ALGEBRA WEEKLY TEST

CHAPTERS COVERED: #ARITHMETIC PROGRESSION & #QUADRATIC EQUATION 

MARKS: 30 

1. Find the first five terms of the following sequences, whose nth terms are given.  t_n = 4n – 3

2. Which of the following lists of numbers are Arithmetic Progressions? Justify.  1, 3, 6, 10, ..... 

3. Find the twenty fifth term of the A. P. : 12, 16, 20, 24, ..... 

4. Find four consecutive terms in an A.P. whose sum is 12 and the sum of 3rd and 4th term is 14.

5. What will be her salary after 20 months? [Ans.]

6. Write the following quadratic equations in standard form ax² + bx + c = 0.  7 – 4x –x² = 0  [Ans]

7. Solve the following quadratic equations by  factorization method.  x² – 5x + 6 = 0 [Ans.]

8. Solve the following quadratic equations by using formula. m² – 3m – 10 = 0 [Ans]

9. Determine the nature of the roots of the following equations from their  discriminants : y² – 4y – 1 = 0 [Ans.]

10. Form the quadratic equation if its roots are : 5 and – 7     [Ans.]

ALGEBRA WEEKLY TEST

CHAPTERS COVERED: #ARITHMETIC #PROGRESSION & #QUADRATIC #EQUATION 

MARKS: 30 

1. For each of the sequence, find the next four terms. 1, 2, 4, 7, 11, .....

2. Find the first three terms of the sequences for which S_n is given below: S_n = n²(n + 1) 

3. Form the quadratic equation if one of the root is  3 – 2√ 5 [Ans.]

4. Find the sum of the first n natural numbers and hence find the sum of first 20natural numbers.

5. Mary got a job with a starting salary of Rs. 15000/- per month. She will get an incentive of Rs. 100/- per month.

6. Which of the following are quadratic equations ?  11 = – 4x² – x³          [Ans.]

7. In each of the examples given below determine whether the values given against each of the quadratic equation are the roots of the equation or not.   x² + 3x – 4 = 0,  x = 1, –2, – 3 [Ans]

8. Solve the following quadratic equations by completing square.  x² + 8x + 9 = 0 [Ans]

9. Find the value of discriminant of each of the following equations : x² + 4x + 1 = 0   [Ans]

10. Find the value of k for which given equation has real and equal roots : (k – 12)x² + 2 (k – 12)x + 2 = 0  [Ans.]

2p2 + 5p – 3 = 0, p = 1, ½, –3

(iv) 2p² + 5p – 3 = 0, p = 1, ½,  –3

Sol. a) By putting p = 1 in L.H.S. we get

L.H.S. = 2(1)² + 5(1) – 3

= 2(1) + 5 – 3

= 2 + 5 – 3

= 7 – 3

= 4

≠ R.H.S.

∴ L.H.S. ≠ R.H.S.

Thus equation is not satisfied.

So,  1 is not the root of the given quadratic equation.

b) By putting p = ½ in L.H.S. we get

L.H.S. = 2( ½)² +5( ½ ) – 3

= 2( ¼ )  + 5/2 – 3

= ½ + 5/2 – 3

= (1+5)/2 – 3

= 6/2 – 3

= 3 – 3

= 0

= R.H.S.

∴ L.H.S. = R.H.S.

Thus equation is satisfied.

So, ½, is the root of the given quadratic equation.

c) By putting p = – 3 in L.H.S. we get

L.H.S. = 2(– 3)² + 5(– 3) – 3

= 2(9) – 15 – 3

= 18 – 18

= 0

= R.H.S.

∴ L.H.S. = R.H.S.

Thus equation is satisfied.

So,  – 3 is the root of the given quadratic equation.

Quadratic equations

Definition

A quadratic equation in the variable 'x' is an equation of the form ax² + bx + c = 0

where a, b, c  are real numbers and a is not equal to zero. 

Nature of roots of a quadratic equation

Relation between roots and coefficients of a quadratic equation

Formation of quadratic equation when roots are given

A car left 30 minutes later than the scheduled time. in order to reach its destination 150 km away in time, it has to increase its speed by 25 km/hr from its usual speed. Find its usual speed.

The base of a triangle is 4 cm longer than its altitude. If the area of the triangle is 48 sq. cm, then find its base and altitude.

The sum of a number and its reciprocal is 5 1/5 . Find the numbers.

Solve by factorization method : 6x2 – 5x – 25 = 0

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0 , find the value of (α+1/β) (1/α+β)

If α and β are the toots of the equation 2x^2 - 3x - 1 = 0, find the values of α^2/β+β^2/α

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of α - β.

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of (α/β) + (β/α)

If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of a^2 + b^2

If the sum and product of the roots of the quadratic equation ax^2 - 5x + c = 0 are both equal to 10, then find the values of a and c.

If one of the roots of the equation 3x^2 - 10x + k = 0 is 1/3, then find the other root and also the value of k.

Find the values of k so that the equation x^2 - 2x ( 1 + 3k) + 7 (3 + 2x) = 0 has real and equal roots.

Determine the nature of roots of the following quadratic equation 2x^2 + 5x + 5 = 0 .

Determine the nature of roots of the following quadratic equation 4x^2 - 28x + 49 = 0

Determine the nature of roots of the following quadratic equation x^2 - 11x - 10 = 0

If α and β are the roots of the equation 3x^2 - 4x + 1 = 0. Form a quadratic equation whose roots are α^2 / β and β^2 /α

Form the quadratic equation whose roots are 7 + √3 and 7 - √3.

Try these Questions Yourself 

Find the values of k for which the roots are real and equal in each of the following equations.

Determine the nature of the roots of real equation. (Try your self)

If α and β are the roots of the equation 3x^2 - 5x + 2 = 0, then find the values of

Form a quadratic equation whose roots are

Find the sum and product of the roots of the following equations.