KCET 2024 Mathematics Question Paper: Download Question Paper with Solution PDF

Analyzing the equation based on the range of x values, it is determined that x values between 2 and 4 are solutions to the function's equation.

The formula for the length of an arc is l = r*theta, where r represents the radius and theta is the angle in radians. For two circles, the relationship between their radii, r1 and r2, and the angles they subtend, theta1 and theta2, is given as r1/r2 = theta2/theta1. To apply this, convert angles from degrees to radians: theta1 is 30 degrees, equivalent to pi/6 radians, and theta2 is 78 degrees, which converts to 13pi/30 radians. Substituting these values in, the ratio of the radii, r1/r2, becomes (13pi/30) / (pi/6), which simplifies to 13/5.

In a right triangle where the right angle is at vertex C, the tangent addition formula states that the sum of the tangents of angles A and B, tan A + tan B, equals ^c2/_ab. This relationship holds because angle C, being a right angle, means that angles A and B are complementary (A + B = 90°).

For the expression to be entirely real, the value of α must be a multiple of π, which ensures that sin α equals zero.

Assume the breadth of a rectangle is b and its length is 5 times the breadth, represented as 5b. The formula for the perimeter of a rectangle is 2(l + b), which in this case becomes 2(5b + b) = 12b. If the smallest possible perimeter is 180 cm, then setting 12b equal to 180 allows us to solve for b, giving b = 15 cm. Therefore, the minimum breadth of the rectangle is 15 cm or more.

The equation 49C3 + 48C3 + 47C3 + 46C3 + 45C3 + 45C4 = 50C4 demonstrates a special relationship between binomial coefficients. It shows that the sum of several consecutive binomial coefficients with specific parameters (in this case, combinations of 49, 48, etc. choosing 3 items, and one combination choosing 4 items) is equal to a single binomial coefficient with different parameters (50 choose 4). This relationship is a consequence of the recursive addition property of binomial coefficients, which allows for the calculation of one coefficient based on the values of others.

The sum of the series 1 + 2 + 3 + ... + n is expressed by the formula n(n+1)2\frac{n(n+1)}{2}. This formula demonstrates how series and combinatorial identities can be effectively used to simplify and calculate the sum of sequential integers.

The original statement describes a geometric series. The ratio of the first term's sum (S1) to the sum of the first two terms (S2) simplifies to 1/(1+r), not 1/(r^n+1).

We have two numbers. We know their average and their geometric mean. Using this, we can find the equation that represents these two numbers.

The slope of the line x + y = 3 is -1. The slope of the line joining (1, 1) and (-3, 4) is -3/4. The angle theta between the two lines is given by tan theta = |(m1 - m2)/(1 + m1m2)| = 0.14285714285714285.

The parabola has its vertex at the origin, represented by the coordinates (0, 0), and it opens towards the right direction. The standard form of the equation for such a parabola is y2=4axy^2 = 4ax. For this specific parabola, the value of aa is given as 6. Substituting 6 for aa in the equation, we get y2=24xy^2 = 24x. This equation defines how the shape of the parabola is formed and its orientation in the coordinate plane.

The limit of [(cos(x) - cot(x)) / (x - π/4)] as x approaches π/4, when approximated using Taylor series for cos(x) and cot(x) around π/4, simplifies to (√2 + 1)/2.

The negation of "For every x" is "There exists an x" where the following condition is negated.

When a constant (k) is added to every data point in a dataset, the entire dataset is shifted by that constant value. However, the spread or variability within the data, as measured by the standard deviation (S), remains unchanged.

The tangent function (tan(x)) has a period of π. The principal value of its inverse, tan⁻¹(1), is π/4. Due to the periodicity of tan(x), all solutions for tan⁻¹(1) can be expressed as nπ + π/4, where n is any integer.

For the function f(x) = x² + 1, x = ±4 when f(x) = 17 and there is no pre-image when f(x) = -3.

If you plug a value into function 'f' and then plug the result into function 'g', you get sin(x). But if you plug a value into 'g' first and then plug the result into 'f', you get (sin(√x))². This tells us that 'f' must be sin(√x) and 'g' must be x².

For the pair (3, 2), the equivalence class includes all pairs (c, d) such that the product of 3 and d equals the product of 2 and c. Finding integer solutions for (c, d) within the set A results in six pairs.

we can solve it Using sin-1 x + cos-1 x = π/2, substituting and solve for 2sin-1 x + 3cos-1 x, which gives (6π - 4)/5.

Expanding (I+A)3(I + A)^3 using A2=AA^2 = A simplifies to I+3AI + 3A. Since A2=AA^2 = A, this becomes 3A3A.

Calculating f(x)f(x) at specific values and taking their product as stated in the expression yields a product of 0.

Using the properties of determinants and adjoints, solving for α\alpha yields α=11 when ∣P∣=16.

Upon differentiating matrix BB with respect to xx and considering the elements within BB, the derivative dBdx\frac{dB}{dx} is found to be 3A3A.

As x approaches 0, the terms in f(x) divided by x2 tend to 0, resulting in the limit being 0.

For any logarithmic function with a base bb greater than 1, the point (1, 0) is consistently found on its graph because log⁡b(1)=0\log_b(1) = 0.

The function ∣cos⁡x∣ is continuous everywhere, yet it has points of non-differentiability at odd multiples of π2, where cos⁡x=0 = 0.

By employing logarithmic differentiation, calculate dy/dx for y = 2x^3x and evaluate at x = 1 to determine dy/dx = 6.

Applying the chain rule and trigonometric identities simplifies the derivative to 1/4.

With both the first and second derivatives equal to zero at a specific point,

Upon Differentiating x^x and analyzing its monotonicity, it is found that the function is strictly increasing for x > 1/e.

By applying principles from geometry and calculus, the maximum volume calculated for the cone is 16√3 π cubic units.

Upon differentiating f(x), we determine that it is increasing over the interval [-1/2, 1].

By employing trigonometric substitutions, the integral simplifies to 

When integrating an odd function across an interval that is symmetric about the origin, the result of the integral is zero. This occurs because the negative and positive areas under the curve cancel each other out.

When the function is simplified using trigonometric identities and then integrated, the result is given by the expression x+2sin⁡x+sin⁡2x+C. This outcome reflects the integral's calculation that incorporates the simplifications made from trigonometric transformations.

When the integral is computed across two specified intervals and then simplified, the total value obtained is 12.

When viewed as a Riemann sum and evaluated as a limit, the expression approaches tan^-1 2.

Integrating the difference between the line and the curve over the interval defined by their points of intersection results in an area of 9/2.

Calculating the integral between the intersection points of the line and the curve results in an area of 0.5 square units.

Solving the differential equation with the specified initial condition leads to the solution y+x−3=(x+1)log⁡(x+1).

The conditions of the problem correspond to the equation xy=C, where C is a constant.

Using vector addition and the Pythagorean theorem, the length of the median through vertex A is calculated to be √33 

By applying the scalar triple product, the volume of the parallelepiped is calculated to be 2 cubic units.

The sum of two unit vectors results in a unit vector if the angle between them is 2π/3, This specific angle satisfies the condition necessary for the resulting vector's magnitude to be one.

Each dot product evaluates to 1, resulting in a total sum of 3.

The dot product of the direction ratios calculates to -10/7, confirming that the lines are perpendicular.

Using the distance formula for parallel planes, the distance between them is calculated as 2/sqrt(29) units.

By applying the sine formula along with the direction ratios and the normal vector of the plane, one can determine specific geometric properties such as angles or distances related to the plane.

The yz-plane and the xz-plane represent two of the coordinate planes that are perpendicular to each other. The yz-plane is defined by all points where x=0, and the xz-plane is defined by all points where y=0.

The equation of a plane can be derived using a given point on the plane and properties of a line parallel to the plane. By utilizing these elements, the normal vector to the plane and a point through which the plane passes are identified, enabling the formulation of the plane's equation.

The minimum value of z is consistent at specific points and remains the same along the line segment that connects these points. This indicates that the function representing z reaches its lowest value across this entire segment.

The probability of obtaining at least one odd number is calculated by subtracting the probability of getting no odd numbers from 1. This approach utilizes the complement rule in probability, which states that the probability of an event occurring is 1 minus the probability of the event not occurring.

The variance is calculated using the values of E(X²) and E(X) which are derived by solving for the constant k. This involves determining kk such that the expected values fit the requirements of the probability distribution, allowing for the computation of variance as Var(X)=E(X2)−[E(X)]2.

The calculation is based on the condition P(X=2)= 9P(X=3) within the framework of the binomial probability formula. This relationship is used to determine the parameters of the binomial distribution, such as the number of trials nn and the probability of success pp, by setting up equations that relate these probabilities according to the binomial formula.