length of the field.
Let the length and breadth of the rectangle be $3 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 3 x \times x=147 \\ \Rightarrow 3 x^{2}=147 \\...
300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of students.
Let the total number of students be $x$. According to the question: $\begin{array}{l} \frac{300}{x}-\frac{300}{x+10}=1 \\ \Rightarrow \frac{300(x+10)-300 x}{x(x+10)}=1 \\ \Rightarrow \frac{300...
If the roots of the equation are equal, prove that
It is given that the roots of the equation $\left(a^{2}+b^{2}\right) x^{2}-2(a c+b d) x+\left(c^{2}+d^{2}\right)=0$ are equal. $\begin{array}{l} \therefore D=0 \\ \Rightarrow[-2(a c+b...
For what values of are the roots of the quadratic equation real and equal?
Given: $3 x^{2}+2 k x+27=0$ Here, $a=3, b=2 k \text { and } c=27$ It is given that the roots of the equation are real and equal; therefore, we have: $\begin{array}{l} D=0 \\ \Rightarrow(2 k)^{2}-4...
Find the roots of the given equation:
$\begin{array}{l} x^{2}-(\sqrt{2}+1) x+\sqrt{2}=0 \\ \Rightarrow x^{2}-(\sqrt{2}+1) x=-\sqrt{2} \\ \Rightarrow x^{2}-2 \times x...
Find the roots of the given equation:
We write, $-2 x=-3 x+x$ as $3 x^{2} \times(-1)=-3 x^{2}=(-3 x) \times x$ $\begin{array}{l} \therefore 3 x^{2}-2 x-1=0 \\ \Rightarrow 3 x^{2}-3 x+x-1=0 \\ \Rightarrow 3 x(x-1)+1(x-1)=0 \\...
Which of the following are the roots of
On subtracting $x=\left(-\frac{1}{2}\right)$ in the given equation, we get $\begin{array}{l} \text { L.H.S. }=3 x^{2}+2 x-1 \\ =3 \times\left(-\frac{1}{2}\right)^{2}+2...
Which of the following are quadratic equation in ?
(i)
(ii)
(i) $\begin{array}{l} (x+2)^{3}=x^{3}-8 \\ \Rightarrow x^{3}+6 x^{2}+12 x+8=x^{3}-8 \\ \Rightarrow 6 x^{2}+12 x+16=0 \end{array}$ This is of the form $a x^{2}+b x+c=0$ Hence, the given equation is a...
Which of the following are quadratic equation in ?
(i)
(ii)
(i) Clearly, $\left(\sqrt{2} x^{2}+7 x+5 \sqrt{2}\right)$ is a quadratic polynomial. $\therefore \sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$ is a quadratic equation. (ii) Clearly, $\left(\frac{1}{3}...
Which of the following are quadratic equation in ?
(i)
(ii)
(i) $\left(x^{2}-x+3\right)$ is a quadratic polynomial $\therefore x^{2}-x+3=0$ is a quadratic equation. (ii) Clearly, $\left(2 x^{2}+\frac{5}{2} x-\sqrt{3}\right)$ is a quadratic polynomial....
The pressure of a 1:4 mixture of dihydrogen and dioxygen enclosed in a vessel is one atmosphere. What would be the partial pressure of dioxygen? (i) 0.8×105 atm (ii) 0.008 Nm–2 (iii) 8×104 Nm–2 (iv) 0.25 atm
The correct option is (iii) 8×104 Nm–2