problem archive
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Show that \(3\sqrt{2}\) is irrational. -
Prove that \( \sqrt{5} \) is irrational. -
Prove that \( 3 + 2\sqrt{5} \) is irrational. -
Prove that the following are irrationals: (i) \( \frac{1}{\sqrt{2}} \) (ii) \( 7\sqrt{5} \) (iii) \( 6 + \sqrt{2} \) -
What is the value of \(p(x) = x^2 - 3x - 4\) at \(x = -1\)? -
Consider first a linear polynomial ax + b, a ≠ 0. You have studied in Class IX that the graph of y = ax + b is a straight line. For example, the graph... -
In general, for a linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point, namely... -
Consider the quadratic polynomial x^2 - 3x - 4. Let us see what the graph of y = x^2 - 3x - 4 looks like. Let us list a few values of y = x^2 - 3x - 4... -
If we locate the points listed above on a graph paper and draw the graph, it will actually look like the one given in Fig. 2.2. -
The value of \(p(x) = 2x^2 - 8x + 6\) is zero when \(x - 1 = 0\) or \(x - 3 = 0\), i.e., when \(x = 1\) or \(x = 3\). So, the zeroes of \(2x^2 - 8x + ... -
Let us take one more quadratic polynomial, say, \(p(x) = 3x^2 + 5x - 2\). By the method of splitting the middle term, \(3x^2 + 5x - 2 = 3x^2 + 6x - x ... -
In general, if \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = ax^2 + bx + c\), \(a \neq 0\), then you know that \(x - \a... -
Find the zeroes of the quadratic polynomial x^2 + 7x + 10, and verify the relationship between the zeroes and the coefficients. -
Find the zeroes of the polynomial x^2 - 3 and verify the relationship between the zeroes and the coefficients. -
Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2, respectively. -
Verify that 3, -1, \frac{1}{3} are the zeroes of the cubic polynomial p(x) = 3x^3 - 5x^2 - 11x - 3, and then verify the relationship between the zeroe... -
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x^2 - 2x - 8 (ii) 4s^2... -
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) 1/4, -1 (ii) sqrt(2), 1/3 (iii) 0, sqrt... -
Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a ring on the items kept... -
Check graphically whether the pair of equations x + 3y = 6 and 2x - 3y = 12 is consistent. If so, solve them graphically.