The students of a class arranged a picnic.
Question: The students of a class arranged a picnic. Each student contributed as many rupees as the number of students in the class. If the total contribution is Rs 1156, find the strength of the class. Solution: Let the number of students be $x$. Hence, the amount contributed by each student is Rs $x$. Total amount contributed $=x \times x=x^{2}=1156$ $1156=2 \times 2 \times 17 \times 17$ $x=\sqrt{1156}=2 \times 17=34$ Thus, the strength of the class is 34....
Read More →If f(x) = x |x|,
Question: If $f(x)=x|x|$, then $f^{\prime}(2)=$__________ Solution: $|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ $\therefore f(x)=x|x|= \begin{cases}x^{2}, x \geq 0 \\ -x^{2}, x0\end{cases}$ Now, $L f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h}$ $\Rightarrow L f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{(2-h)^{2}-2^{2}}{-h}$ $\Rightarrow L f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{4-4 h+h^{2}-4}{-h}$ $\Rightarrow L f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{(-4+h) h}{-h...
Read More →1225 Plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows.
Question: 1225 Plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row. Solution: Let the number of rows bex. Therefore, the number of plants in each row is alsox. Total number of plants $=(x \times x)=x^{2}=1225$ $x^{2}=1225=5 \times 5 \times 7 \times 7$ $x=\sqrt{1225}=5 \times 7=35$ Thus, the number of rows is 35 and the number of plants in each row is 35....
Read More →Find the smallest number by which 2925 must be divided to obtain a perfect square.
Question: Find the smallest number by which 2925 must be divided to obtain a perfect square. Also, find the square root of the perfect square so obtained. Solution: Resolving into prime factors: $2925=3 \times 3 \times 5 \times 5 \times 13$ 13 is the smallest number by which the given number must be divided to make it a perfect square. New number $=2925 \div 13=225$ $\sqrt{225}=3 \times 5=15$...
Read More →In figure, tangents PQ and PR are drawn to a circle
Question: In figure, tangents PQ and PR are drawn to a circle such that RPQ = 30. A chord RS is drawn parallel to the tangent PQ. Find the RQS. Solution: PQ and PR are two tangents drawn from an external point P. $\therefore \quad P Q=P R$ [the lengths of tangents drawn from an external point to a circle are equal] $\Rightarrow \quad \angle P Q R=\angle Q R P$ [angles opposite to equal sides are equal] Now, in $\triangle P Q R \quad \angle P Q R+\angle Q R P+\angle R P Q=180^{\circ}$ [sum of all...
Read More →Find the smallest number by which 252 must be multiplied to get a perfect square.
Question: Find the smallest number by which 252 must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained. Solution: Resolving into prime factors: $252=2 \times 2 \times 3 \times 3 \times 7$ Thus, the given number must be multiplied by 7 to get a perfect square. New number $=252 \times 7=1764$ $\therefore \sqrt{1764}=2 \times 3 \times 7=42$...
Read More →Find the smallest number by which 252 must be multiplied to get a perfect square.
Question: Find the smallest number by which 252 must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained. Solution: Resolving into prime factors: $252=2 \times 2 \times 3 \times 3 \times 7$ Thus, the given number must be multiplied by 7 to get a perfect square. New number $=252 \times 7=1764$ $\therefore \sqrt{1764}=2 \times 3 \times 7=42$...
Read More →If f(x) = x |x|,
Question: If $f(x)=x|x|$, then $f(-1)=$____________ Solution: $|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ $\therefore f(x)=x|x|= \begin{cases}x^{2}, x \geq 0 \\ -x^{2}, x0\end{cases}$ Now, $L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{f(-1-h)-f(-1)}{-h}$ $\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-(-1-h)^{2}-\left[-(-1)^{2}\right]}{-h}$ $\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-\left(1+2 h+h^{2}\right)+1}{-h}$ $\Rightarrow L f^{\prime}(-1)=\lim _{h ...
Read More →If f(x) = x |x|,
Question: If $f(x)=x|x|$, then $f(-1)=$____________ Solution: $|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ $\therefore f(x)=x|x|= \begin{cases}x^{2}, x \geq 0 \\ -x^{2}, x0\end{cases}$ Now, $L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{f(-1-h)-f(-1)}{-h}$ $\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-(-1-h)^{2}-\left[-(-1)^{2}\right]}{-h}$ $\Rightarrow L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{-\left(1+2 h+h^{2}\right)+1}{-h}$ $\Rightarrow L f^{\prime}(-1)=\lim _{h ...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:17424 Solution: Resolving into prime factors: $17424=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 11 \times 11$ $\therefore \sqrt{17424}=2 \times 2 \times 3 \times 11=132$...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:15876 Solution: Resolving into prime factors: $15876=2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7$ $\therefore \sqrt{15876}=2 \times 3 \times 3 \times 7=126$...
Read More →In a right angle ΔABC is which ∠B = 90°,
Question: In a right angle ΔABC is which B = 90, a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at PQ bisects BC. Solution: Let O be the centre of the given circle. Suppose, the tangent at P meets BC at 0. Join BP. To prove $B Q=Q C$ [angles in alternate segment] Proot $\angle A B C=90^{\circ}$ thangent at any noint of circle is nernendicular to radius through the point of contact] $\therefore \ln \triangle A B C$, $\angle 1+\angle...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:11025 Solution: Resolving into prime factors: $11025=3 \times 3 \times 5 \times 5 \times 7 \times 7$ $\therefore \sqrt{11025}=3 \times 5 \times 7=105$...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:9216 Solution: Resolving into prime factors: $9216=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$ $\therefore \sqrt{9216}=2 \times 2 \times 2 \times 2 \times 2 \times 3=96$...
Read More →Solve this
Question: Let $f(x)=\left\{\begin{array}{cl}a x^{2}+3, x1 \\ x+\frac{5}{2}, x \leq 1\end{array} .\right.$ If $f(x)$ is differentiable at $x=1$, then $a=$___________ Solution: The given function $f(x)=\left\{\begin{array}{cl}a x^{2}+3, x1 \\ x+\frac{5}{2}, x \leq 1\end{array}\right.$ is differentiable at $x=1$ $\therefore L f^{\prime}(1)=R f^{\prime}(1)$ $\Rightarrow \lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{-h}=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ $\Rightarrow \lim _{h \rightarrow 0} ...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:8100 Solution: Resolving into prime factors: $8100=2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5$ $\therefore \sqrt{8100}=2 \times 3 \times 3 \times 5=90$...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:7056 Solution: Resolving into prime factors: $7056=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7$ $\therefore \sqrt{7056}=2 \times 2 \times 3 \times 7=84$...
Read More →The function f(x)=|sin x|,
Question: The function $f(x)=|\sin x|, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is not differentiable at $x=$________________ Solution: We know that, $|x|$ is not differentiable at $x=0$. Therefore, $|\sin x|$ is not differentiable when $\sin x=0$. $\sin x=0$ $\Rightarrow x=n \pi, n \in Z$ $\Rightarrow x=\ldots,-2 \pi,-\pi, 0, \pi, 2 \pi, \ldots$ Now, the only value of $x$ lying in given interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ at which the function $f(x)=|\sin x|$ is not di...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:4096 Solution: Resolving into prime factors:$4096=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ $\therefore \sqrt{4096}=2 \times 2 \times 2 \times 2 \times 2 \times 2=64$...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:2025 Solution: Resolving into prime factors: $2025=3 \times 3 \times 3 \times 3 \times 5 \times 5$ $\therefore \sqrt{2025}=3 \times 3 \times 5=45$...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:1296 Solution: Resolving into prime factors: $1296=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$ $\therefore \sqrt{1296}=2 \times 2 \times 3 \times 3=36$...
Read More →Two circles with centres 0 and 0′ of radii 3 cm and 4 cm,
Question: Two circles with centres 0 and 0 of radii 3 cm and 4 cm, respectively intersect at two points P and Q, such that OP and 0P are tangents to the two circles. Find the length of the common chord PQ. Solution: Here, two circles are of radii OP = 3 cm and PO = 4 cm These two circles intersect at P and Q. Here, $O P$ and $P O^{\prime}$ are two tangents drawn at point $P$. $\angle O P O^{\prime}=90^{\circ}$ [tangent at any point of circle is perpendicular to radius through the point of contac...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:729 Solution: Resolving into prime factors: $729=3 \times 3 \times 3 \times 3 \times 3 \times 3$ $\therefore \sqrt{729}=3 \times 3 \times 3=27$...
Read More →The function
Question: The function $f(x)=\cos ^{-1}(\cos x), x \in(-2 \pi, 2 \pi)$ is not differentiable at $x=$________________ Solution: $f(x)=\cos ^{-1}(\cos x)= \begin{cases}x+2 \pi -2 \pi \leq x \leq-\pi \\ -x, -\pi \leq x \leq 0 \\ x, 0 \leq x \leq \pi \\ 2 \pi-x, \pi \leq x \leq 2 \pi\end{cases}$ Let us check the differentiability of the function at $x=-\pi, x=0$ and $x=\pi$. At $x=-\pi$ $L f^{\prime}(-\pi)=\lim _{x \rightarrow-\pi^{-}} \frac{f(x)-f(-\pi)}{x-(-\pi)}$ $\Rightarrow L f^{\prime}(-\pi)=\...
Read More →Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:441 Solution: By prime factorisation: $441=3 \times 3 \times 7 \times 7$ $\therefore \sqrt{441}=3 \times 7=21$...
Read More →