Class 10 Mathematics (Standard) 2022 Board Exam

(a) Find the value of $m$ for which the quadratic equation $(m-1)x^2+2(m-1)x+1=0$ has two real and equal roots.
OR
(b) Solve the following quadratic equation for $x$ : $\sqrt{3}{x}^2+10x+7\sqrt{3}=0$

The product of Rehan's age (in years) $5$ years ago and his age $7$ years from now, is one more than twice his present age. Find his present age.

Solve the quadratic equation : $x^2+2\sqrt{2}x-6=0$ for $x$.

Solve the quadratic equation : $x^2-2ax+(a^2-b^2)=0$ for $x$.

(a) A 2-digit number is such that the product of its digits is $24$. If $18$ is subtracted from the number, the digits interchange their places. Find the number.
OR
(b) The difference of the squares of two numbers is $180$. The square of the smaller number is $8$ times the greater number. Find the two numbers.

(a) The sum of two numbers is $34$. If $3$ is subtracted from one number and $2$ is added to another, the product of these two numbers becomes $260$. Find the numbers.
OR
(b) The hypotenuse (in cm) of a right angled triangle is $6\text{cm}$ more than twice the length of the shortest side. If the length of third side is $6\text{cm}$ less than thrice the length of shortest side, then find the dimensions of the triangle.

(a) Find the sum of first $30$ terms of AP : $-30,-24,-18,\dots$.
OR
(b) In an AP if $S_n=n(4n+1)$, then find the AP.

Case Study - 1
In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in different figures.
One such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression :
Figure 1, Figure 2, Figure 3
(a) Write the AP for the number of triangles used in the figures. Also, write the $n^{\text{th}}$ term of this AP.
(b) Which figure has $61$ matchsticks ?

(a) Find the sum of the first twelve 2-digit numbers which are multiples of $6$.
OR
(b) In an AP, if $a_2=26$ and $a_{15}=-26$, then write the AP.

(a) Which term of the A.P. $-\frac{11}{2},-3,-\frac{1}{2},\dots$ is $\frac{49}{2}$ ?
OR
(b) Find $a$ and $b$ so that the numbers $a, 7, b, 23$ are in A.P.