IAS Prelims test Math for IAS Prelims Math Challenge

Solution

Correct: B

Given a = 2, b = 3, c = 4. Put the values of a, b, c in (a + b + c)/(ab + bc + ca). (2 + 3 + 4)/(2*3 + 3*4 + 4*2) = 9/(6 + 12 + 8) = 9/26.

Solution

Correct: B

By trigonometric identity, we know that sin^2(x) + cos^2(x) = 1. This is true for all real values of x.

Solution

Correct: A

2^x + 2^(x+1) = 24, we can rewrite 2^(x+1) as 2^x * 2. Then 2^x + 2^x * 2 = 24, 2^x(1+2) = 24. 2^x * 3 = 24, 2^x = 24/3, 2^x = 8, 2^x = 2^3, so x = 3.

Solution

Correct: A

First, find the slope of the line (y2 - y1)/(x2 - x1) = (5 - 3)/(4 - 2) = 2/2 = 1. The slope is 1. Now use point-slope form y - y1 = m(x - x1). Use point (2, 3) and m = 1. y - 3 = 1(x - 2), y - 3 = x - 2, y = x + 1.

Solution

Correct: A

This is a geometric progression where first term(a) = 2^0 = 1, common ratio(r) = 2 and number of terms(n) = 11. The sum of n terms of a G.P. is given by Sn = a(r^n - 1)/(r - 1). So, (2^0 + 2^1 + 2^2 + ... + 2^10) = 1*(2^11 - 1)/(2 - 1) = 2^11 - 1 = 2048 - 1 = 2047.

Solution

Correct: A

Since (0, 0) is on the circle, the equation has the form x^2 + y^2 + Dx + Ey + F = 0. Plugging in (0, 0), we get F = 0. Plugging in (0, 4), we get 16 + 4E = 0, so E = -4. Plugging in (6, 0), we get 36 + 6D = 0, so D = -6. The equation is x^2 + y^2 - 6x - 4y = 0, but none of the options match this equation.

Solution

Correct: A

To solve |3x - 1| > 5, we must solve two inequalities: 3x - 1 > 5 and 3x - 1 < -5. Solving 3x - 1 > 5, we have 3x > 6, x > 2. Solving 3x - 1 < -5, we have 3x < -4, x < -4/3. So the solution set is x < -4/3 or x > 2.

Solution

Correct: B

We have (x^3 + 2x^2 - 7x - 12)/(x^2 + 4x + 4). Given x = -2, substituting in the equation: ((-2)^3 + 2(-2)^2 - 7(-2) - 12)/((-2)^2 + 4(-2) + 4) = (-8 + 8 + 14 - 12)/((4 - 8 + 4)) = -8/0 is undefined, however factoring the numerator (x + 1)(x + 2)(x - 3) and the denominator (x + 2)(x + 2) then ((x + 1)(x + 2)(x - 3))/((x + 2)(x + 2)) = (x + 1)(x - 3)/(x + 2) then when x = -2 we get (-2 + 1)(-2 - 3)/(-2 + 2) = (-1)(-5)/0 which is undefined. The expression is not defined for x = -2, but factoring the numerator and simplifying we get (x^2 - 2x - 3)/(x + 2), so when x = -2 we get ((-2)^2 - 2(-2) - 3)/(-2 + 2) = (4 + 4 - 3)/0 = 5/0 which is undefined, the given equation is not defined at x = -2.

Solution

Correct: A

The derivative of x^n is n*x^(n-1). Applying this rule, the derivative of x^3 is 3x^2, the derivative of -4x^2 is -8x, the derivative of x is 1, and the derivative of 1 is 0. Combining these gives f'(x) = 3x^2 - 8x + 1.

Solution

Correct: A

To find the area of a triangle given its vertices (x1, y1), (x2, y2), (x3, y3), we use the formula: area = (1/2)*|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. Plugging in the values: area = (1/2)*|0(3 - 0) + 2(0 - 0) + 4(0 - 3)| = (1/2)*|0 + 0 + 4(-3)| = (1/2)*|-12| = 6.

Solution

Correct: A

To solve x^2 - 5x - 6 = 0, factor the left-hand side: (x - 6)(x + 1) = 0. This gives two equations: x - 6 = 0 and x + 1 = 0. Solving these gives x = 6 and x = -1.

Solution

Correct: A

The equation of a line with slope m passing through point (x1, y1) is y - y1 = m(x - x1). Given slope m = -2 and point (2, 3), we have y - 3 = -2(x - 2). Simplify: y - 3 = -2x + 4, y = -2x + 7.

Solution

Correct: C

Substitute x = -2 into the equation f(x) = x^2 - 4: f(-2) = (-2)^2 - 4 = 4 - 4 = 0.

Solution

Correct: C

The given expression is (x^2 - 4)/(x - 2). Factoring the numerator gives ((x + 2)(x - 2))/(x - 2). The expression is undefined when the denominator (x - 2) equals zero, which occurs when x = 2.

Solution

Correct: C

The diameter of a circle is twice its radius. Given the radius is 4, the diameter is 2*4 = 8.

Solution

Correct: C

Given 2^x = 16, we notice that 16 is 2^4. Therefore, x must equal 4 for the equation to be true.

Solution

Correct: D

The area of a rectangle is given by length times width. So, area = 5 * 7 = 35.

Solution

Correct: A

The general equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Given the center (0, 0) and radius 5, we substitute h = 0, k = 0, and r = 5 into the equation to get (x - 0)^2 + (y - 0)^2 = 5^2, simplifying to x^2 + y^2 = 25.

Solution

Correct: A

The given equation a^2 - b^2 = 15 can be rewritten as (a + b)(a - b) = 15. Since 15 is a product of 15*1 or 5*3 (ignoring negative factors), possible values for (a + b) and (a - b) are (15, 1) or (5, 3). Solving simultaneously: for (a + b) = 15 and (a - b) = 1, adding both equations gives 2a = 16, a = 8, then 8 + b = 15, b = 7. For (a + b) = 5 and (a - b) = 3, adding both gives 2a = 8, a = 4, then 4 + b = 5, b = 1. Hence one of the possible pairs could be a = 4, b = 1.

Solution

Correct: D

This is a geometric progression where first term(a) = 5^0 = 1, common ratio(r) = 5 and number of terms(n) = 11. The sum of n terms of a G.P. is given by Sn = a(r^n - 1)/(r - 1). So, (5^0 + 5^1 + 5^2 + ... + 5^10) = 1*(5^11 - 1)/(5 - 1) = (5^11 - 1)/4.

Solution

Correct: A

First, find the slope (m) of the line: m = (y2 - y1)/(x2 - x1) = (4 - 2)/(3 - 1) = 2/2 = 1. The slope is 1. Now use point-slope form y - y1 = m(x - x1). Use point (1, 2) and m = 1: y - 2 = 1(x - 1), y - 2 = x - 1, y = x + 1.