Quadratic Equations in One Variable - ICSE STD 10 Mathematics Questions

Q 1[1 Mark Question Answer]1 Mark

Find the nature of the roots of the following quadratic equations: $x^2-2 \sqrt{3} x-1=0$ If real roots exist, find them.

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Q 2[1 Mark Question Answer]1 Mark

Find the nature of the roots of the following quadratic equations: $x^2-\frac{1}{2} x-\frac{1}{2}=0$

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Q 3[1 Mark Question Answer]1 Mark

Discuss the nature of the roots of quadratic equations :$2 \sqrt{3} x^2-5 x+\sqrt{3}=0$

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Q 4[1 Mark Question Answer]1 Mark

Discuss the nature of the roots of quadratic equations : $-2x^2 + x + 1 = 0$

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Q 5[1 Mark Question Answer]1 Mark

Discuss the nature of the roots of quadratic equations :$x^2-\frac{1}{2} x+4=0$

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Q 6[2 Mark Question Answer]2 Marks

Five times a certain whole number is equal to three less than twice the square of the number. Find the number.

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Q 7[2 Mark Question Answer]2 Marks

At an annual function of a school, each student gives the gift to every other student. If the number of gifts is $1980$, find the number of students.

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Q 8[2 Mark Question Answer]2 Marks

Find the value (s) of k for which each of the following quadratic equation has equal roots : $kx^2 – 4x – 5 = 0$

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Q 9[2 Mark Question Answer]2 Marks

Without solving the following quadratic equation, find the value of ‘p’ for which the given equations have real and equal roots: $x^2 + (p – 2)x + p = 0.$

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Q 10[2 Mark Question Answer]2 Marks

Without solving the following quadratic equation, find the value of ‘p’ for which the given equations have real and equal roots: $px^2 – 4x + 3 = 0$

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Q 11[3 marks sum]3 Marks

In a certain positive fraction, the denominator is greater than the numerator by 3. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by$\frac{1}{14}$ Find the fraction.

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Q 12[3 marks sum]3 Marks

Find three consecutive odd integers, the sum of whose squares is $83$.

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Q 13[3 marks sum]3 Marks

Find three successive even natural numbers, the sum of whose squares is $308$.

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Q 14[3 marks sum]3 Marks

There are three consecutive positive integers such that the sum of the square of the first and the product of other two is 154. What are the integers?

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Q 15[3 marks sum]3 Marks

The difference between the squares of two numbers is 45. The square of the smaller number is 4 times the larger number. Determine the numbers.

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Q 16[5 marks sum]5 Marks

The length (in cm) of the hypotenuse of a right-angled triangle exceeds the length of one side by 2 cm and exceeds twice the length of another side by 1 cm. Find the length of each side. Also, find the perimeter and the area of the triangle.

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Q 17[5 marks sum]5 Marks

A piece of cloth costs Rs. 300. If the piece was 5 metres longer and each metre of cloth costs Rs. 2 less, the cost of the piece would have remained unchanged. How long is the original piece of cloth and what is the rate per metre?

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Q 18[5 marks sum]5 Marks

A shopkeeper buys a certain number of books for Rs 960. If the cost per book was Rs 8 less, the number of books that could be bought for Rs 960 would be 4 more. Taking the original cost of each book to be Rs x, write an equation in x and solve it to find the original cost of each book.

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Q 19[5 marks sum]5 Marks

Solve the following equation by factorisation :
$
\sqrt{3 x^2-2 x-1}=2 x-2
$

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Q 20[5 marks sum]5 Marks

Solve the following equation by factorisation :
$
\sqrt{x+15}=x+3
$

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Q 21[4 marks sum]4 Marks

Sum of two natural numbers is 8 and the difference of their reciprocal is 2/15. Find the numbers.

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Q 22[4 marks sum]4 Marks

If twice the area of a smaller square is subtracted from the area of a larger square, the result is $14 cm^2$. However, if twice the area of the larger square is added to three times the area of the smaller square, the result is $203 cm^2$. Determine the sides of the two squares.

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Q 23[4 marks sum]4 Marks

Two years ago, a man’s age was three times the square of his daughter’s age. Three years hence, his age will be four times his daughter’s age. Find their present ages.

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Q 24[4 marks sum]4 Marks

The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find the present age.

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Q 25[4 marks sum]4 Marks

Paul is $x$ years old and his father’s age is twice the square of Paul’s age. Ten years hence, the father’s age will be four times Paul’s age. Find their present ages.

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Q 26MCQ1 Mark

The value(s) of k for which the quadratic equation $2x^2– kx + k = 0$ has equal roots is (are)